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The publication of THE DOCTRINE OF COMPOUND INTEREST
has met with unavoidable delay, in consequence of the destruc-
tive Fire on MR. WILSON's premises, where it was in progress
of printing.
The present edition is offered to SUBSCRIBERS only, who are
requested to apply to the Author, at No. 5, Hercules-Buildings,
(near the Asylum,) Lambeth.
Subscription - price, L.2.-There are a small number of
Library Copies, Quarto, at L.4.
←
}
THE
Doctrine of Compound Interest,
ILLUSTRATED AND APPLIED TO PERPETUAL ANNUITIES,
TO THOSE FOR TERMS OF YEARS CERTAIN, TO LIFE-ANNUITIES,
AND GENERALLY TO PROSPECTIVE TRANSACTIONS;
WITH
AUTHOR OF
New and Compendious Tables,
SOLVING THE QUESTIONS THAT DEPEND ON COMPOUND INTEREST, ACCRUING
ANNUALLY, HALF-YEARLY, OR QUARTERLY ;
AND
OTHER ORIGINAL TABLES,
EXPRESSING, WITH REFERENCE TO COMPARATIVE SYSTEMS, THE PROBABILITIES AND THE EXPECTATIONS
OF LIFE AT EACH AGE, ALSO THE CORRECT VALUES OF ANNUITIES ON SINGLE
AND ON JOINT LIVES.
“TEMPUS AGER MEUS.”
BY
·
LONDON-1825.
FRANCIS CORBAUX, 176 9 ?
1890
“An Inquiry into the National Debt, and into the Means and the Prospect of its
Redemption, with a Plan for redeeming that Debt, upon the principle
of Terminable Annuities," &c. &c.
W
PRINTED for SUBSCRIBERS,
AND TO BE HAD AT THE AUTHOR'S, No. 5, HERCULES-BUILDINGS, (NEAR THE ASYLUM,) LAMBETH.
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INDEX.
PREFACE…….
INTRODUCTION :-
Invention of Money
Progress of the Money-system.....
The quantity required, of Circulating-medium, depends on the convenience
afforded for the rapidity of its transfer.-Bank establishments.......... Ibid.
Capital consists of the unappropriated masses of Money
Merchantable character of Capital.....
17
Ibid.
18
The association of Capital, with Industry, is the foundation of Interest
The principle of that association is compromised, in rating the Interest of money. Ibid.
Principle of contracts for Loans at Interest
19
20
Credit generally defined: Government-security, and Public-credit
Circumstances by which the rate of Interest is governed
21
The proportion between supply and demand influences the value of Capital …….. Ibid.
The unlimited increase of Capital is of minor consideration, towards the
permanency of National power
The Rate of Interest is compounded of Remuneration for the Capital employed,
•
+
D
and of Compensation for the Risk…………………
The consideration of Risk has been discarded, in the enactments restraining the
rate of Interest..
23
The original motive of Usury-laws is not applicable to the present state of Society. Ibid.
Equal degrees of security may admit some difference as to the Rate of Interest.
Interest of money, and Available income, are not the same thing...
24
25
Definition of Compound-Interest
26
The operation of Compound-Interest may be affected with various considerations
of Probability
The character of Annuities discriminated
Page
9
a
15
16
2.2
410705
Ibid.
Perpetual annuities, Funded Stock, and Irredeemable debts
Annuities for terms of years certain, and Annuities settled on Life
The ratio of Terminable Annuities is compounded of the rate of Interest, with
a surplus adequate to recompose the Capital at the expiration of the term..・ 28
Definition of a Sinking-fund
Definition of a Redemption-fund......
29
30
Data required, for solving the questions that depend on Compound-Interest,... Ibid.
The transactions relative to Life-interests have a different character, when
between private individuals, and when assumed on an extensive scale
by incorporated societies
Ibid.
Ibid.
27
Ibid.
Moral inconvenience attending private spéculations on Life and Death
Moral Expectation is to be discriminated from Mathematical Expectation; but
they coincide in the multifarious transactions of incorporated societies.. Ibid.
31
Ibid.
主
​ii
INDEX.
THE DOCTRINE OF COMPOUND INTEREST: Part First, (divided
into 64 sections,) containing the application of that doctrine to
Perpetual Annuities, and to Terminable Annuities the period of which
is unconditional
M
Page
Sect. I.-Definitions, in which five quantities, bearing an invariable relation
with each other, are considered…………..
II.-Progression according to which any Principal is improved by the
operation of Compound-Interest
III. From the general expression of the Improved Principal, four
equations are derived, each of which solves a specific question…….. Ibid.
IV.—Example, in the question concerning the Improved Principal
V.—Example, in the question concerning the Present or discounted value
35
of forthcoming capital
Ibid.
... Ibid.
VI.-Example, in the question concerning the Rate of Interest...
VII.--Example, in the question concerning the Period of time required for
the improvement of Capital, in any given ratio...
VIII. Observation respecting analogous cases
IX.-Equation expressing the Present value of Terminable annuities
X.-Equations establishing the constant relations between any Perpetual
annuity, its Present value, and the Rate of Interest
XI.-Exemplification of the three questions relative to Perpetual annuities.
XII.--When the first instalment of any annuity is paid at the outset,
this
Ibid.
operates only as a diminution of the invested Principal
.......... Ibid
XIII.-Equation generally expressing the Amount of any Annuity, accu-
mulated and successively improved by Compound-Interest, at
each stage of income………..
XIV.-Modification of the preceding equation, when each instalment of
the annuity is paid by anticipation ....
XV.-Deductions from the fundamental equations (Sect. IX and XIII);
establishing that any three quantities being supplied as the data,
out of the five stated in section I, the remaining two are ascertain-
able. Tuble of the Equations that solve the questions concerning
each of those two unknown quantities, in ten cases susceptible of
arising in consequence of the separation of 2 objects from 3 others. 39
XVI.-Selection of three amongst the preceding ten cases; whence six
questions arise, the solution of which, only, are capable of
becoming the subject of Tables for habitual reference
··
33
Ibid.
Ibid.
36
Ibid.
T
34
Ibid.
37
XVII. Those Tables, immediately solving six questions, out of 20 that
may arise from the consideration of Compound-Interest, further
supply an easy method of solving the other 14; with a single
exception, of the question in which the unknown quantities con-
sist of the Rate of Interest and the Period of time
XVIII. First question exemplified; that of the Improved Principal, with
its solution by the Tables
XIX.-Second question exemplified, and solved by the Tables; that of
the Ratio of the Annuity, or the proportion which that annuity
should bear with the Principal it represents...
38
42
Ibid.
43
Ibid.
INDEX.
?
:
Sect. XX -Third question exemplified; that of the Discounted value of forth-
coming Capital, with its solution by the Tables
(
XXI.-Fourth question exemplified, and solved by the Tables; that of the
Redemption fund, or the proportion of periodical savings, for
producing by their accumulation and improvement any Capital
required at a future period .....
..
XXVIII.-Case 6th exemplified, with the solution of questions concerning the
Term of the Annuity, and its represented Principal; when the
data consist of the Annuity itself,-its Cumulated amount,—and
the Rate of Interest
►
Page
XXII.—Fifth question exemplified and solved by the Tables; that of the
Present Value of the Annuity
XXIII.-Sixth question exemplified, and also solved by the Tables; that
of the Cumulated Annuity, or of the capital ultimately created by
the savings of Income, progressively improved
XXIV. The Tables of Compound-Interest further described
XXV.-Use of the Tables, towards mediately solving the questions that
arise in cases 4th-5th-6th-8th-9th and 10th, described in
section XV; the 7th case being an exception
XXVI.-Case 4th exemplified with the solution of questions concerning the
Annuity and its Terminating period; when the data consist
of the Principal or present value,-the Improved Principal or
Cumulated amount of the Annuity,-and the Rate of Interest .. Ibid.
XXVII.-Case 5th exemplified, with the solution of questions concerning
the Term of the Annuity and its Cumulated amount; when the
data are the Annuity,—its Present value,—and the Rate of
Interest
XXIX.-Case 7th exemplified, the questions being the Term of the Annuity
and the Rate of Interest; when the data possessed are-the
Annuity,its Present value, and its Cumulated amount; with
the solution of those questions...
XXX.-Case 8th exemplified, with the solution of questions concerning
the Annuity and the Rate of Interest; having for the data-the
Principal, or original equivalent of the Annuity,—the Improved
Principal or the Cumulated annuity, and the Period either of
Improvement or of Accumulation
XXXI.-Case 9th exemplified, with the solution of questions concerning
the Rate of Interest and the Cumulated annuity; the data con-
sisting of the Annuity,-its Term,-and its Present value
XXXII.-Case 10th exemplified, with the solution of questions concerning
the Rate of Interest and the Principal, or original equivalent
for the annuity; when the data are-the Annuity,—its Term,—
and its Cumulated amount
XXXIII.—Instructions for using the Tables of Compound-Interest
XXXIV.—Solution, by the Tables, of the question concerning the Improved
Principal.......
iii
3 •
44
45
46
47
Ibid.
48
50
51
52
53
54
57
58
60
I
iv
INDEX.
Page
Sect. XXXV.-Solution, by the Tables, of the question concerning the
Discounted Principal
XXXVI.—Solution, by the Tables, of the question concerning the
Cumulated Annuity
XXXVII.-Solution, by the Tables, of the question concerning the
Redemption-fund or Annuity for cumulation ………….
XXXVIII.-Solution, by the Tables, of the question concerning the
Value of the Annuity
XXXIX.-Solution, by the Tables, of the question concerning the
Ratio of the Annuity, or the proportion between the Annuity
and its represented Principal
XL.-The excess of the Ratio of the Annuity, over the Redemption-
fund, is always equal to the Ratio of Perpetual Interest .... Ibid.
XLI. The product of the Ratio of the Annuity and of the Value of that
annuity, multiplied into each other, is always equal to unity.
XLII.-The Value of the Annuity, as expressed in the Tables, is the
quotient of the Cumulated Annuity divided by the Improved
Principal; and it is equal to the sum of the discounted instal-
ments of such Annuity
XLIII.-Differences in the Value of the Annuity, according as its instal-
ments may be understood, Annually,Half-yearly, or Quarterly Ibid.
XLIV. —The intervals between the instalments of an annuity are, by the
Tables, always understood to correspond with the intervals at
which the Interest is supposed to accrue
XLV. Hypothesis, of the Interest accruing Momently...
XLVI.-Acceleration of the progression of accumulation and improve-
ment of the Annuity, when its instalments come nearer to
each other
G
•
wit
60
L.-Exemplified rule, for ascertaining the Comparative Values of
an Annuity, in the several cases of its accruing Annually,
Half-yearly, Quarterly, at any other intervals, and even
Momently....
LI.-Leasehold property assimilated with Annuities for terms of years
certain; whence the fines due, on the renewal of Leases, are
ascertainable by the Tables of Compound-Interest
LII.—Indications for solving, by the Tables, the questions that depend
on Compound-Interest, when the rate is either lower or higher
than the limits assigned to those Tables; and Examples of the
61
Ibid.
62
Ibid.
63
Ibid.
65
Ibid.
66
XLVII.—The manner in which accelerated instalments operate on the
value of Annuities is reversed, in case of the latter accruing at
the beginning-instead of the end-ofeach Year, Half-year, or
Quarter....
XLVIII.—The Value of the Annuity bears, with the Cumulated Annuity,
the same proportion as the original Principal does with the
quantity expressing its Improvement
Ibid.
XLIX.-Correspondence between the Value of Funded Stock and the Rate
of Perpetual Interest
Ibid.
67
68
70
INDEX.
V
Page
six questions solved, under the supposition of the Rate of
Interest being inferior to that of three per cent. per annum
Sect. LIII-Examples of the six questions solved by the Tables, in case of the
rate of Interest being superior to that of six per cent. per annum.
LIV.-Questions respecting the Value of the Annuity and its Cumulated
amount, when the instalments are either more proximate, or more
distant, than those available for the Improvement at any stated
rate of Compound-interest ..
LV.-Comparative value of Funded Stock, when the Principal is re-
deemable at an assigned period, and when the market-price of
that Stock should be above par.
LVI.-Value of Deferred annuities
LVII.—Investments in Foreign securities, at high rates of Interest:
Three questions concerning the period at which the extra-risk
may terminate...
LVIII.-Questions concerning the Period at which the sums due, at equidis-
tant instalments, may be discharged by the single payment of
a stated sum; and concerning the sum required for that dis-
charge at any specific period……………….
LIX.—When the Improved Principal is stated, with reference to any two
periods of time, the Original Principal and the Rate of Interest
may be ascertained, supposing them both unknown....
LX. When any Principal is stated, and also its Improvement at the
expiration of a given period, the Corresponding Improvement of
the same or of any other Principal, together with the unknown
Rate of Interest, may both be ascertained
LXI.-When the Principal, its Improvement, and the Period of that im-
provement are given, though the Rate of Interest be kept out
of view, the period at which the same or any other Principal
will improve to a specific amount, and at the same unknown
rate, may be ascertained.
LXII.-Solution of five questions, in the case of Deferred Annuities
LXIII.-Solution of five questions that may arise in the case of a Debt,
the Interest of which has remained unpaid; when the Principal,
together with its increase, are to be converted into a Perpetual
annuity
LXIV.-Solution of five questions, arising in case of any sum-whether
more or less than the Interest due-being paid by the Debtor at
each instalment; and also in the contrary case, of the Debtor
receiving an additional sum at each instalment, instead of his
paying any interest
71
74
77
79
81
82
84
86
87
88
89
Ibid.
90
THE DOCTRINE OF COMPOUND-INTEREST: Part Second (divided
into 42 sections), in which that doctrine is applied to Life-annui
ties, and to other transactions affected with Contingency………….
Sect. I.-General considerations.....
93
Ibid.
II.--Operation of Compound-Interest, viewed as subordinate to Probability. Ibid.
vi
INDEX.
Sect. III.-Chance and Probability :-the latter assigns to the former a moral
value:—Mathematical Probability is a Relative Certitude..
IV.-FIRST GENERAL PRINCIPLE, whence is derived the mathematical
expression of Probability :-Illustration of that Principle ...
V.-Corollary of the first Principle.-Caution against misapplying the
rule......
Page
XIII. Of the natural Law of Mortality, generally considered
XIV. Of the information conveyed by the observations made in various
countries, respecting the Law of Mortality
XV. Discrepancies of the general results exhibited by different Tables
of Mortality; and discrimination required in applying those
93
Tables
XVI.-Conditions prescribed, for obtaining correct returns of the
Mortality relative to any population
XVII.-Method of inferring, from any register of Mortality, a Table of
the Decrements and Probabilities of Life ...
XVIII.-Mathematical expressions of the Absolute and of the Relative
averages of forthcoming years of life
XIX.-Method of obtaining a correct expression of the Absolute average
94
VI.-SECOND GENERAL PRINCIPLE:-The Probability, that two or more
events will result from distinct hazards, is equal to the product
of their several Probabilities.-Illustration of this position...
VII.—THIRD GENERAL PRINCIPLE; relative to the Probability of the con-
currence of two or more events, when under such mutual
dependence, that the occurrence of either must influence the
probabilities of the other events.-Illustration of that Principle.
VIII.-FOURTH GENERAL PRINCIPLE; which relates to the Value of
Expectations depending on Probability
IX.-Question, the solution of which exemplifies the application, alto-
gether, of the first four principles
Ibid.
X.-Fifth GENERAL PRINCIPLE; which relates to the Value of Expecta-
tions depending on a plurality of probable events, when any
advantage is to result from the occurrence of each
XI.—Sixtii geneRAL PRINCIPLE; relative to the Value of Expectations
depending on a plurality of probable events, when the occurrence
of some is to produce a gain, and that of some others a loss
XII. Of various other Principles, concerning the Doctrine of Probabi-
lities:-but on the first six, already exposed, are founded all the
rules of calculation relative to prospective transactions in which
the uncertain duration of life is implicated
96
97
98
99
100
101
Ibid.
102
103
104
105
106
Ibid.
1
of life
XX.-Method of obtaining a correct expression of the Relative average
of Life
XXI.-The Probable term of forthcoming years is to be discriminated
from the relative average, commonly called Expectation of life. 108
XXII.-Method of ascertaining, from any Table of Decrements, the amount
of a population, classed in quantities of persons existing of each
age
107
Ibid.
Ibid.
INDEX.
vii
\
Page
() {
Sect. XXIII.-Inferences deducible from a Table of Decrements and of Morta-
lity, in respect to the probabilities of Life and of Survivorship,
and to the value of Expectancies depending on them ......
XXIV.—Application of the second general principle, and of the corollary
(V), for calculating the Probabilities of survivorship
XXV.-The transactions assumed by societies incorporated for the pur-
pose of granting Life-annuities, of Assuring lives,—-of
purchasing Reversionary property,-of endowing Widows,
Children, or persons in old age,-and of accomplishing any
other prospective objects, are founded upon applications of the
six General Principles here set forth, combined with the
operation of Compound-Interest; and upon a judicious.
selection of appropriate Tables of Decrements
XXVI.-Calculations that relate to prospective transactions, affected with
contingency, are to proceed on the principle of reducing to
a positive value each object of distinct consideration
XXVII.-Method founded upon the 1st, the 2d, and the 4th general princi-
ciples, for correctly ascertaining the Value of Life-annuities.. Ibid.
XXVIII.-Discrimination of the values of Life-annuities, according to the
optional intervals of their instalments
-
109
110
Ibid
111
112
XXIX. Easy and expeditious method afforded, by the Tables of
Compound-Interest, for calculating the Values of Annuities
on single lives according to any Table of Decrements,-to any
Rate of Interest,—and to any intervals of instalment: that
method of approximation exemplified
XXX.-The value of an Annuity, on any single life, is always inferior
to that of an annuity for a term of years certain, and equal
either to the probable term of that life, or to the average of
forthcoming years belonging to it....
XXXI.-Rule deduced from the 4th principle, for ascertaining the value
of an Annuity the term of which is limited, both by the dura-
tion of life, and by any assigned period of time
XXXII.-Rule for ascertaining the value of Life-annuities, when their
enjoyment is deferred
XXXIII.—Assurances on life may be compensated in different ways.-
Rules for ascertaining the equivalent, either by a present
capital, or by an annual premium...
XXXIV.—Dr. Price's method of deducing, from the correct value of a
Life-annuity, as ascertained according to any Rate of In-
terest, the value of that annuity according to any other Rate. 117
XXXV.—Annuities depending on a plurality of lives may be settled either
on their joint continuance, or during the continuance of the
longest amongst those lives; and accordingly, are either worth
less than if settled on the worst life, or more than if settled
on the best life, taken singly
XXXVI.-Principle and rule, for ascertaining the value of Annuities pay-
able during joint continuance of a plurality of lives
113
115
Ibid.
Ibid.
116
118
Ibid.
viii
INDEX.
Page
Sect. XXXVII.—Rules for ascertaining the value of Annuities, when settled on
the life of the last survivor amongst a plurality of persons ..
XXXVIII.-The value of an Annuity on the longest of two or more lives,
of the same age, is always superior to that of an annuity for
a term of years certain and equal either to the Expectation,
or to the probable term of one of those lives; but, admitting
unequal ages, the value of the life-annuity may be less than
if for a term of years equal to the longest of the individual
Expectations.....
XXXIX —Value of Reversionary property, whether Capital, Annuities, or
Estates,-and whether to be entered upon conditionally, or
unconditionally; also, value of Reversionary expectations
depending on survivorship, for each party concerned, when
several concur to the contingent exclusion of each other—Mr.
Morgan's method of solving the problem of the probabilities
of Absolute survivorship between A and B..
Ibid.
123
XL —Review of the general subject of this Second Part………
XLI.-Notice concerning the 4 synoptical Tables hereafter inentioned. Ibid.
XLII.—An account of various sets of Observations, towards ascertain-
::
119
ing the Law of Mortality amongst mankind………………………...
TABLE (No. I),—of the Decrements and Probabilities of Life, as deduced
from various sets of observations distinctly considered ...
TABLE (No. II),—of the Averages of forthcoming years, or Expectations of
life appertaining to each age, according to the preceding
Table
TABLE (No. III),-of the Correct Values of Annuities on Single lives, for each
age, and according to the statements in Table No. I; ad-
mitting 4 per cent. per ann, as a standard rate of Interest.
Table (No. IV),—of the Values of Annuities on Joint continuance of two
lives;-First, admitting an equality of the ages,—and
secondly, admitting different ages of those lives.....
TABLES, solving the questions that depend on Compound Interest, accruing
either Annually, Half-yearly, or Quarterly
1 to 53.
ERRATA.
are known"; read " and p, are known."
Page 53, Sect. XXX. “ and
´Page 61, (XXVII,) read (XXXVII.)
Page 62, At the end of the last line, the word "it" is omitted.
120
126
132
134
136
138
PREFACE.
OUR MONEY TRADE embraces the national concerns of the
greater part of the civilized world, and exceeds in magnitude
any other branch of British commerce. The increased compe-
tition in that trade renders it impossible to be prosecuted with
advantage, except through great acuteness and a judicious
application of mathematical powers. Hence a work destined
to remove every difficulty and chance of error in calculation,
-by exhibiting at a single glance, and under the considera-
tion of Compound Interest, all the possible relations be-
tween Capital and Income,-can scarcely fail to be acceptable
to persons engaged in transactions of that description.
The operation of Compound Interest has been illustrated by
expert calculators, who have published sets of Tables, establish-
ing, with reference to various rates of Interest, the proportions
according to which any capital may be improved from year to
year, the comparative value of Capital forthcoming at the
expiration of any given number of years,-the value of Annuities
for terms of years certain,-and the Capitals susceptible of
being formed in process of time, by the savings of Income
b
+
X
PREFACE.
W
progressively improved; but the practical utility of those ta-
bles is limited, and inadequate to the present demands. They
are even inaccurate, as they are founded on a supposition that
the Interest of money invariably accrues in one single and
annual payment; whereas in almost all cases, especially when
Interest is derived from investments in the public funds, and
though nominally expressed as an annual prorata, it accrues
Half-yearly, and on many occasions is susceptible of accru-
ing by Quarterly instalments: an important consideration by
which the series of calculations ought to be governed.
The TABLES now offered have reference to certain rates of
Interest, between the inclusive limits of three and of six per
centum per annum, proceeding by differences of one quarter per
cent.; to specific periods, from year to year, until the hundredth
inclusive, but proceeding as far as the sixteenth by successive
portions of years; to distinct considerations of the Interest
accruing either annually, half-yearly, or quarterly; and ac-
cording to those data, they immediately solve six different
questions.—First, that of the increase of Capital, supposing it
improved by the operation of Compound Interest at a uniform
rate; Secondly, that of the present value of forthcoming capital,
as reduced by the Discount, taken in a converse ratio of the
improvement at Compound Interest; Thirdly, that of the
capital ultimately formed, by a periodical Income accumulated
and improved as before mentioned; Fourthly, that of the
Redemption-fund, or the requisite proportion of annual savings
out of any income, in order to create, by means of their accu-
mulation and improvement, the Capital that may be wanted at a
prescribed period; Fifthly, that of the Capital represented by
*
F
1
PREFACE.
xi
any annual Income, or the present value of that income, the
last instalment of which is forthcoming at the period referred
to; Sixth and lastly, that of the proportion which an Annuity,
terminating at a specific period, should bear with present Ca-
pital, for constituting an equivalent.
Besides those principal questions, arising from the data
above described, the same Tables further solve, mediately and
with the assistance of simple indications, not only a great
variety of incidental questions naturally growing out of the
general subject, but also those in which the considered rate of
Interest may be either higher or lower than the limits assigned
to the Tables. These again yield an approximate solution
of the questions concerning the Rate of Interest itself, or con-
cerning the specific time, when failing either of those par-
ticular data any other may be introduced; such calculations
always requiring three different data, the third of which is, in
the stated answers to the six questions first propounded, im-
plicitly understood to consist of unity, whether considered as a
present or a future Capital, or as a corresponding Income.
-
The text is divided into Two Parts, preceded by an Introduc-
tion, where some leading and important points are discussed.
The First Part contains an exemplification of all questions
to the solution of which the above-mentioned Tables are ade-
quate; and reuniting every consideration relative to the
operation of Compound-Interest, referable to periods of time
unconditionally determined, it will constitute, in all respects,
a complete theory of that branch of knowledge.
•
xii
PREFACE.
The Second Part, exclusively devoted to Life-Annuities, is
nearly reduced to an abridgment of what has already been
exposed on that subject by authors of great merit, and whose
immediate province any attempt to invade could scarcely be
justified, except upon grounds not here assumed.
}
Desirous, however, to extend as far as possible the practical
utility of this work, a few more Tables have been superadded;
stating from authentic sources, and from observations carefully
collected during long periods, the distribution of a whole popu-
lation into relative quantities of individuals of different ages,
and the progression of mortality amongst them, according to
a systematic classification of certain general circumstances of
human existence. Whence deducing, as indispensable data,
the Expectation of life for each respective age, together with
the various probabilities attached, whether to a present life's
further enduring any determinate period of years, or to the
joint continuance of any two or more lives during such pe-
riod, the questions susceptible of frequent recurrence concern-
ing Life-Annuities are enabled to be solved; and more parti-
cularly, the scale of value of those annuities-the ascertain-
ment of which is generally considered to be a matter of some
difficulty is here established, with sufficient accuracy, and
under comparative points of view.
As the claim of a production of this kind, to public favour,
must chiefly rest upon the correctness of the results set
forth, nothing has been spared towards securing this point
in the utmost degree. The calculations have undergone a
rigorous verification; and typographical errors have, by
PREFACE.
xiii
repeated collations, been sedulously guarded against. If, not-
withstanding the extreme care bestowed in these respects, any
errors should have escaped detection, (amongst upwards of
thirty thousand results exhibited,) they will be but few and
always confined within the limits of two units, at most, of the
last decimal figure; errors which could never be of real impor-
tance. It may be confidently advanced that the results, as ex-
pressed in the Tables, are safely to be confided in for all practi-
cal purposes. The means, however, of corroborating their
accuracy, are contained in the text that precedes them, and in
those developments of mathematical reasoning that properly
constitute The Doctrine of COMPOUND INTEREST.
W
A
INTRODUCTION.
THAT which is called MONEY, from the latin moneta, signifying
"to give notice," has been invented to serve as a common
standard of reference, in expressing the relative values of other
commodities; and for facilitating the barter of those commo-
dities, which the money should represent in a general sense.
At an early period of historical record, the more precious
metals-Gold and Silver-were preferably selected for per-
forming that office, in consequence of their apposite properties,
such as Solidity, Malleability, and Durability; and also from
their comparative uselessness towards the ordinary purposes of
life, though not obtainable from the bowels of the earth with-
out a considerable expense of human labour. During many
ages, the denominations of Money continued to be identical
with those of Weight; and each article of necessity or of com-
modity was assimilated, in respect to general value, with some
quantity of uncoined gold, silver, or even inferior metal.
When afterwards these became shaped into Coin, the integrity
of which was vouched for under public authority, that metallic
currency assumed a character more or less independent of the
xvi
INTRODUCTION.
consideration of Weight, and remained liable to be arbitrarily
dealt with at the hand of power. For a while, Money was
issued as the invariable representative of certain things, and
principally Cattle, of which the pieces of coin bore the respec-
tive impressions; whence from pecus has been derived pecunia.
The history of the Circulating Medium, which more exten-
sively to retrace would be an unnecessary deviation from our
main object, informs us respecting not only the frequent varia-
tions that have taken place in the reciprocal valuation of Gold,
Silver, and Copper, as established by their coinage at different
periods and in different countries; but also respecting the
numberless frauds, adulterations, and abuses of every kind, that
have been practised from time to time upon the Money-system,
by barbarous, crafty, and profligate despotism. Ultimately,
and by a prodigious enlargement of that system, the represen-
tative metallic currency has admitted its own and boundless
representation, by means of paper tokens, and sub-tokens, under
a certain conception of Public Credit that seems not yet to
be precisely defined; thus generating various national as
well as private abuses, never before imagined, and giving
occasion to diversity of opinion respecting the future results of
so bold an experiment. The fruitful principle of Generaliza-
tion, though applied with the greatest success to some elevated
branches of science, should perhaps not without caution be
extended to Political Economy; in which the bias of human
propensities, and the versatility of the passions, are to be taken
into consideration as combined with other operative causes.
The quantity of money that may be required, for the single
INTRODUCTION.
xvii
purpose of a circulating medium, is confined within limits
depending in some degree on the convenience afforded, by the
materials, the form and divisions of that money, for the rapidity
of its circulation. In large and commercial cities, existing
under a liberal and protecting government, and where the
transfer of money is likely to be often called for, even in con-
siderable sums at each time, the establishment of Deposit-
Banks, of which that of the city of Hamburgh offers a very
good specimen, is calculated for promoting that object; the
transfer in a general Account-book being operated with much
less waste of time, than it could be by a physical delivery from
hand to hand. In Great-Britain, that office is imperfectly
executed by Banks of circulation and by private Banking-
houses, whose transactions are referable to the fictitious
currency already mentioned. The best that can be said of our
money-system is, that it has hitherto worked its way,
notwithstanding great and palpable inconveniences.
Money, from its promiscuous and unequal modes of circula-
tion, has a tendency to accumulate into masses, more or less
considerable in different hands. Those masses, during the
intervals in which they remain unemployed, assume the deno-
mination of Capital.
Increase of Capital is the constant object of Industry
and Economy. When the increase of capital, in any coun-
try, has attained a certain height, by the profits of an exten-
sive foreign trade, such capital may in process of time form
an important branch of the national wealth; but that in-
crease would be unavailable, and its advantages would be
•
C
GRO
xviii
INTRODUCTION.
neutralized in consequence of a corresponding advance in the
prices of all other commodities, if the Capital were not also to
acquire a merchantable character, both at home and abroad.
Capital, alone, is however a precarious and but a secondary
species of wealth. The fate of countries, -as exemplified by
Spain, for the last three centuries,—that relinquish creative
industry merely to pursue the acquisition of silver and gold,
must inevitably be to subsist in a continual and wretched state
of dependance upon other countries, for those necessaries and
commodities of life, which, being assuredly reproduced in pro-
portion with the demand for their consumption, constitute both
the principal source and the real substance of wealth.
Capital, unaided, remains an inert and useless mass. To-
wards becoming productive, its association with Industry is an
indispensable requisite; and then it is entitled to share, with
the latter, in the profits derived from their combined powers.
The proper destination of Capital is to stimulate industrious
pursuits, to favour useful enterprize, to promote beneficial
improvements, and ultimately to participate in the various
advantages thence resulting. This is the true foundation,-
the equitable motive of what is termed Interest of Money.
2
The implied share of profits, appertaining to Capital, is
according to established custom compounded for in a certain
ratio of the capital employed; which ratio is understood as
referable to an annual period, and multipliable in proportion
with the quantity of years, or otherwise the mere fraction of a
year, during which such employment may continue. What
is commonly called Rate of Interest is only a different mode
---
INTRODUCTION.
xix
1
of expressing, by duplication of terms, that ratio which is a
simplified mathematical term; and thus explaining that 3, 4,
5, or any other quantity of money, whatever may be its deno-
mination, is to remunerate the hire or use for one year, of a
quantity uniformly expressed-namely one hundred-of money
similarly denominated and of equal value. The Interest of
money, although specified at an annual rate, is payable at such
regular periods as may be agreed upon; whether by Annual,
by Half-yearly, or by Quarterly instalments, or even at shorter
intervals; each payment being always proportionate with the
annual rate.
In almost all the money transactions that do not exactly.
consist in the purchase or the sale of commodities, and more
particularly in cases of Loans between individual persons, or of
those made to Governments under specific contracts; and
whether, according to the terms of such contracts, the reim-
bursement of the capital advanced be secured or not by any real
pledge, or whether the loan be made on simple Credit, with
a symbolical pledge only; it is to be understood, in those
various cases, that the above definition of Interest becomes a
mere fiction, inasmuch as no association actually takes place
between the borrower and the lender, for any purpose common
to both parties. Yet all those transactions are uniformly
regulated on the principle we have described, because it is
obvious that no justifiable motive for borrowing can possibly
exist, except upon the supposition of some profitable or truly
beneficial purpose contemplated in the intended application of
the money.
↓
XX
INTRODUCTION.
Transactions that have Interest of money for their object
are founded upon what is called Credit, or the faith with
which the expectation of reimbursement and remuneration is
accompanied.
Credit may be yielded either to substantial pledges, to
individual persons, to certain societies acting collectively, or
to those who are entrusted with the administration of the pub-
lic finances; and in the latter case, the credit rests on what is
usually denominated Government-Security. When it is yielded
to real pledges, their imperishable character is a requisite no
less essential than their adequate value; when to indivi-
duals, the estimation of their moral integrity possesses greater
weight than the merely external appearances of wealth;
and when to societies collectively, their permanency is a
paramount consideration. It is the exclusive privilege of
these, and especially of Governments, to be susceptible of
that permanent character, enabling them confidently to
look forward to remote periods for the accomplishment of
rational purposes, through the developement of resources
almost inexhaustible, and that seldom fail to be supplied
by lapse of time aided by a wholesome administration:
nevertheless, a decayed state of things, resting its present
claims on antecedents only, not unfrequently obtains Credit
upon illusory grounds. The concerns of incorporated
societies may be so interwoven with the financial system
of the Government, as to preclude the separation of their
respective credits; that of the former being rendered su-
bordinate to that of the latter, on the faith of which only
depends the existence of those societies. Public Credit is
C
INTRODUCTION.
xxi
not, however, the same thing as the credit directly possessed
by the Government: it consists in the generality of mutual
facilities obtained in money transactions, throughout any
country, from a universal feeling of confidence and security,
which a wise and a sufficiently protecting government alone
can convey. When that foundation is shaken, Public Credit
is at an end.
From the natural tendency of things, abstracted from all
interference of the civil laws in regulating the Interest of
money, the rate of that interest, when freely stipulated
between man and man, can be governed by no other than the
following three circumstances :-the competition existing in
the money-market, the degree of risk attending any par-
ticular investment of capital,-and the more or less facility
for reconverting at pleasure into Capital the proposed mat-
ter of that investment.
First:-The prices of all commodities being liable to fluc-
tuate, according to the proportions between the supply of
and the demand for those commodities at the time being, this
circumstance possesses a similar influence upon the value of
Capital, considered in its relation with the periodical income it
should produce; whence Capital becomes less valuable, under
this point of view, in proportion with the increase of that par-
ticular description of wealth; and all Interest must cease,
where no demand for Capital exists. From these premises we
may infer that, in all thriving countries, the Government,
as well as the great landed proprietors whose estates are
susceptible of improvement, will probably be benefited in con-
M
xxii
INTRODUCTION.
sequence of their constantly employing Capital taken at
Interest; the rate of which Interest is not unlikely to sustain
successive reductions, whilst the nominal Capital, thereafter to
be repaid, will probably have sunk a part of its original and
comparative value. But in the absence of adequate demand,
and of a sufficiently remunerating employment for Capital in
the country where it may abound, the deficiency, in that
respect, is capable of being supplied by productive employment
of that capital in foreign countries; the risk attending which
is to be duly appreciated.
This dependance, however, proves that the unlimited increase
of Capital, which is rather the symbol than the reality of
wealth, could not without inconvenience and ultimate disad-
vantage be aimed at, as the principal means of establishing the
strength of a nation, and its permanent influence abroad. Nor
does it appear, from the lessons of History, that extensive
dominion, when chiefly founded upon commercial advantages
and accumulation of wealth, is susceptible of maintenance
beyond a limited time.
FOR
Secondly-The risk of a total or only of a partial loss,
at which risk any capital may be invested, cannot be al-
ways the same; and it is obvious, that if the proportion of
profit, contemplated in the association of Capital with Industry,
should be sufficient for compensating that risk and for
warranting its being incurred, the share of such profit, which
in the shape of an annual rate of Interest is to devolve upon the
Capital, ought in that case to be more considerable than the
requisite share on a supposition that the Capital were subjected
INTRODUCTION.
xxiii
{
either to a less risk, or to no risk at all. But in a mathematical
sense, the risk is not in any case to be considered as absolutely
nul; although it may often be disregarded on the score of
presumed exility. Therefore, any stipulated rate of Interest
is implicitly compounded of two parts ;-remuneration for the
Capital employed, in proportion to the period of that employ-
ment, and compensation for the risk: a distinction abundant-
ly exemplified by the very different rates of Interest, at
which the loans of British capital are granted, at home or to
foreign countries.
Nevertheless, and discarding all consideration of the various
degrees of risk attending the application to different purposes
of borrowed Capital, Laws have been enacted, at almost all
times, and nearly in all countries, for restraining the rate of
Interest on loans of money.
Let us remark, that in the origin of what is called Usury
Laws, the necessary protection of the absolute poor against
tyrannical extortion, was the only view entertained. Very
little reflection will then be necessary for discovering, that
however well adapted those laws may have been to the state
of society in former times, the case is widely altered; money
having become as much a merchantable article as any other,
and having an open market in every great city of Europe. In
these days, such laws serve occasionally to obstruct mutual
accommodation, upon terms justified on one hand by fair com-
petition, and on the other hand by a due consideration of the
greater or less risk that may attach to the intended appli-
cation of any Capital; they are not less unjust than impolitic,
xxiv
INTRODUCTION.
:
inasmuch as they fail to operate according to the principle of
reciprocity; they are in fact a direct infringement on the
right of property, to free disposal, and to equal protection, of
whatever description such property may be; they could not
any longer be retained, except for the purpose of favouring
particular classes of persons, to the disadvantage of other
classes; and they should be considered as one amongst those rem-
nants of barbarism, which we are always slow in eradicating.
Thirdly :-The effect of competition, in the money-market,
is not only to establish the lowest possible rate of Interest, in
those cases where the nature of the investment affords the
utmost degree of security for the reimbursement of Capital
borrowed; but also to introduce, between different modes of
investment equally secure, another important distinction in the
mutual relation of Capital and Interest. If the investment
should be of such a description, that the Capital continued
available at all times, and at a short notice,-every chance
being retained of its more profitable or more eligible invest-
ment, there exists in that case a sufficient inducement for
preferring this mode of investment, even at a rate of Interest
further reduced. But if any proposed investment, however
substantially secure, should require the Capital to remain
absolutely locked up during a considerable time,-precluding
all favourable chances until the expiration of that time,-then
the rate of Interest ought to be more elevated than in the
preceding supposition. In a country abounding with floating
capital, there are various motives in constant operation, tending
to establish a remarkable difference between the ratios of
Interest depending on this third circumstance. The position
?
INTRODUCTION.
XXV
here advanced is corroborated by the comparatively low rate
of Interest yielded by investments in the British Funded-Debt,
which are accompanied with the convenience of commanding
the Capital at any time, by means of re-selling in the public
market the Stock of that debt, though at some contingent
profit or loss, in consequence of the perpetual fluctuation in
its price; and by the still lower rate of Interest afforded by
investments in Exchequer Bills, the integral amount of which
is equally available at all times, without incurring any such
contingency. When the rate of Interest is remarkably low,
the difference alluded to becomes more sensible; but if
uncommonly high, the lender may on the contrary consider it
an advantage to secure, during a long period, the employment
of his Capital at that higher rate of Interest.
It follows from all this, that Interest and Income, though
both relative to an investment of Capital, are not synony-
mous expressions; that the truly available Income, arising
from any Capital laid out at Interest, is always in a less propor-
tion than the rate of Interest stipulated for; and that, mathe-
matically speaking, the present value of any sum of money,
receivable at a future time, is not ascertainable by merely
subtracting from it what is called Discount, considered as an
equivalent for the Interest of Capital employed from the present
to that future time, but by the further subtraction of another
quantity, equal to that first residue multiplied by the fraction
expressing the probability of failure in the expected receipt *.
* All mathematical probabilities are represented by fractional quantities,
expressing the exact proportions of those probabilitics compared with unity, taken
as the expression of certitude.
d
xxvi
INTRODUCTION.
When any number of successive instalments of the Interest,
according to a stipulated rate, or one implicitly under-
stood, is not regularly paid to the Creditor; but by mu-
tual consent is allowed to remain in the hands of the Debtor,
who agrees that each instalment falling due shall be considered
as a new loan, on the same terms at which the Capital of the
original loan was granted to him; there arises, from this
specific mode of transacting for Loans, what is termed Compound
Interest; by which is implied, as a necessary consequence, the
constant increase both of the Capital and of the Interest, in a
geometrical progression, and for any length of time during
which the same system may continue to operate.
From this last consideration is generated a series of combina-
tions, the more susceptible of being complicated, as they involve
in their generality many transactions of frequent recurrence,
and in which the relation between Capital and Income is made
to depend even upon uncertain events, such as the duration of
human life, and therefore ascertainable only in the shape of
their respective probabilities.
Annuities are to be discriminated, as either perpetual or
terminable. A Perpetual annuity is the sum of money propor-
tioned with the Capital of any loan, and covenanted for annual
payment by the borrower to the lender, at certain intervals-
usually of 12, 6, or 3 months from each other, and in due
proportion at each of those instalments, according to a stipulated
rate of Interest, until the reimbursement of that Capital; but
without any other limitation of time. A Terminable annuity
differs from the preceding, inasmuch as its duration is limited
INTRODUCTION.
xxvii
>
either to a certain term of years, or by the occurrence of some
anticipated event; when in either case the Annuity ceases,
and the Capital of debt which it represented becomes
cancelled.
Perpetual annuities that originate in loans to the British
Government, and represent the Capital of those loans, are issued
to the public at large, with the conveniences of being divisible
at pleasure, and transferable with ease and without attendant
expense; they are thus sold in the public market, under the
general denomination of Stock, the price of which, always sub-
ject to fluctuation, is expressed by the sum given for one hun-
dred pounds share of that Capital; but similar annuities
resulting from contracts between individual persons are
divested of those accompanying conveniences. Perpetual
annuities issued by the Government may, by a special clause of
contracts for loans, be constituted irredeemable during a certain
term of years; by which is understood that the indebted party
relinquishes, until the expiration of that term, (from whatever
consideration it may be,) his natural right to discharge the
capital of his debt; thereby assuring to the proprietors of Stock,
of that description, the continuance of a rate of Interest con-
sidered as being advantageous for them. It is obvious, that
the higher the constituted rate of Interest, the more valuable
to the debtor is the privilege of redeeming his debt at pleasure.
Loans may be contracted for, and debts of other descriptions
may be created, in such a shape as to supersede the obliga-
tion of refunding the capital at any time. This is when annui-
ties are constituted payable, either during a specific term of
ભગ
xxviii
INTRODUCTION.
years, during the natural life of any individual person, or
during the longest amongst the lives of several nominated
individuals. In the last two cases, the probable term of such
life or lives,-which by certain rules is deducible from the
respective ages of the nominees, combined perhaps with other
considerations,— becomes one of the elements of calculation,
instead of any period of time precisely determined; but in
all cases of Terminable annuities, their ratio, or proportion
with the Capital they represent, must be more elevated than
the ratio of the corresponding Perpetual annuities, or simple
rate of Interest, that should otherwise have been stipulated
and payable until the reimbursement of the Capital.
From the nature and tendency of contracts for Terminable
Annuities of any kind, it is immediately perceptible that the
excess of their ratio, over and above the ratio of simple Interest,
ought to be neither more nor less than a just equivalent for the
Capital, considered as forthcoming when the corresponding
Annuity attains its period of termination. If therefore we
suppose four per centum per annum the rate of Interest
stipulated for a loan; and that, independently of paying such
Interest, the borrower annually set apart an additional one per
cent. of the Capital, improving such surplus by the process of
Compound Interest at the same rate of four per cent., until the
accumulation of a sufficient sum for discharging his debt,
this object would be attainable at the expiration of forty-one
years, as will be seen in page 20th of the TABLES here annexed.
Whence follows, that an Annuity of five pounds, limited to forty
years duration, may be considered as the equivalent of
another of four pounds only but payable perpetually, or other-
one
INTRODUCTION.
xxix
wise until the reimbursement of one hundred pounds Capital
represented by the latter Annuity. And it further follows, that
the same purpose of redemption may be accomplished by the
Debtor, at the expiration of forty-one years, in consequence of
his paying at the rate of five per centum annually to the creditor,
during that limited period, as it would have been in case that he
paid him in the proportion of four per cent. only, and undertook,
by the personal management of the above mentioned surplus, to
create the Capital ultimately required in that case for discharg-
ing his debt. If we should suppose any other rate of annual
Interest than that of four per cent., or any other proportion of
annual surplus than that of one per cent, and applicable in the
same manner, the results would not otherwise differ from the
preceding, than as the surpluses in a greater proportion, or their
improvement at a higher rate of Interest, would respectively
contribute to abridge the period of redemption; and as that
period would on the contrary be protracted, under the suppo-
sition of the rate of Interest being lower, or of the surplus
being less.
Proceeding upon this general principle, of applying the
savings from a periodical Income to the future redemption of
Debt, the Governments of some indebted nations have instituted
what is called Sinking-funds; consisting of annual sums set
apart from the public revenue, or sometimes appropriated from
merely fictitious resources,-and through the real or the
simulated improvement of which, at Compound Interest, a
reduction in the Capitals of such National debts is professed
to be successively obtained. We have elsewhere enlarged on
this important subject.
XXX
INTRODUCTION.
A Sinking-fund assumes the denomination of Redemption-
fund, when the savings of Capital, of Income, or of both,
and destined for improvement at Compound Interest, with the
intention to redeem at the expiration of a stated period any
specific Capital of Debt, are with sufficient exactness
proportioned with that object; so as to insure its attainment
according to all moral probability.
Transactions between individual persons are liable also to be
framed, in a variety of shapes, upon the same principle. When
they relate either to the mere improvement of Capital at Com-
pound Interest, to the purchase or to the grant of Annuities
for specific terms of years certain, or to the accumulation of
savings out of a certain income for creating any determinate
capital at the end of a given time, the solution of the questions
concerning those transactions depends on the possession of any
three amongst the following five data: first, the period of
time; secondly, the rate of Interest; thirdly, the present or
the future Capital; fourthly, the quantity of the terminable
Annuity; fifthly, its cumulated amount at the expiration of
the period referred to. All other quantities involved in the
question are then capable of being deduced, with more or less
facility, and by known rules of calculation, from those three
data only. But the transactions that involve the further consi-
deration of uncertainty, as to the duration of lives, are to be
viewed under a different aspect. They possess a peculiar
character; and the solution of questions concerning Life-assu-
rances and Life-annuities, reversions of Capital and reversionary
annuities, survivorships, endowments, or other analogous trans-
actions, requires the additional introduction of specific data.
-
INTRODUCTION.
xxxi
We have already made the remark, in a preceding publica-
tion, that the collective results of a multitude of transactions
relative to life interests, embraced by Governments or by
incorporated societies, exhibit only inconsiderable variations
from what had been previously ascertained as the result of cal-
culation upon satisfactory data; but that the case is widely
different, when solitary transactions of this description are
ventured between one individual and another, or even when the
quantity of assimilated transactions is too small for affording
sufficient probability that the average results shall in a certain
degree approximate the mathematical anticipations; and there-
fore, that such solitary transactions not only constitute a species
of gaming, but scarcely retain the character of equitable con-
tracts, particularly when it is considered that an accurate
ascertainment of the respective values of Expectancies, duly
discriminated, is often unattainable for the contracting parties.
•
Besides, the speculations upon life and death, between
private individuals, are attended with great moral inconve-
nience. This entirely disappears in the multifarious transac-
tions of societies, whose principal object is to secure, by a
wholesome administration of their affairs, certain average and
equitable results.
Nor should the profits derived to those societies, if moderate,
be considered as a disadvantage for the individuals contracting
with them. In human concerns, individually considered, there
is always a moral and appreciable disadvantage attending
uncertainty. The real value of property, if the whole be
exposed to any single risk, is by no means equal to the integral
11****
xxxii
INTRODUCTION.
amount of that property, minus the mathematical proportion at
which the risk incurred might be accurately ascertained; and
there may possibly exist a great difference between the value
of the Mathematical Expectation and the value of the Moral
Expectation, both deduced from the probability of the result.
The relative difference will be the more considerable, and
therefore the value of the latter Expectation-which is chiefly
to be considered-will be less, in proportion as the risk is pre-
sumed to be small; as for instance the ordinary risk of Fire.
But admitting, instead of a single risk incurred by the whole
undivided property, a division of that property into portions
exposed to separate risks although of similar kind, the relative
difference between the two values will then be reduced; and
it would ultimately vanish altogether, if an infinite division
of risks could be calculated upon. Whence follows, that the
values of those two Expectations very nearly coincide, in
respect to the societies alluded to; and that each individual
assured by them, under just regulations, gains in reality, by
the mere exchange of his uncertainty against a determinate
and convenient object of certitude.
1
THE
DOCTRINE
OF
COMPOUND INTEREST.
Part First;
CONTAINING THE APPLICATION OF THAT DOCTRINE TO PERPETUAL ANNUITIES, AND
TO TERMINABLE ANNUITIES THE PERÌOD OF WHICH IS UNCONDITIONAL.
THE doctrine of COMPOUND INTEREST chiefly consists in the developement of a
very few mathematical principles.
Added
(I.) FIVE quantities are here to be considered, as bearing an invariable rela-
tion with each other, under all their possible combinations; and when describing
those quantities in a general point of view, we shall affix to each a conventional
sign, or initial representative, for the purpose of future reference, as also for that
of facilitating certain expressions in an algebraic form, and immediately poisting
out the arithmetic operations required for numerically solving the various questions
susceptible of arising on this subject, when the indispensable data are afforded.
1
First:-Let p signify the Principal; implying either any present capital
destined for improvement at compound interest,-the present value to which is
reducible any other capital more considerable, and forthcoming at a future.
period, or else the equivalent for any annuity payable at certain instalments
computed from the present time, and until a stated period when such annuity is to
terminate.
Secondly:-Let a signify that Annuity, or one of the equal sums successively
forthcoming at the expiration of equidistant periods, whether consisting of one
year, of six months, of three months, or of any other interval of time commencing
from the present.
Thirdly:-Let n signify the quantity of those forthcoming sums, each of
which equal to a.
e
34
THE DOCTRINE OF
..
m
نفر
she
Lake
Fourthly:-Let r signify the Ratio, or the fraction expressing a standard
proportion of interest, produced by every unit comprised in whatever invested
capital, and for any one of those periods the'quantity of which is expressed by n.
Fifth and lastly:-Let m signify either the Improved Principal, originally
P
and invested at compound interest in the ratio r, during the time implied
by n,—any capital forthcoming at a future period, and which by discount under
the consideration of compound interest would be reducible to p as the present
value of that capital,—or the cumulated amount of the successive sums a, added
together, with their respective improvements at compound interest according to
the ratio r from each instalment to the next, and until the period of termination
of the annuity by the receipt of the last of those sums.
(II.) It follows from those definitions, that by the operation of Compound
Interest during any period n, the improvement of every unit comprised in the
principal p must, at the respective stages of such period, accord with the
successive terms of this geometrical progression
÷ (1+r)¹ : (1+r)² : (1+r)³: (1+r)': &c.………...……..to (1+r)";
and therefore that the present value, by discount of every unit comprised in any
sum forthcoming at those different stages respectively, must also accord with the
successive terms of this other progression
=
Questions:
1
1
1
1
(1+r)² · (1+r)² · (1+r)² (1+r); : &c.
'
(III.) Having thus observed that unity, considered as the principal, will, at
Compound Interest in the ratio r and at the expiration of a period n, improve
into a quantity (1+r)", any ascertained principal, or present capital expressed
by p, will consequently improve into p(1+r)"; whence, and expressing by m
this improved principal, if any three of the four quantities, n, r, p, and m, be
known, the fourth will be ascertainable by resolving into a numerical expression
some one amongst the following four, which are mere transformations of the
above original equation, identical with each.
m (the improved principal) = p(1+r)”.
p (the original principal, or present capital) =
.to
log.m-log.p
n
or log. (1+r)
n (the quantity of successive instalments) =
=
1
(1+r)"
r (the ratio of interest, corresponding with each interval of time)
m
(1+r)"
m
= {"} ==
·log. m-log. p
log. (1+r)
-1;
COMPOUND INTEREST.
35
1,
It
may be expedient to illustrate these positions by appropriate examples.
(IV.) Suppose the principal p to be £12500, 3 per cent. the nominal rate
of annual interest at which that present capital may be improved by half-yearly
instalments, the ratio r in that case 0·0175; and that a term of 25 years were
prescribed for the improvement, n being consequently equal to 50 half-yearly
instalments of the interest. We have the improved principal m = p(1+r)”
= 12500 X (10175) 50; and reducing this to a logarithmic expression,
log. m log. 12500+50 (log. 1.0175) 40969100 + 50 (0-0075344179)
= 4·4736309. This last logarithm corresponds with £29760, the sum into
which such principal p is improvable at the expiration of 25 years.
(V.) Suppose m a sum of £40000 required at the expiration of 20 years, and
to be obtained by the improvement at Compound Interest of a present capital p,
invested at an interest of 4 per cent. per annum, accruing by quarterly. instal-
ments; in which case r = 0·01, and n 80. That capital to be invested, or
40000
the discounted value of the future capital m, being p =
(1+r)" (1·01) 80
will be ascertained by an operation analogous to the above; and we have
log.plog.40000-80 (log.1.01)=4.6020600-80(0-0043213738)=4.2563501,
which last quantity is the logarithm of £18045, the capital to be invested.
m
(VI.) Suppose a principal of £10000, intended for improvement at Compound
Interest during 37 years, so as to produce at the expiration of that period a
capital of £40000; and the question, in that case, what the rate of annual interest
should be, under the stipulation of half-yearly instalments. The data being
p10000, m = 40000, and n 75 (half-years), we have log. (1+r)
log. m―log. p_log. 40000—log. 10000
n
75
=0·0080274666;
this last logarithm corresponds with 1·018656 = 1+r; whence r 0·018656
is the ratio of half-yearly interest, the nominal rate of which per annum is therefore
0·037312, or a very small fraction less than 34 per centum.
i
4.6020600—4
75
whence n
log. m log. p
log. (1+r)
0.00485853
very nearly 144 quarters, or 36 years, the requisite term.
=
(VII.) Suppose £25000 invested at an interest of 4 per centum per annum,
payable quarterly, with a view to accumulate in process. of time, and by the
operation of Compound Interest from each quarter of a year to the succeeding
quarter, a capital of £125000 ultimately required; and be it asked the period
n, computed from the present time, and at the expiration of which the desired
improvement will be accomplished. In this case, we have p = 25000,
m = 125000, and r =
0.045
4
= 0.01125;
5.0969100 4.3979400
www.
=143.864, being
•
36
THE DOCTRINE OF
(VIII.) Questions analogous to either of the preceding four may be put in
various shapes; but the similitude of cases,-as it chiefly rests upon the possession
of any three data amongst the general quantities m, p, r, and n-can scarcely
fail to be easily discernible.
(IX.) But in order to establish, invariably, the relation between each of the
five general quantities susceptible of being considered in questions of this kind, and
the other four of those quantities, let us now substitute, for unity, any quantity a
of periodical income terminating at the expiration of a time expressed by n; such
quantity a to be compared with the Principal p, and necessarily discriminated
from the ratio or standard rate of interest. It clearly follows again, that p can
be no other than the addition of the respectively discounted quantities a, under-
stood as the sum forthcoming at each stage of that period n. Those discounted
quantities will then accord with the successive terms of the progression
: &c.....to
a
a
÷
(1+r)i · (1+r)² ·
the united value of which
P =
a
a
a
a
+
(1+r); + (1+r)² + (1+r) +
a
(1+2)"
a
a
(1+r)³ (1+r)*
+
a
(1+r)*
a
a (1+r)" — a
r (1+r)²
(by the general rule for summing the terms
of any geometrical progression) is the Present value of the Terminable annuity.
modified, will instead of p =
Present value of the Perpetual annuity.
a
(1+r)"
;
(X.) The quantity a, in its relation with p (the Principal), has hitherto been
considered as an Annuity, or as the portion of an annuity terminating at a period
n that indicates the precise quantity of forthcoming instalments of such terminable
annuity. But admitting, in compliance with general custom, the fiction of
Perpetuity; by which is understood that the annuity shall continue, either
without end, or until the redemption of the capital it originally represented;
the last term
of the preceding progression, and consisting of a then
a
(1+r)"
↑
a
+ &c. .... to + (1-
a
(1+r)"
divided by an infinite quantity, becomes null, as it is no other than the
discounted value of a, a quantity infinitely remote and in fact never forthcoming
at all. Hence the addition of the successive terms of the above progression, thus
(1+r)™
a
be p; expressing the
Under this hypothesis, in which the quantity n is discarded from consideration,
if
p, a, and r, be successively admitted as the unknown quantities, they are
readily ascertainable by resolving into numerical expressions the three algebraic
COMPOUND INTEREST.
37
a
NIE
the last two of which are mere transformations of the first.
expressions (now extremely simplified,) p =
apr, and r =
It is self-evident that, under this same hypothesis, the quantity m, considered
as questionable so long as n could be retained as a limited quantity, must now
be discarded also; because any sum, whether p or a, and however small,
necessarily becomes infinite,—that is to say, greater than any assignable quantity,--
when proceeding from its uninterrupted improvement during eternity; and there-
fore no question can possibly arise in the present case, as to any ascertainable
quantity m.
(XI.) Examples of the three questions relative to Perpetual Annuities.
First:-Let £5000 be the principal p, or the value of an annuity payable by
quarterly instalments, the first of which falling due in three months from the
present time; and let the nominal rate of annual interest be 5 per centum, or in
the ratio r = 0.0125 for one quarter. Each of those quarterly instalments will
then be a pr= £5000 X 0.0125 £62. 10s.; four times which, or £250,
will be the Annuity. The multiplication, at once, of £5000 by 0·05 or by the
rate, per annum, yields the same result.
a
r
ཅ |ིག
Secondly:-Let the principal be £5000, and the Annuity nominally £275,
a
137.5
but in reality £137. 10s. half-yearly; then the ratio r —
=0.0275,
p
5000
corresponding with each half-yearly instalment; and 0·055, otherwise 5½ per
cent., will be the nominal rate of Interest per annum, according to which such
annuity is paid. The same result is obtained by merely dividing 275 by 5000.
75
0.01125
Thirdly:-Let the annuity be nominally £300, but accruing by quarterly
payments of £75 each; and suppose 44 per cent. per ann. the nominal rate of
Interest by which the calculation is to be governed, which reduces the quarterly
ratior to 001125; the present value of a perpetual annuity, to the above
£6666.13s.4d. And if the whole
amount, will then be p
annual sum of £300 be divided by 0.045 or the rate of annual interest, the same
quantity £6666. 13s. 4d. will exhibit the capital represented by that Annuity.
(XII.) Should the first instalment of an annuity, whether terminable or
perpetual, be paid at the outset of the constituting transaction, instead of being
payable at the expiration only of the first year, the first half-year, the first quarter,
or at the expiration of any other specific term, computed from the period of
58
THE DOCTRINE OF
investment of the corresponding principal p; such anticipated payment, of any
sum a, ought in that case to be considered in no other light than as a reduction
of the invested principal itself. This is indeed the simplest and clearest view to be
taken of the matter; whence the relations between the several quantities a, r, n,
and p, need not be established by any other rules than those already set forth;
requiring only to substitute p-a, for the principal, instead of p, and to prosecute
the calculations accordingly.
(XIII.) If any future capital m is to result from the accumulation of each por-
tion a of income, improved at Compound Interest until the expiration of a period
n, that quantity m can be no other than the addition of the several quantities a,
respectively improved until that term of expiration. And if it be further
considered, that the first portion a will be forthcoming only at the end of one
of the stipulated intervals computed from the present time, and each of which
produces interest in the ratio r; such portion of income consequently benefitting
no more than by a quantity n-1 of successive stages of improvement, whilst
the last portion a, forthcoming at the complete expiration of the period n, is
not susceptible of any improvement at all; we shall have by analogy with the
progression (IX)
m=a+a(1+r)¹+a(1+r)²+a(1+r)³+a(1+r)*+ &c. ... to + a(1+r)*~¹,
which by the rule of summation resolves into
m
a(1+r)”~'. (1+r) — a
r
-
m
produce = (1+r)",
p
a (1+r)"
go
G
a
; and this quantity is the
Cumulated annuity, or what the authors have technically termed the Amount of
the annuity.
It is immediately perceptible that the quantities p and m, the former of which
has been analized in section IXth, stand in such mutual relation as constantly to
(XIV.) But independently of m, we may admit another quantity M, similarly
formed of the same quantity n of consecutive and equal portions a of income,
the first of which realized at the moment of the supposed investment of the
principal p, all the other portions forthcoming at equidistant intervals, and the
whole respectively improved during n stages; in such manner, that the last
realized of those sums a participated, during one of those stages or intervals of
time, in the general process of improvement embracing a whole period equal to
that already considered in forming the quantity m. It is obvious that, in such
case, M will exceed the latter by a single quantity a(1+r)", resulting from the
improvement at Compound Interest of one of those sums a realized at the outset;
that quantity'a (1+r)" diminished however by another quantity a that will not,
COMPOUND INTEREST.
39
as in the former case, be forthcoming at the close of the period indicated by n.
That is to say, that M and m are so related as invariably to produce M =
m+a(1+r)”—a: the two cases differing from each other, in consequence only of
the first instalment of the periodical income a being, in one case, exclusively
realized at the outset of the transaction, whilst an equal quantity is, in the other
case, exclusively realized at the terminating period of the operation of improvement.
Having thus inferred, from a simple mode of reasoning, the equation
a (1+r)” —a+ar (1+r)"—ar, there
a(1+r)”—a
M =
+ a(1+r)”—ɑ =
j
r
now remains to establish this position by direct calculation analogous to those
that have produced the equations p and m (IX and XIII.)
From the definition, this quantity M is equal to the successive terms of the
following progression added together:-
}
a(r)': a(1+r)²: a(1+r)³: a(1+r)': &c. .... to a (1+r)" inclusive ;
the summation of which, by the general rule, produces M a (1+r)*+¹ — a(1+r),
go
identical with the preceding equation.
jo
Should this identity not appear at a first glance, it will be sufficient to observe,
a
that dividing by each of those assimilated results, they become
(1+r),
(1+r)" +r(1+r)"
(1+r) = (1+r)*+¹
whence (1+r)" + r(1+r)” = (1+r)*+¹;
and again dividing them by (1+r)", the identical results (1+r) are obtained.
C
(XV.) Being now possessed of two fundamental equations, establishing the
respective values of any present capital p, and of any future capital m, in their
relation together and with the other quantities a, r, and n; we are enabled,
through the transformations of which those equations are susceptible, and provided
that any three of those five quantities are supplied as the data, to ascertain the
remaining two.
:
Any five objects afford ten combinations, by which two of those objects may
be separated from the three others; hence there are also ten different modes,
according to which the questions may possibly be framed; the unknown
quantities being either p and a, p and r, p and n, p and m, a and r, a and n,
a and m, r and n, r and m, or n and m.
40
THE DOCTRINE OF
Waiving, as unnecessary towards practical purposes, the mathematical
demonstrations that would point out the path leading to our conclusions, it may
immediately be proceeded to exhibit the various theorems solving, in each of
those ten cases, the respective questions of the two unknown quantities.

CASES. DATA.
I.
II.
III.
IV.
V.
VI.
VII.
n, p, r.
IX.
m, n, r.
a, n, r.
m, p, r..
a, p, r.
a, m, r.
a, m, p.
VIII. | m, n, p.
a, n, p.
X. a, m, n.
a =
α
a
m
SOLUTION OF THE FIRST UNKNOWN QUANTITY.
pr (1+r)"
(1+r)” — 1
n =
P
m =
you
log. m-log. p
log. (1+r)
ap
a-pr
am
a + mr
a m
m p
g
r = {() = -1.
p
b
=
d =
C
-
e
mr
(1+r)" — 1) *
a [(1+r)^-1]
p.
6
n— 1
(뜸​)
- 1).
ƒ = b-2(d-1),
f
n+1 ? r = b−√ (bf) =
6
n+1
(mm)
an
g=2(e-1)+cj
==
$
r = √(eg)—c=
a m
:
(~~—1).
·1)
m\p
m
— c = 1/2(-1).
mp
SOLUTION OF THE SECOND UNKNOWN QUANTITY.
mp (1+r)".
P =
p
a
n
11
n =
ཆ་
H
a =
m
(1+r)"
p =
a [(1+r)" -1]
~· (1+1')"
mr
=)
Ky miqda
1)
mr
(1+r)” — 1) *
log. a-log. (a—pr)
log. (1+r)
mr
log: (a+mr) — log. a
log. (1+r)
log.m-log.p
log. (1+r)
(1+r)" —I'
m = p (b +1 - √bƒ)" =
m
[√(cg)+1−c]"
ap
(a—pr)
m
(1+r)*°
+
COMPOUND INTEREST.
41
In the first six cases of the preceding cathegory, and in which the quantity r
or ratio of interest is one of the data supplied, the equations that point out the
respective solutions of the unknown quantities are obtained through very simple
transformations of the equations already established (IX and XIII); and each of
the two questions concerning those unknown quantities may be solved inde-
pendently of the other. But the last four cases, where r is one of the unknown
quantities, involve these peculiarities; that in the 7th and 8th, r must be solved
before the other unknown quantity can be ascertained,—and that in the 9th and
10th cases, the unknown quantities are not strictly commensurable, each admitting
only an approximate solution in consequence of the elevated degree n of the
indicative equations.
It may not be easily perceived, in those last two cases, how the conclusions are
arrived at; which, in the ninth case, is as follows.
From p =
whence
an
t
an
P
=1+
1+
Binominal theorem, we have
−
(n − 1) r²+n
2
1 + nr + n
a(1+r)”—a
r(1+r)²
n+1
2
n+1
2
nr (1+r)"
(1+r)" — ī 1
(n−1)
2
r+
is immediately deduced
r +
% +
; and developing the series, according to the
(n−1). (n−2)
2X3
(n+1). (n-1)
2X2X3
(n−1). (n−2)
2X3
therefore, if we further admit
P
n+1
n+1 (n+1). (n−1) p² +
(1+r)²=1+ r +
2
2×2×2
an
2+1
there will be (1+r+%) 2 = 1 +
p2
But it is observable, by means of a similar developement, that
r² +
n+1
2
(n−1). (n−2). (n−3)
2X3X4
(n+1). (n-1)
2X3X4
the first two terms of which are identical with those of the preceding series 19 and
= (1+r+z)™+',
1
r(1+r)"
p a(1+r)*—a
r³ + &c.
2
-
~ + 2 + 11
n+1
2
which compared with the above first series produces
(n+1). (n−1), __ (n + 1) . (n−1)
2×2×2
2×2×3
f
(n+1). (n−1). (n−3)µ³+&c.,
2×2×2×2×3
-M
+
•r$+&c.
3+&c.
(n+1). (n−1)
2X2X2
r²; whence z
r²+&c.;
n—1
-p2.
12
42
THE DOCTRINE OF
Consequently, (1+r — (2—1) r²)
12
1+r
n— i
12
r² =
an\nFi
()
and as the first four terms in the developement of (1+
identical with those arising from the developement of the first series above, is
6
6
1° - √(27) × [ 1 - 2 (d—1)] = b— √(bƒ).
1)
6
X
22-1
N
_n—
ultimately deduced r =
2+1
2
an
p
or otherwise
=d, or again r²
12r
12
= (1-d); from which,
n— 1 n—1
n-1
12
p2
n+1
are
An analogous process leads to the conclusions in the tenth case.
(XVI.) The practical questions most frequently recurring are those in which r,
or the ratio of interest, and n or the quantity of instalments forthcoming of such
interest, or of any annuity until its terminating period, are known quantities;
those unknown and questionable, under that supposition, being limited to p, m,
and a. Admitting then the third datum to be p, there may be two questions, the
one concerning m, and the other concerning a ; admitting the third datum m, the
questions may in the same manner concern either a or p; and admitting a as the
third datum, jointly with r and n, the questions susceptible of arising may be
concerning p or concerning m. All those questions apply to the first three only,
of the ten cases exhibited in the preceding section, where their respective solutions
are given.
Those six questions are selected as the proper subject of general TABles,
applicable to daily use; and in order to facilitate their construction, in such
manner as to avoid either confusion or obscurity, it has been necessary to
substitute unity for the third quantity supposed to be known. The solution of
each question, thus referable to every unit contained in that known quantity-
whether p, m, or a,-is susceptible of being expressed by a numerical quantity
referable also to every joint indication of those represented by r and by n.
(XVII.) From this description of the Tables, their insufficiency towards the
solution of questions in many other cases should not however be inferred.
Whenever one at least, of the two quantities r and n, forms part of the data
supplied, those tables will afford an easy method of solving, with sufficient
approximation towards all practical purposes, the question relative to the other
quantity, whether r or n, and supposed to be the unknown. The various positions
here implied shall, in due progress, be illustrated by appropriate examples.
Amongst the ten cases aforementioned, the seventh, in which r and n are both
COMPOUND INTEREST.
43
unknown quantities, exhibits in that respect a solitary instance, in which the
method of direct calculation must inevitably be resorted to, according to the
equations pointed out in page 40th, for numerically solving the two questions
then susceptible of arising.
The six questions comprised in the first, the second, and the third cases,
and of which the solution in numerical quantities is obtained by the Tables, are
as follows:
(XVIII.) FIRST QUESTION.-Being given the ratio r, either of annual, of half-
yearly, or of quarterly interest; as also the quantity n of those instalments,
intervening until a final period when such interest is to cease; and substituting
unity for the principal p, or for any present capital supposed to be invested;
m = p (1+r)” is then reducible to the more simple expression m = (1+r)*,
indicating the Improved principal, or the quantity to which every unit comprised
in the original principal will amount, by its improvement at compound interest,
at the expiration of the stated period.
EXAMPLE.-Suppose 31 per cent. per ann. the nominal * rate of interest, at
which a present capital of £16500 may be improved half-yearly and during a
period of 36 years; whence r = 0·0175 is the half-yearly ratio, and n = 72 is
the quantity of successive instalments for the accretion of interest. We have
m = (1+r)” = (1·0175)7² = 3·4872098; which, as appears in page 12th of the
tables, is the improvement of every unit contained in the principal £16500;
and therefore, multiplying the latter into the preceding quantity, we find
£57538.96, or very nearly £57539, to be the improvement of that principal
at the expiration of 36 years.
?
(XIX.) SECOND QUESTION.-Being given the ratio r of interest at which any
principal may be invested, and also the quantity n of instalments of a terminable
annuity proposed as the equivalent for that principal, or present capital; if unity
be substituted for such principal, the equation
a
pr (1+r)"
(1+r) 1
indicating the Ratio of the annuity, or a fraction comparable with unity, and
expressing the proportion which the annuity, intended to terminate with the
will in that case become simplified into a =
r (1+r) n
(1+r)" -1'
* This rate is here called nominal, in contradistinction with that of 1 per cent., the propor-
tionate and true rate of interest accruing every half-year, and added to the principal at each
stage of its improvement.
† The reader is always supposed to possess a clear notion of decimal arithmetic, without
which we could not possibly make ourselves understood.
44
THE DOCTRINE OF
period n, should bear with that capital, or with the price for which the said
annuity is to be purchased.
EXAMPLE.-Suppose 34 per cent. per ann. to be the rate at which money might
be laid out at interest, with the condition of its accruing by half-yearly
instalments; and that a capital of £25000 were proposed for the purchase of
an annuity limited to 13 years' duration, and payable also by half-yearly
instalments, the last of which falling due at the expiration of that term. In the
www.com
}
present case we have r = 0·01875, and n = 27; whence a =
(1.01875)27 X 0.01875 (1.6512993)27 × 0.01875
(1.01875)27-1
0.6512993
=
.I
is the half-yearly ratio, and therefore 0.0950772 the whole ratio according to
which the annuity should be paid as an equivalent. By referring to the
table (page 15th), this result of calculation will immediately be found; and as it
expresses the annuity corresponding with one pound supposed principal, it follows
that 0.0950772 × 25000 2376·93, or say £2377, should be the annual sum
paid during thirteen years and one half.
r(1+r)*
(1+r)" -1
=0·0475386, which
In all questions of this description, it is invariably to be understood (without
which the conclusion must be false) that the instalments of the proposed annuity
must, in due proportion, take place at the same intervals as would otherwise be
stipulated for the corresponding rate of perpetual interest. Various cases, in which
a deviation from uniform principles is admissible, are generally-and in order to
avoid confusion-reserved for consideration hereafter.
(XX.) THIRD QUESTION.-Any capital m is forthcoming at a future period,
the interval to which, computed from the present time, being divided into n
equidistant instalments, whether annual, half-yearly, quarterly, or any other,
each of them is productive of interest in the ratio r; query the present value P
of such future capital, reduced by the adequate discount in which the operation
of Compound interest is considered. The solution of this question has
already been given under the general expression p =
be substituted for m, that expression becomes p
=
m
(1+r)”
; but if unity
1
indicating the
(1+r)"
Discounted Value of every unit comprised in the capital forthcoming at the
stated period.
EXAMPLE.-A sum of £6425: 10s. is forthcoming at the expiration of 7
years; the rate of annual interest at which the investment of capital might now
COMPOUND INTEREST.
45
be made is 4 per cent., accruing however by quarterly instalments of one per cent.
each; and it is inquired the present value of that future sum, discounted in regard
to the lapse of 30 quarters of years. The value of every unit contained in such
future capital is p=
be seen in the Table (page 19th); therefore the total present value is.
6425.5 X 0.7419229 4767-225, or £4767. 4s. 6d.
$
1
1
1
(1+r)" (1.01)30 1.3478489
=
******
(XXI.) FOURTH QUESTION.-Having ascertained the ratio r, of interest accruing
either by annual, by half-yearly, or by quarterly instalments, and according to
which ratio it is admitted that money may be uninterruptedly improved at com-
pound interest during a period of certain extent or a quantity n of such instalments;
and supposing that any capital m, required at the expiration of that period, were
then to result from the accumulation of that same quantity n of equal sums ɑ,
successively forthcoming or periodically saved, and in both cases improved during
a =
mr
the specified period, from each instalment to the next; the equation a
(1+r)"-1
will, in case of substituting unity for m or the future capital required, be simpli-
=0.7419229, as will
για
fied into a = (1+), indicating the proportion which such periodical
Redemption-fund should bear with that future capital; or in other words,
r is the fractional expression of the annuity, or of the portion of
(1+r)-1
annuity corresponding with each interval between the successive instalments, and
required to be laid out for accumulation, so as ultimately to produce unity at
the time when the capital m will be wanted.
¿
EXAMPLE. A sum of £125000 will, at the end of 41 years from the present
time, be required for redeeming a debt to that amount, and then only demandable;
and it is inquired the adequate annuity accruing by quarterly instalments,-the
first of which becoming due in three months from the present time, or otherwise
the adequate savings to be made at similar periods out of any income, for the
purpose of accumulation and improvement at Compound Interest, towards the
intended redemption, and admitted to be practicable throughout that period of 41
years; or 164 quarters, at the nominal rate of 4 per cent. per annum,
but really
in the quarterly and proportionate ratio 0.010625. We have in that case
= 0·002280364, the
0.010625
(1.010625)16-1
r
0.010625
4.6593442
(1+r)” — 1
quarterly prorata, and therefore 0·0091215 the prorata of annual savings required,
46
THE DOCTRINE OF
as will also be seen in the Table (page 25th); and which quantity multiplied
into £125000 produces 1140-182, or very nearly £1140. 2s. as the annual
redemption-fund to be appropriated.
This fourth question is stated in terms that perhaps may appear unneces-
sarily minute; but it is sometimes assumed on insufficient grounds, that a certain
rate of Interest, available for the time being, will thus continue during a very
protracted period; in consequence of which miscalculation, important errors are
not unlikely to result. When under peculiar and transitory circumstances, the
rate of interest happens to be unusually high, or else unusually low, it is indis-
pensably necessary, for calculating any anticipated improvement of Capital at
Compound Interest for a great length of time, to adopt such moderated rates of
Interest as may safely be relied on; and this calls for the exercise of great
sagacity, as the error may lie on either side of the argument.
(XXII.) Fifth QUESTION.—Any quantity n, of equal sums a, is forthcoming
at equidistant intervals of time computed from the present; the ratio r of
Interest is ascertained with reference to one of those intervals, whether consisting
of years, half-years, quarters of years, or any other; and it is inquired the present
equivalent p of those forthcoming sums, considered in the aggregate and relatively
to the operation of Compound Interest. This question, of the Value of an annuity,
is the counterpart of the 2d, in the same manner as the third question is the
counterpart of the first. If unity be then substituted for a, or for one of those
equal sums, the equation p =
a[(1+r)"—1]
r(1+r)"
We have p =
(1+r)*-1
r(1+r)”
becomes p =
-
expressing the equivalent, in present capital, that should be given for every unit.
contained in the quantity a of periodical income; or otherwise indicating what is
usually called the number of years' purchase of an annuity, when a being any
other than one whole year's income, a due proportion relative to the latter is
substituted instead of it.
(1+r)"-1
r(1+r)"
EXAMPLE.—Suppose an annuity of £750, forthcoming in one single payment
at the expiration of the year now commencing and of every succeeding year to
the 35th inclusive; and the rate of interest, at 4 per centum per annum,
assumed to be available during the whole of that period; it is required the number
of years' purchase that should be paid as the equivalent for that annuity.
(1.045)35 1
(1·045)35 X 0·045
S
;
PAR
thus
3.6673478
4.6673478 X 0·045
= 17-461, or very nearly 17 years' purchase; and this value of the annuity is
accordingly found in the Table (page 28th.)
COMPOUND INTEREST.
47
p =
OTHER EXAMPLE.-Suppose an annuity of £1200, payable by quarterly
instalments during 45 years, the first quarter now commencing to run; and the
rate of interest, admitted to govern the calculation, being 42 per cent. per
annum, but subject also to quarterly payments; query the present value of that
annuity? In this case we have the quarterly ratior 0011875, and the
quantity of forthcoming instalments 2
180; whence
(1.011875)180-1
(1.011875)180 X 0·011875
here m =
As the latter quantity multiplied into the true and quarterly income would
constitute the required equivalent, it follows that 18.53802 is the quantity of times
the nominal and annual income, or in other words the number of years' purchase;
and the same result is exhibited in page 32d of the Tables.
Tag
(XXIII.) SIXTH QUESTION.-The quantities r and n are given, as in the 4th
question of which this is the counterpart; and a is a sum applied, at each of the
successive and equidistant instalments (in quantity n), to create a capital at the
expiration of the whole term :-Query, what will be the amount of that capital?
Substituting unity for a, or for the sum thus appropriated at each instalment,
the equation m =
; indicating
the future capital which every unit contained in the quantity a of periodical
savings will create, or in other words the Cumulated annuity.
a[(1+r)]”—1]
r
7.3721287
8-3721287 × 0.011875
(1+r)^-1
yo
======
will become m
EXAMPLE. The rate of Interest, subject to half-yearly payments, is 5 per cent.
per annum, and it is assumed that it will thus continue available during an
indefinite period; a sum of £350 is each year, and in the proportion of £175 at
the expiration of every half year computed from the present time, to be applied as
a Redemption-fund, towards the ultimate discharge of debt, after the accumulation
of that fund at compound interest until the 60th year expired; the question
being, what capital of debt may then and by that means be redeemed. We have
(1+r)".
7"
(1·025) 120 — 1
0.025
: 74.15209.
J
18.35815
0.025
= 734·326, the
quantity of times the half-yearly savings, and consequently 367.163 times their
annual proportion; being the same result announced by the Table, (page 36th).
The capital of debt which, in that case, may be paid off at the expiration of the
60th year, from this time, will be £350 × 367·163, or £128507.
(XXIV.) The general intent, the limits, and the practical use of the Tables
that conclude this book, have already been set forth in a short prefatory discourse.
Those Tables are thirteen in number; each embracing an extent of four pages,
48
THE DOCTRINE OF
and referable to a certain standard rate of interest per annum. They exhibit
under the several heads of Improved principal, of Discounted principal, of
Cumulated annuity, of Annuity for cumulation (or Redemption-fund), of Value of
the annuity, and of Ratio of the annuity, specific and numerical solutions of the
six questions expounded in the preceding sections (XVIII to XXIII). All
other details concerning the construction of those Tables would be superfluous.
But it is necessary now to explain the further use to which they are applicable,
for easily-though mediately-solving the questions that may arise in the 4th,
the 5th, the 6th, the 8th, the 9th, and the 10th cases specified in page 40th;
the 7th case being an exception, as already mentioned (XVII).
(XXV.) The clearest mode of proceeding, to that effect, will be to exemplify
the questions susceptible of arising in each case, together with their respective
solutions by means of the Tables. It should however be borne in mind, that in a
great number of instances it will be found more expeditious to solve the
questions by direct calculation, and according to the algebraic formulæ exhibited
in page 40th; with the further advantage of rigid accuracy, which cannot
otherwise be obtained in the results.
(XXVI.) CASE FOURTH, in which m, p, and r, are known quantities:
Query?
A principal of £25000 has improved by compound interest, at the rate of 3 per
cent. in single payments annually, to a sum of £51500; and it is inquired in what
period of time.
Reducing to unity the first of those sums, the proportion of Improved principal
is 2.06; the Table (page 2d) indicates 2.0327941 as the improvement during 24
years, and 2.0937779 as that improvement extended to a 25th year: difference
0.0609838 for the last year. Now, 2:06 exceeds 2·0327941 by 0·0272059 only,
which in proportion with the preceding difference is 0-446, or of the 25th
163
365
year; whence the whole period of improvement has been 24 years and 163 days.
The direct calculation is
log. m―log. p
log. (1+r)
The same data, m, p, and r, being given: Query a?
n
N =
-
4-7118072-4.3979400
0.0128372
=24-4498.
An annuity stated to have been purchased for a capital of £13333. 6s. 8d., at
a time not mentioned, is now at its terminating period by the accretion of the
COMPOUND INTEREST.
49
[
last and half-yearly instalment of that annuity; and in consequence of the whole
having been accumulated and improved at 3 per centum annual interest, but
accruing also half-yearly from the time of purchase to the present, has created a
capital of £66000. It is inquired, either the annuity or its half-yearly proportion,
which by such accumulation and improvement has produced this result.
The principal £133333 having ultimately produced £66000, the proportion of
improvement is from unity to 4·95, which at 3 per cent. per ann., or in the
half-yearly ratio 0.01625, must have embraced a period between 49 and 50 years,
according to the Table (page 8th). If 49 years, the improvement would have
been 4.8534966; and if 50 years, 5·0125169; whence the difference, 0·1590203,
is the improvement during the 50th year. But as the actual improvement has
exceeded, by 0.0965034 only, that which corresponds with 49 years, this excess
compared with the preceding difference represents an additional 0·6069, making
the whole term 49.6069 years. We find in another column of the same Table,
(page 9th,) that an annuity for the term of 49 years should be in the proportion
0·0084339, for every unit of the future capital it is intended to create by
accumulation; and that the proportion of a 50 years' annuity should be only
0·0080997: difference for the last year, 0·0003342. The additional fraction
0.6069 of one year, being multiplied into this difference, produces 0·0002029 for
the difference in the ratio, corresponding with that fraction of a year; and the
latter quantity is to be subtracted from 0·0084339, as a longer term of the annuity
must diminish its ratio or proportion with the capital. Hence 0.008231 will be
the proportion of annuity, corresponding with every unit contained in £66000; and
in that proportion, the latter capital is found to result from the accumulation and
half-yearly improvement of an annual sum of £543.246, or very nearly £543. 5s.
But if it be otherwise considered, that the fraction 0.3931 is wanting towards
the completion of a term of 50 years; then by taking a similar proportion of
the difference 0.0003342 already found, we have 0·0001314, which added to
0·0080997 (the ratio for 50 years), yields the same result 0·008231 as above.
It is observable that we have sought the precise Annuity, or Redemption-fund,
corresponding with £66000 capital ultimately created; whereas the result of
calculation would have been the same, if we had sought the Ratio of annuity
corresponding with the original sum invested, say £13333. 6s. 8d., and proceeded
in the same manner; and again if we had sought the Value of the annuity, or
the quantity of years' pnrchase, according to the Table (page 8th, column 6th).
The direct calculation, however, is a =
mr
[カ​]-1
مد
66000 X 0·01625
4·95 — 1
1072.5
3.95
= £271-519; this being accurately the half-yearly proportion of
f
the annuity, the whole of which therefore £543·038.
50
THE DOCTRINE OF
1
(XXVII.) CASE FIFTH, in which a, p, and r, are known quantities: Query n?
A present capital of £1250 is given as the equivalent of an annuity of £132. 10s.,
by quarterly instalments, and the term of which now commencing to run; the
rate of interest, admitted to govern the calculation, is 31 per cent. per annum,
but understood as also accruing quarterly; and it is inquired the term of that
annuity, or the quantity of its quarterly instalments that should be forthcoming.
Dividing 1250 by 132.5, we find 9.433962 years' purchase, the price of that
annuity; and by the Table (page 10th, column 9th), an annuity at 9.433772
years' purchase (which differs from the above price but in a very insignificant
fraction) should be of 11½ years' duration, or 46 quarters.
If on the contrary 132.5 be divided by 1250, there will be found 0.106 the
exact ratio of the annuity compared with the principal; and in the same Table
(last column of the 11th page), that an annuity constituted in the ratio 0·1060021
should continue during 114 years: same result as above.
1.5201559—1.3461084
0.0037835
log. a-log. (a-pr) log. 33 125 — log. 22·1875
log. (1+r)
log. 1.00875
= 46·0017 quarterly instalments forthcoming.
By direct calculation n =
The same data, a, p, and r, being given: Query m?
A capital of £50000 is to be invested, at 25 years' purchase, in an annuity
payable by half-yearly instalments, and the term of which is agreed to be
regulated by calculation, founded upon 32 per cent. rate of interest per annum,
also accruing half-yearly; and it is inquired the sum to which that annuity,
accumulated and improved at the said rate of compound interest, 'will amount
at the period of its termination.
It is obvious that the ultimate sum, inquired after, should be the same as the
improvement of the represented principal £50000, laid out at compound interest
until the expiry of the annuity; and this period may be ascertained as in the
preceding question.
The Table (page 16th) indicates the present value of a 74 years' annuity to
be 24-96074, and that of a 75 years' annuity to be 25·02295 years' purchase ;
the difference in value for the last year of this term being consequently 0·06221.
But the deficiency of the former value is only 0.03926, corresponding in due
proportion with 0.631 additional fraction of a year; whence the term of the
annuity should be 74-631 years. The Table further indicates, that any principal
COMPOUND INTEREST.
51
laid out at the above rate of interest should improve in 74 years to 15-631765
times its amount, and to 16.223452 times in 75 years, the difference being
0.591687; and if a part of the latter quantity, say 0·373354, in proportion
with the fraction 0·631 of the 75th year, be added to 15.631765, we shall find
16-005119 for the ratio of improvement at the expiration of 74·631 years, or
74 years and 230 days. Hence, and by multiplying £ 50000 into that ratio, we
find £800256, the future capital to result from the accumulation and improvement
of a £2000 annuity accruing by half-yearly instalments of £1000 each.
The direct calculation, in algebraic form, does not require the term of the
annuity to be previously ascertained: it gives at once, and with perfect exactness,
= £800000.
M
ap
a-pr
1000
1000 X 50000
50000 X 0.01875
F
Some inaccuracy is observable in the result obtained by the indirect method.
It proceeds from having successively admitted certain proportional parts of value,
to correspond with fractions of time in similar proportions, which is not strictly
correct; whence the smallest deviation from rigid accuracy, in any one instance,
may generate errors more considerable. Such, however, cannot acquire real
importance in a practical point of view; but the direct method is always preferable.
(XXVIII.) CASE SIXTII, in which a, r, and m are known: Query n?
An annuity of £365, accruing by yearly instalments, accumulated and
improved by compound interest at the rate of 4 per cent., has ultimately produced
a capital of £10660: it is inquired in what time?
10660
365
it is found that the annuity has accumulated in the pro-
Dividing
portion of 29-205479 for every unit; and by the Table (page 18th, column 2d),
that the accumulation during 19 years should be in the proportion of 27·671229,
whilst that during 20 years in the proportion of 29.778079; difference 2.10685.
But the first proportion is short of a quantity only 0.5726, to correspond with a
20 years' accumulation; and :: 2-10685: 1 :: 0·5726: 0.27178, which last
is the fraction wanting to the accomplishment of the 20th year; whence 19-72822
years is the required period of improvement.
If on the contrary we divide
365
10660
the annual redemption fund, or the proportion between the annuity and its
cumulated amount; comparing this quantity with 0.0361386 and with 0.0335818,
by the same Table (p. 19th, 2d col.), and calculating the differences in the same
manner as above, the result will be 19-74249 years.
the quotient 0.0342402 then expressing
>
52
THE DOCTRINE OF
J
1
But from the cause already assigned in the preceding section, neither of those
results are strictly accurate, and it is best to proceed with the direct calculation
as follows:-
N
1
=
de Gang
log. (a+mr) log. a
log. (1+r)
10446.25
1.8880266'
G
log. (791.4) log. 365
Jog. (1.04)
and 267 days, the exact term of the accumulation and improvement.
The same data, a, r, and m, being given: Query p?
An annuity of £500, purchased at a time not specified, and the last half-
yearly instalment of which is presently paid, has been accumulated and constantly
improved at an interest nominally 41 per cent. per annum, but really of 21 per
cent. half-yearly, so as now to have become a capital of £10446. 5s.; and it is
inquired the principal, or the original cost of that annuity, according to the
above mentioned rate of interest.
<
p =
log. (365 +10660 × 004)
log. (1.04)
am
a+mr
= £5533. 3s. 2d.
0.3361031
0.0170333
It is clear, from the principal and the annuity being originally equivalent,
that the former must be such as should have produced £10446. 5s., by its
improvement at compound interest during the implied term; which term,
ascertained as in the preceding question, is 15 years and 40 or 41 days,
say 15 years. The improved principal, according to the Table (page 22d),
is 1-8791528 at the expiration of 15 years, and 1-9190848 at the expiration
of 15 years; the difference for the last half-year being 0.0399320. If to
1.8791528 is added 0·0088738,-being the proportion of that difference cor-
responding with of a year,—the result 1·8880266 will be the improvement of
every unit contained in the principal; which therefore is found to have been
or £5532. 188.
log. 365
By the direct method, this question will be more accurately solved, even
without previously ascertaining the term " of the annuity. We then have
19-732 years, or 19 years
250 X 10446·25
250 + 10446.25 × 0·02125
?
2611562.5
471.9828125
(XXIX.) CASE SEVENTH, in which a, m, and p, are known quantities;
whilst r and n are both unknown.
COMPOUND INTEREST.
53
晶
​In this case, as already observed, neither of the questions arising can be solved
by means of the Tables, which are expressly founded on the possession of the
two data r and n; and in exemplifying those questions, their solution will be
easily obtained by the direct method, as pointed out in page 40th. It is further
observable, first, that the ratio r must be ascertained previously to the period of
timen; and secondly, that the stipulation of instalments, whether annual, half-
yearly, or quarterly, become a matter of indifferent consideration, provided that
they be the same for the annuity a and for the ratio r of interest.
EXAMPLE.-An annual fund of £4000 is supplied for accumulation and
improvement at compound interest until it shall have created a capital of One
million; that capital is, then only, to be applied for the discharge of present
debts amounting to £86000, but admitted in the meantime to increase at
compound interest also; and it is inquired, first, the rate of that interest at which
the annual fund should be improved, in order that the debts may have accumu-
lated to an equal amount of One million at the period of payment; and secondy,
how far that period is removed from the present time.
a m
r = — (7572 — 1) =
m
ԴՈՆ՝P
n=
4000
1000000
log. m
log.p
log. (1+r)
SOLUTION.
1000000
86000
7211
2400
which is the ratio of interest, exceeding by a very trifling fraction the rate of
41 per cent. per annum :
log. 1000000
__1)
waddin
B
10.627907
250
log. 86000
log. 1.042511628
=0·042511628,
1.06550155
0.01808091
= 58.93,
or very nearly 59 years, the period of reimbursement of the debts, as accrued
by compound interest, and without gain or loss on either side.
(XXX.) Case EighTH, in which m, n, and, are known: Query r?
A principal of £2400 has, by improvement at compound interest, quarterly
instalments, produced £7211 at the expiration of 24 years: and it is inquired
the nominal rate of that interest per annum.
The division quotes 3.0045833, the improvement of every unit contained
in the principal; and by the Table (page 26th), 2-9269771 is the improved
principal when the rate of interest is 41, and (page 30th) 3·1058405 when it is
42 per cent.; the difference of improvement 0.1788634, corresponding with
per cent. difference in the annual rate of interest. But the proportion of actual
improvement has exceeded, by 0·0776062 only, the quantity relative to 4 per
54
THE DOCTRINE OF
cent. ; and, as 0·1788634: 0·0025 :: 0·0776062: 0·0010847, the latter quantity
added to 0.045 gives 0.0460847, the annual ratio of improvement, more accurately
specified 0.0115212 the ratio per quarter.
By the direct method, r =
E
0.0115257, is a =
the quarterly ratio, and therefore 0.0461028 the annual ratio; which last two
results are correct, the above being only an approximation.
The same data, m, n, and p, being given: Query a?
A 24 years annuity, subject to quarterly instalments, has been purchased for
£2400; which annuity cumulated and improved at compound interest, also
quarterly, until the expiration of its term, has produced a capital of £7211;
and it is inquired the ratio of that annuity, or its proportion with the principal,
when understood that the purchase and the improvement were governed by the
same rate of interest.
That rate of interest is 0-0460847, ascertained as in the preceding question.
The ratio of an annuity for 24 years should be 0.0683526, when the interest is
at 4, and 0.0700563 when at 44 per cent., according to the Table (p. 27th and
31st); difference 0.0017037, corresponding with per cent. difference in the
annual rate of interest. The proportion :: 0·0025: 0·0017037 :: 0·0010847
: 0.0007537 indicates that the latter quantity should be added to 0.0683526,
whence 0.0691063 must have been the ratio of the purchase; and multiplying this
ratio into £2400, we find £165. 17s., the annuity purchased and improved as
above.
[]
I
12
]*-
The direct calculation, after having ascertained the quantity r to be correctly
7211X0.0115257
2.0045833
£41·4609, the quarterly
mr
The division
1 = (3·0045833) — 1 — 0·0115257,
(7)
instalment of the annuity; the whole of which therefore is £165. 16s. 10d.
1
(XXXI.) CASE NINTH, in which a, n, and p, are known: Query r?
An annuity of £1750, the whole of which sum forthcoming at the expiration
of the year now commencing and of every succeeding year to the 35th inclusive,
is purchased for £28208. 8s. 6d., as its just equivalent; and it is inquired the
rate of annual interest with which that purchase corresponds.
28208-4271
1750
quotes 16.11910, the quantity of years' purchase of
COMPOUND INTEREST.
55
that annuity; and
1750
28208-4271
quotes 0·0620382, its ratio or proportion with
every unit contained in the principal. The departure may be taken from either
of those results, but it is better to proceed upon the former. It is then perceived
by the Table (page 36th), that an annuity such as the above is valuable at
16.37419 years' purchase, taking the annual rate of interest at 5 per cent. in a
single payment, and (page 40th) that the same annuity is worth 15-87025 years'
purchase when the interest is at 5 per cent.: difference 0.50394, corresponding
with that of per cent. in the ratios of interest. If it be further considered that
the purchase has been rated at 16:11910, differing by 0-25509 only from the
valuation at 5 per cent., the proportion :: 0·50394: 0·0025 :: 25509 : 0·0012655
indicates that the ratio of interest, according to which the purchase was governed,
must have exceeded by the latter quantity the ratio 0.05, or the rate of 5 per
cent.; therefore 0·0512655, or 5.12655 per cent. per annum, (exceeding 5 by
a very trifling fraction) is the rate inquired after.
The direct mode of calculation is somewhat more intricate in this than in the
preceding cases, and requires first to ascertain three quantities, b, d, and ƒ.
According to the demonstration in section XVth, we have
6
2017
n-
2
(—), ƒ = b − 2 (d − 1), and consequently
f
—
r = b—√(bf), which will afford an approximate solution of the question.
b =
>
d =
We find b=
6
n~1
2
an
= (=)==
P
d =
6
34.
61250
:(:
28208,4271)'s
and ƒ — b−2 (d—1) = 0·17647059-0·08803152
=
whencer = b = √(bf) = 0·17647059
=0.051543.
=0·17647059,
I
IS
(2·1713399) — 1·04401576,
(0.17647059
0·08843907 ;
0.08843907)
It is worthy of observation that the true ratio of interest should be exactly
0·05125, or 5 per cent., as proved by admitting this datum and stating the
converse questions.
This 9th case, as well as the 10th hereafter exemplified, are two cases of
exception, in which the questions they involve may be solved with a nearer
approach to rigid accuracy by means of the Tables, than by the direct method;
as instanced in the discrepancies of the two results above. The first, 0·0512655,
exceeds the exact quantity by a very trifling fraction only; and by explaining
56
THE DOCTRINE OF
the cause of this difference, however small, it will account for all analagous
deviations from exactitude in the results indirectly given by the Tables, in the
seven cases from the 4th to the 10th inclusive.
The value of a 35 years annuity, interest at 5 per cent., is 15.87025, and at
5 per cent. it is 16-37419, the difference being 0-50394. But we also find, by the
Table (p. 32d), that the value increases to 16.90390 when interest is at 4 per
cent. only; the difference from the value at 5 per cent. being then 0·52971,
which is more considerable than the former difference. Now, if, for two
equal differences of interest the corresponding differences in value of the
annuity are generally unequal, it is evident that the proportion set forth in the
preceding page, and producing for its fourth term the quantity 0·0012655,
cannot be strictly correct; and it is evident also that this quantity must be rather
too considerable, because one half of the difference 0.50394, or 0.25197, is a
reduction too small from the value 16.37419 of an annuity calculated at 5 per
cent. for corresponding with an increase of per cent. in that rate of interest,
considering that we have perceived the differences in value to increase in a
progressive ratio, with reference to corresponding but equal reductions in the
rate of interest.
The same data, a, n and p, being given: Query m?
An annuity of £1750, during 35 years, is purchased for £28208. 8s. 61, as
above mentioned; and be it inquired the amount of which it is susceptible, by
accumulation and improvement at compound interest until the expiration of that
term; the rate of such interest not specified, but understood to be the same that
established an equivalent in the purchase.
The amount, by accumulation of an annuity purchased for £28208. 8s. 6d.,
will at the end of its term be the same as this represented principal improved at
compound interest during that term. But, towards solving the question by means
of the Tables, a reference is required to the ratio of interest 0.0512655 already
found, although we are aware of some inaccuracy in this result. According to
the Table (p. 36th), the improvement of every unit in the principal is 5·5160154
at 5 per cent., and (p. 40th) is 5·9947856 at 5 per cent.; difference 0.4787702.
Taking therefore a due proportion of that difference :: 0·0025: 0·4787702 ::
0·0012655 : 0·2423535; and adding the latter quantity to 5.5160154, we
obtain for the assumed improvement (at the ratio of interest 0.0512655) a
quantity 5-7583689, which multiplied into £28208. 8s. 6d. produces £ 162435,
the Cumulated annuity.
By the direct method of calculation, we have m = p [b + 1 — ✓✓ (bƒ)]*
=28208-4271 (1·051543)35 = 163804·3, or £163804. 6s.
COMPOUND INTEREST.
57
t
The rate of interest being exactly 5 per cent., as already observed, we
should have obtained (exactly also) 162214.3108, or £162214. 6s. 24d., for
the Cumulated annuity, if the formula had admitted a rigorous solution.
(XXXII.) CASE TENTH and last; in which a, m and n, are known:
Query r?
An annuity of £450, during 45 years, and the 90th or last half-yearly instalment
of which is now received, has by uninterrupted improvement at compound
interest, from the commencement of that period, accumulated to a capital of
£ 82706. 10s.; and it is inquired the rate of interest productive of this result.
The division
quotes 183.79222, as the proportion according to
which every unit contained in the annuity has increased by accumulation and
improvement. The Table also indicates (p. 40th) that the proportion should be
177-11941 when interest is at 5 per cent., and (p. 44th) 190-74879 when it is
at 5 per cent., accruing half-yearly in both cases; the difference 13.62938
corresponding with that of per cent. in the rate of interest. But the excess, of
183.79222 over the proportion relative to 5 per cent., is only 6·67281; whence
:: 13.62938: 0·0025 :: 6·67281: 0·001224, and the latter quantity added to
0·0525 establishes at 0·053724 the ratio of annual interest productive of the
improvement announced.
n
The following is the direct calculation, in which 90 the quantity
of half-yearly instalments, and a = 225 the portion of annuity accruing at each.
C =
с
We have first to ascertain three quantities; c, e and g, as already observed
in Section XVth:-
e =
6
n+1
82706.5
450
m
an
6
91
n-1 -
= 0·06593407,
82706.5
2
89
225X90) = (4·084271605)
~ = (4·084271605)~ — 1·03212646,
and g = 2(e-1) + c = 0·06425292 + 0·06593407 = 0·13018699;
whence is ultimately obtained
r = √✅/ (cg) — c=√/0·06593407 0·13018699—c―0·09264857—0·06593407
=0·0267145, the half-yearly, and therefore 0·053429 the annual ratio of interest
{
58
THE DOCTRINE OF
The same data, a, m and n, being given; Query p?
}
Let the case be the same as in the preceding question, and be it inquired the
sum for which the annuity of £450 should have been originally purchased;
calculating on the same ratio of interest. whence resulted £82706. 10s., the
Cumulated annuity, though that ratio is kept out of view.
The use of the tables requiring a reference to the ratio of interest, it must first
be ascertained; and having as before mentioned found that ratio 0.053724,
exceeding by a quantity 0·001224 the ratio 0.0525, we proceed as follows:-
I
¿
By the Table (p. 40th), an annuity for 45 years, when the rate of interest at
5 per cent. per ann. is accruing by half-yearly instalments, has the value of
17.19812 years' purchase, and (p. 44th) it has the value of 16-59958 when the
interest is 51 per cent. ; the difference corresponding with 4 per cent., or with 0·0025
in the ratio, being 0-59854. The proportion :: 00025: 0-59854 :: 0·001224
: 0.29305 indicates that the latter quantity should be subtracted from 17.19812, as
the value of an annuity decreases when the rate of interest increases; whence
the annuity of £450 was originally valuable at 16-90507 years' purchase,
therefore represented a principal of £7607. 5s. 74d.
By the direct method (pointed out in Section XVth, page 40th) we have
m
[√(cg)+1—c]"
82706.5
(1·0267145)90
P =
82706:5
10.726856
82706.5
0.06593407)⁹0
(1.09264857
= 7710-2275, or £7710. 4s. 6d.
But it has been observed in the foregoing part, that even this direct method of
calculation affords only an approximate solution; the present and the preceding
cases being, in that respect, two exceptions amongst the series of ten cases. The
rate of interest should have been exactly 53 per centum per annum, or 0·026875
for the half-yearly ratio; and the principal should have been (also exactly)
£7602. 10s. 4d., corresponding with an annuity of £450, so as to produce by
accumulation a capital of £82706. 10s. at the expiration of 45 years; all which
may easily be proved by the converse calculation.
(XXXIII.) Before proceeding to elucidate various other questions, some of
which assume a complicated character, and that may occasionally arise from the
consideration of compound interest in its multifarious modes of operating, it
is proper to revert to the six questions comprised under the first three general
cases set forth in Section XVth, and the solution of which only could be
COMPOUND INTEREST.
59
1
exhibited in the shape of tables destined for habitual reference. As the purpose
of those tables, that form the concluding part of this work, is exclusively practical,
their construction and use require to be clearly explained; which will render
their application easy for the most simple understandings.
On opening the book at any part of those tables, eighteen columns, each
containing a series of results of calculation, obviously following a progression
whether of increasing or of decreasing quantities, offer themselves in a synoptical
view, with reference to a specific rate of annual interest, there indicated, and
which is always one of the three data required towards solving any question.
Independently of those eighteen columns, each page begins with a smaller column
of quantities regularly increasing, and indicating certain periods of time, for the
purpose also of reference as a second datum; and it should be distinctly under-
stood that this second reference governs, for the whole horizontal line at the
head of which it is placed, the nine results of calculation there contained.
From this construction and disposition of the tables, as also towards their
intelligence and convenient application, it became necessary that the third indis-
pensable datum should be, altogether, uniform and such as the mind could retain
without effort: unity has therefore been adopted for invariably representing that
third datum, of whatever description it might be; and with the greater reason,
as variable data could not in this instance have found any place for insertion. If
then it be borne in mind, that whatever the third and variable datum,-as
supplied by the statement of the question itself, it must always carry the same
proportion with the quantity inquired after, as the proportion existing between
unity and the quantity expressed by the table in the appropriate column, and in
the line indicated by the period of years referred to, no difficulty can remain
in obtaining at a first inspection into the proper table the answer to any of the
six questions it is intended to solve.
!
}
-
By successively exemplifying the solution of those questions, this matter will
De placed in the clearest light possible. But it is necessary first to observe, that
each page of the tables being divided into three parts, similar in their general
indications, yet diversified in the results they exhibit with reference to the same
periods of time, and according as the specified rate of interest per annum may
be understood to accrue-by annual-by half-yearly-or by quarterly instal-
ments, there is always one of those three divisions which must exclusively be
referred to for the required answer; avoiding, in all instances, to confound that
proper division with either of the two others.
60
THE DOCTRINE OF
(XXXIV.) FIRST QUESTION:-The Improved Principal.
If it were inquired the sum to which a present capital of £3600 is susceptible
of being improved at the expiration of 123 years, by compound interest accruing
quarterly, and at the nominal rate of 3 per cent. per annum; it would be found
by the table (page 2d, column 7th), that every unit of that capital should then be
improved into 1.4638541; from which is to be inferred, in the same proportion,
that the whole £3600 ought to improve into a sum of £5269. 178. 6d.
But supposing the question to be concerning that improvement at the expiration
of 120 years, a period exceeding the limits of the tables; it is then evident, that
whatever the sum to which every unit contained in the original principal will have
improved at the end of half that period, or at the expiration of 60 years only,
every unit contained in that improved principal will again have improved in the
same proportion at the end of a second and similar period. Therefore, and as
we find (page 4th) that the improvement for the first 60 years should be in the
proportion of 6.0091516 for every unit of the original principal, it must be
inferred that this quantity multiplied into itself, the product of which is 36-109903,
expresses the proportion of improvement corresponding with 120 years; and
consequently, the original principal £3600 will then have improved to the sum
of £129995. 13s., admitting the same rate of interest to have been constantly
available. The same result would be obtained, if the proportion of improvement
corresponding with 100 years were multiplied into, the proportion corresponding
with the 20 years remaining, or generally by multiplying the respective proportions
of improvement corresponding with any two periods forming together that of
120 years.
+
(XXXV.) SECOND QUESTION :--The Discounted Principal.
This question is exactly the converse of the preceding.
Be it inquired the present value of £10000, forthcoming at the expiration of
30 years; the interest supposed at 34 per Cent. per ann., and accruing half-yearly.
It will be found (page 9th, column 4th), that every unit comprised in
that forthcoming capital is, with reference to the specified period, reducible to
she proportionate fraction 0-3801601; whence £3801-601, or £3801. 12s. 04d.,
is the present value of the £10000 above stated.
But if the proposed discount should relate to any period exceeding the limits
assigned to the tables, then multiply into each other the proportionate decimal
COMPOUND INTEREST.
61
fractions, as there expressed with reference to any two periods into which the
given period is susceptible of being divided; and the further reduced fraction,
thence resulting, will express the discounted value of every unit contained in the
capital forthcoming at the remote period supposed.
(XXXVI.) THIRD QUESTION :-The Cumulated Annuity.
A sum of £500 is to be annually received in a single payment, the first of
which at the expiration of the year now commencing; that annuity is to be
accumulated and improved at 3 per cent. compound interest during 25 years,
being the period at which its last instalment will be forthcoming; and it is
inquired the Cumulated amount of such annuity, or the capital it will have created
when its term is expired.
Assuming that the improvement at compound interest is also to take place from
one year to another, and not half-yearly, nor quarterly, (which otherwise would
complicate the question,) it will be found in the Table (page 10th, column 2d,)
that on the accomplishment of the 25th year, every unit of the annual income
must have created 38.949857 capital; whence £500 of such income will, in
due proportion, have created a capital of £19474. 18s. 6d.
XXXVII.) FOURTH QUESTION:-The Annuity for Cumulation, or the
Redemption-fund.
The question here exemplified is converse of the preceding.
A capital of £125000 is required at the expiration of 45 years from the pre-
sent time; and it is questioned the annual provision that should be made, by
half-yearly instalments, the first commencing six months hence, so as to create
the above mentioned capital by means of those savings improved at compound
interest, also half-yearly, and at the annual rate of 34 per cent.
We find in the Table (page 17th, column 5th), that every unit of capital to be
provided at the stated time requires an annual saving of 0.0086763; whence
and from multiplying £125000 into that fraction, it follows that £1084 5375,
or £1084. 10s. 9d., is the Redemption-fund required to be annually appropriated
to the above effect, by half-yearly instalments of £542. 5s. 4 d. each.
62
THE DOCTRINE OF
(XXXVIII.) FIFTH QUESTION :-Value of the Annuity.
Supposing the rate of interest 4 per cent. per annum, but available by quarterly
instalments; it is inquired the capital which should be given as the equivalent
for an annuity of £ 480, also accruing quarterly, and to terminate at the expira-
tion of 14 years from the present time, its last instalment then forthcoming.
It will be found, in the last column of the table (page 18th), that every unit
of the annuity of that description has the value of 10.96216, which is commonly
called the quantity of years' purchase; whence the annuity of £480 is valuable
according to the product of those two quantities multiplied into each other, which
is £5261. 16s. 9d., the present equivalent.
(XXXIX.) SIXTH QUESTION :-Ratio of the Annuity.
This is the converse of the preceding question, it being here required the
proportion which a terminable annuity should bear with the price at which it is
purchased.
Suppose £13333. 6s. 8d., the principal proposed for investment in an annuity
payable by half-yearly instalments, the first of which in six months from this
time, and the last at the expiration of 36 years; suppose also the given rate of
interest, for ascertaining the value of the annuity, to be 41 per cent. per annum,
but accruing half-yearly; and it is inquired the annuity itself, or the income
limited to 36 years, corresponding with the above mentioned capital.
By the table (page 25th, column 6th), it is found that every unit of the prin-
cipal should produce 0·0544896, as the corresponding annuity; which proportion
multiplied into £13333. 6s. 8d., the stated principal, indicates £726. 10s. 6d.
as the equivalent and annual income that it should yield.
(XL.) It is obvious that the ratio of a terminable annuity, however
extended the duration of the latter, should always be superior to the ratio of
perpetual interest which, under the admission of an equal security, a certain
capital might command at the time being; although in case of very long
annuities the difference of the ratios may be only inconsiderable. If it be
further considered, according to the principles from which we took our departure,
that an annuity limited to a certain term of years is not any thing else than the
adjunction, with the simple interest, of a periodical surplus adequate to produce,
by accumulation and improvement at compound interest until the expiry of that
annuity, another capital equal to the one sunk in the original investment;it
COMPOUND INTEREST.
. 63
necessarily follows that, in all the Tables applicable to the solution of the fore-
going questions, each of the quantities expressed in the column indicating the
Annuity for cumulation (meaning the Redemption-fund) should differ from the
corresponding quantity as stated in the column expressing the Ratio of Aunuity,
by a quantity never more nor less than the precise rate of simple and annual
interest of money; and this coincidence will always be observable throughout
the Tables, although the calculations have been executed for both columns
independently of each other.
(XLI.) On the other hand, the Ratio of the annuity, and its counterpart the
Value of the annuity, are reciprocal quantities, such as invariably to produce
unity by multiplying them into each other; and therefore such, also, as that the
division of unity by either should yield for its quotient the other quantity.
Thus, in the same manner as the column entitled Ratio of the annuity exhibits
a series of decreasing quantities, whilst the corresponding periods of reference
increase, so the other column entitled Value of the annuity, exhibits a series of
quantities increasing simultaneously with the enlargement of the periods of time.
to which they refer.
!
(XLII.) The Value of the annuity is a quantity having further an imme-
diate dependence upon other quantities expressed by the Tables. First, it is
the quotient of the Cumulated annuity, divided by the Improved principal; and
secondly, it is the sum total of the discounted values of the successive portions
of annuity forthcoming to its last instalment inclusive. This last position has
been demonstrated (Section IXth), and is exemplified by summing up the quan-
tities expressed in the column entitled Discounted principal, for any period of
time where the instalments are supposed annual; but the same verification is
impracticable with reference to half-yearly and to quarterly instalments, as the
intermediate quantities corresponding with portions of years, and which should
also be taken into the account, are omitted in the tables beyond the 16th year.
It should be observed, that what is there termed the Discounted principal is
precisely the same as the discounted instalments of the annuity; both referable to
unity, considered as the forthcoming sum.
(XLIII.) It is obvious again that the value of an annuity, accruing by full
yearly payments, 'must be more or less inferior to the value of a similar annuity
accruing by half-yearly instalments; and that the latter value must also be less.
than the value of an annuity to the same amount but accruing in due proportion,
at the expiration of each quarter of a year. That such differences whatever
should exist, in the value of similar annuities, is immediately accounted for in
consequence of the earlier accretion, during each succeeding year, of certain
portions of the annuity, in one case than in another.
•
64
THE DOCTRINE OF
Respecting the law of those differences, Dr. Price has inadvertently made the
erroneous remark, that they continually decrease as n (the period of time) increases,
till at last it vanishes, when n becomes infinite or the annuity is a perpetuity.
Dr. Hutton has come nearer to the truth, when observing that for any given rate
of interest, those differences of value first increase as the term n increases, unto
some certain finite term when the difference is a maximum; and that afterwards,
´as n increases more, that difference will continually decrease to nothing again. Mr.
Baily merely observes on this subject, but without elucidating the facts, that he
has no where seen them set in their proper point of view; and the writer of a
late pamphlet, of no very moderate pretensions*, though exulting in the detection
of those and of other discrepancies, whether real or only apparent, equally fails
to clear up the points of contention.
The reality of this case is, however, that the difference or excess of value of
any annuity for terms of years, when the instalments approximate each other,
over the value of a similar annuity the instalments of which are more distant,
continually increases until a period ʼn of years equal to the quantity of years'
purchase of a perpetual aunuity according to the given rate of annual interest, or
1
(mathematically expressed) until a period n = be this annual ratio r what
r
it may; the progression of which increase, or the second order of differences,
proceeds by a series of decreasing quantities. After that period, the difference of
* INVESTIGATION of the errors of all writers on Annuities, in their valuation of Half-yearly and
Quarterly payments; including those of Sir Isaac Newton, Demoivre, Dr. Price, Mr. Morgan, Dr.
Hutton, &c. &c.; by WM. ROUSE, London, 1816.
This writer has taken great pains merely to prove truisms. According to his hypothesis,
4 per cent. annual interest does not imply that any division of this ratio should mathematically
correspond with a similar division of the year, so as in the present instance (for example) to
justify the expression of one per cent. interest quarterly; and he insists, that such quarterly or
other partial payments of the interest should be such as to improve unity (the principal) exactly
to 1'04 at the expiration of the four quarters of the year, whence the quarterly ratio ought in
that case to be expressed by 0'0098533, and not by 0·01.
Nothing can be more concordant than this mathematical position; and it is very certain
that unity, improved in the ratio 1.0098533 at the end of a first quarter, will in the same ratio
become 1.0198039 at the end of the second, 1'0298524 at the end of the third, and exactly
1.04 at the expiration of a whole year. Even the most humble of Sir Isaac Newton's
successors could not be unaware of this. But the respectable authors animadverted upon with
uncalled for severity, in that pamphlet, could not have made themselves understood had they
not admitted, according to the expressions generally received, the quarterly ratio of 1 per cent.,
as also the half-yearly ratio of 2 per cent., equally to correspond with a nominal ratio of 4 per
cent. per annum, although these are three different things. Their calculations, in this respect,
neither involve nor lead to any error on the part of those who rightly understand the
positions: a different view of the case inducing different results, however, in figures.
COMPOUND INTEREST.
65
which we are speaking commences to decrease, thus proceeding during another period
of similar extent, whilst the second order of differences forms a series of increasing
quantities; and after that second period accomplished, the differences of value
continue to decrease until the remotest term imaginable, or until the annuity
becomes perpetual, but the progression of decrease, or the second column of
differences, then proceeds by a series of quantities decreasing also. It is thus,
and in consequence of a somewhat more considerable acceleration in those pro-
gressions, (particularly the latter,)-when the differences considered are those
between the half-yearly and the quarterly, than when considered between the
annual and the half-yearly,-that all such disproportions ultimately disappear,
and that a perpetual annuity becomes valuable in the same proportion, exactly,
whether the instalments be paid annually, half-yearly, quarterly, or
momently; provided always, that the ratio of interest, which governs the
calculation, be admitted to accrue by instalments similar with those of the annuity
with which that ratio is brought into comparison. But if otherwise, and
supposing (for example) the value of an annuity by half-yearly or by any shorter
instalments, to be calculated from a rate of annual interest accruing in one single
payment, it is very certain that instead of assimilating at the period of perpetuity,
with the value of an equal annuity accruing in one single and yearly payment-
such value calculated according to the same stipulation of interest,-the respective
values of the similar annuities, but subject to different intervals of instalment, would
then diverge ad infinitum.
even
K
(XLIV.) In the same manner as any half-yearly or quarterly ratio of interest
is admitted, by common consent and without any alteration in those proportions,
to correspond as aliquot parts of a certain ratio of interest per annum,-though
the latter ratio should then be understood as nominal only,—so the ratio of
annuity, as expressed by the Tables, should only be considered as nominal when
subject to the condition of being paid either by half-yearly or quarterly propor-
tions. We cannot too much impress our readers with the necessity of keeping
these and other analogous distinctions constantly in view; by which may some-
times be avoided very gross mistakes.
1
(XLV.) The simplest reasoning, no less than the inspection of the Tables,
shews that quarterly rates of interest improve a capital faster than it could be
improved by half-yearly rates; and still faster than by undivided annual
rates. Should we suppose such interest to accrue daily, still maintaining a
corresponding proportion, the progression of improvement would then be further
accelerated; and following up the same principle, even to the fiction of interest
accruing momently in a proportionate ratio, this would have the effect of carrying
to its utmost limit the rapidity of that progression. To indulge in contemplations
such as the latter may be divested of practical utility, but those researches of
analogical truth, though otherwise useless, have a general tendency to rectify
i
#
66
THE DOCTRINE OF ·
and to invigorate the powers of ratiocination; hence shall presently be considered.
some of the results attending the momently accretion of interest.
(XLVI.) In like manner with the proportion of increase of the Improved
Principal, by lapse of time, the proportion of corresponding increase of the
Cumulated Annuity (which the authors have technically called the Amount) is
remarkably greater when the instalments are brought nearer to each other. For
example: One million annually appropriated, as a fund for the redemption of
the national debt, and improved at the rate of 4 per cent. compound interest,
would, in case of that sum being undivided, and of the progression of
accumulation and improvement taking place from year to year,-be adequate
to discharge at the expiration of the 90th year 828 millions capital of debt;
if invested and improved half-yearly, a capital of 858 millions; and if quarterly,
£873,741,030 of such national debt.
Šp
(XLVII.) We have already noticed the corresponding effect of accelerated
instalments upon the value, in present capital, of annuities constituted for terms
of years. But it may be worthy of remark, that all those effects are reversed
when the comparison is made between annual, half-yearly, quarterly, and even
momently instalments, with the condition of advancing each instalment of the
interest or annuity; that is to say, that the income should accrue at the beginning
instead of the end of the year, half-year, &c., respectively. It is obvious then, and
in consequence of such early accretion of income being greater as the instalments
are farther removed from each other, that the Improved principal, as also the
Value of the annuity, must be more considerable when the instalments are
annual than when half-yearly, more so in the latter case than when the instalments
are quarterly, and still more so than when momently. In this last supposed case,
there would be no acceleration whatever of income; the two modes coinciding.
•
It is observable, however, that the consideration of advanced instalments of
the annnity may be wholly discarded in treating the questions of compound
interest; as such advance of 12, 6, or 3 months income, operates not otherwise
than as a reduction from the represented principal, or from the price at which the
annual income may have been purchased.
(XLVIII.) Notice has been taken (XL, XLI and XLII,) of certain invaria-
ble relations between each of the quantities expressed in the Tables, under the
heads either Ratio of the Annuity or Value of the Annuity, and other quantities
expressed with reference to the same period of time and rate of interest. It is
further observable, that each of the quantities there expressed under the head
Cumulated Annuity, is so related with others expressed in the same horizontal
line, as constantly to maintain, between the Value of the Annuity and the
Cumulated Annuity, a proportion identical with that subsisting between the
COMPOUND INTEREST.
67
I
eriginal Principal and the quantity of its Improvement at compound interest.
This identity of. proportion, p: m :: 1: (1+r)", follows indeed from the
definition of those general quantities; whence is to be remarked, that the sum
into which, any principal may be improved-whatever the time allotted for
such improvement and whatever the rate of interest-is equal, in all cases,
to the capital susceptible of being created at the expiration of the stated period,
by the accumulation and improvement of the annuity which might have been
purchased with the same original principal. In consequence of this, if the
Cumulated amount of a quantity whatsoever of successive instalments of the
annnuity should be reconverted into another annuity, to endure a similar period,
commencing when the last of those instalments became due, the two annuities
would bear with each other the same proportion as the original principal bears.
with its improvement at Compound interest at the expiration of the term of such
accumulation; provided that the rate of interest and the intervals of the instal-
ments were to continue as before. Any of the Tables selected at pleasure, with
reference to whatever period of time, will corroborate this position.
Suppose the rate of Interest 34 per cent. per annum, and the instalments
half-yearly; an aunuity of One pound will by accumulation and improvement.
during 25 years, create a capital of £4084049, which converted into another
annuity limited to a similar term, and valuable at 16-1328 years' purchase,
establishes the quantum of the latter annuity at
= £2.53152, being
also the improvement at Compound interest of One pound Principal, at the
expiration of 25 years (Table, page 14th).
40.84049
16.1328
(XLIX.) At the head of each table of Compound Interest, and with refer-
ence to the annual rate according to which that table has been calculated, is
specified the corresponding value of public securities, or of any Stock supposed
to be funded at the several rates of 3, 32, 4, and 5 per centum, perpetual interest
per annum; but it should be understood that those assimilations of respective
value essentially rest on various admissions, without which the comparisons
become illusory.
First, it must be admitted that the instalments of periodical income, whether
annual, half-yearly, or quarterly, are the same in both instances compared ; as
otherwise we should fall within the series of anomalous cases, requiring distinct
and complicated calculations for solving the question.
Secondly, the compared price of Stock must be exclusive of any acquired
claim, arising from the time elapsed since the last dividend became due. As
each quoted price of Stock is compounded of two parts,—the portion of interest
accrued since the last dividend took place, and the price of the bare Stock,
68
THE DOCTRINE OF
•
considered as an investment of capital producing interest from the date at which
it is made; it follows that at any intermediate period between two dividends,
a proportionate subtraction should be made from the quotation, in order to
establish the real price exclusive of any portion of future dividend then acquired,
and the whole of which dividend it is customary to transfer with the Stock
itself.
Thirdly, the compared values necessarily rest on the admission that the holders
of the Stock are indefinitely secured against any disadvantage depending on the
possibility of its early redemption at par. Without this last consideration, and
supposing (for example) a Stock funded at 4 per cent., being presently valuable
above par; if such Stock were redeemable at pleasure by the Government,-or
if not being actually so, it were to become redeemable at the expiration of a limited
period of forthcoming years, the proportion of its corresponding value, as stated
in the Tables, would then be only imaginary, and could not be received as a
just comparison.
(L.) The differences in the present value of annuities, according to the
specific distribution of their instalments, have already been considered (XLIII.
S
and XLVII.) But reverting to the equation p =
p
(XV. and XXII.) as a general expression of the value of annuities, a decomposition
of that general expression will exhibit at a single glance the comparative values,
in the several cases where the annuities may be accruing annually, half-yearly,
quarterly, or otherwise, and even momently; the latter, though merely a fiction,
shewing the utmost value of which an annuity is susceptible, according to the
specific rate of interest admitted to govern the calculation.
1
a [(1+r)" − 1]
r (1+r)"
If we take a = 1; so that the value, as expressed by p, may be exactly the
quantity of years' purchase of the annuity; the above-mentioned equation will
r
(1+r)” — 1)
r(1+r)
1
T
then be simplified into p =
1
5 and will always express the present value of the
r (1+r)*
Perpetual Annuity, or its relative quantity of years' purchase. Considering r as
the ratio of interest to correspond with a whole year; further admitting any sub-
divisions of that ratio (thus become merely nominal) proportionately to correspond
with similar subdivisions of the year, whether into half-years, quarters, &c. even
into indivisible moments; and again if we admit, instead of the general expres-
sion p of value, the more particularized expressions Pa Ph, Pq, Pz, and Pm
it is evident that we shall comparatively obtain—
set forth
which is identical with
COMPOUND INTEREST.
69
1
1
Pa
717) the value when the Annuity and the Rate of interest
9
r(1+r)”
are both considered in one single and yearly payment;
Ph =
1
r(1+÷)²™¹
interest are both considered by half-yearly instalments;
T
Px =
1
VIF
{
1
1
P, = — — 7(1+-ju, the value when the Annuity and the annual rate of
interest are both considered by quarterly instalments;
1
1
ጥ r(1+1)™n
the value when the Annuity, as well as the annual
rate of interest, are both considered to accrue by instalments dividing each year
into a quantity whatever x of equal parts;
and Pm =
?
1
rn
1 + +
1
1
the value when the Annuity, as also the rate
r(1+1)=n,
of yearly interest, are equally considered as accruing in similar proportion at each
indivisible moment*.
2
the value when the Annuity and the annual Rate of
The equation Pm, expressing the utmost relative value which any annuity can
possess, in consequence of admitting it to accrue even by momently instalments,
is reducible to a more simple expression through the developement of the series
involved in the binominal quantity (1+2). That developement produces
r² n²
доз n3
pt no
+
1X2 1X2X3 1X2X3X4
+
+
p5 ns
1×2×3×4×5
+ &c....
ad infinitum; the summation of which series yields precisely the quantity that
has rn for its Hyperbolic logarithm, or in Neper's system, and equally has for its
ordinary and tabular logarithm rn X0·43429 44819 03251 82765... &c., accord-
ing to Briggs's system of Logarithms, having the latter and infinitèsimal fraction
for its modulus. Therefore, admitting N to represent such implied quantity,
* We must be permitted here to remark, on Dr. Price's theorems analogous to the above,
that from his omitting to state the indispensable requisite (with which it should be supposed
he was acquainted), of assimilating the instalments of the valued annuity, with those of the
governing rate of interest, many of his readers are not unlikely to be betrayed into false
conclusions.
70
THE DOCTRINE OF
we obtain P
M
Pm =
1
0.04
11
Pm =
2
duration, subject to momently instalments, and the rate of interest at 4 per cent.
accruing in the same manner, we find .
1
0.035
1
; and supposing, for example, an annuity of 25 years'
rN
1
0.04N
= 15·80301 years'
purchase*. Thus the value of a 50 years' annuity will also be found 21·61662;
and that of an annuity for 100 years, 24.54211 years' purchase, at the same
rate of interest and under the similar condition.
1
0.035 N
= 25
For a last example, let us suppose an annuity for 48 years, and the rate of
interest 3 per cent. per ann.; the value of that annuity being 23.09125, if in
a single yearly payment, 23.16852 if by half-yearly instalments, and 23-20740
if quarterly, (all which results are stated in page 12th of the Tables); its value,
in the hypothesis of momently instalments, will be
25
2-71828 18284 59..
**
200
7
1
200
7X5.365556
quence of 5.365556 being the hyp, log, of
7X48
200
=23-24646, in conse-
-1·68 - rn.
Those, and analogous calculations, prove that only an inconsiderable increase,
as to the real value of annuities, is obtainable by reducing the intervals of their
instalments to a less time than 3 months.
(LI.) Leaseholds and various other descriptions of property, when their
annual income is susceptible of ascertainment, or of being reduced to a valuation
founded upon satisfactory probability, may be assimilated in all respects to
annuities for terms of years and which have been hitherto the subject of our
consideration. A question not unfrequently recurring is that of the fines due,
at an equitable valuation, for the renewal of unexpired leases; and the rule for
solving such questions is of the greatest simplicity. The renewed lease is
valuable as an annuity for the whole term of that lease, and its value may be
immediately ascertained by the Tables of Compound-Interest, after having
determined the standard rate of interest by which the calculation is to be governed.
On the other hand, the unexpired part of the old lease is valuable on the same
principle; and the value of the latter being subtracted from that of the former,
* Dr. Price has erroneously stated the result as 15·8010; and has also stated 15·7694, instead
of 15.75722, as the value of a 25 years' annuity, at 4 per cent, interest, quarterly instalments.
An
71
COMPOUND INTEREST.
exhibits a remainder which is the value of the renewal, or the pecuniary consi-
deration justly due.
FIRST EXAMPLE.-A leasehold or equivalent property, producing a clear
annual income of £350 and accruing by half-yearly instalments, has 7 years
unexpired; the renewal is proposed for 21 years, computed from the present time;
and it is inquired the fine that should be paid on that transaction, the interest of
money being considered at 5 per cent. per ann., also by half-yearly instalments.
The Table (page 34th) shews the new lease to be worth exactly 12·91030, and
that the relinquished portion of the old lease is worth 6-190689; whence the
difference, 6.719611 years' purchase, multiplied into £350, exhibits the sum of
£2351. 17s. 3d. as the fine to be paid.
SECOND EXAMPLE.-The lease of a property rendering £1750 annually, in a
single payment, has 12 years unexpired; the renewal is proposed for a term of
99 years, and the rate of interest is agreed to be taken at 4 per cent. as a standard.
The Table (pages 26th and 28th) exhibits in this case the two values 9·118581
and 21.93760 years' purchase; and the difference 12.819019, multiplied into
£1750, produces £22433. 5s. 8d., being the value of the renewal.
(LII.) It is now requisite to point out the use of the Tables, in cases where the
rate of annual interest (constituting a principal datum towards solving the questions.
that depend on Compound-Interest) may be either lower than three per centum,
or higher than six per centum; being the two limits within which those Tables
are circumscribed. The introduction of an example, appropriate to each general
case of that description, will be the clearest method of proceeding.
EXAMPLES of application of the Tables, when the Annual Rate of Interest is less
than Three per centum.
FIRST Supposing the rate of 24 per cent., in one single and annual payment;
be it required the Improved principal.
The progress of that improvement will then be, for every succeeding year, the
same as for every succeeding half-year when the rate is 5 per cent. per ann., and
accruing by half-yearly instalments. We therefore find in the Table (page 42d),
that each unit of the original principal will improve, at 22 per cent., to 1·7204284
at the expiration of 20 years, and to 3.8823218 at the expiration of 50 years;
being respectively the improvements for similar quantities of half-years, when the
interest is 5 per cent. And if the interest at 24 per cent. per annum were to
accrue by half-yearly instalments, the progress of improvement would then
•
72
THE DOCTRINE OF
correspond with the quarterly instalments at 5 per cent.; whence would be found,
1.7267708 and 3.9182011, instead of the above results.
SECONDLY: Supposing the rate of interest 2 per cent. annually, in a single
payment; be it required the present value of a future capital.
If we refer to the Table relative to 5 per cent. interest; and assimilate each
half-year, in the progress of discounted principal at that rate, with each full year
at the rate of 2 per cent.; we find (p. 35) that each unit of capital forthcoming
at the expiration of 25 years, and according to the latter rate, is reducible to
0.5393906, and (p. 37) to 0.1387046 at the expiration of 80 years; those
reductions respectively corresponding with the discounts for 12 and for 40 years,
at 5 per cent. And if we had to consider the annual rate of 21 per cent. as
accruing half-yearly, we then should find the results 0.5373390 and 0·1370234 ;
in consequence of the decreasing progression being the same, for every succeeding
half-year at 24 per cent. annually, as for every succeeding quarter at the rate of
5 per cent.
}
THIRDLY: Supposing the rate of interest 24 per cent. per ann. in one single
payment; be it required the Cumulated annuity.
The results, by the Table corresponding with the annual rate of 4 per cent.,
exhibit for every succeeding half-year the progress of accumulation and improve-
ment at Compound-Interest of a portion of annuity equal to 0.5, and therefore
referable to each half-yearly instalment; the whole of that annuity being repre-
sented by unity. But if the annual rate of interest should be admitted at 24 per
cent. only, those results would exhibit the similar progress of an annuity no more
than 0.5, received and improved during as many full years as the quantity of
half-years considered with reference to the double rate of interest; those results,
as expressed by the Table, differing however from exactitude inasmuch as they
require to be doubled, in order to correspond with the integral unity of annual
income, instead of corresponding as they do with 05 only. Hence if it be
inquired, (for example,) what amount of capital will be created at the expiration
of 30 years, by means of any annual income accumulated and improved from
year to year at 24 per cent., we find (page 26th of the Tables) that amount to be
in the proportion of 21.097632 times the half unity of periodical income during
30 successive half-years, or 15 years as there expressed; from which we con-
clude 42.195264, or double the preceding quantity, to be the proportion of
accumulation when unity is taken for the whole yearly income. And if the latter
were specifically £1000, it follows that its accumulation and improvement at
2 per cent. annual interest, in one single payment and during 30 years, would
produce a capital of £42195-264, or £42195. 5s, 3d.
COMPOUND INTEREST.
73.
Further supposing the same rate of annual interest, 24 per cent., but to accrue
half-yearly; if it were inquired the cumulated amount, at the expiry of a 54
year's annuity of £600, we should find (page 28th) the proportion 52-166864
corresponding with 27 years, by quarterly instalments at 4 per cent. per annum,
and corresponding in the same manner with 54 years by half-yearly instalments
at 24 per cent. only; whence the cumulated amount would be £600 multiplied
into the double of that proportion, producing therefore £62600. 4s. 9d.
FOURTHLY: Supposing the interest at 21 per cent., in a single yearly payment;
be it inquired the Redemption-fund that should annually be supplied, as adequate
to the purpose of discharging, at a specific period, the capital of a certain debt.
The considerations are here precisely the reverse of what they were in the
preceding case.
We must, in the same manner, resort to the Table (pages 23d and 25th) cor-
responding with a double rate of interest, or that of 41 per cent. per annum;
and there look for the series of results relative to half-yearly instalments, in the
column entitled Annuity for Cumulation. If the period assigned for the redemp-
tion should be 21 years, we find 0·0765553 the required proportion referable to
21 half-yearly instalments, or to a period of 10 years; and if the assignment
should be 72 years, we also find the proportion 0·0119896 relative to a period of
36 years, by half yearly instalments. But as those proportions refer alike, not
to unity, but to the quantity 2 as the future capital (because of the case converse
from the above), we must reduce them to one-half; whence 0.03827765 will
be the proportion of the annual redemption-fund for producing any required
capital at the expiration of 21 years, and 0.0059948 will be the proportion of
that annual fund for creating any capital at the expiration of 72 years. Let that
capital be supposed £100,000; the annual redemption-fund will then be, either
£3827. 15s. 3d. during 21 years, or £599. 9s. 7d. during 72 years.
Supposing again that the annual rate of 2 per cent. were understood by
half-yearly instalments, we should refer in that case to the column of results
corresponding with quarterly instalments at 44 per cent. per annum; and pro-
ceeding as above, the specific results would respectively be found £3803. Os. 1d.
and £593. 8s.
FIFTHLY: Supposing the interest at 2 per cent. per annum, in a single pay-
ment; be it required the value of an annuity for any specific term, and also to
accrue by instalments of successively undivided years.
Referring to the Tables (pages 18th and 20th), where the value of the annuity
is stated with reference to interest at 4 per cent., it is necessary only to observe that
k
74
THE DOCTRINE OF
*
half years are there to be taken as full years, subject to interest at 2 per cent. only;
that quarters, at the former rate, are likewise to be assimilated to half-years at the
latter rate; and that the Value, as expressed by the Tables, is consequently to be
doubled.
For exemplification, let two cases of annuities be specified, the one for
20 years and accruing annually, the other for 45 years and accruing by half-
yearly instalments, both at 2 per cent. per annum. The value of the first will
be, according to the Table (page 18th), twice 13.67774 or 27.35548 years' pur-
chase; and the value of the other (page 20th) twice 20 8304, or 41·6608 years'
purchase.
SIXTHLY: Supposing the annual rate of interest 17 per cent. in a single pay-
ment; be it required the Ratio of the annuity, or the proportion which that annuity
should bear with its represented principal.
This question being exactly converse of the preceding; refer to the Table
corresponding with the double rate of interest, or 3 per cent.; substitute years
for half-years, as also half-years for quarters; and reduce the Ratio to one half
of the fractional quantity expressed in that Table. Thus it will be found
(page 15th) that a 25 years' annuity in a single payment, when interest is at
1 per cent., should be in the proportion 0.03099275, being one-half of
0.0619855, for every unit of the represented principal; and (page 17th) that a
50 years' annuity, half-yearly instalments, and calculated at the same rate of
annual interest, should be constituted in the proportion of 0.0221815, being one
half of 0.044363. If the capital to be invested were £10000, the Annuities
would therefore be, respectively, £309. 18s. 6d. and £221. 16s. 3d.
(LIII.) EXAMPLES of application of the Tables, when the Annual Rate of Interest
exceeds six per centum.
FIRST: Supposing the annual rate 6 per cent., to accrue by half-yearly
instalments; be it required the Improved principal.
The progress of improvement will be, for every succeeding half-year, the same
as for
every succeeding year at the rate of 34 per cent.; consequently, we find in
the Table (page 6th) that each unit of principal improves at the latter rate, and at
the expiration of 25 years, into 2.2245978, answering also to 12 years by
half-yearly instalments, when that rate is doubled or 64 per centum. And if the
interest at this annual rate were to accrue quarterly, the corresponding improve-
ment for the same term of 12 years would be 2-2388651, the same as 25 years'
improvement at 34 per cent. interest accruing at each half-year.
I
COMPOUND INTEREST.
75
SECONDLY: Supposing the annual rate of interest, at 7 per cent., to accrue
half-yearly; be it inquired the present value of future capital.
The considerations in this case being converse of the above, and reverting to
the Table that corresponds with 3 per cent. annual interest, we find (page 13th)
that every unit of capital forthcoming at the expiration of 30 years is reducible to
0.3562784, by discount at the latter rate in a single payment; which same
result is produced by the discount of 30 half-yearly instalments comprised in a
term of 15 years only, when the interest is doubled or rated at 7 per cent. per
annum. Quarterly instalments, at this last rate, will equally answer to half-yearly
instalments at 3 per cent. annual interest.
THIRDLY: Supposing 7 per cent. interest per annum, and subject to half-yearly
instalments; be it inquired the Cumulated annuity.
This being converse of the 3d case exposed in page 72d, every consideration
there set forth accordingly applies in the present; and it may be sufficient, in
addition, to state a specific instance of that application.
If an annuity of £500 and paid by half-yearly instalments, when the interest
is at 71 per cent. per annum and accruing half-yearly also, be accumulated and
improved during 20 years at that rate of Compound Interest; it will create, at the
expiration of such period, a similar capital to that which would be produced by
the accumulation and improvement at an interest of 32 per cent. only, in undivided
yearly payments, of an annuity equal to no more than half of the first, or to
£250, but extended to 40 years' duration. Therefore, as we find in the Table
(page 16th, column 2d) that every unit of the annuity will at the expiration of
this last period have created 89.6101 capital, the conclusion is, that the capital
created at the expiration of the prescribed term of 20 years will be £250 multi-
plied into 44.80505, being one half of the former quantity indicated by the Table;
producing in either way £22402. 10s.
If again a £500 annuity were to accrue by quarterly instalments, during
the period of 20 years, that annuity would assimilate with another of £250
only and paid half-yearly during 40 years; when by the same Table (page 16th
column 5th) we find that every unit of sach income would create capital in the
proportion of 91-1966; therefore the whole annuity a capital of £22799. 3s.
or £500 multiplied into half that proportion.
FOURTHLY: Supposing 8 per cent. annual interest, by half-yearly instalments;
be it inquired the periodical Redemption-fund.
This is the counterpart of the 4th example of the preceding series (page 73d), tl:
reasoning on which occasion equally applies here.
f
76
THE DOCTRINE OF
*
If the question should be of the savings required out of any income, at the
expiration of each succeeding half-year, and until the 25th year accomplished,
with a view then to redeem £100,000 of debt, by means of the accumulation
and improvement of those savings at the above-mentioned rate of compound
interest; it is evident that they must be the same as would be required at the
expiration of each succeeding year until the 50th accomplished, upon a supposi-
tion of the improvement taking place at only one-half that rate of annual interest,
or at 4 per centum. Hence the table (page 21st, column 2d,) informs us that
the proportion of savings, either annually in the latter supposition, or half-yearly
in the supposition of 8 per cent. interest,—and for every unit of the required
capital, is 0.0065502; consequently £655-02 will be the half-yearly, or
£1310-04 (say £1310.0s. 5d.) the annual redemption-fund. And if required by
quarterly instalments, it would be in the proportion of 0.0064055 for every unit,
corresponding with half-yearly instalments at 4 per cent. per annum; therefore
£320. 5s. 6d. the quarterly, £640. 11s. the half-yearly, and £1281. 2s. the
annual redemption-fund, during 25 years.
M
FIFTHLY: Supposing the annual interest 8 per cent., with the condition
of its accruing half-yearly; be it inquired the Value of the annuity.
If we further suppose that annuity to be of 17 years' duration, and also paid
by half-yearly instalments, its value will be the same as that of another annuity
of half the sum, but payable at the expiration of each succeeding year during
double that time, or until the 35th year accomplished, on the supposition of
44 per cent. interest only. The table (page 24th, column 3d) indicates the latter
annuity to be valuable at 18.04729 years' purchase; whence the former, being
for 35 half-years, or 17 years only, will be valuable at exactly one-half, or at
9-023645; and if by quarterly instalments, the rate of interest understood to
accrue in the same manner, the value will be 9.064875 years' purchase, or
one-half of 18·12975, corresponding with half-yearly instalments of the interest
rated also at 44 per cent. per ann., as indicated in the same table (column 6th).
SIXTHLY: Supposing the annual rate of interest 9 per cent., to accrue half-
yearly; be it inquired the Ratio of the annuity, or the proportion that any annuity
should bear with the invested capital.
The present question being converse of the preceding, if we further suppose
the implied annuity to endure for a term of 25 years, and to accrue by 50 half-
yearly instalments, it will correspond with another annuity equal to one-half only
of the first, but extended to 50 full years' duration, and accruing by undivided
annual payments, upon the supposition of an annual rate of interest no more
than 4 per cent. and accruing in the same manner. We then find, according
to the Table (page 29th, column 3d), that the latter annuity should be constituted
COMPOUND INTEREST.
77
in the ratio of 0·0506021 for every unit of the invested capital, whence the
former in the ratio of 0·1012042; and therefore, admitting £10,000 capital, the
annuity of 25 years' duration should be £1012. Os. 11d., calculating from 9 per
cent. rate of interest. Quarterly instalments would, according to the same table
(page 29th, column 6th), reduce the ratio to 0.109038, being the double of
0-0504519, ratio corresponding with 4 per cent. interest per annum, by half-
yearly instalments.
(LIV.) When the anuuity, or income whatever, set apart for creating by its
accumulation any capital at a future period, accrues at shorter intervals of
instalment than those at which the improvement by compound interest at any
given rate is obtainable; or when, on the contrary, the income or annuity
accrues by instalments more distant than those available for the improvement;
the question concerning the progression of such accumulation and improvement,
as also the question concerning the value of the annuity, become complicated.
In both cases, and for solving each of those questions, it is necessary to revert
to the primary equations stated in Sections IXth and XIIIth, where is generally
established m=
a(1+r)*.
go
a
2
as also p =
r
a (1+2). (1+r)²; whence m =
a(1+r)" — a
r (1+r)"
by analogy, as follows.
The question being of the Cumulated annuity, and supposing the instalments
of such annuity to be shorter than those that apply to the rate of interest; if we
then signify by 4a the whole annuity, by a each of its quarterly instalments,
by n the forthcoming quantity of those instalments, and by r the half-yearly
ratio of interest, we obviously find the cumulated amount m composed of these
two series of progressive quantities:-
;
n-2
m = a+a(1+r)+á(1+r)² + a(1+r)³ +a(1+r)*+ &c....... to a(1+r) +
2
+ a(1+2)+a(1+2). (1+r)+a(1+).(1+r)²+a(1+).(1+r)²+&c. to
[a+e(1+4)] × [(a+r)²=1]
and then to proceed
n
28
a (1+r); — a + (1+2). (1+r); — a(1+5)
r
In order to exemplify this solution, if we suppose the annuity £100, by
quarterly instalments of £25 each, the term 16 years or 64 quarters, and the
78
THE DOCTRINE OF
1
half-yearly rate of interest 2 per cent., or ratio 0·02; the latter equation, reduced
to numerical quantities, will then be m=
50-25(0.8845406)
0.02
or £2222. 8s. the Cumulated amount. Each unit of the annuity having therefore
created 22.224082 capital at the expiration of 16 years, we find by the table.
(page 18th), that if the instalments of that annuity had been half-yearly, the
capital produced would have been only 22-113515, and that it would have been.
22-261547 on the different supposition, of quarterly instalments applied both to
the annuity and to the rate of interest.
Further supposing (as another example) an annuity of £1, its term extended
to 100 years or 400 quarterly instalments, and the same ratio of interest as
above to govern the calculation, we obtain
m =
50.25 [(1.02)2001]
0.02
= £1293·5581; being
the Cumulated amount, comparable with 1287·1225, if the instalments of that
annuity were half-yearly, and with 1313-1029 if the instalments of interest were
quarterly, as appears by the same table (p. 20th).
Under the same supposition of shorter instalments of the annuity, and fol-
lowing by analogy we have, in consequence, of p
p =
50.25 × 51·484899
0.02
of such annuity generally expressed by p =
p =
22.224082
1.8845406
=2222-4082,
1293.5581
52.484899
m
ท
グ
​2
[a+a(1+ " ")] × [(1+r) −1 ]
r (1+r) =
whence, by admitting the above two cases of annuities as examples of applica-
tion, and further admitting 4a 1, in order that the results may refer to unity
of the annual income, we obtain in case of the annuity limited to 16 years,
71
(1+r)*
the value
= 11.79284, and in case of that annuity extended to 100 years
=24.64630; being the respective values, comparable the
former with 11.73417 and 11·77572, and the latter with 24·52367 and 24·53292,
results found in the same table and according as the instalments may be taken,
either half-yearly or quarterly. And be it observed that this excess of value, in
both cases, does not proceed from any error; but is the natural consequence of
COMPOUND INTEREST.
79
any present capital, improvable by half-yearly instalments, being relatively less
valuable than the same capital susceptible of quarterly improvement.
Should on the contrary the annuity, or any other income, accrue by instal-
ments more distant than those available for the improvement at compound interest,
we have then, and by similar analogy with the primary equations,—
m = a + a(1+r)² + a(1+r)*+a(1+r)° + &c. to a(1+r)" — a(1+r)" — a
M
(1+r)²-1'
and p =
m=
To exemplify in a particularized case, let it be supposed £1 annuity, re-
ceivable at the expiration of the year now commencing and of every succeeding
year, to the 25th inclusive; the rate of interest, considered in the improvement,
being nominally 4 per cent. per annum, but accruing by half-yearly instalments;
whence r = 0·02, and n = 50; we shall then have
= 41 87099, or £41. 17s. 5d., being the
amount of that annuity accumulated and improved at compound interest until
the period of its expiration, which amount is comparable with 41.645908 and
with 42.289701, as indicated by the table (page 18th); and
15.55624 years' purchase, as
its present value, comparable also with 15.62208 and with 1571180, the values
expressed in the same table.
p =
a(1+r)”—a
[(1+r)² —1] × (1+r)”
a(1+r)”—a
(1+r‍)² — 1
1.691588
0.0404
a(1+r)”—a
[(1+r)²—1] × (1+r)”
and p =
41.87099
2.691588
If we further suppose the same annuity extended to 100 years, the results of
51.484899
0.0404
1274-3787, the Cumulated annuity,
= 24-2808 the Value of such annuity; the former of
calculation will be m =
1274-3787
52.484899
which results is comparable with 1237·6237 and with 1287.1225,—the latter
with 24.505 and with 24.52367, as expressed by the same table (page 20th).
And again it is to be observed, that this apparent deficiency of value is the con-
sequence of a capital, improvable half-yearly, being relatively worth more than
the same capital improvable only by annual instalments.
(LV.) When a Stock originally funded by perpetual annuity, at a higher
interest than the current rate at the present time, is redeemable at the debtor's
pleasure, that stock, notwithstanding its superior nominal rate of interest, can
possess no greater value than another perpetual stock funded at the present rate
80
THE DOCTRINE OF
of interest; except in consequence of assuming, either that the latter rate is not
entitled to consideration as a permanent one, or else that the faculty of redemption
will continue during more or less time without being acted upon.
?
But if any stock funded on the principle of perpetuity should, from the condi-
tions of the contract, be irredeemable until some assigned and future period, it
will, under circumstances similar to the above in other respects, possess a market-
able value superior to par; and the exact value, to which it is entitled, may then
be accurately ascertained. To this effect, let it be considered that any Stock of
that description is compounded of two parts, having each its proper value; first,
that of a terminable annuity, to continue at its original and superior ratio until
the period available for the redemption of that annuity at par, and secondly the
value of a perpetual annuity constituted according to the present reduced rate of
interest, but from which last value should be subtracted that of a terminable
annuity, at the same reduced rate, and supposed to endure until the assigned
period of redemption. A single example of the solution of this question, by
means of the tables, will be sufficient.
Suppose a Stock funded at 5 per cent. per annum, by half-yearly instalments,
the last of which had just fallen due, such stock to continue irredeemable
until the expiration of 7 years from the present time; and that another stock
funded at 3 per cent., accruing also by half-yearly instalments, being at par, it
were inquired the present value of the former, relatively to that of the latter stock.
From the value of the perpetual stock, stated at par
subtract the value of an annuity at 3 per cent., supposed to endure
for 7 years only, or 14 half-yearly instalments, and which by the
table (page 10th) is 6:161002 years' purchase; this last quantity
multiplied into 3 is
remains....
and adding the value of an annuity at 5 per cent. for the same term
of 7 years, which is equal to 5 multiplied into the same quantity
6.161002
100.
21.56351
78.43649
30.80501
together....
109-24150
or very near 1094; which is the value of every 100 of stock at 5 per cent., but
irredeemable during the next seven years.
The same result is to be obtained by merely adding to 100, or to the price of
stock at par, the difference 1 of the two rates, multiplied into 6-161002; which
last is the quantity of years' purchase at which an annuity limited to 7 years
should be valued, when interest is at 33 per cent. and the instalments half-yearly.
COMPOUND INTEREST.
81
I
(LVI.) The preceding question leads us to consider also that of the value
of deferred annuities; which of their description may be either perpetual or
terminable.
The value of a deferred annuity, in both cases, is that of an equal annuity
commencing to run from the present time, minus the discounted value of the
instalments withheld,
Supposing a perpetual annuity constituted at 4 per cent., and the term of
which should not commence to run until the expiration of 10 years from the
present time, its first half-yearly instalment consequently falling due six months.
afterwards; be it inquired the present value of that deferred annuity, admitting
the same standard rate of interest, 4 per cent. per annum, also to accrue half-
yearly. The table (page 18th) indicates 8-175717 years' purchase, as the value
of the 20 instalments withheld and comprised in the suspensive term of 10 years,
which quantity subtracted from 25, the value of perpetuity, leaves for the required
value 16-824283 years' purchase; and as the ratio of that annuity is 4 per
cent., the value of £100 stock of that description will be £4 multiplied into
16.824283, or £67. 6s. The capital represented by annuities that are altogether
deferred and terminable will, on the same principle, be ascertained by the
difference between the present values of the long and of the short term; which
position does not require any specific exemplification.
We may further suppose, however, the case of a perpetual and deferred
annuity such as the above, but constituted at some earlier period with the
condition of its continuing irredeemable during the first 20 years, or until the
expiration of 30 years from this time; and that the rate of interest, at present
available, were reduced to 34 per cent. per annum, also half-yearly instalments.
Admitting this rate of interest as a permanent standard, and applying the rules
already set forth, we proceed in the following manner to ascertain the value of
each £100 stock funded as here mentioned.
From the present value of a 31 per cent. stock, being at par.……. £ 100.
subtract the value of the first 30 years of an annuity in that ratio,
which by the Table (page 8th) is 19-07200 years' purchase, and
multiplied into £ 34, is
Remains the present value of a perpetuity after 30 years
Add to this, the difference between the value of a 30 years' annuity
and that of a 10 years' annuity, both of £4; or between 19.07200
and 8-47947, according to the Table (pages 6th and 8th), that
difference being £4 multiplied into 10.59253 years' purchase, or..
61.98400
£ 38·01600
42.37012
Together........ £ 80·38612
the value of £100 stock, or about 80% per centum.
1
Un
82
THE DOCTRINE OF
From those examples it will be easily perceived how to operate in all analogous
cases.
(LVII.) British capital being frequently invested in the securities of foreign
governments, and at very elevated rates of interest comparatively with the rates
afforded by investments at home,-such surplus interest compensated however
by an additional risk incurred and depending on political considerations,-it
may be desirable to ascertain the period at which that additional risk is likely to
terminate; considering that the capital originally put to hazard is gradually
recoverable, by means of periodical savings out of the income, and of their im-
provement during that period. The simple question, thus stated, is easily solved
by the Tables.
Supposing the annual interest at home to be 34 per cent., assumed as a per-
manent rate, subject to half-yearly instalments, and that a loan were granted to
a foreign government at 5 per cent. per annum, payable in the same manner;
the extraordinary risk will be extinct whenever the saving of 14, annual sur-
plus has, by accumulation and improvement at the rate of 31 per cent. compound
interest, recomposed the original capital £100; although the ordinary income,
at that rate, will have been available in the meantime. This being the period
at which unity of income will, in a corresponding proportion, have produced
£ 57·142857 capital, it is immediately ascertained, by the Table (page 8th), to
be a period of 32 years or thereabouts. Therefore, if the probable consequence
of a war with that foreign country should happen to be the confiscation of its
enemy's tangible property, such a transaction could scarcely be considered as
prudent, unless the balance of chances were in favour of uninterrupted peace
during 32 years at least.
But instead of allowing the risk of foreign contingencies to diminish from
each half year to the next, in consequence of exempting from that risk the
reinvestment of the savings; if such reinvestment were also made abroad and
there improved at 5 per cent. per annum compound interest, assumed to be a
permanent rate, the Table (page 36th, column 5th) indicates that 27 years, or
thereabouts, will then be sufficient for recomposing the capital integrally exposed
to extraordinary risk until the accomplishment of that period. The Cumulated
annuity being 55.878498 for 27 years, and 59-719847 for 28 years, those quan-
tities respectively multiplied into 14 (the annual surplus) produce 97-787372 and
104.509733, capital at the end of those terms.
Let any foreign investment be further supposed at 7 per cent. annual interest,
accruing half-yearly, therefore in the proportion of 34 per cent. at each instalment ;
and the whole of that income to be successively reinvested at the same rate.
The Table (page 14th) shews that in such case, and after 19 instalments
***
COMPOUND INTEREST.
83
embracing a period of 94 years only, every unit of that half-yearly income will
have grown, by accumulation and improvement, to a capital of 27·00471;
whence 33 to 101.26766, or even exceeding in the proportion of 1·26766 to
100 the original capital at stake. The accurate solution, however, is indicated
by the 6th case amongst those enumerated in Section XVth, where the term of
log. (a-mr)-log. a
log. (1+r)
the annuity is generally expressed by n =
=
which numerically constructed, gives n =
log. 2
log. (1.0375)
instalments required.
log. (0·0375+0-0375)-log. (0-0375);
log. (1·0375)
= 18.82827, the quantity of half-yearly
0.301029995664
0.01598811
And lastly, let us suppose a question of the more extensive period at which
the same annuity would, in consequence of a similar accumulation and improve-
ment, repay the original capital, together with its increase by compound interes
rated only at 3 per cent. per annum; still admitting half-yearly instalments of
both rates.
It is perceived, by the table (page 16th), that unity improved during 33 years,
at 33 per cent. in entire annual payments, produces 3.3698260, and 3-4961945
if thus improved during 34 years; which quantities, respectively taken in the
proportion of one half, are reducible to 1·6849130, corresponding with 33 half-
yearly instalments, and to 1-7480972 corresponding with 34 similar instalments,
admitting the interest at 7 per cent. per annum,—difference 0·0631842 for the
last half-year's improvement; and it is also perceived, that when the annual
interest is rated at 3 per cent. by half-yearly instalments, unity should improve
to 1-6750118 at the expiration of 16 years, and to 1.7298920 at the expiration.
of 17 years; difference 0.0548802 for the whole 17th year's improvement.
Now, of this difference subtracted from the preceding quantity leaves 1-716172,
for 16 years' improvement; whilst subtracting one balf of the former difference
from 1.7480972 there is left 17165051 for the improvement during the same
term, or 33 half-years, when the annual interest is rated at 7 per centum. The
coincidence of those results shews that the required period is 163 years, according
to the nearest approximation attainable by this circuitous method; and it is
obvious that the original capital will then be reimbursed, together with constant
interest at the minor rate.
›
But the direct calculation yields a more accurate solution of this question,
which partly falls within the 3d general case (Section XVth); the Cumulated
amount of the annuity being there expressed by m =
[(1+r)” — } ] × ɑ
a
ร
84
THE DOCTRINE OF
It has already been ascertained, by the solution of the preceding question, that
at a period n = 18.82827 half-years the original capital would barely be repaid;
and there remains to ascertain an additional period n' sufficient for producing,
by the accumulation and improvement of a half-yearly income of 0·0375 in
this same ratio of compound interest, a quantity equal to the accumulation
and improvement of a half-yearly income of 0·01625 only, during a period
n+n'.
Therefore we have, on one part, m
[(1·0375)”—1] × (0-0375), and on
0.0375
the other part, m =
[(1·01625)"+"'—1] × (0·01625)
0.01625
each of those equations, and as m is an identical quantity in both, is simplified
into (1·0375)"' = (1·01625)"+"', or (1·0375)"' = (1·01625)+18-62827; whence is
successively deduced, (n'+18·82827) × log. (1·01625) — n' × log. (1·0375),
18-82827 X 0·00700056 + n' × 0·00700056 n' X 0·01598811,
×
n(0.01598811 -0.00700056)
0·00700056) = 18-82827 × 0·00700056,
18-82827 X 0·00700056
0.00898755
and ultimately n =
which, as ar in
= 14.66575.
Adding this last quantity to 18.82827, we have n+n' 33·49412, a period
very near 33 half-years or 16 years, at the expiration of which the whole
capital risked in the foreign loan would be repaid, together with compound
interest at 31 per cent. per annum ; and beyond that period, all further income
derived from the transaction would be a clear profit.
(LVIII.) In case of any series of equal sums, payable at equidistant instalments,
the quantity of which expressed by n, if it were required to ascertain the
intervening period when the united amount of those sums might be discharged
in a single payment, without injury either to the debtor or to the creditor;
having ascertained the present value of those sums, respectively discounted and
considered as a terminable annuity, the period at which that present value
would, if improved at compound interest, amount to the same total sum as
before the discount, must be the period required.
EXAMPLE. A portion of the National Debt of this country consists in certain
terminable annuities, to the amount of £1,338,837, the last half-yearly
instalment of which will become due the 5th of January 1860; and after dis-
charging the instalment due on the 5th of January 1825, it is inquired the
COMPOUND INTEREST.
85
period at which the total sum of £46,859,295, being the unreduced amount of
the 70 instalments then forthcoming, might equitably be discharged in one single
payment; calculating on 34 per cent., as a standard rate of annual interest, but
supposed also to accrue in half-yearly proportions.
The table (page 8th) immediately solves this question. It is there stated
that a 35 years' annuity, accruing by half-yearly instalments, is exactly worth
20.8134 years' purchase; and as the improvement of this last quantity at compound
interest, into the original quantity 35, is in the proportion of unity to 1·681609,
the same table (page 6th) further indicates that such improvement should take
place between the 16th and the 17th year. At the expiration of 16 years it
will be 1.6750118, and at that of 17 years 1.7298920,-difference 0.0548802
for the 17th year; and as the required improvement exceeds that for 16 years by
a quantity only 0.0065972, we find by taking the proportion of those differences
that the latter corresponds with 44 days; whence follows, that the payment of
the whole sum £46,859,295 might equitably be effected at the expiration of 16
years and 44 days after the 5th of January 1825, or on the 18th February of the
year 1841.
Under the supposition of deferred annuities as already described (LVI.), the
present value of the withheld instalments should first be subtracted from that of
the complete annuity; proceeding afterwards as above directed, to ascertain
the period (computed from the present time) at which such discounted value is
improvable into the original and total sum.
But it might be proposed to discharge such a debt as the Long-annuities by the
payment of any other specific sum, whether more or less than the unreduced amount
of their forthcoming instalments. Let it then be inquired the respective periods.
at which either the sum of 40 millions, or that of 50 millions, should be paid
for the discharge of those annuities. Having ascertained their present value at
20-8134 years' purchase, if the latter quantity be multiplied into £1,338,837,
that value will be £27,865,750; the required improvement of which sum into
that of £40,000,000 is in the proportion of 1-4354539 for every unit, and into
£50,000,000 is in the proportion of 1.7943174 for every unit also contained in
the expression of the present value. The table (page 6th) then indicates that
the sum of 40 millions should be paid at the expiration of about 11 years, or
else that of 50 millions at the expiration of a little more than 18 years, as a fair
equivalent. The method of proceeding, by direct calculation, to obtain results
strictly accurate, have already been sufficiently explained.
If we further suppose a specific period assigned for the discharge, by any sum
once paid, of an annuity whatsoever; it is easily perceivable that such sum must,
in all cases, be that into which the value of the annuity is susceptible of being
86
THE DOCTRINE OF
improved at compound interest, from the present time to the assigned period;
and the Tables will readily answer all questions of this description.
(LIX.) When the sum to which any original and unknown capital has
improved by compound interest, in a ratio also unknown, may be stated with
reference to any two periods of that improvement, those data are sufficient for
ascertaining both the original principal and the rate of interest.
According to the first case of the series set forth in Section XVth, the
statement of this question supplies two similar equations, mp (1+r)", and
M = p (1+r); admitting m as the improvement of the principal p during
the period n, and M as its improvement during the period N. From those
log. m
log.P
equations is deduced, log. (1+r) =
whence log. p =
N log. m
N
G
Ginangan
M
log. M -
N
n log. M
n
in consequence of substituting for p its value in the preceding equation determine
also the quantity r, or the ratio of interest.
EXAMPLE. A principal laid out at compound interest has, at the expiration
of 24 instalments, improved to a sum of £2484. 12s. 6d., and at the expiration
of 50 instalments it has further improved to a sum of £ 4292. 6s. 11d.: query
that principal, and the ratio of its improvements.
We have next, log. (1+r) =
Reducing shillings and pence to decimal fractions of one pound, we have
4
N log, m
log. p=
N
n log. M
50 × log. 2484-625— 24 × log. 4292·346
50 24
n
log. P
; and this last, being resolved, will
50 (3·3952609) — 24 (3·6326947) 169-763045 87.184673
26
log.m
Sugg
N
T
n
26
which last quantity is the logarithm corresponding with £1500, the original
principal inquired after.
M
log. p
= 3.1760912,
3.3952609 — 3·1760912
24
= 0·00913207, being the logarithm of 1·02125; whence the ratio is 0·02125,
or 2 per cent., rate of interest corresponding with each instalment, which, if
understood half-yearly, would establish the annual rate at 44 per cent. It might
COMPOUND INTEREST.
87
indifferently have been proceeded with log. (1+r)
the same result.
(LX.) Any principal sum being specified, with its increase by compound.
interest at the expiration of a given period, the corresponding increase either of
the same or of any other principal, at the expiration of a period different from
the first,—and also the rate of interest supposed to be unknown,-may both be
deduced from those data; admitting uniformity in that unknown rate of interest.
EXAMPLE, in which the principal is considered different in the two clauses of
the statement.
A capital of £ 2000 p has increased to £ 10641. 18s. 9d. = m, by
improvement at compound interest until the accomplishment of 168 = n
instalments; it is then inquired, first the corresponding improvement of £3550
=P at the expiration of 100 N similar instalments, the ratio of interest being
the same in both cases; and secondly, what is that ratio r of interest appertaining
to each instalment.
M = P(1+r)²; whence, log. M =
100 (log. 10641.9375
SOLUTION.-Admitting M as the improved principal required, we have (in
common with the preceding question) these two equations, m =p (1+r)”, and
K
100 X 0·7259907 + 168 X 3.5502284
168
log. (1+r) =
Bangging
The same equations being further transformable into
log. (1+r)=
log. M
log. P
N
N (log. m
log. 2000) + 168 (log. 3550)
168
log. M-log.P, yielding
N
2
the logarithm of 9602-1 or of £9602. 2s., the sum proceeding from the improve-
ment of £3550 original capital, at the expiration of the 100th instalment.
we may indifferently take log. (1+r) =
168
4.0270207 3.3010300
S
"
log. p) + n log. P
N
and into log. (1+r) =
3.9823667, which quantity is
; therefore M =
log. m
-
n
3.9823667 3.5502284
100
log. P.
2
or
either of which yields the result
88
THE DOCTRINE OF
;
0.00432137, logarithm corresponding with 1011+r, whence the ratio
r001; and if the instalments were understood as quarterly, the annual
ratio would be 0·04, or the rate of interest 4 per cent.
The solution would be obtainable by the same method, if the two original
capitals differed not from each other; in which case, we should have
log. M =
N (log. m−log. p) + n log. P¸
n
(LXI.) When it is stated that a certain principal has improved at compound
interest to a certain sum at the expiration of a certain period, although the ratio
of such improvement be unknown, those data are sufficient for ascertaining the
period at which the same or any other principal would improve to any specific
amount, provided that the same unknown ratio of interest be understood in both
instances.
In the present question there are two results, either N =
when the two principals are alike, or N ≈
n (log. M —
log. m
SOLUTION. N=
n (log. M-
log.m
63 (log. 64641·1104- log. 10000)
log. 4950-4312 log. 2750
log. P)
log. P
from each other, those results immediately deduced from the equations above
stated; and in the following exemplification, the latter supposition is proceeded
upon.
question; having log. (1+r) =
EXAMPLE. A capital of £2750 has produced another of £4950. 8s. 7 d.,
by improvement at compound interest until the expiration of a 63d instalment;
and it is inquired the period at which an original capital of £10,000 would
be improvable to £64,641. 2s. 24d., at the same rate of interest, which is kept
out of view.
when they differ
log, p)
log. p
63(4.8105088-4)
3.6946430
200, the period of instalments required. And the ratio of interest, corres-
ponding with each of those instalments, will be found as in the preceding
log. 64641 1104-log. 10000
200
3.4393327
0.8105088
200
= 0·004052544, which is the logarithm of 1.009375, therefore r = 0·009375;
and if the instalments were understood quarterly, the annual ratio would be
4 × 0·009375 = 0·0375, or 32 per cent. the rate of interest.
COMPOUND INTEREST.
89
(LXII.) Embracing in a general view the series of questions that depend on
the consideration of deferred annuities, we find their solution in the transforma-
tions of which one single fundamental equation is susceptible.
Thus, let p, a, and r, retain the significations respectively assigned to them
in the foregoing pages; let t signify the period during which the annuity is to
remain suspended, and n the subsequent period of its actual payment; the
present value of that annuity will then be expressed by
P
a
a
ɑ
(1+r)'+1 + (1+r)²+s + (1+r)²+3 + &c. to
+2
p =
a
in consequence of which, any four of the five quantities p, a, t, n, and r, being
given, the remaining one may be deduced from those data, and the following
five equations respectively solve the different questions.
n
a[(1+r)" - 1]
r(1+r)”.(1+r)*
pr(1+r)”.(1+r)"
(1+r)" — 1
;
log. [ 1 -
[ 1 =
t =
log. (1+r)*+ log. a - log. [(1+r)”—1] — log. (pr)
log. (1+r)
;
¸´ pr(1+r)*
a
-log. (1—r)
"]
]
21+n+1
2
>
C =
с
a
(1+r) + "
12 b
n² - 1
=
in
a(1+r)".
r(1+r)+”
Lastly, after admitting b=
we find r = n ✓ (ec), by analogy with the 9th case of Section XVth, and
demonstrable in the same manner.
2
an
d= (™™)³, e=c—2(d−1),
(LXIII.) If we suppose any principal p originally due, and subject to interest
in the ratio r; which debt, in consequence of the interest remaining unpaid
until the expiration of a period n, had increased by compound interest to a
total m, then proposed to be converted into a perpetual annuity a; those five
quantities, p, r, n, m, and a, will in all cases answering to the above description
be invariably related according to the following general expressions, obtained
by mutual transformation:
90
THE DOCTRINE OF
a = rp(1+r)" = rm;
a
m = p(1+r)" = =/;
n =
be z =
P =
r =
a
log. m-log. p
log. (1+r)
·(1+r)"
a
m
=
=
log. ȧ-log. p-log. r
log. (1+r)
;
(1+r)"
;
a
P(1+r)n; or log. (1+r) =
The first four of those questions are solvable without any difficulty; but as
regards the latter, it is to be observed that the quantity r cannot become the
subject of the question, except in the case of m being also unknown, thus rendering
that question very complicated. It then admits an approximate solution by the
following method, for which we are indebted to Sir Isaac Newton:-
Take r', and r=r'+z; substitute these in the equation r(1+r)” =
log. m-log. p
n
there will ultimately be obtained r =
p
which produces (r'+z).[1+(r'+x)”] = r'; and discarding, in the development
of the binominal quantity, all the terms affected with z², z³, &c., there will
r'—r′(1+r')
(1+r')”-¹ × (1+r'+nr')'
Let the latter quantity be expressed by r" and again substituted in rr"+%,
which gives r =
and r'+z =
dia
p
r'+nr' × r'(1+r)^−1
(1+r')~~¹×(1+r'+nr')'
r'+nr" × r" (1+r')"−1
(1+r"')"−¹ × (1+r"'+nr"); and signifying this last by 7",
r'+nr'' × r'''(1+r''')~~!
(1+~")~~¹× (1+r""'+nr")
By further continuing the same process, the value of r may be obtained by
approximation in any required degree.
(LXIV.) Suppose p the principal of a debt, with r the ratio of its interest;
and that instead of pr, or the amount of such interest corresponding with each
instalment, being regularly paid, the debtor should pay each time, and during a
period n of instalments, any other sum a, whether greater or less than pr. If by
m be signified the balance remaining due at the expiration of that period, under
the consideration of compound interest; by m' the sum to which the debt
would have increased, in case that the debtor had not paid any thing; and by m"
COMPOUND INTEREST.
91
the amount of the periodical sums a, accumulated and improved at compound
interest in the same ratio r; we shall then have m m' —m", m' = p(1+r)",
a(1+r)"—a
=
and m"
= p (1+r)" — [a(1 + r
r
7-a7.
p
Consequently, any four of the five quantities m, p, a, n and r, being given,
the remaining one may also be ascertained as follows:
m = p(1+r)”—a
Ɑ*****
a
13
; whence m =p(1+r)”
r =
-a[(1+r)" — 1],
↑
mr+a(1+r)”—a)
r(1+r)*
r[p(1+r)”—m]
(1+r)*-1
log. (mr—a)—log. (pr—a)
log. (1+r)
a(1+r)"—a
p(1+r)”—m
+
"].
But this last can only be solved by approximation, and in the manner pointed
out in the preceding section.
1
If the sum a, paid by the debtor at each instalment, were such as ultimately to
produce mo; by which is meant, that the original debt should be exactly
cancelled at the expiration of the term n; the case would then fall within the
series of positions stated in Section XVth,
END OF THE FIRST PART.
Again, let us suppose that instead of the debtor paying any sum a at each
instalment, he received that sum additionally from his creditor, the five questions.
then susceptible of arising would be solvable according to the equations above
stated; with this difference only, that a should be made a negative instead of a
positive quantity; substituting + for, and for, in all those terms
which are affected with it.
1
THE
DOCTRINE
OF
COMPOUND INTEREST.
IN WHICH THAT DOCTRINE
N
Part Second;
IS APPLIED TO LIFE-ANNUITIES,
TRANSACTIONS AFFECTED WITH CONTINGENCY.
AND TO OTHER
(I.) ANNUITIES on single or on joint lives; Life assurances; Reversionary
payments, by annuity or otherwise; Expectations depending on survivorship;
Endowments for children, for widows, or for persons in old age; with other
prospective transactions, equally founded upon the operation of Compound-
interest combined with the chances affecting the duration of human life, collec-
tively form a copious subject, already enlarged upon by able writers of this and
of other countries. The intention, here, is therefore to treat that subject with
great brevity; merely stating the general principles on which it rests, with so
much of their application as will be sufficient for establishing the basis of calcula-
tions relative to the transactions above described; and superadding a few Tables,
which, occasionally consulted, may so far guide the unpretending reader as at
least to prevent his being betrayed into gross errors, upon a matter beset with
difficulties and involving considerations of a delicate and complicated nature.
(II.) In the first part of this work, the various modes according to which the
power of Compound Interest is susceptible of operating have been traced with
reference to specific and absolute periods of time: we have now to view a similar
operation connected with the constant uncertainty that attends the continuance of
life; whence all calculations, in this instance, are subordinate to the consideration
of Probability.
(III.) When, in any stated order or system of events, the general laws that
govern them in the aggregate are directly known, or at least may be inferred from
an adequate series of correct observations; whilst the secondary laws, to which
those events are individually and variously liable, not only remain concealed
from our view, but also escape all power of observation, there results from this
94
THE DOCTRINE OF
.
state of things what is called Possibility, or Chance; whence is derived the idea
of Probability, which implicitly or explicitly assigns to the former a moral value.
Probability, in its vulgar acceptation, implies only a certain undefined notion of
likelihood, or preferable chance, concerning the realization of any future and
admissible event; but Mathematical probability is not confined to that vague
meaning, as it numerically specifies the degree, or value, of each probability
compared with Absolute certitude, the latter being the maximum of probabilities.
Whence mathematical probability may be considered in the light of Relative
certitude.
(IV.) FIRST GENERAL PRINCIPLE. Towards establishing any mathematical
probability, it is requisite comprehensively to embrace all the possible events that
depend on a stated order of things; to discriminate and classify those events,
resolving them into a certain quantity of cases, whether individual or collective,
to which an equal possibility attaches; and to ascertain the quantity of those
cases that militate in favour of the realization of the specific event which is the
matter of consideration. Then, the fraction of which the latter quantity is the
numerator, and of which the former quantity (or the whole of the possible cases)
is the denominator, will express the probability of that event; at the same time as
another fraction, consisting of the difference between the preceding one and unity
(taking this as the expression of Certitude,) will also express the contrary proba-
bility, that such an event will not take place. Further, the addition of all the
individual probabilities, existing under any order or system of possible events,
will amount to neither more nor less than unity; as it is evidently certain of any
event in expectation, that it will be realized, or that it will not; and it is no
less certain, that if any series of numbers are mixed together in a wheel, from
which one of them is to be extracted at hazard, such eventual number will be
oneof those that the wheel contained.
To exemplify the foregoing positions, let us suppose a quantity of events, A,
B, C, D, &c., all of which ascertained to be possible in an equal degree; and that
the realization of any one amongst them, to the exclusion of the rest, is consigned
to the decision of chance; then the expression of the probability, indiscriminately
attached to each of those events, will be unity divided by their whole quantity.
If two events only, A and B, the probability of each will be; if three, A, B
and C, it will be; if four, A, B, C and D, it will be ; and so on, for any
other quantity of equal chances; whilst the sum of all the probabilities will amount
to unity, in each of those instances.
But let us, on the contrary, suppose some ascertained and specific inequalities of
chance attached to the pending events, A, B, C, &c. If the quantity of events is
limited to A and B, and if the chances between them are so distributed that the former
COMPOUND INTEREST.
95
1
has only a single case favourable to its realization, whilst the latter has two favour-
able cases, the three individual cases being possible in an equal degree; then the
probability of A will be 3, and the probability of B will be . If three events,
A having one, B having two, and C having three cases, respectively favourable to
them, with an equal chance or possibility attached to each of those six cases; the
probability of A will be, that of B will be 2, and that of C will be 3; altogether
amounting to g, or to unity, as before. And if we further suppose A, signifying
one Black ball, B signifying two White balls, C signifying three Red balls, and
D signifying four Blue balls; all mixed together, and one only of which to be
extracted at hazard; it is clear that the probability of extracting any specific ball,
amongst the whole ten, will be exactly; at the same time as the probability of
a Black ball will also be, that of a White ball, that of a Red ball, and that
of a Blue ball (indiscriminately considered).
6
When under those or any analogous supposition, the probability attached to a
specific event is exclusively considered, then the sum of probabilities militating
in favour of all the other events belonging to the same system of things, and
collectively taken, constitutes a probability directly opposed to the former. Thus,
in the last stated supposition, the probability of a black ball not being extracted
must be understood to be; that of a white ball not being extracted,; that of
a red ball not being extracted,; and the probability that not one of the blue
balls will be extracted, only.
उठ
What has been hitherto said, towards illustrating our first general principle,
no less applies to regularly constituted hazards of all other descriptions imaginable.
From the casting of two dice, in which each face of the one is susceptible of com-
bination with the six faces of the other, there results 36 chances, that may be
taken as distinct events in one system; whence the probability of each event
will be. But in another system, deduced from especially considering the
sum of the two points that may eventually be turned up, there are only the points.
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12; limiting the quantity of events to 11,
which then possess different probabilities, according to the unequal distribution of
the 36 particular cases that have each an equal possibility, and are severally favour-
able to some one of the events considered. In the same manner, the casting of
three dice affords 216 positions or individual cases equally probable, with not
more than 16 events amongst which those 216 positions are capable of being
distributed; namely, the points comprised between 3 and 18 inclusive; and in
either of those systems, or in any other analogous system, the mathematical proba-
bility of each event is always equal to the sum of the cases which, being possible
each in the same degree, may be favourable to the realization of that event, the
said sum divided by the totality of cases.
Extending the same principle, with its self evident inferences, to the considera-
96
THE DOCTRINE OF
tion of those chances by which the duration of human life is affected; further
supposing it a fact ascertained from an adequate course of observations, that,
amidst any given population, the quantity of coexisting individuals aged 21 years
is generally reduced to one half of that quantity remaining alive after a lapse of
35 years more; and that out of a similar original quantity, the survivors after the
expiration of 44 years are only one third part thereof; if all other probabilities
which that particular order of things may involve are discarded, it will necessarily
follow that each person at the age of 21 has a probability equal to of attaining
the age of 56, and a probability equal to of attaining that of 65 years
revolved. A consequence of which is, that in case of any sum of money
staked upon each of those respective events, and admitting the same to be ulti-
mately realized, those sums should produce, as a just equivalent for the unfavour-
able chances incurred,—in the former instance an increase equal to twice the
staked sum multiplied into the proportion of its improvement at Compound
interest during 35 years,—and in the latter instance, an increase equal to thrice
the staked sum, together with 44 years improvement also at Compound interest.
However important the consideration, that the probabilities attending the continu-
ance of life may be very different, for individuals of similar age, yet existing under
various conditions of health and various other circumstances capable of especial
influence upon the probabilities of life, this consideration may be dismissed in the
present application of a fundamental principle of the Doctrine of Probabilities ;
our view being here confined to generalities. The cases requiring discrimina-
tion will be duly noticed hereafter.
(V.) It is to be stated only as a corollary of our first principle, that the probability
attached to the realization, indifferently, of either amongst two or more possible
events depending on any single trial of chance, is equal to the sum of their
separate probabilities, then considered as merely relative to different cases favour-
able to an identical event. Thus in a preceding supposition of ten balls, of four
different colours-A, B, C and D, the probability of indifferently extracting A
or B is; that of indifferently extracting A or C is; that of C or D is;
that of B, C, or D, indifferently, is ; &c. &c. But care must be had not to
misapply this rule, by extending it to any plurality of cases that might depend
on trials of chance independent of each other. For example: if on an average
of all persons aged 21 years, 3 out of 5 live to accomplish their 50th year, thus
establishing the probability that any one amongst such persons will attain the
latter age; it is not thence to be inferred that the probability of attaining the age
of 50, for either of two persons aged 21 and considered indiscriminately, should
be, a fraction even exceeding unity or the expression of absolute certitude.
Such an inference must always be absurd, as more or less probability evidently
attaches to the contrary event; and it might so happen, that not only two, but
many more lives of persons taken at the age of 21 years each, were wholly
extinct before the accomplishment of a period of 29 years. The false
COMPOUND INTEREST.
97
inference would then arise from mistaking the events that depend on distinct trials
of chance, for those that are dependent on a single trial.
(VI.) Second General PrincipLE. When any two or greater quantity of
possible events are independent of each other, so that their coincident occurrence
must be the result of distinct hazards assigned for producing those events; or in
the similar case of successive repetitions of any particular description of hazard,
in which each event, exclusive of any other, is made to depend on a specific trial;
then the probability attached to the coincidence of those different events, in
whatever quantity, is exactly equal to the product of their respective probabilities
multiplied into each other.
To exemplify this position, suppose two equal chances A and B, the probabi-
lity of each being, and suppose any quantity n of successive trials of those
chances; the probability that either A or B will be consecutively realized 1, 2,
3, 4, 5, &c... to n times, as the result of trials especially assigued, will be respec-
1 1 1 1 1
tively
and similar probabilities will attach
2'4' 8' 16' 32
to the corresponding realization of both chances A and B, the same quantity
of times between them, either alternately or in any other predetermined order.
By analogy, if the probabilities were for A and for B, the probability of
repeatedly realizing either A or B any stated quantity of times, or of realizing
both those chances the same quantity of times in any successive order whatsoever,
would be easily determinable by the same rule; and also in case of any other
expressions of individual probability.
1
&c... to ;
2n
If we suppose a perfect cube, such a die, the probability favourable to each
of its six faces will be ; and consequently, that of repeatedly producing any
one of those faces a given number of consecutive times, or otherwise an equal
number of times various faces of the die in any pre-assigned order of succession,
1
1 1
1
36 216 1296
6
will be according to the progression
But the probabilities would be very different from those, under the supposition
of casting more than one die at each time or trial of chance; unless a specific
face were assigned to be eventually turned up by each of the dice in particular,
which then would be equivalent to casting any single die successively.
: &c. ... to
n
1
6"
It may be sufficient lastly to exemplify our second general principle in a ques-
tion concerning the uncertain duration of human life. Suppose that amongst any
given number of coexisting persons, each of whom aged 15 accomplished years,
one third part were the ascertained proportion of those whose life should extend
to the accomplishment of their 64th year; and that the proportion of survivors,
after a similar lapse of 49 years, were likewise ascertained to be one fifth part of
98
THE DOCTRINE OF
any quantity of persons taken amongst those aged 23 years; the probability that
two persons, the one aged 15 and the other 23 years, will both continue alive until
the former attained 64 and the latter 72 years of age, must be expressed by
multiplied into, or by the fraction, the true expression of such probability.
(VII.) THIRD GENERAL PRINCIPLE. When two successive events are under
mutual dependance, so that the occurrence of either, in the first instance, should
influence the probability relative to the second event; then the initial probability
of the double event is represented by the product of the probability relative to
the event first expected, and multiplied into the remaining probability of the second
event, the latter probability calculated on the admission of the first event having
occurred. And if any more than two events under similar dependancy; the
probability, that any stated combination of those events will be realized, is equal
to the product of the initial probability of the event first expected, that proba-
bility multiplied into the remaining probabilities, each of which calculated from
successively assuming the actual realization of all the events that are to precede,
and whose respective probabilities have been pre-established.
Suppose A, B and C, to contain each one ball; two of them are white balls,
a third is a black ball; and it is inquired the probability that the two white balls.
are contained in A and B. The probability that A contains a white ball is
evidently; and assuming the fact, there remains the probability of B's
containing a white ball also, as the uncertainty will be equally divided between
B and C ; whence the probability of the double event is × =}. If we
423
further suppose A, B, C and D, to contain three white balls and one black ball ;
the probability that all the white balls are contained in A, B and C, will be X
34/35/35 X 3/1/20 1. Indeed it is clear, that in the event required by the first supposed
case, C only could contain the single black ball, the probability of which was };
and that in the event required by the latter case, D only could contain that black
ball, the equally evident probability of which was 1.
J
(VIII.) FOURTH GENERAL PRINCIPLE. When the object of expectation is a
Sum of money, depending upon any event, however doubtful, but the probability
of which is susceptible of ascertainment; or when it depends upon the coincidence
of a plurality of events independent of each other, or again upon the successive
occurrence of events under any mutual dependance, and even when the occurrence
of any one amongst them should necessarily exclude the possibility of one or
more others; the value of such expectation is, in all cases, equal to that sum
of money multiplied into the fraction expressing either the initial probability of
the single event, or the ultimate probability resulting from a combination of the
individual probabilities relative to the various events together to be considered.
This value is referable to the period or periods of time at which the events are
susceptible of accomplishment; otherwise it is reducible to a present value, by
COMPOUND INTEREST.
99
discount according to the consideration of Compound-interest at a certain rate;
and this last value is the sum for which the expectation might justly be compro-
mised, in case that the expectant preferred a composition, rather than to abide
by his uncertainty.
(IX.) The following question involves an application, altogether, of the four
general principles hitherto exposed.
By admitting certain Tables of Mortality, and by applying the first general
principle (IV) to the consideration of the proportion of survivors after a stated
period, out of any quantity of co-existing persons respectively of certain ages at
the commencement of that period, it is inferred these successive probabilities;
0.8725, that a person aged 20 will attain 30 years accomplished, 0·8430 that he
will afterwards attain 40 years, 0·8042 that he will thence attain 50 years, 0·7189
that he will subsequently attain 60 years, 0-5509 that he will further attain 70
years, 0.2950 that he will from this last age attain that of 80 years, and 0·11036
that he will ultimately attain the accomplishment of his 90th year. Having pre-
mised those data, it is further supposed A aged 60, B aged 40, also C aged 20
years; and that a certain advantage is to accrue in the event, altogether,--of A,
B and C's co-existing during a period of ten years,-of B and C's co-existing
during ten years more, and of C's life continuing ten years longer than this last
period, or until he attains 50 years of age. Then, and by applying the second
general principle (VI), we haye 0-5509 × 0.8042 X 0·8725 0.3865, expressing
the probability of A, B and C's lives jointly continuing until the expiration of a
first period of ten years; next, and by combining the application of the 2d and 3d
principles, we have 0·3865 × 0·7.189 × 0·8430=0·2343, expressing the pro-
bability that the lives of B and C will jointly continue ten years after they had
co-existed with A during a first similar period; and further we have 0·2343 ×
0·8042 0:1884, expressing the probability that C,will live ten years after having
co-existed with B during the ten preceding years, as also with A and with B
during ten years before that time. Whence, and supposing a sum of £ 10,000
forthcoming in case of those events being accomplished, that expectation might
fairly be compromised at the outset, for a certainty of receiving, at the expiration
of 30 years, a sum of £1884, resulting from £10,000 multiplied into the
probability 0.1884; or else for immediately receiving the discounted value of
£1884 forthcoming after 30 years, which reduces the latter sum to that of £574,
admitting compound interest at the rate of 4 per cent. per annum and half yearly
instalments, as may be seen in page 21st of the Tables.
But reversing the order of the conditions specified in the above question,
and supposing the advantage of £10,000 to depend on the event of A, at
first aged 60, attaining 90 years of age, after having co-existed to the age of 80
jointly with B who is twenty years younger than him, and to the age of 70 jointly
*
Un
100
THE DOCTRINE OF
YO.
with B and with C, the latter being twenty years younger than the preceding and
forty years younger than A; the probabilities, in that case, would be 0.3865 (as
above) of A, B and C's co-existing the first ten years, 0.3865 X 0·7189 X
0·2950 = 0·08198 that B and A co-existed during the next ten years, and
0·08198 × 0·11036 0·009047 that A continued alive ten years beyond the
accomplishment of those first two periods under the conditions attached to them.
Whence, and under the supposition we departed from, the value of such expecta-
tion would be either £ 90-47 receivable as a certainty at the end of 30 years, or
else immediately its discounted value, which is 90-47 × 0·30478 = £27.11s. 6d.
according to the Table of Compound-interest already referred to.
(X.) Fifth geneRAL PRINCIPLE. The present value, reduced to certainty, of
any advantage collectively expected as depending on a plurality of events, when
the occurrence of each is to produce some partial and distinct advantage, will be
equal to the sum of the several products of the probabilities of those events, respec-
tively multiplied into the specific advantages attached to their accomplishment;
and this rule applies in either case, of those events excluding each other or not.
Suppose two equal chances Y and Z; that the event Z, if realized at a first
trial, should produce an advantage of £20; and that if realized at a second
trial of chance, the event Y being realized at the first, it should produce an
advantage of £50; the whole expectation will be worth £20 multiplied into,
or into the probability attached to the first condition, plus £50 multiplied into or
the probability of the second condition; together £10 + £12. 10s. = £22.10s.,
for the certainty of which the aggregate of those expectations may be compro-
mised. But supposing, that independently of £20 advantage attached to the
realization of Z at a first trial of chance, an additional expectation of £20 were
to depend on the event of Z being unconditionally realized at a second trial;
then the value of those expectations, considered in the aggregate, would be
£20 x = £10, plus £50 x = £25, together £35.
X
Supposing again, as in the preceding section, A, B and C, respectively aged 60,
40, and 20 years; that an expectation of £1000 depended on the joint conti-
nuance of their lives during ten years; and that £2000 further depended on the
joint lives of A and of B only, but to continue ten years longer; the value of the
first expectation would be £1000 × 0·3865 = £ 386. 10s.; that of the second
expectation, depending on the probability 0.1628 of A (aged 60) living 20
years, combined with the probability 0.5781 of B (aged 40) living the same time,
which makes the probability of their joint lives during that period 0-09394,
-would be £2000 × 0·09394 £187. 17s. 6d.; and the united amount of
those expectations therefore £574.7s. 6d., reducible to a discounted value under
the consideration of Compound-interest at a certain rate, in case of exchanging
the two expectancies against a sum of money payable at the outset.
a
COMPOUND INTEREST.
101
.
But we may, on the contrary, suppose a case in which £1000 depended on
the event of either life, A or B, dropping before the expiration of ten years; and
in which £2000 depended on the event of A, B and C, jointly co-existing for
twenty years those two expectations being mutually exclusive. Calculating
from the same general principle, we find (the probability being 1-0-3865 that
either of the lives will drop) £1000 × 0·6135 £613. 10s., value of the first
expectation; and £2000 × 0·07918 (the probability of the three lives co-existing
for 20 years) £158. 7s. 8d., value of the second expectation; the united
value of both, independently of the consideration of discount relative to the periods
of 10 and of 20 years, being £ 771. 17s. 8d.
Observe, however, that in those various cases no positive loss is admitted to
be incurred by the expectant, in failure of either object of his expectation; which
otherwise would bring those cases under the application of the next general
principle set forth.
(XI.) Sixth GENERAL PRINCIPLE. When considering in the aggregate any
series of probable events, from the accomplishment of some amongst which is to
result a gain, and from that of some others a loss, the value of the whole expectation
may be either positive or negative; as it is equal to the sum of products of the
probabilities relative to those events which are favourable to the expectant,
respectively multiplied into the specific advantage attached to the accomplishment
of each,-minus the sum of products of the probabilities relative to such of those
events as are unfavourable to him, and in like manner multiplied into the loss or
disadvantage which is to result from each.
;
This principle is obviously no more than a corollary of the preceding, and
scarcely requires any illustration.
If amongst three events jointly considered, the accomplishment of X is to
produce a gain of £ 50, that of Y is to produce a gain of £75, and that of the
event Z is on the contrary to produce a loss of £100,-those events being
equally possible, and their mathematical probabilities therefore being uniformly };
the whole expectation, taken together, will be worth exactly £8. 6s. 8d., equal
to the excess of 50+75 over 100. But if the probabilities, instead of being on a
3
3
2
3
6
par with each other, were for X, for Y, and for Z; then the whole
expectation would be valuable only as 50 75 100
+
£16. 13s. 4d.,
which negative quantity indicates a disadvantage in the aggregate. All ana-
logous cases are to be treated in the same manner.
2
Cabin
(XII.) The Doctrine of Probabilities further rests on other principles, the
development of which is the bess called for on this occasion, as their application,
5+
102
THE DOCTRINE OF
sometimes involving mathematical operations of a transcendant description,
chiefly relates to philosophical questions, and therefore would be foreign to the
purpose of this book.
On the foregoing six general principles, and on their necessary inferences,
are founded the rules exposed at great length by Dr. Price and by other
authors, for executing the calculations concerning Life-annuities, Reversionary
payments, Survivorships, Endowments, and generally all transactions referable
at the same time to the operation of Compound-interest and to the uncertain
duration of human life. We refer our readers to those rules, admitting. them
without discussion; and now proceed to explain what must be understood by
the Laws of Mortality,-in what consists the information obtainable by consulting
the various Tables of mortality that have been published, the useful application
of which they are susceptible,—and the caution necessary against being led into
error through their indiscriminate consideration.
(XIII.) Amongst any stated and considerable number of persons born, the
proportions according to which their lives successively drop at several stages of
existence, or from year to year, until the whole number becomes extinct, are not
alike in different climates, nor even in different countries under similar degrees of
latitude; neither are they, in the same countries, exactly the same with reference
to each sex; and they are especially dissimilar, in respect to different circum-
stances of human existence that may be discriminated in classifying any entire
population. The differences in those proportions are great, between a cold
climate and a temperate one; but respecting the influence of remarkably elevated
temperatures upon the annual decrements of life, and upon the average of its
duration, no sufficient data is possessed for establishing any calculation.
Without going farther than Europe, the average term of human life is some-
times very different between countries subject to nearly the same temperature,
though such difference may not be ascribed to any known cause; as (for exam-
ple) between the Duchy of Milan and France, the average of life is ascertained
to be 3 years less in the former than in the latter. Even in any one country,
the observable proportions, in the annual decrements of life, differ not inconsi-
derably with regard to the inhabitants of mountainous parts or those of val-
leys, to the residents on a dry soil or those on marshy land, and to those who
dwell near the sea or otherwise in tracts remote from it. Amidst various
other causes, eminently favourable to the protraction of human life, are however
to be considered, independently of a primary healthy constitution, the habitual
breathing of salubrious air, the adequate supply of wholesome food, the exemp-
tion from extreme vicissitudes and incessant cares,-from intolerable labour of
body or mind, from the exercise of dangerous professions,-from excesses,-
from vice and its consequences. Whence obviously follows, that in large cities
COMPOUND INTEREST.
103
and in many manufacturing towns life is comparatively short; of which fact
London (that immense receptacle of all human miseries) affords, by its Bills of
Mortality, a most remarkable illustration.
On the other hand, so great is sometimes the difference between the probable
Expectations of forthcoming years of life, for persons even of the same age, that
the ultimate result of a selection in considerable number amongst the best lives at
5 accomplished years, when that Expectation is a maximum, would exhibit one
half of that number attaining their 60th year, one quarter their 75th, and one
eighth part fully accomplishing 80 years of age; whilst similar proportions do
not respectively outlive their 41st, 59th and 70th years, upon an average of all
London lives, originally taken at the same age of five years accomplished.
If to those considerations be added that, from obvious causes, the lives of
females, particularly at an advanced stage, are generally better assured than those.
of the males; and that an extension of three years, to the average of life such as
it has been heretofore calculated, would probably result from the total extirpation,
by means of vaccination, of a destructive disease,-admitting no other to be intro-
duced instead; it will justly be inferred that, concerning the human race, there
is no law of Mortality admissible as universal under all circumstances, or
susceptible of being indiscriminately applied, towards calculating with a suffi-
cient degree of accuracy either the value of Life-annuities or that of expectations.
depending on survivorship, nor towards guiding with a near approach to perfect
correctness any operations relative to the investment of capital, in whatever
mode or shape in which the uncertain duration of life is implicated; and that all
private transactions of that description, between one individual and another,
require to be proceeded upon with greater caution than is usually practised.
(XIV.) Notwithstanding that the natural law by which, at each stage of
human existence, the average period of its further endurance, together with the
proportion of individuals destined to attain any other specific stage, may seem to
elude all means of detection, the deficiency of accurate knowledge, in that respect,
is capable of being in a great measure supplied by experience, during a long
course of well-directed observations. We are thus informed that, in the same
countries or districts, the general results of those observations have continued
very nearly the same for many successive generations; proving at least the con-
stancy of nature's laws of mortality, upon which in a multitude of instances we
are called upon to place an implicit reliance. We are informed, secondly, that in
each country, or particular district, the series of observations have also yielded
similar results at various times, regarding any selection of individuals in consider-
able number, and collectively considered as living under certain general circum-
stances susceptible of assimilation. In very extensive countries, those observations
could seldom, and for any length of time, embrace the whole population. In
t
i
104
THE DOCTRINE OF
.....
Sweden and in France, however, the means of accurate observation have been
generally afforded throughout their extent; whilst the defective character of the
birth registers, in England, has hitherto deprived us of the like benefit. In this
as well as in some other countries, the observations alludeď to, and however
attentively they may have been proceeded upon, were therefore confined to certain
towns or districts of very limited population; otherwise to select classes, such as
Government-annuitants, and sometimes to persons in monastic retirement. But
although the laws of mortality, as concluded from experience under that variety
of aspects, exhibit corresponding differences equally remarkable; in the whole, a
sufficiency of statistical observations is possessed for enabling, by a comparison
of them, to establish the proportions according to which the decrements of life are
operated from one year to another, under the consideration of general circum-
stances duly discriminated, and also to establish with tolerable accuracy the actual
state of a given population in its manifold subdivisions.
(XV.) It was not long before the close of the seventeenth century, that the
Doctrine of Chances and of Probabilities began to attract the special attention of
scientific men, who then applied that doctrine to the consideration of human life,
and to various questions of statistics and of political arithmetic. With a view to
those objects, Dr. Halley published the first Table of Mortality that is on record,
and which he deduced from a series of experimental observations made at Breslaw
in Silesia: other observations have since been collected in different parts of
Europe, the compared results of which, reduced to Tables, exhibit very great
discrepancies. Various considerations may prescribe the preferable application of
either of those tables. For example: when the Government or any incorporated
society proceed to grant Life-annuities, it is always probable that they will
have chiefly to deal with a selection of the best assured lives, whence it becomes
expedient to establish the calculations upon such Tables as exhibit the slowest
decrements of life that may be admissible; and on the contrary, in case of receiving
certain sums, either in a single payment, or else in the shape of an annual premium
or any other periodical payments depending on the continuance of life, as the
supposed equivalent either for capital to be paid at the period of death, or for
annuities whatever and then to commence, it is requisite, from reversing the above
consideration, to adopt some table of mortality generally exhibiting rapid decre-
ments of human life; without which distinction no society incorporated for those
purposes could subsist for any length of time, and no government could assume
transactions of a similar description without a notable disadvantage, which has
sometimes occurred.
Many such tables, with results of calculation thence deduced concerning the
value of Life-annuities and other prospective transactions, are introduced into
Dr. Price's and other works. Before it is proceeded to place in a comparative
point of view those and other valuable tables, not generally known in England
COMPOUND INTEREST.
105
at the period of the earlier publications just now alluded to; with comparative
statements, also, of their various results and deductions susceptible of practical
application; it may be proper to explain the method of constructing any Table
of Mortality, and that of inferring from it, altogether, the average of forthcoming
years of life belonging to persons of any stated age, the probabilities attached
to any expectation of life's continuing during any assigned period,—and a classi-
fication of the proportionate quantities of persons co-existing at each age respec-
tively, with reference either to an entire population or to any selection from it.
(XVI.) Correct Tables of Mortality, relative to any country or district, should
originate in the legal obligation, strictly enforced, to inscribe within the shortest
delay, and in registers of unquestionable authenticity, all the births and all the
deceases that occur; with the precaution, amongst others, of referring each
decease to the identical person whose birth at a precise period has been previously
ascertained. Nothing short of legislative interference could in a sufficient degree
secure the regularity of those proceedings; and it is only after their uninterrupted
continuance, during a very long period, that the conclusions deduced from them
may be such as to warrant any reliance. In England, amidst the enormous
neglects tolerated by established custom in respect to the records of life and death,
-the registers of baptism being tainted with more or less uncertainty as to the
time of each birth, whilst a great part of the population have scarcely a distinct.
notion of their age,—any attempt at constructing a general table of mortality
applicable to the whole country, or even such a table of that description as with
satisfactory approach towards exactitude might refer to any considerable portion
of its inhabitants, would be absurd: it is an object in all probability unattainable
for a century to come. At Northampton, at Chester, and in a few other towns
and parishes, registers of the births and of the deceases have been kept with all
practicable accuracy during many years, for which we are exclusively indebted
to the laudable attention of some gentlemen possessed of local influence; but the
circle of their observations has been too limited for answering general purposes.
The Northampton tables, however, are usually considered as a specimen of
regular decrements, applicable to a quality of lives rather inferior to their general
standard in England; whence those tables are chiefly used, as the safest, for
regulating the premiums of Life-assurances. On the contrary, the tables deduced
from the registers kept for the town of Chester, and for the parish of Holy-cross
(near Shrewsbury), afford an eligible specimen of the decrements amongst the
best lives in this country.
It were desirable that the British government adopted effectual means of ob-
taining, in process of time, accurate returns concerning the population divided
into certain classes, and concerning the law of mortality to which each of those
classes is subjected: the results would be fully worth the incurred expense.
O
3
106
THE DOCTRINE OF
When any regular and complete system of registring births and deceases is
acted upon,
it becomes easy to construct a table of life-decrements from each stage
of human existence to the next, and to deduct from that table all other data
requisite towards the objects under our present consideration. The greater the
inscribed number of births and of corresponding deaths, the more accurate will
be the general results inferred from them, and the greater safety also in applying
those results to the calculations concerning Life-annuities, reversions, and other
prospective transactions; but the number of inscriptions should always be con-
siderable, without which the results could not be entitled to credit.
The mode of proceeding is as follows:-
(XVII.) After having ascertained the quantity of individuals who respectively
died before the accomplishment of their first, their second, their third, or other
more advanced years of life; the successive subtraction of those quantities, be-
ginning with the first, from the total quantity of births, will leave a series of
quantities representing those of the individuals who remained alive at the com-
mencement of each year. Let that total of births be noted, as corresponding
with the year O of existence; let the results of the above-mentioned subtractions
be also noted to correspond with a first, a second, a third, &c., accomplished
year; and considering the rapid waste of life during the first two years, it will
be proper to discriminate the half-yearly results of observation referable to that
early stage. This process continued, until the whole quantity of lives originally
inscribed be disposed of, will exhibit a Table of Mortality, or of the decrements
of human life, admissible as average results, with reference to any country at
large, to any town or district, or to any particular class of persons selected from
the general population as the subject of that table.
(XVIII.) The aggregate amount of years, during which all the individuals
inscribed in the Table of Mortality, from the period of their birth or at the age 0,
have continued to live,-being divided by the number of those individuals, yield
a certain quantity of years which is the absolute average of life. Considering,
in the same manner, the aggregate amount of years, during which any number
of persons thus inscribed, as having attained any specific age, have continued to
live beyond that age; and dividing such aggregate amount by that number of
persons, the quotient will express, in a certain quantity of years also, the relative
average of life, referable to individuals having already attained the specific age
first considered.
From this mode of framing any Table of Mortality, it is obvious that the cal-
culations pointed at are executed by unavoidably substituting, for absolute quan-
tities, certain mediate quantities viewed as sufficiently approximative.
COMPOUND INTEREST.
107
(XIX.) Therefore, the quantity of individuals who, taken at the age O, have
not lived until the accomplishment of their first year, may be considered as having
averaged one half of a year's existence; the quantity of those who accomplished
their first, but not their second year, may be averaged at 14 years; the quantity
of those who accomplished their second, but not their third year, may likewise be
averaged at 2 year's each; and so on, for every succeeding age. By multiplying
those quantities into their respective averages, 1, 1, 2, 3 years, &c., until the
whole number of original lives be disposed of; and by summing up those various
products, the aggregate alluded to, in the first clause of the preceding Section,
will be obtained. In case of half-yearly returns of mortality for the first two
years, as already recommended, there will be, instead of and 1, the several
averages, 4, 14 and 12 years of life; into which last quantities, certain corres-
ponding numbers of individuals should then be respectively multiplied. A shorter
process, however, will lead to the same result of expressing the absolute average
of life; it is that of summing up the quantities of living persons of all ages, as
inscribed in the Table, dividing the whole by the quantity of births, and lastly
subtracting one half of a year from the quotient; but the latter mode of proceeding
excludes the inscription of half-yearly returns of mortality, which would induce
a considerable error in the result.
The Absolute average of life is of little import in a practical point of view. It
has scarcely any other utility than as it further expresses, under the consideration
of a stationary population (or one in progress neither of increase nor of decrease),
the proportion subsisting between the whole quantity of people and that of the
annual births amongst them, supposed equal to that of the annual deaths.
(XX.) The Relative average of life is one of the requisite data for establish-
ing the value of Life-annuities, according to Mr. Simpson's abridged method
of approximation; and it is calculable according to the foregoing principle.
Taking the departure from any quantity of individuals inscribed in the Table
as coexisting of the same age; admitting one half of a year as the average.
life of those who do not attain one full year beyond that age, 1½ years as the
average of those who accomplish a first but not a second year beyond the first
age, and subsequently admitting 2, 3 years, &c., as the respective averages of
lives extending between 2 and 3 years, between 3 and 4 years, &c., beyond that
stated age; multiplying those averages into the quantities of individuals with
which they respectively correspond; and lastly, dividing the aggregate of those
products by the quantity of individuals originally considered, the result will be
the average of forthcoming years of life, belonging in common to all those
individual persons.
The aforementioned shorter process may equally be applied in the present
case: add to the number of the living at any specific age, that of the living at all
108
THE DOCTRINE OF
ages more advanced, divide the whole by the first number, subtract one half of a
year from the quotient, and the remainder will be the average of forthcoming
years of life appertaining to those individuals of the age specified.
(XXI.) That average of forthcoming years, computed either from the birth or
from any other stage of human existence, is what most of the British authors have
called the Expectation of life. But this denomination is inconsistent and improper,
inasmuch as the period of rational expectation cannot be any other than that
which admits an equal probability, for the expectant, of attaining it or not.
That term of well-founded expectation, applicable to any given age, is the same
at which a stated number of persons of that age should, by the annual decrements
(according to the adopted Table of mortality), be reduced to exactly one half of
the same number. It may, as depending on the specific age, be either more or less
than the average quantity of forthcoming years of life, to which all those of that
age are entitled. For new-born infants, that term of probable life or of just Expec
tion is about twenty years, whilst the average quantity of years' existence indiscri-
minately falling to their lot is between 28 and 29, at least in England and in
France. For individuals having accomplished their first year, and for all those
under 40 years of age, such term of probable life is more or less superior to that
of the relative average; from 40 to 45, those two terms are nearly alike; and
above the latter age, to the most advanced age of man, the average of forth-
coming years of life is greater than the term of probability, and the difference
increases with the respective ages.
(XXII.) The amount of population (supposed to be stationary), classed into
certain quantities of individuals of each different age, is easily deducible from a
table accurately exhibiting the annual decrements of life. To this effect, admit,
as the quantity of existing persons between the ages of 0 and of 1 accomplished.
year only, one half of the united quantities of deceases that occurred at both
those ages; admit in the same manner, as the actual quantities of the living between
the ages of 1 and of 2 accomplished years, one half of the united quantities
of deceases that also took place at both those ages; and continue, by a similar
process, to establish the quantities of the living at the respective ages between
2 and 3, between 3 and 4 accomplished years, &c. to the most advanced term of
life; then the total of those quantities of living persons, thus ascertained, will
constitute a fair estimate of the whole population under its various divisions
relative to age. Inscribe that total quantity in another column of the Table, and
in the horizontal line corresponding with the age 0; inscribe in the next
horizontal line, to correspond with the age of 1 accomplished year, that same total
diminished by the quantity of the living between the ages 0 and 1, which will
exhibit the whole quantity of individuals actually existing of the ages of 1 full
year or more; and by proceeding to the end, with analogous and successive
subtractions of the quantities ascertained as above, of the living at each age,-as
COMPOUND INTEREST.
109
also by inscribing each remainder in its proper order,-there will result from
this a series of quantities corresponding with each accomplished year of life, and
generally expressing the respective quantities of those persons who have either
attained or surpassed any stated age.
But, if any constant excess of the annual births, over the annual deceases,
should denote a progressive increase in the whole population; then,-and although
the proportions in the relative quantities of the living at each age, together with
their respective probabilities of life's continuance, will not be disturbed,-the
total of those quantities, which otherwise would represent the true state of the
population, will no longer do so. For obtaining this true representation, it will
be necessary further to add, to that total, the annual excess duly portioned out
amongst the living at each age; and the contrary operation must be performed,
towards the same object, if the population appeared to be constantly declining.
the
(XXIII.) With the assistance of a Table thus constructed, and exhibiting,
relatively to each age of life, altogether the quantities of the living,
respective quantities of those who have attained that age only,—and the respective
quantities of individuals who died within the last year, which would otherwise
have accomplished that precise age,-all the probabilities concerning any forth-
coming period either of individual life or of coexisting lives, and those concern-
ing the various casualties of survivorship, are susceptible of being with sufficient
accuracy calculated; as also the value of any expectancy, whether single or
periodical, and depending on those events; by applying to the data thence
possessed, as the case may require, one or more of the six general principles
exposed in Sections IVth to XIth, with reference to the Doctrine of Probability.
C
In a table of the foregoing description, the series that expresses the annual
decrements of the living at each age constitutes of itself, and in all respects, a
table of the probabilities attending the expectation that may be entertained, at
a stated age, of the further continuance of life during any assigned period; and
when any question involves the consideration of a plurality of lives, those pro-
babilities admit every requisite combination, upon the general principles already
set forth.
The first of those principles (IV) establishes that the probability of any
single event, in expectation, is represented by a fraction of which the denomi-
nator is the whole number of events assumed to be possible in an equal degree,
and the numerator is that number only-amongst those events-which is under-
stood will in fact be realized; therefore the probability that any person, actually
of the age x, will live until he attains the further age X, is equal to the fraction
resulting from the quantity of the living at the latter, divided by the quantity of
the living at the former of those ages.
13
110
THE DOCTRINE OF
The second general principle (VI), applied in the same manner, shews that
the probability of two or more persons continuing to coexist during any stated
period,-whether they be of the same age or of different ages,-is equal, in all
cases, to the product of their individual probabilities of living throughout that
period, multiplied into each other. And upon this principle, it is always easy to
ascertain the probability that any marriage, or other life-association, will endure
a certain length of time.
The remaining four general principles, set forth in Sections VIIth, VIIIth,
Xth, and XIth, are equally applicable to a variety of other questions that depend
either on the continuance of single or of joint lives, or on the eventuality of sur-
vivorship; but the greater or less facility of that application, exemplified in section
IXth, depends much on the clear understanding and discrimination with which
those questions may be stated.
(XXIV.) That second general principle is especially applicable in calculating
the probabilities of survivorship, between any two or greater number of persons.
Thus, the probability that Paul, aged x years, will continue in existence during
a period y computed from the present time, and then have survived Thomas who
now is aged X years, is equal to the probability of Paul's living from the age x
to the age x+y, multiplied into the probability that Thomas will not continue to
live from the age X to the age X+y.
By the corollary (V), the probability that at the assigned period either Paul or
Thomas will be alive, but not both, is equal to the sum of the last mentioned
probability added to that of Thomas (on the contrary) surviving Paul; calculating
the latter on the same principle as the former probability.
And supposing two persons of the same age x, the probability of survivorship
will be, for each, one half of those united probabilities; which assimilates with
the probability that any one person of the age x will continue to live until the age
+y, multiplied into the complement of that probability, or its difference from
unity, which difference expresses the probability that a person of that age will
not continue to live throughout the period y.
(XXV.) It has been established by the fourth general principle (VIII), that
when the object of expectation is a sum of money, its real value is equal to the
product of that sum, multiplied into the fraction expressing the probability of the
event upon the occurrence of which its payment is to depend. But this considera-
tion must be combined with that of the operation of Compound-interest; whence
such real value, referable only to a future period, is further reducible to a present
and discounted value, upon the principle set forth in the First part of this book
and exemplified by a series of Tables, according to different rates of Interest.
COMPOUND INTEREST.
111
C
Upon an application of the general principles before mentioned, to the probabi-
lities of life and of survivorship, as deduced from Tables of Mortality assumed
to be sufficiently correct, are founded all the transactions undertaken by societies
incorporated for the various purposes of granting Life-annuities,—of Assuring
lives, (by which is meant to incur the obligation of paying certain sums at the
decease of individuals, in consideration of receiving on the contrary, and under
the denomination of premium, certain annuities terminating with the lives of those
individuals respectively),—of purchasing reversionary property, whether con-
sisting of capital or of income,—of endowing either widows, children, or persons
in old age, and generally with any other prospective views; all of which, under
such conditions and regulations as may be deemed expedient. But it should
not be inferred, from the uniformity of the general principles to be proceeded
upon in other respects, that any specific Table, exhibiting the decrements of life
as deduced from select observations, could indiscriminately be adapted to such
a variety of transactions, frequently having an opposite tendency. The laws of
Mortality appear under a very different aspect, when relative to that class of
persons who usually stipulate for Assurances on their lives, and when relative to
the opposite class, or to the generality of persons that stipulate for receiving
annuities during life. And here again it is proper to remark, respecting stipulations
for Life-annuities between individual persons, that the one who receives an
annuity of this description, whether on his own or on any other life, constitutes
himself the assurer of such life; with this difference only, that he yields for the
annuity a present equivalent, instead of a capital payable at the uncertain period
when the assured life may happen to drop; whilst the engagement to pay any
Life-annuity, as a solitary transaction of its kind, is at best an inconsiderate one,
especially if the calculation be not established upon an appropriate Table of
Decrements. A variety of those tables, that either are or may be used, as occa-
sions require, will hereafter be adverted to.
(XXVI.) The transactions engaged into, by the societies mentioned in the
preceding Section, depend sometimes on the solution of complicated questions, to
be proceeded upon with attentive discrimination; for example, in the exchange
of reversionary property depending on contingencies, against capital or income
made also to depend upon uncertain events, or on the uncertain time of their
The most proper and simple mode, of executing the calculations
relative to such transactions, consists in reducing each object of distinct considera-
tion to a value either in present capital, or in capital referable to any other common
period; whence the calculations may be completed by mere additions and sub-
tractions. It is, doubtless, the method practised by those societies.
occurrence.
:
(XXVII.) The present value of any Life-annuity is calculable, according to
the fourth general principle, from a Table of the Decrements of life. It is evident
that such value can be no other than the discounted values, under the consideration
1
112
THE DOCTRINE OF
of Compound-interest, and added together, of an equal sum accruing at the expi-
ration of each successive year, and multiplied into the probability of its eventually
becoming payable for that year; and it is also evident, that the probabilities
attached to those respective payments will be expressed by a series of fractions,
having for their numerators the quantities of persons living at the expiration of
the first, of the second, of the third, of the fourth year, &c., computed from the
present age of the life in expectation, and until the last accomplished year of
that life; all those fractions having for their common denominator the quantity
of persons living at the present time, or at the commencement of the first year;
which comparative quantities of living persons, at each age, may be taken from
any Table of Decrements thought proper to be adopted. In case of annuities on
joint continuance of life, the 2d general principle will further apply.
Tables of the assumed value of Life-annuities, calculated from various Tables
of Decrements selected as eligible, and according to various rates of the annual
interest of money, have been constructed upon the above-mentioned principle;
they are to be found in many works, but more particularly in that of Dr. Price ;
which, valuable as it is, would be still more so and easier to consult, if the
abundant matter it contains were methodically classed.
In all those Tables, the value is stated as corresponding with every unit con-
tained in the supposed annuity or sum forthcoming, in a single payment, at the
expiration of each successive year during the continuance of life. That mode of
expressing the value of Life-annuities indicates certain quantities of years' purchase,
in like manner as the Tables of Compound-Interest hereto subjoined indicate, in
the appropriate column, the value of annuities for terms of years certain; such
value comparable, in either case, with the value of a perpetual annuity. What is
commonly called the Value of a Life is identical with that mode of expression.
When the value of a Life-annuity is calculated on the foregoing principle, it is
with an invariable understanding that the annuity is not otherwise payable, than
at the expiration of every successive year, computed from the period when it
was constituted; and that it ceases to be payable after the last full year is com-
pleted. But when, by express stipulation, it is additionally payable, in due
proportion, for the portion of time intervening between the last accomplished
year and the precise period of extinction of the life,-in which case it is usually
called an Annuity secured on land,—its value is increased by the amount of half
a years' income, discounted from the probable period of termination.
(XXVIII.) Dr. Price has entered at some length into the consideration of
the value of Life-annuities, discriminated according to their optional stipulation,
either by annual, by half-yearly, by quarterly, or by any other instalments; but
it may be unnecessary here to touch upon that subject. Whenever the value of
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113
1
a Life-annuity is established, relatively to annual instalments and to any given.
rate of Interest, the differences may safely be taken, for shorter instalments, in
similar proportions with those observed in the Table of Compound-interest at the
same rate, and for an equal or a nearly equal value of the annuity for terms of
years certain.
(XXIX.) The principle to be proceeded upon, for correctly ascertaining the
value of any Life-annuity, according to a stated rate of interest, has been ex-
plained in section XXVIIth; and the Synoptical Tables that terminate this Second
Part and exhibit the values of annuities of that description, with reference to
various sets of observations on the decrements of life, have been calculated from
a direct application of that same principle. But, the Tables of Compound-Interest
having already supplied the correct values of annuities for terms of years certain,
—at whatever rates of interest, accruing either annually, half-yearly, or quarterly,-
an easy and expeditious method is afforded, through that medium, for calculating
the values of annuities on single lives and according to any rate of Interest; at
the same time as the results will refer to optional instalments of those annuities.
This method has indeed been pointed out by other writers; but without the
assistance of the last-mentioned Tables, it could avail only in a small degree,
towards diminishing the labour of calculation. It is founded upon an hypothesis,
originating with Mr. Demoivre (a contemporary of Sir Isaac Newton),-that of
Decrements numerically equal for each year of life; and therefore it cannot, with
the utmost correctness, yield results expressive of the value of Life-annuities.
Yet as that hypothesis comes near the truth, with regard to ordinary lives, between
20 and 70 years of age, those results will make a sufficient approach to exactitude
for answering usual purposes, when the method is applied within those limits and
particularly to middle ages. It must however be considered inapplicable, without
great inaccuracy, to persons either in the earlier or in the later stages of life; as
the decrements, in both cases, are very unequal from one year to another.
This method is very simple. Take the Relative average of forthcoming years
(commonly called Expectation of life) belonging to the given age, according to
any adopted Table of Decrements, and admit double that quantity of years as
the Complement of life*; seek in the Tables of Compound-Interest the value of
an annuity for a term of years certain and equal to that Complement, such value
corresponding with the instalments of the supposed Life-annuity, whether annual,
half-yearly, or quarterly; multiply that value into unity increased by its interest
for one year, for half a year, or for a quarter of a year (as it may be); divide
the product by that of the Complement multiplied into the fraction expressing
the ratio of interest for a whole year,—and subtract the quotient from the value
* The Complement of life is an imaginary quantity, frequently used for abridging the
calculations.
P
4
114
THE DOCTRINE OF
of a perpetual annuity according to the stated rate of Interest: the remainder
will be the present value of that Life-annuity.
FIRST EXAMPLE. Suppose the age 45 accomplished years, the rate of Interest
31 per cent. per ann., the instalments yearly, and the adopted decrements of life
to be those deduced from the observations on the Male population of Sweden.
The Table hereafter (No. II.) shews the Expectation, for a life of that age and of
that class, to be 20-712 years; whence the Complement is 41.424 years. The
Table of Compound-interest (page 12th) indicates, as the value of an annuity for
this last term of years certain, 21.69907 years' purchase, when payable by full
yearly instalments (taking the due proportion between 21.59910, the value for
41 years, and 21.83488 the value for 42 years). Multiplying 21.69907 into
1.035 (unity increased by its Interest for one instalment), the product is 22.45854.
Dividing the latter by 1.44984 (the Complement 41.424 multiplied into 0·035
the annual ratio of Interest), the quotient is 15.49036; which subtracted from
28.57143, the value of Perpetuity (indicated by the last line of the same page
12th), leaves 13.08107 years' purchase, which is the value of an Annuity payable
by full yearly instalments, and on a life described as above.
SECOND EXAMPLE. Suppose the age 60, the rate of Interest 4 per centum
per annum, but accruing half-yearly, and the Decrements of life to accord with the
Northampton observations. The Expectation will be found 13.216, and therefore
the Complement 26-432 years. The value of an Annuity for this last term of
years certain, and admitting half-yearly instalments, is 16-22266 (by Table
page 20th, and consequently to having added to the value of a 26 years' annuity
the proportion 0·432 of the difference between that value and the value for 27
years). The quantity 16-22266 multiplied into 1.02 (unity increased by one
half year's interest) is 16 54711, which divided by 1·05728 (or the Complement
multiplied into the Ratio of interest for one full year) yields the quotient 15-65064.
Subtracting the latter from 25, which is the value of perpetuity at 4 per cent.,
leaves 9.34936 years' purchase, as the present value of an annuity payable by
half-yearly instalments, and settled on a life aged 60 accomplished years.
THIRD EXAMPLE. Suppose the age 30, the Interest at 5 per centum per annum,
accruing quarterly,-and the Decrements to accord with the observations on
the general population of France. The Expectation then is 28-5177 and the
Complement 57·0354. The value of an annuity for the latter term of years
certain (by Table page 36th) is 18.84271 (in consequence of 18.82250, the value
for 57 years, and of 18.87858 the value for 58 years). That value 18·84271,
multiplied into 1.0125, yields the product 19.07824, which divided by
2-85177, (the Complement 57·0354 multiplied into 0.05 the annual ratio of
Interest,) quotes 6'68996; and this quotient subtracted from 20, the value of
COMPOUND INTEREST.
115
Perpetuity, leaves 13.31004 years' purchase, as the present value of an Annuity
to be settled on the described life, and to accrue by quarterly instalments.
It is only of minor importance, in such calculations, to discriminate the values
of Life-annuities according to the specific intervals of their instalments; in con-
sequence of what has been remarked on that subject, in Sect. XXVIIIth; and
further because results strictly accurate, in respect to those values, may always
be considered unattainable, as a series of data merely probable are inevitably
proceeded upon.
(XXX.) It is immediately perceptible, that the value of an annuity on any
single life, as expressed by a quantity of years' purchase,-must, in all cases,
be inferior to the value of an annuity for a term of years certain and equal either
to the probable term of the annuitant's life, or to the average of forthcoming
years that may appertain to his age; because the proportions according to which.
the lives successively drop in, as observable from any table of decrements, are
not the same for every year of life, but are less considerable for the years preceding
that term of probability than for those that follow,—and because those proportions
of annual decrement increase with every year of life's protracted duration beyond
that same term.
Samsun
In other respects, and in like manner with what has been observed (Part I.
Sect. XL.) concerning annuities for terms of years certain, the Ratio according
to which any Life-annuity is paid, ought to differ from the ratio of a perpetual
annuity, by neither more nor less than the fractional quantity which accumulated
and improved at compound interest, during the probable period of its payment,
would at the expiration of such period create the same capital as was originally
invested.
(XXXI.) The equivalent, in present capital, for an annuity limited to a certain
term of years supposed n, and at the same time depending on the continuance of
life during that term, is (by the same 4th principle) equal to the sum of the
different products of each annual payment in expectation,- and reduced to its
present value by discount under the consideration of compound interest,—such
discounted values respectively multiplied into the corresponding probabilities that
the life of the annuitant will endure a first, a second, a third, &c. to the quantity
n of accomplished years. If unity be admitted to represent the annuity, the
calculation will be much simplified by taking the successively discounted values
of unity from the Tables of Compound-interest; and then the equivalent,
inquired after, will be expressed by a quantity of years' purchase, called the
Value of the life for n years.
;
(XXXII) The grant of an Annuity, payable during life, may be subjected
116
THE DOCTRINE OF
to the condition that its term shall not commence to run until the expiration of
a certain period of years, expressible by n. The equivalent, in present capital,
is then equal to the equivalent referable to the expiration of the suspensive term
n, and calculated according to the preceding directions (XXVII and XXIX);
this last equivalent reduced to its discounted value corresponding with that term,
and multiplied into the fraction expressing the probability that the life in expec-
tation will endure until the term of the annuity commences to run.
The same result will be obtained by subtracting from the value of a Life-
annuity, the term of which should commence to run from the present time, the
present value also of an annuity depending on the same life, but limited to a term
of n years; which last value is calculable as above (XXXI.)
(XXXIII.) The Assurance of a life, by which is meant the undertaking to
pay a certain capital at the demise of the life assured,-or rather at the expiration
of the last year accomplished after that event and computed from the period of the
assurance,—may be compensated in two ways; either by an inferior capital once
and presently paid, or by an annual premium, the payment of which by the
assured party is exactly the reverse of a Life-annuity enjoyed by himself. It is
evident that such present capital should be precisely the value of the latter
annuity, the transaction amounting to this; that the assured party borrows a
capital, against an annuity upon his own life, which capital is re-invested until
required with the amount of its improvement at Compound-interest.
following is an easy rule of calculation.
The
Ma
Let the excess of the value of perpetuity, over the value of the assured life, be
multiplied into the interest which the capital in expectation should produce for
one year; let the product be divided by unity increased in the ratio of one year's
interest; and the quotient will exhibit the present capital to be paid as an equi-
valent for the assurance. The division by 1+r, as last mentioned, arises from
the consideration that the eventual payment is only to take place at the expiration
of the year fulfilling the whole term, whilst the equivalent is paid at the outset.
It follows again, that dividing such present equivalent by the value of the life,
expressed in a quantity of years purchase (which value the Tables may shew),
the quotient will exhibit the annual premium otherwise exigible.
Various motives may induce the Assurance of a life for a limited term only.
In that case, ascertain (as in Sect. XXXI) the value of the given life, for that
term, and also its value for one year less than that term; add unity to the latter
value; divide this sum by unity increased by its interest for one year; from the
quotient, subtract the value of the life for the full term of the assurance ; and
the remainder will be the equivalent, payable at the outset, for every unit con-
}
COMPOUND INTEREST.
117
tained in the capital intended to be assured. This rule, with the preceding and
many others, applicable to individual cases into which the present general question
branches out, have already been stated and exemplified in Dr. Price's work, edited
by Mr. Morgan (who directs the transactions of that excellent institution,-the
Society for Equitable Assurances on Lives and Survivorships): it is a sufficient
reason for here abstaining from any enlargement upon this particular subject.
The prospective views of persons who, under appropriate circumstances,
stipulate for the Assurance of Capital, or of any equivalent for Capital, to accrue
after their demise and in behalf of their successors, are attended with this moral
effect;-that the fruits of industry and of economy, during a period equal to the
probable term of any such person's life, may thereby be secured, as certain and
at all events.
(XXXIV.) Dr. Price has further supplied a method, founded upon analogy,
of deducing with sufficient approximation, from the correct values of any single
life or joint lives,-as calculated at any rate of interest, and from any table of
observed decrements,-the correct values also at other rates. We must state this
method, which may occasionally ease the process of comparative calculations.
He proposes a preliminary problem relative to each case, of a single life and
of two joint lives,-assuming, for greater simplicity, the decrements of life to
be equal for each succeeding year*; the following solutions of which problem,
merely intended for this single purpose, should not however be mistaken for
correct ones, nor be otherwise applied.
FIRST. The Expectation of a single life being given, as also the rate of
interest, the value of that life is thus to be calculated by approximation. Multiply
the value of an annuity for a term of years certain, and equal to twice the
Expectation, into the value of perpetuity increased by unity; divide the product
by that term of years; subtract the quotient from the value of a perpetual
annuity at the given rate of interest, and the remainder may be taken as the value
of the single life.
SECONDLY. The Expectations of any two lives being given, with the rate of
interest, the value of the joint lives (meaning that of an annuity during joint
continuance of those lives) is then to be calculated as follows. Ascertain the
difference between twice the Expectation of the youngest life, and twice the
Expectation of the oldest life increased by unity and by twice the perpetuity;
multiply that difference into the value of an annuity certain for a term of years
equal to twice the expectation of the oldest life, and let the product be divided by
* An hypothesis introduced by an early author, already mentioned in Sect. XXIXth.
118
THE DOCTRINE OF
that same term of years,-reserving the quotient. Then, from twice the perpetuity,
subtract the reserved quotient; multiply the remainder into the perpetuity
increased by unity; and dividing this last product by twice the expectation of
the youngest life, the quotient subtracted from the value of perpetuity will leave
the required value of the joint lives. It has been here assumed, that twice the
expectation of the youngest life would be found less than that of the oldest life also
doubled and increased by unity and by double the perpetuity; but if it should
happen to be greater, the reserved quotient-instead of being subtracted from
twice the perpetuity-must be added to that quantity; and the sum, instead of
the difference, must be multiplied into the perpetuity increased by unity.
Either of those preparatory solutions being obtained, with reference to any two
different rates of interest; and admitting a correct value* of the life-annuity to
have been previously ascertained, with reference to either of those rates, the value
according to the other rate of interest will be obtained, either by adding to the
preceding value, or by subtracting from it (as analogy will point out), a quantity
equal to the difference between the results of the aforementioned calculations.
Caution is to be observed, against an error which the argument from induction
only might lead to. Although the value of a Life-annuity, according to any
specific rate of Interest, will coincide with that of an annuity referable to the
same rate and for a term whatever of years certain; it should not thence be inferred
an equality of the respective differences between each of those values and the
corresponding value calculated according to any other rate. Those differences
may vary, even to the extent of one half a year's purchase, though the Rates of
Interest differed no more than one per cent.
(XXXV.) Annuities that depend on a plurality of lives are susceptible of two
different descriptions; they may be payable either during the joint continuance
of any two or more lives, or during the longest of those lives. In the former
case, the annuity is of less value than if it were constituted on either of the lives
in particular, because the probability of each is affected with a contingency
respecting the other life or lives; and in the latter case, it is of greater value than
that of an annuity on the best life amongst the whole, because the probability
attached even to that best life is then augmented by the probability—however
small it may be of the worst life eventually surviving.
(XXXVI.) The value of annuities of the first description, or on joint lives, is
seldom of any practical consideration, except for the purpose of subsequently
calculating either the value of an Anuuity on the life of the last survivor, or
else the value of expectations depending on survivorship. The value of two
* This expression refers to the direct method of calculation stated in Section XXVIIth,
and (however tedious) indispensable when great accuracy is aimed at.
COMPOUND INTEREST.
119.
or more joint lives is ascertainable on the same principles as those already set
forth as applicable to any single life. The fundamental principle is, always,
that the present value of any sum in expectation, as eventually payable at a
certain period, is equal to the discounted amount of that sum multiplied into the
probability of the event upon which its payment is made to depend; therefore,
by multiplying into each other the respective probabilities attached to the several
lives, of their further enduring one full year, two full years, three full years, &c.
until the last year when the annuity may possibly be payable, the product will
express the probability of the joint continuance of those lives until the accom-
plishment of each of those periods distinctly considered; and the sum of the
different products of those probabilities, respectively multiplied into the annuity
discounted at compound interest for one year, for two years, for three years, &c.
until the remotest possible term, will yield the present value required.
(XXXVII.) The value of annuities on the longest of two or more lives, or
payable until the extinction of all those lives, is calculable on the afore-mentioned
general principles; first ascertaining the different probabilities, that the whole
of the considered lives will not be extinct at the expiration of a first, of a second,
of a third year, &c. respectively; and the rest of the calculation then to be
proceeded upon as directed in the preceding Section.
For greater clearness, let us first suppose the annuity to be settled on the longest
of two lives only, A and B.
Subtracting, from unity, the probability that A (according to his age) will live
one full year, there remains a fraction expressing the probability of his not
outliving that first year; and calculating, in the same manner, the probability that
B will not outlive the same period, the product of those probabilities multiplied
into each other will be the probability that neither A nor B will then be alive.
The difference between unity and the latter quantity will express the probability
that either A or B, indifferently considered, will continue alive at the expiration of
a first year; and the latter probability multiplied into the discounted value of the
sum then forthcoming, but depending on that event, will yield the present value
of that first instalment of the annuity. Let similar calculations be executed, with
reference to the expectation of the annuity becoming payable at the expiration of
a second, a third year, &c. to the last year of possible expectation; and the united
amount of those partial values will be the present value of the annuity, constituted
on the longest of the two lives A and B.
In case of three or any greater number of lives, the probabilities of their
severally not continuing until the expiration of a first, a second year, &c., are pre-
viously to be ascertained; those probabilities must be multiplied into each other
as above mentioned, with reference to the expectation of each successive year
120
THE DOCTRINE OF
distinctly considered,-which will yield the probability that the whole number of
lives will then be extinct; and the latter probability subtracted from unity will
leave a remainder, expressing on the contrary the probability that one at least
(indifferently) of those lives will continue to endure. The rest of the calculation
may be proceeded with as in the preceding instance of two lives only.
This method of calculating the value of the longest life amongst two, three, or
any other quantity, is immediately deduced from the fundamental principles;
but it ultimately resolves into an abridged method, producing similar results.
It requires that there should be previously ascertained, the value of each life
taken singly, the value of the joint continuance of all the lives,—and lastly,
the value of as many collections of joint lives as can be formed by successively
excluding from consideration each of the lives in particular.
If two lives only, A and B; subtract the value of their joint lives from the
united values of their single lives, and the remainder will be the value of the
longest life.
Suppose three lives, A, B and C. From the sum of their distinct values,
subtract that of the values of A and B, of A and C, and of B and C, each of
which jointly considered; then add to the remaining quantity the value of joint
continuance of all the three lives; and the result will express the value of the life
of the last eventual survivor.
Further supposing four lives, A, B, C and D; add the respective values of
those single lives; subtract, from the whole, the sum of the four values of joint
continuance, ABC, ABD, ACD and BCD; add to the remainder the value of
joint continuance of all the four lives, and the result will be the value of the
longest life amongst them.
Those abridged rules have been first promulgated by Mr. Simpson.
(XXXVIII.) The value of an Annuity, settled on the longest of two or
more lives of similar age, is always superior to the value of an annuity for a term
of years certain, and equal either to the Expectation or to the probable term of
any one of those lives; which probable term extends to about two years beyond
that of the Expectation, at the age when the latter is greatest. But if the ages
are very unequal, the value of such an annuity may be less than that of an
annuity for a term of years equal to the longest of the individual Expectations
or Probabilities of life.
(XXXIX.) The value of Reversionary property, or of any expectation in
the event of survivorship, depends on the conditions attached to such reversions,
COMPOUND INTEREST.
121
or to the expectations entertained. The one and the other may consist either of
Capital, of Annuities, or of Estates.
The present value of Reversionary Capital, unconditionally forthcoming upon
the failure of any given life, is the same as the equivalent actually payable for an
Assurance on that life, as ascertained in section XXXIII. But in case of the
reversion being subject to the condition of survivorship, the present value will be
equal to the product of the Interest which the expected capital should produce for
one year, multiplied into the excess of the value of perpetuity, over the difference
between the values of the life in expectation and of the joint lives.
The present value of a reversionary Life-annuity, to a survivor, is equal to
that of an annuity on the life in expectation, diminished by the value of the same
annuity during continuance of the joint lives.
The present value of the reversion of an Estate, to be certainly entered upon
after the failure of two lives, is equal to the product of the clear annual income
of that estate, multiplied into the excess of the value of a perpetual annuity, over
the value of an annuity on the longest of the two lives. But, if the Estate is to be
enjoyed upon the failure of either of two lives, the present value of such a re-
version will be the product of the income multiplied into the excess of the value
of perpetuity, over the value of the joint lives.
The latter question branches in two others, arising from the consideration
that either of the two lives may happen to be the longest. Thus, that question
may concern the value of the reversion in respect either to A's expectation, or to
B's; and supposing them of the same age, or assuming an equal value of their
lives, the object of certainty will then divide into equal probabilities in favour of
them respectively. But supposing their ages different, or assuming their lives to
be of unequal value; the value ascertained as above, of the reversionary Estate,
being multiplied into the fraction expressing the probability of A surviving B, will
yield the value of the former's expectation in particular; and if multiplied into
the fraction expressing the contrary probability, or that of B surviving A, the
product will in the same manner shew the exact value of B's share in the common
expectation. Either of those probabilities, which, added together, will amount
to unity, or to the expression of Certitude, is ascertainable; and Mr. Morgan
having supplied, in his seventh edition of Dr. Price's Work (vol. i. page 406 &c.)
very valuable tables of the probabilities of survivorship between two persons
of all ages, we need only refer the reader to them.
The method of ascertaining the respective probabilities of survivorship
between A and B, without reference to any specific period of time, has been so
clearly stated and demonstrated by the same Mr. Morgan, that it will be quite
q
122
THE DOCTRINE OF
sufficient to give, in a note, the extract from a paper of his upon that subject, and
which has been read at the Royal Society, in May 1788*.
Supposing any sum of money, depending on a contingent reversion of the last
description, to be assigned to some third person, in consideration of another
sum presently advanced to the expectant, whether A or B; it follows from the
preceding solution, that those two sums should bear, with each other, the
same proportion as that which the present value of the absolute reversion,
multiplied into the probability of the unfavourable contingency, bears with the
integral object of expectation; independently of the further and indispensable
consideration of the reduced value of the future sum, according to the operation
of compound-interest.
* Let a represent the number of living persons (according to the Table of Decrements) at
the age of A, supposed the younger of the two lives. Let a', a", a""', a""", &c. represent the
decrements of life at the end of the 1st, 2d, 3d, 4th, &c. years from the age of A. Let b
represent the number of persons living at the age of B the older of the two lives, and
c, d, e, f, &c. the number of persons living at the end of the 1st, 2d, 3d, 4th, &c. years from
the age of B. Supposing now it were required to determine the probability of B's surviving A
in the first year. It is manifest that this event may take place either by A's dying before the
end of the year and B's surviving that period, or by the extinction of both the lives, restrained
however to the contingency of B's having died last. The probability that A dies in the first
The probability that both the
•
year, and that B survives it, is expressed by the fraction
ac
a b
a.(b −c)
ab
lives die in this year is expressed by the fraction
; and as it is very nearly an equal
chance that A dies first, this fraction should be reduced one-half, and then it will become
a²
Hence the whole probability of B's surviving A in the first year will be
•
a'c
a b
(b - c)
2ab
+
a.(b-c) a' .(b+c)
2ab
2ab
In the same manner the probability of B's surviving A in the
· (e+ƒ), &c. re-
2ab
a" (c+d)
2ab
spectively; therefore, the whole probability of B surviving a will be
= 1 / (0 += a + c + du "+d+e
eta all tate all teta", &c.)
b+c
ab
2
2
2
2d, 3d, 4th, &c, years may be found =
•
a" .(d+e)
2ab
·
Q"""
Having thus found the probability of B,
the elder life's, surviving A the younger; the probability of A's surviving B is the difference
between the foregoing series and unity.
The sum of this series might easily be determined from tables of the expectations of single
and joint lives; but no such table as the latter having ever been computed, it will not be a
laborious undertaking to compute a table of the probabilities of survivorship between two
persons of all ages immediately from this series, without having recourse to the expectations
of life. For if the probability of survivorship between any two persons be found, the proba-
bility between two persons one year younger is obtained with little difficulty; and by pro-
ceeding in this manner a whole table may be speedily constructed.
26
The above series, expressing the probability of survivorship, may be easily found to be
B. (K—AK)—c. (C—AC)
; K denoting the expectation of a life one year younger
than B; C the expectation of a life one year older than B; AK and AC the expectations of
the joint lives of A and K, and of A and C, respectively; and 6, the number of persons
living in the table of observations at the age of K.
COMPOUND INTEREST.
123
The questions.concerning survivorships and reversionary interests may occur in
many other shapes, but they will always exhibit sufficient analogy, with some one
of those above described, for inferring with more or less facility the method of
solving them, from such solutions as are already pointed out.
(XL.) It has here been avoided, as far as possible, to reproduce details upon
which various authors have enlarged on the present subject; endeavouring, at the
same time, clearly to expose the fundamental principles not always reverted to
by them, respecting the laws of Probability, and upon which only could rest the
illustrations from which we have abstained. The analogies of the cases to which
those principles however have been applied, with other cases omitted to be
noticed though susceptible of frequent recurrence, will be easily discernible
when the questions are stated with proper discrimination. The inadequacy of
the means hitherto employed, especially in this country, towards obtaining
satisfactory returns of the population according to its eligible subdivisions, hast
left the Doctrine of Life-annuities and of Reversions in a very imperfect and
perplexing state; and it may be superfluous to add, that absolute accuracy in the
results of calculation, regarding these matters, could but unsuccessfully be aimed
at, in consequence of the substitution of merely tolerable approximations,
for the more correct data that fail to be supplied. Therefore, in the practical
application of the rules laid down for calculating the values of life and of rever-
sionary interests, or for obtaining sufficiently accurate results concerning any other
prospective transactions, it is always of paramount importance to class the lives
according to their comparative qualities; a consideration that cannot too much be
impressed on the reader's mind, and without which those transactions themselves
must often be at variance with equity, if not even with common sense.
To the effect of facilitating the necessary classification of lives,-as they may
greatly differ in value, though of similar ages,—there now remains to be exposed
a series of deductions from actual experience and observations collected in various
parts of Europe, concerning the laws of mortality and the decrements of human
life at its successive stages. When about to constitute any Life-annuity, the
great importance of discriminating the quality of the life or lives, during the con-
tinuance of which such annuity is to be paid, will be clearly evinced, from the
considerable discrepancies observable in the results of calculation founded upon
different sources of information.
(XLI.) By offering those results in the shape of Synoptical Tables, the chief
purpose of facilitating habitual references will better be answered. This second
part of our book will therefore terminate with four of those Tables; and these
aresusceptible of a few explanations.
The FIRST Table exhibits, as deduced from each set of local observations,
124
THE DOCTRINE OF
the assumed proportions amongst the quantities of living persons at each different
age; exposing the natural law by which the waste of human life appears to
be constantly regulated. The results there brought into comparison, relatively
to each age, have been calculated upon the principle and according to the rules.
set forth in Sections XVII and XXII, If it be considered, that the difference
between the quantity of living persons at each age in particular, and the quan-
tity of those living at that age increased by one year, exactly represents (at least
under the supposition of a stationary population) the quantity of the deceases
during such last year, it will become obvious that those proportionate quantities
of the living, at different ages, also constitute a Table of the probabilities that
are attached, for any given age, to the continuance of life during successive
periods of forthcoming years.
In order to render more effective the means of comparison which it is intended
by that first table to afford, it were desirable that the various decrements there
exposed could have originated with equal numbers of births; but from the
many causes productive of irregularity, and of uncertainty, in the statements
relative to the earliest stages of life, it was apprehended that such accommoda-
tion of the calculations to each other would cause the whole to participate
more of illusion than of reality, under a general aspect; and it has been con-
sidered preferable to adapt a common Radix to the age of five accomplished
years, at which the Expectation of forthcoming years of life is almost always at its
maximum. It may be sufficient further to remark, concerning local observations
upon the law of mortality in any metropolis, or in any great commercial city, that
the inferences from them are liable to be more or less vitiated, consequently to the
population of such places not being equally permanent with that of the country at
large. Their population is partially and variously recruited, whether from the
surrounding country or from the accession of strangers, whilst a considerable
proportion of the individuals born in those cities are subsequently removed; all
which circumstances, occurring without constant regularity, may render very
inaccurate the generalized deductions from any actual distribution of population,
or from any register of Mortality, as regards the Expectation of forthcoming years
of life, especially at its early stages.
The SECOND Table is a deduction from the first; and it exposes, in the
same manner, a comparative statement of the Expectations or Relative averages of
forthcoming years of life, belonging to each age and referable to the different
Decrements of life already brought under consideration. This Table has been
calculated according to the principle and rule set forth in Section XX; and one
of its purposes is to facilitate the application of an easy method, described
and exemplified in Section XXIX, of calculating the value of life-annuities,
conformably with any given rate of the Interest of money, and in either case
of the instalments of the annuity accruing yearly, half-yearly, or quarterly.
COMPOUND INTEREST.
125
The THIRD Table gives the values of Annuities on single lives, as deduced
from the data supplied by the first Table. Those values are calculated upon
the principle set forth in Section XXVII, and according to a uniform rate of
Interest, that of four per centum, understood in a single payment at the expi-
ration of each year. The excessive labour of such calculations has required
that the statement of those values, strictly correct, should be confined to a cor-
respondency with the most important systems, only, amongst those previously
exposed of the Decrements and Expectations of Life; but even with this
limitation, the intelligent reader will find the Table amply sufficient for all
practical purposes; and the more so, as other means of comparing the values are
abundantly afforded.
-
The Northampton Table being often consulted, although far from offering a
fair specimen of the annual decrements of life, we have thought it worth while to
re-calculate, with the utmost care, the values of annuities according to that
Table of Decrements; extending the results to five places of decimals, instead of
three only, to which those results were limited in all previous calculations*.
The correct values of lives are further stated in correspondency with Mr.
Duvillard's Law of Mortality accurately observed, respecting the general popu-
lation of France; and these values, together with the preceding, are brought
into a comparative point of view with others supplied from respectable
authorities.
From a prevailing ignorance on these matters, innumerable contracts for Life-
annuities are entered upon, at variance with equitable principles; and the value
of those annuities, commonly, is much underrated, whilst they possess a specific
character of convenience that should render them the more valuable for the
annuitants themselves.
The FOURTH Table gives the values of Annuities supposed to be payable
during the joint continuance of two lives, whether of the same or of different
ages; and from various motives, that statement of values has been circum-
scribed within comparatively narrow limits.
When the value of three joint lives is required, it may correctly be found by
an ingenious method described by Mr. Morgan, and adverted to by Dr. Price,
(vol. ii. page 211). But a more abridged and easier method of obtaining that
value, with tolerable approximation, has been pointed out by Mr. Simpson: it is
* The unimportant differences, observable between Mr. Morgan's results and those we have
obtained, are satisfactorily accounted for, without impeaching the character of accuracy of
either calculations; if (as we apprehend) he has employed logarithms limited to sir places of
decimals, and has further abandoned the decimal parts of his results, when even amounting
to a full half unity belonging to the last figure retained.
!
126
THE DOCTRINE OF
}
as follows." Let A be the youngest, and C the oldest of the three proposed
"lives; take the value of the two joint lives B and C, and find the age of a
single life D of the same value; then find the value of the joint lives A and D,
"which will be the answer." Either of those methods may be applied, in
correspondency with any Table of the Decrements of Life.
66
The value of Annuities during Joint continuance of Life is but seldom re-
quired for direct purposes: it constitutes, however, an indispensable datum for
establishing the value of Annuities payable during the longest of any two or
more lives, according to the rules stated in Section XXXVII; and also for
calculating the value of Reversions, on the principle exposed in Section XXXIX.
References to the last two Tables, and for practical purposes, will not be
attended with much inconvenience arising from the uniform rate of Interest
applied to all the calculations of which those tables exhibit the results; an easy
method of concluding from the value of a Life-annuity, as ascertained according
to any one rate, its corresponding value at any other rate of Interest either
superior or inferior to the first, having already been supplied in Section XXXIV.
The computation of Correct Tables of the Values of Joint lives, both of
similar and of dissimilar ages, according to comparative systems of Decrements,
and to various rates of the Interest of money, would consume the whole life of
an expeditious calculator. It has been hitherto executed only partially and
piecemeal. There will be no inducement to attempt any thing complete of that
description, until governments are mindful of the expediency of providing satis-
factory data, upon which those laborious calculations may be proceeded.
(XLII.) We may now dismiss the subject of COMPOUND-INTEREST
applied to Transactions affected with Contingency, after further giving a brief
account of those local observations which, at different times and different places,
have been made for the purpose of discovering the natural law that governs the
gradual waste of human life; and according to which observations the four
Tables last mentioned have been constructed.
FIRST.-The series of regular observations, earliest recorded, consists in a
Register of Mortality for the city of Breslaw, in Silesia. That register had been
carefully proceeded upon during a whole century, commencing with the year
1633, when Dr. Halley constructed from that source the first known Table of
Decrements; but as it is not exempt from error, chiefly produced by a cause
already adverted to in the preceding Section (p. 124), that, table is seldom
referred to at present. It has induced, however, the admission by Mr. Demoivre
of a convenient hypothesis respecting the decrements of life. His hypothesis
assumes that, out of 86 living persons aged less than one accomplished year, a
COMPOUND INTEREST.
127
single life annually fails; that of the last survivor being consequently extinct
after the accomplishment of the 85th, and before that of the 86th year of his
age; thus producing equal decrements for every year throughout that period.
From that assumed equality, the Average of forthcoming years in Expectation is
made exactly to coincide with the true Probability, thence inferred, of life at each
age; which Expectation, therefore, is always considered as one half of the differ-
ence between the number of accomplished years of any person's life and the
supposed term, 86 years, of the longest life; the whole of that difference being
taken as the Complement of life, which (however suppositious) is one of the
requisite data towards calculating by an abridged method the value of Life-
annuities, the labour of which calculations may thus be considerably diminished.
Although the proportions of annual decrement, as also the averages or expec-
tations of forthcoming years of life, are very defective under Mr. Demoivre's
hypothesis, as regards both the earlier and the later stages of life; yet they are
acknowledged to correspond with the Decrements and Expectations resulting
from the Breslaw observations, relative to the ages comprised between 20 and
70 years or thereabouts, in a sufficient degree for warranting their admission
within those limits. A similar correspondency has since been noticed, between
the results of admitting that hypothesis and those deduced from the Northampton
Observations further mentioned hereafter. But, from the preceding remark, it is
obvious that the hypothesis of which we are speaking must be utterly inapplicable
towards calculating the values of Reversionary property, depending on survivor-
ship; an exception that has not escaped the sagacity of Dr. Price.
•
SECOND.—The observations supplied by the Bills of Mortality of London,
during the ten years ending with 1737, have been the foundation of Mr. Simpson's
deductions and calculations, consigned in the works of that celebrated writer.
Analogous deductions were afterwards made from the same source of information,
during the ten years ending with 1768, and again during a similar period ending
with the year 1780. The general results, inferred from those three series of
observations, are so nearly alike as not to indicate any material alteration in the
degree of healthiness of this metropolis, between the first and the last of those
periods; in consequence of which, we have preferred, in the construction of the
first and second Tables hereafter, to proceed upon an average of the results of
observation, as supplied at those different periods, relatively to each successive
year of life.
It has already been remarked by Dr. Price, that the Expectations (Average)
of Life, as deduced from the London observations, come out generally too high
for the first 35 or 40 years; that they are not at all to be depended upon for the
first 20 years, and also that the Bills of Mortality, for London, exhibit an
instance of constant waste of human life the most rapid at all its stages. The
first two circumstances are doubtless ascribable to the great mutability in the
population of this metropolis, especially in early life.
128
THE DOCTRINE OF
THIRD. From a Register of Mortality instituted for the town of Northamp
ton, and correctly kept during a period of 46 years ending with 1780, has been
deduced a Table of Decrements considered as one of those most to be relied on
for regularity. In its general aspect, it very nearly agrees with the Breslaw
observations, and like the latter it affords a specimen of lives rather inferior in
quality to the common standard of those in England; whence the Northampton
Table is principally adopted in this, and even in other countries, as a safe guide
for calculating Life-assurances and Survivorships. It follows that the Expecta-
tions of Life, as resulting from that table, are lower than they should appear in
consequence of a table of true decrements deduced from observations supposed to
embrace a more extended selection out of the population of the kingdom at
large; and that the values of Life-annuities, stated in the corresponding column
of the third Table hereafter, could not be indiscriminately admitted in the
practice, without probably yielding an undue advantage to Annuitants placed
under circumstances more favourable to the protracted duration of life. Nume-
rous tables of various descriptions, and elaborately calculated from the North-
ampton observations, have been supplied by Mr. Morgan, and inserted in the 2d
Volume of Dr. Price's Observations on Reversionary payments, &c. &c.
FOURTH.-A specimen, on the contrary, of the best lives susceptible of being
brought under observation in England, as averaged from any mass of population
not absolutely inconsiderable, has been supplied by two other Registers kept in
this country.
The one through the care of the Rev. Mr. Gorsuch, for the
parish of Holy-cross (near Shrewsbury), and embracing a period of 20 years,
ending with 1770: the other through the care of Dr. Haygarth, for the town of
Chester, and during the ten years terminating with 1781. The former exhibits
rather the best quality of lives; but considering the limited number of the
population to which those sets of observations are referable, it has been pre-
ferred, in the comparative Tables hereafter, to admit the corresponding deduc-
tions at an average of the results produced by each set, relatively to every age of
life. It will be sufficient further to remark, that the values of Life-annuities, as
deduced from the observations at Chester and Holy-cross, come out generally
too high for their indiscriminate application, in the practice, to the common
standard of lives: they might, however, be a safe guide for dealing with State-
annuitants, until the Government has taken proper measures for being supplied
with more correct data, respecting the Law of Mortality amongst that specific
class of persons.
FIFTH-A more eligible medium, perhaps, is instanced by the Bills of Mor-
tality of Norwich, from the year 1740 to the year 1769, and by the results of Dr.
Aikin's observations on the mortality at Warrington (in Lancashire) and its
vicinity, from the year 1773 to the year 1781. The average results of those two
sets of, observations, admitted as a single experiment, are also brought into com-
parative view in the following Tables.
COMPOUND INTEREST.
129
SIXTH-An industrious observer and well-informed German writer, Mr.
Sussmilch, has supplied, about the middle of the last century, very valuable docu-
ments concerning the Mortality and the Decrements of life, in a considerable part
of the Prussian dominions. From those data, tables have been constructed, which,
as they are extensively used in that country, will also form a branch of the compa-
rative system hereafter exposed. Mr. Sussmilch's Decrements, for Brandenburgh,
&c., may be considered as applicable to lives of the middling quality.
SEVENTH.-A specimen of the Decrements of life, under general circum-
stances rather favourable than otherwise to its protracted duration, and deduced
from a series of correct observations upon an extensive scale, is afforded by the Re-
gisters of Births and of Mortality in Sueden. Their results for a period of 45 years,
ending with 1776, embrace with very few exceptions the whole population of
that kingdom. Mr. Wargentin's scientific reports upon that subject, and in which
the circumstances have been attentively discriminated as relating to the Male and to
the Female lives in that country, supply more authentic and satisfactory information
concerning human mortality, and concerning the classification of an entire popula
tion, than had been obtained from any other quarter until that period. In
consequence of distinct returns for the city of Stockholm in particular, Dr. Price
has made the remark, that "The Tables for Sweden at large, compared with
"those for Stockholm the capital, confirm in a very striking manner all that has
"been said of the difference between the duration of life in great towns and in
"the country."
The proportions of the decrements, resulting from the Swedish observations,
might admit some modification, in order to render them quite applicable to
countries under a more southern latitude; considering that the progression of
mortality, through the different stages of life, is in more or less degree influenced
by the climate, even under the supposition of an equality in the absolute average
of forthcoming years. The Swedish decrements are, however, and with the con-
fidence to which they are justly entitled, frequently resorted to for calculating the
values of annuities on good lives; and in our comparative representation of them,
the distinction between male and female lives will not be omitted; where it may
further be remarked, that the progression of mortality is sensibly less rapid with
respect to the latter sex, whose Expectation of life is therefore greater at all
ages, as appears indeed to constitute one of Nature's general laws.
EIGHTH.—Whatever general law may govern the gradual waste of human life,
in a community of which the members are considered without any distinction of
classes or conditions of health, it is evident that such law must be subject to a
considerable modification with regard to any particular selection, either of the
best, or of the worst lives amongst the whole. Annuitants in general, and
especially State-annuitants, may properly be viewed, in all countries, as a select
r
130
THE DOCTRINE OF
body of the former description; and experience has abundantly proved that the
Decrements of life, amongst any collection of annuitants, proceed according to a
progression remarkably slow. The progression of those decrements might be
ascertained in a very satisfactory degree, as regards this country, if the Govern-
ment, together with certain incorporated societies to whom much valuable
experience has doubtless been imparted, would publish the data they possess.
But such information being withheld, the experience of other countries must be
consulted, towards supplying the deficiency; and in the first place, there is to
notice the Table of Decrements given by Mr. Kersseboom, and deduced from a
Register of Mortality kept during 135 consecutive years, with reference to the
State-annuitants of both sexes in Holland. Although no distinction is there made,
between male and female lives, Mr. Kersseboom has remarked that the female
annuitants have, amidst all accidents of age, lived about three or four years longer
than the males.
NINTH.—Of the same description with the preceding are the Tables of
the Decrements and Probabilities of life amongst the State-annuitants of France,
as published by Mr. Deparcieux, somewhat less than a century ago. Those tables,
and the several deductions from them, may usefully be brought into comparison
with the corresponding results inferred from the Dutch-annuitant register.
TENTH and last. Subsequently to the works of other authors of any
note, concerning this general subject, Mr. Duvillard of Geneva published, in the
year 1806, some amongst the results of his laborious and scientific researches on
the population of France at large. Having been supplied, under government-
authority, with adequate means of information, embracing a very extensive
portion of the French territory, the statistical data he has been able to collect,
together with his deductions from them, are at present, and indisputably, admitted
as more correct and exempt from objections than any other documents existing of
the same kind. In bringing into view the results of Mr. Duvillard's researches,
comparatively with those noticed in the foregoing paragraphs, there are sufficient
grounds for assuming that they exhibit a very fair specimen of the law of
Mortality indiscriminately applicable to the kingdom of France, such as it existed
previously to the year 1790; and it is probable that, were our institutions in
England such as to afford local information equally authentic and extensive on
the same subject, very little difference would be found to exist between the two
progressions, expressive of the Decrements of human life for the one country and
for the other.
The correctness of Mr. Duvillard's Decrements of Life may the more be relied
on, as he has applied to them a very ingenious method of rectifying the inaccu-
racies with whichan assumed Law of Mortality, derived from any course of ob-
servations, will probably be tainted, consequently to the casual character of the
COMPOUND INTEREST.
131
events observed in limited number. His method reduces the general expression
of a correct Law of Mortality to that of a Parabolic curve; in the equation of
which the Ordinates represent the quantities of the living, at the respective
ages represented by the Abscissa. Without such previous rectification of any
Table deduced from actual Observations, even a tolerable approach to accuracy
in the ultimate results, referable to very advanced years, and expressing the
Expectations of life, together with the values of Life-annuities, could only in
vain be expected. In fact, Mr. Duvillard's Table of Decrements, is the only
one in existence that may, in any degree, be depended upon for that purpose.
END OF THE SECOND PART.
I
+
132

BUCKSKILA
AME PRAKAS
TABLE (No. I.), exhibiting the Decrements of Life from year to year, and the Quantities of
living persons of each age, in proportion with certain assumed totals of population ;
according to observations collected in different parts of Europe, relatively to various
circumstances of human existence, and the results of those observations accommodated-
for the purpose of comparison-to a uniform radix, of 10,000 individuals supposed to exist
each of five accomplished years of age, when the Expectation of life is generally at its
maximum; this TABLE also exhibiting, with reference to each age, the Probability of life's
enduring until any other specific age; such probability expressed by a fraction, of which
the numerator is the quantity of the living at the latter age, and of which the denominator
age departed from.
is the quantity
of
the
living
the
years of age.
Accomplished
0
1234
5678
DEC 149 4998 72** PONO OOM! 3220 5800 Ho
10
11
12
13
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
33
34
35
36
37
38
39
40
41
42
43
44
AAAAAA
45
46
47
48
of France.
State-Annuitants
of Sweden.
Female population
10000
9810
10000
9820
9652 $662
9515
of Sweden.
General population
9389 9429
9283
9193
9117
at
9534 9522
of Sweden.
Male population
15461 15022 15235 15449 16031 14523
11518 11882 11891 11896 12969 11670
10959 11104 11115 11123 11722 11151
10540 10578 10591 10602 10912 10685
10232 10203 10218 10232 10377 10301
7823
7738
7655
10000 10000
0814 9807
9654 9645
9506
9410 .9390
9359 9316 9289
9261 9234 9204
9192 9165
9135
9051 9132
8990
9101 9065
9079 9043 9004
8933 9027 8987 8942
8875 8974
8931 2882
8513
8662 8608 8548
8431 8597 8536 8471
8532 8463 8393
8262 8466 8390 8308
8347
8176 8399 8314
8088
8331 8237
8256 8158
8223
8138
8053
7998
7910 8178 8076 7968
and Holy-Cross.
Chester
8811
8821
8800
8914 8870
8743 8854 8809 8761 8725
8670 8791 8745 8693 8646
8593 8726 8679 8625 8569
10000
9718
9504
9349
7169 7365 7262
7092 7281
7150
7057
7174
7016 7197 7086 6964
6942 7110 6994 6872
of Holland.
State-Annuitants
9253
9378
9182 9284
9126
9079
10000
9824
9647
9471
9025
9031
8980
8952
8927 8880
8871 8807
7699
7596
7489 7735 7633 7520
7409 7637 7537 7428
7328 7539 7441 7335
7248 7449 7351 7242 7048
9191
9168 ·
8486 8382
8403 8299
8318 8216
8223 8122
8126 8008
8019 7884
7912 7749
7805
7624
8734
8662
8568
8475
8095 7992 7881
8013 7908 7800
7500
7376
7922 7818 7706 7509
7571 7832 7727 7613 7424 7127
7251
6866 7012 6895 6771 6537 6183
6790 6900 6784 6660 6429 6089
6713 6785 6665 6536 6322 5996
6634
6671 6545 6413 6207 5902
of France.
General population
6554
5809
6559 6427 6289 6090
6472 6450 6312 6166 5972 5705
6388 6350 6199 6042 5855
6303 €252 6089 5918 5738
412269 | 418098 414951 411539 408401
5602
5198
Warrington.
Norwich and
17148.02 20395
13161.68 15172
11520.76 12313
10711.95 11110
10266.83 10440
Brandenburgh.
8897.57
8805 65
8710.42
8612.11
9525.60 9139 9154
9450.75 9020 9071
9378.16
8923 9005
9305.14 8833 8939
10000.00 10000 10000 10000 10000
9826.36 9666 9685
9706 9699
9482 9454
9305 9290
9703.11
9507.20
9453 9453
9262
9287
8667
9230.03 8748
9152.18
9070.88 8587 8740
8986.01 8506 8673
15887
18643
17201
22561
12587 13842 13661 15362
11426 11655 11680 12574
10796 10851 10902 11363
10365 10315 10383 10549
Northampton.
8872
8806
➡
Breslaw.
5728.74 5343
5604.78 5230
5479 52
5124
5352.79 5023
397770 406264.19 | 394587
|
8838
8743
8758 8661
8678 8579
8598 8497
8418 8607 8513
8322
8415
8421
8333
8541
8474
8222
8320
8251
8126
8408
8213 8169
8510.95
8025 8325 8097 8087
8407.21 7928 8242
7977 8005
8301.15 7827 8159 7857 7910
8193.02 7727 8076 7737 7828
8083.09 7626 7993 7617
7971.58 7518 7894
7858.72 7401 7794
7744.73 7280 7695
9061
9177 9153
9081 9030
8998 8921 8825
8918 8825 8708
8942
Bills of Mortality.
London
7629.78 7159 7595 7137
7514.06 7047 7496 7017
7397.70 6943 7396 6897
7280.84 6834 7297
6777
7163.57 6730 7197 6657
6926
7337
7242
6160
6013
7045.96 6621 7098 6537 6817
6882
5719
7002
6898
7147 6795
6928.08 6513 6998 6417 6694 5866
6691 €809.95 6401
6297. 6571
6949 6587 6691.56 6285 6766 6177 6448
6848 6483 6572.91
6165 6650 6057 .6325
6747 6380 6453.96 6045 6534 5937 6202 5270
6642 6276 6334.62 5921 6418 5817 6079
5571
5424
5107
10000
9664
7363
7251
7145
7036
6214.83 5805 6302 5695 5956
6094.47 5689 6186 5572
5973.44 5576 6070 5447
5851.57
5459 5954 5322
5833
5697
5560
9404
9219
8596
8490
8388
8287
8180
8074
7966
7858
7746 7261
7497 7650 7134
7377 7555 7008
7.257 7459 6878
7749
7632
7514
7388
6736
6594
6419
6305
4950
4785
4620
4463
4305
5837 5198 5423
5721 5073 5287 4150
5605 4948 5150 3995
5189 4823 5014 3840
400445 389530 | 392867 | 376960
133

years of Age.
Accomplished
2858 8529 5808 58** 2868 8222
49
50
51
53
54
55
56
57
59
60
61
62
63
64
6.5
66
67
69
70
71
72
73
74
3456
75
76
F008
77
78
79
81
82
*** *95* 8852 3338 528 2020 202
100
101
103
104
105
106
107
108
109
of France.
State-Annuitants
5840 5689
5731 5561
5614
5489
412269 418098 414951 411539
6217 6154 5978 5795
6129
6049 5859
6037 5937 5730
5942 5816
5664
5597
of Sweden.
Female population
5357 5178 4906
5221 5043 4761
5092 4908 4616
4934 4757 4464
5433
5306
1447
1255
4126 3874
3586
3956 3694 3406
3785 3514 3227
3612 3333 3042
1078
912
761
625
3437 3153 2853
3261 2973 2665
3081
2778 2471
2897 2567 2268
506
404
318
246
2704
2342
2102
2063
1849
2500
2287 1876 1651
2067 1666 1107
181
124
1852 1471 1292
1645
1290 1132
1125
975
****
4011
4780 4592
4623 4415
4450
4235
4299
4126
3841
3946 3663
4295 4054 3766 3485
of Sweden.
General population
832
697
569
449
5460
5323
336
254
5184
5046
194
149
114
87
65
47
32
20
11
5
ર
987
850
713
585
471
372
288
219
166
125
..
..
94
72
50
32
TABLE (No. I.)—continued.
of Sweden.
Male population
17
8
3
2
5517
5370
5223
5076
4930
4783
4636
4482
4327
4173
3308
3130
2952
2767
2574
2381
2188
1994
1137
998
867
743
627
519
419
326
249
187
141
107
8000 DOR
59
40
26
15
6
and Holy-Cross.
Chester
2
408401
1809 2224 2106
1624 2040 1961
1454 1868 1815
1292 1718 1660
5616 5373
5497 5259
5378 5135
5260 5000
5143 4876
4895
5018 4751
4627
4776 4502
4657 4367
4232
4530
4399
4098
4263 3963
4112
3828
3944
3693
3776 3558
3607 3413
3452 3268
3312 3122
3175 2977
3042
2832
2907 2687
2760 2541
2596 2396
2410 2251
1032
904
779
656
of Holland.
State-Annnitants
541
441
1577 1504
1439 1349
1302 1193
1167
1037
358
280
231
188
149
119
397770 | 406264.19
89
61
32
16
4
!
902
778
664
571
467
373
290
218
156
104
73
52
31
21
10
6
5
4
2
Totals 547244543821 532038 520536 526144 507871
of France.
General population
394587
400445
5224.41 4917
5373
5094.21
4812
5257
4962.03 4699 5124
4827.69 4586 4992
4410.40
4266.17
3504.76
3344.83
4119.25 3026 4229
3969.61 3789 4063
3817.28
3648 3881
3662.29 3508 3698
3182.72
3018.69
4691.06 4465 4859 4180 4276
4552.00 4336 4710 4049 4126
4200 4561 3917
4063 4395 3786
2183.78
2017.58
1353.21
1691.44
951.92
824.08
1534
1533.12
1379.11 1390
1230.30 1249
1087.61
1108
704.91
595.13
495.35
406.07
327.64
260.23
203.81
158.18
122.87
97.23
80.35
65.68
53.05
42.29
and Warrington.
Norwich
33.24
25.71
19.55
14.58
2853.08 2756 2786
2686.28 2595 2604
2518.76 2439 2421
2351.06
2282 2239
2129 2056 2100 2076
1976 1874 1972 1940
1823
1708
1679
1843 1790
1715 1639
1542
10.64
7.59
5.27
3.55
2.31
1.45
0.87
0.49
0.27
0.13
0.06
0.03
0.01
3363 3516
3219 3333
3066 3151
2913 2968
Brandenburgh.
1376
1211
1061
929
976 813
713
852
614
531
736
632
542
475
407
339
288
237
186
153
119
85
51
17
17
448
381
315
265
216
182
149
116
::::
88888
99
83
66
29
22
18
14
9
Northampton.
4
2
-1
1
389530 392867 376960
4698 4877
4572 4727
3698
3558
3418
4442 4576
4311 4426
3278
3130 3169
2999 3033
2869
2896
2740 2760
3655 3716 2638
3524 3579 2523
3393 3443 2408
3261 3306
2623
2612
2484
2486
2356 2350
2228 2213
1587 1489
1459 1339
1331 1202
1203
1066
Breslaw.
929
1080
963 792
855
669
751
560
..298
232
178
133
650 464
383
554
462
374
::
:
3989
3852
99
74
*** CON
54
38
26
14
6
314
260
Bills of Mortality.
London
205
150
109
68
41
14
3143
3014
2887
2763
2289
2169
2058
1946
1834
1722
16.13
1504
1395
1296
1196
1103
1011
919
840
764
688
614
540
482
384
335
287
244
202
175
138
109
82
65
49
40
31
23
17
12
7
3
1
501811.46 487169 495453 478789 480789 | 438475
CAREERAZTI 10/30
134

Fo
TABLE (No. II.), exhibiting with reference to each age the Average of forthcoming years,
commonly called Expectation of Life, as deduced from each set of observations respecting
the annual Decrements of the Living; with the addition of those Expectations according
to Mr. Demoivre's hypothesis of equal decrements.
N. B.—The true Expectation, or Probable term of life, is the quantity of years after
which the living, at any specific age, will be reduced precisely to one half their number,
according to the progression of Decrements that may be adopted, as a natural law, amongst
the different progressions set forth in the preceding Table; the probabilities being equal,
either for or against the event of any person of that age continuing alive beyond such term.
But these Averages are important data, towards calculating the value of life-annuities, by an
abridged method of
approximation.
years of age.
Accomplished
44
0 34.896 35.702 34.423
|
1 45.669 44.003
2
46.975 46.052
3 47.780
4 48.245
7
8
DOEL **** ***&
10
11
of France.
State-a nnuitants
12
9 47.379 46.801 45.619 44.472
|
46.917 46.259 45.076 43.947
46.370 45.634 44.473
45.751 44.973 43.802
18
of Sweden.
Female population
**** HONG DANA 320 1829 ****. ****
5
45 904
46.222
44.454 43.4002
44.243 43.1583
48.352 48.003
46.798 45.625
6 48.280 47.875 46.675 45.512
48.061 47.648 46.446
46.446 45.269
47.748 47.280 46.080 | 41.924 | 46.007 | 43.853 42.1215 | 41.368
41.273
41.684
41.609
46.249 44.043 | 42.7001
of Sweden.
General population
20 40.288 39.153 38.027
21
22
23 38.430
24
37.820
of Sweden.
Male population
42.963 | 41.960
44.925 43.839
47.316 46.125 44.967
48.039 | 46.790 | 45.579
45
46
47
48
39.66338.442
39.044 37.727
33.194 32.314
and Holy-Cross.
Chester
39.145 37.5
38.492 37.
41.008 42.755 40.360 38.5
40.540 42.142 39.782 38.
39.978 | 41.448
39.377 40.752
43.107 41.998 42.556 | 40.901 38.7419 | 28.756 | 40.053
14 44.383 43.520 42.379 41.282 41.793 40.229 38.0674 38.113 39.351
42.770 | 41.642 40.564 41.036 39.553 | 37.4041
37.466 38.646
42.018 40.902 39.836 | 40.295 | 38.875 | 36.7507
36.81737.938
36.817
13
45.084 44.266
15
16
43.664
42.949
22.615
21.913
17 42.255 41.298
41.298 40.179 | 39.105 | 39.61638.194 | 36.1130 | 36.198 | 37.226
41.580 40.574 39.454 38.370
19 40.924 39.862 | 38.739
38.952 37.510 35.4839 35.610 | 36.512
38.303 36.914 34.867235.037 | 35.794
37.643 | 36.315
37.643 36.315 34.2596 34.445
34.445 35.072
37.666
36.959
36.959
37.340 36.289
36.649 | 35.603
37.009 35.961 | 34.938
36.294 35.270 34.290
of Holland.
State-annuitants
45.483 | 43.289 | 41.4780
44.829 42.715 | 40.8025
43.348 44.103 42.144 40.1155
42.674 | 43.326 | 41.524 39.4254
34.470 28.7632 23.386
32.743
30.686 25.182
38.82741.774 | 36.3499 | 30.266
37.600
41.904 42.694 | 40.4253 36.177 40.369 37.794
43.977 | 43.538 42 4409 39.039
39.555
41.697
45.216 44.141 43.2591
40.512
40.585
42.410
30.272 29.301 28.402 29.861
31.613 29.653 28.671 27.750 29.244
35 30.952 29.030 28.034 27.095
27.095 28.623
36 30.289
30.289 | 28.375| 27.371 26.435 28.022
33 32.266
34
25 37.215
26
36.615
27. 36.020
28 35.415
29 34.802
32.838 31.898 31.004 32.358 31.400
30 34.179 32.170 31.231 30.330 31.791 30 935
33.547
31.531 30.584 29.695 31.153 30.444
32.911 30.888 29.953 | 29.050
30.504 29.967
31
32
37 29.617 27.694 26.698 25.771 27.414
38 28.934 27.008 26.021 25.130 26.811
39 28.241 26.318 25.339 24.431
40 27.538 25.634 | 24.663 | 23.753
41 26.836 24.984 24.010 23.098
42 26.130 24.384 23.396 22.476
43 25.425 23.786 22.804
24.722 23.184 22.212
44
France.
Population of
and Warrington.
Norwich
Brandenburgh.
35.582 34.589 | 33.640 34.552 | 33.279 31.3354 31.538 31.761
34.866 33.904 | 32.986 34.008 32.797
30.7668 | 30.985 | 31.155
34.179 33.228 32.328 33.462 32.359 30.2014 30.466 30.547
32.915 31.879
33.501 32.562 31.668
|
27.222
27.222
26.681
26.205 26.075
|
25.612 25.498
29.491 | 26 8395
28.927
28 361
27.793 | 25.1552
24.01922.574 | 21.615
20.712 22.703 22.346
23.31621.944 | 20.998 20.117 22.139 21.743
21.573 21.136
21,003 | 20.526
21.284 20.370 | 19.687
20.609 19.72818.916
Northampton.
37.006 35.714 | 33.6562 | 33.873 | 34.416 | 32.904 32.5
36.366 35.066 33.0700 33.279 | 33.758 32.391 32.
35.732 34.415
35.139 | 33.805
32.4861
31.9082
32.700 33.099
32.120 32.430
hypothesis.
Demoivre's
29.0776
29.0776 | 29.460 | 29.321
28.5177 28.921 28.704
27.9584 | 28.349
27.3991 27.790
đ đà
42.939 40.849
41.073
43.320
40.5
40.
43.372 41.031 39.5 41.523 36.335
40.798
43.133
39. 41.247 36.055
27.214 26.832
26.651 26.201
26.2791 26.651
25.7177 £6.086 25.566
25.1552 | 25.535 | 24.989
43.
27.452
18.935
42.5 33.435 26.574
42. 38.020 31.355
41.5 39.701 33.646
41. 40.661 35.202
37.836 | $6.5
37.177 30.
36.515
35.850
35.5
35.
30.851
30.5
30.337 30.
29 823
29.5
29.6386 29.963 29.935 29.308
| | |
29.
Breslaw.
Bills of Mortality.
London
41.197
36.107
41.458 36.346
35.202 34.5
36.114 31.098
34.585 34. 35.464 30.502
33.998 33.5 34.811 29.908
34.156 29.312
33.435 33.
20.0478 20.707 19.656
19.4802 20.143 19.045
18.9140 | 19.547 18.429 19.514
18.3500 18.942 | 17.808 | 19.006
40.855 35.663
40.405 35.143
39.894 34.602
39.320 34.059
33.497 28.717
32.834 28.150
31.878 31.5 32.225 27.585
31.365 31.
31.558
27.045
38.684 33.496
38.046 32.910
37.404 32.303
36 760 31.691
28.792
28.5 28.387 24.416
28.276 28. 27.807 23.932
28.083 27.759 27.5 27.225 23.458
27.459 | 27.242
27.
26.640 22.984
30.886
30.266
26.509
25.971
29.643
25.433
29.016 24.902
24.647
24.124
24.5
24.
23.833 20.679
23.286 20.227
24.0263
24.5914
24.5914 24.997 24.409
24.473 23.827
23.4599 23.949 | 23.241 23.602 23.5 22.738 19.805
22.8924
|
22.8924 | 23.443 22.652 23.079 23. 22.188 19.418
22.512
26.724 26.5 26.052
26.206 26. 25.462 22.051
25.687 25.5 24.920 21.591
25.167 25. 24.377 21.134
22.5
25.017 24.875 22.3240 22.900 | 22.061 22.561
21.635 19.020
24.426 | 24.249 21.7550 22.357 21.465 22.049 22. 21.088 18.658
21.891
23.832 | 23.619 21.1857 21.797 20.866 21.542 21.5 20.374 18.307
21.303 23.265 | 22.985 20.6165 | 21.253
21. 20.068 17.935
20.263
21.036
20.5 19.560 17.573
20.529
20.022 20. 19.053 17.210
19.5 18.545 16.858
19.
16.518
18.037
·135

years of age.
Accomplished
1
Cena 5008 5883 8368 8072 REDO POR: 7*** ***: 2878 8338
53 18.448
17.787
55 17.149
56
16.529
54
49 21.211 19.928 19.086
50 20.509
19.265 18.463
51 19.828 18.621 17.869 17.180
52 19.123
17.979
17.280 16.635
66
76
78
81
82
83
84
85
86
89
13.593 12.542 12.098 11.722
13.040 12.025 11.587 11.219
12.526 11.516
11.092 | 10.739
64 11.961 11.005
10.597 10.261
96
97
98
99
100
of France.
State-annuitants
101
102
103
104
105
106
of Sweden.
Female population
107
108
109
17.389
16.776
16 159
15.536
of Sweden.
General population
87 2.722 3.229
88
of Sweden.
Male population
16.702
16.119
15.537
14.950 14.432
9.318 8.437 8.179
8.795 7.918 7.724
8.278 7.439 7.289
7.772 7.008 6.896
18.309
17.721
15.922 14.906 14.363 13.873 15.833 15.661
15.324 14.293 13.784 13.334 15.263 15.144
14.700 13.673 13.201 12.792 14.703 14.626
14.156 13.081 12.634 12.247 14.154 14.167
7.291 6.634 6.533 6.534
6.846 6.336 6.229 6.223
6.438
6.036 5.916 5.892
6.068 5.735 5.597 5.570
5.714 5.431 5.288 5.258
5.371 5.119 4.964 4.921
5.037 4.799 4.619
4.591
4.730 4.460 4.283 4.271
11.429 10.494 10.104 9.785
11.892 11.566
10.899
9.982 9.610 9.312 11.371 11.080
10.370
9.468 9.118 8.842 10.840 10.596
9.813 8.953 8.640 8.401 10.325 10.114
aud Holy-Cross.
Chester
::
::::
of Holland.
State-annuitants
2.054 2.829 2.435 2.346
1.773 2.552 2.053 2.026
1.536 2.268 1.712 1.731
1.350 1.952 1.405 1.383
20.446
19.990
19.879 | 19.413
19.309 | 18.871
18.729 18.367
16.089
16.089
18.144 17.823 15.5814 15.974
15.540 17.581 17.277 15.0421 15.435
14.987 17.014 16.728 14.5090 14.922
16.426 16.177 13.9827 14.407
13.4635
13.892 12.680
12.9522 13.376 12.178
12.4491 12.872 11.727
11.9548 12.368 11.280
7.994 9.781 9.634
7.602 9.276 9.156
7.229 8.834 8.681
6.880 8.473 8.208
13.657 13.586
13.218 13.064
12.784 12.540
12.357 12.052
3.328 3.402 3.235 3.165 4.141
3.043 3.346 3.090 3.015 3.973 3.103
2.922 2.885 3.771 2.846
2.379 3.056 2.719 2.645 3.685 2.629
··
France.
Population of
17.7886
17.2305
16.6762
16.1263
3.315 2.480
3.008 2.470
2.650 2.314
2.196 2.040
1.698
1.000
0.500
9.6294
9.1962
8.7746
8.3648
··
7.9673
7.5824
7.2106
6.8524
1.200 1.643 1.227 1.000
1.134 1.346 1.101 0.749
1.775*2.067 3.2831
1.369 1.849 3.0975
0.858 1.071 0.998 0.500 1.127 2.200 2.9164
0.526 0.833 0.500
0.754 2.334 2.7401
0.500
0.500
and Warrington.
Norwich
11.4697 11.878 10.840
10.9942 11.389 10.406
10.5287 10.932 9.979
10.0737 10.480 9.561
3.382 *4.1572 3.500
4.2123
4.2790
4.2756
17.131 15.966
16.541 15.377
2.5683
2.4024
2.2398
2.0837
Brandenburgh.
18.327
17.182 18.498 18.5
17.717 16.550 17.996 18.
17.506 | 17.5
17.5
17.024 17.
1.9331
1.7883
1.6505
1.5174
4.428 4.139 4.011 3.968 5.202 4.738 4.4228
4.875 4.908 4.416 2.5
4.140 3.845 3.779 3.690 4.864 4.416 4.2853 4.500 4.671 4.095
3.865 3.595 3.574 3.456 4.565 4.089 4.1914 4.167 4.553 3.805 1.5
3.599 3.423 3.393 3.296 4.328 3.676 4.1476 3.900 4.313 3.581 1.
1.3902
1.2692
1.1667
1.1000
1.0000
8.423
8 037
7.669
7.286
10.046
9.638
9.155
8.761
9.226 8.384
8.825 8.026
4.0688
1.929
3.8661 1.500
3.6675 1.167
3.4053 *1.500
0 500
14.783
14.236
13.685
13.183
8.141 7.740 6.5084 6.926
6.524
7.329
7.841 7.276 6.1791
6.595 6.319 6.925
7.506 6.818 5.8663 6.280 6.172 6.543 5.5
7.117 6.411 5.5703 6.013 5.982 6.186 5.
6.710 6.022 5.2931 5.760 5.765 5.835 4.5
6.302 5.659 5.0366 5.529 5.500 5.482
5.914 5.332 4.8035 5.322 5.311 5.116
5.541 5.057 4.5974 5.114 5.063 4.756
Northampton.
16.542 | 16.5
16.5
16.062 | 16.
15.584 15.5
15.106 15.
hypothesis.
Demoivre's
14.631 14.5
14.157 14.
13.685 | 13.5
13.216 13.
12.749 12.5
12.285 12.
11.817 11.5
11.353 11.
4.193 3.376 0.5
3.143 3.864 3.190
2.864 3.612 3.013
2.389 3.500 2.861
3.001 2.661
2.501 2.413
2.001 2.088
1.501 1.750
1.501
1.375
0.999 1.056
0.999
0.500
•
3.5
0.750
0.500
Breslaw.
10.885 10.5
9.859
10.420 10. 9.374
9.959 9.5 8.890 9.178
9.503 9. 8.407 8.858
7.691 9.052 8.5 7.923 8.498
7.391 8.607 8.
7.063 8.170 7.5
6.769 7.743 7.
7.451 8.161
7.034 7.808
6.633
7.476
6.5
17.528
17.069 15.752
16.613 15.376
16.161 15.010
Bills of Mortality.
London
15.715 14.631
15.265 14.239
14.771 13.843
14.277 13.443
!
13.783 13.053
13.290 12.628
12.798 12.204
12.306 11.815
16.134
11.815 11.438
11.324 11.032
10.835 10.637
10.347 10.255
6.252
5.898
5.511
5.154
4.838
4.586
4.337
4.085
0.833
0.500
9.888
9.523
7.170
6.796
6.425
6.083
5.755
5.471
5.077
*5.242
4.936
3.824
3.536 4.675
3.196 4.419
2.763 4.231
2.367
2.045
3.796
3.677
1.625 3.523
1.300
3.520
3.334
3.226
2.827
2.526
2.200
1.818
1.434
1.056
0.752
0.500
* The decreasing progression of the Expectations appears sometimes interrupted, with regard to the later
stages of life. In the London lives, it arises from a limitation to the 79th year, of the observations on the Bills of
Mortality from 1728 to 1737; whence the Decrements in the preceding Table, for the ages of 80 and upwards, are
stated on an average of two subsequent periods only, and also of 10 years each. The probable cause of this irregu-
larity in two other instances, those of the Norwich lives and of the Dutch Annuitants, is the very small number
of lives exceeding 92 and 93 years, to which the observations were necessarily confined; but a similar irregularity
as to the Population of France, for the ages of 85, 86, 87 and 88, must be otherwise accounted for, the correctness
of Mr. Duvillard's calculations being fully ascertained. It is not unlikely that the comparatively few whose lives
extend to 85 years, or thereabouts, may be indebted for this distinction to specific causes subsequently continuing
in operation, so as to afford those persons a better security of life, at least during a small number of years, than
is possessed by the generality of individuals existing at even less advanced ages.
136

TABLE (No. III.)-Of the VALUES of ANNUITIES on SINGLE LIVES, at each age,
according to various systems of the Decrements and Probabilities of Life as stated in
TABLE NO. I.; those values expressed by quantities of Years' Purchase, and ascertained
upon the principles referred to in Section XXVII.; admitting as a Standard Rate of
Perpetual Interest that of four per centum, to accrue by full yearly instalments,—and
the Annuity to be payable at the expiration of each successive year, until the last
accomplished previously to the demise of the annuitant.
years of Age.
Accomplished
0
1234
167∞
5
8
9
10
11
12
13
14
15
16
17
18
19
20
22** **** 2272 m 2020 10**
21
23
24
25
26
27
28
29
30
32
34
35
36
37
38
40
42
43
44
France.
of
State-Annuitants
...
...
···
18.24246
18.55893
18.74921
18.87658
18.95347
18.99571
19.02190
19.00758
18.94924
18.84375
18.73423
18.62049
18.50233
18.37954
18.27496
18.16664
18.05439
17.93804
17.84073
17.74037
17 63682
17.52994
17.41957
17.30556
17.18773
17:06591
16.93990
16.80952
16.67454
16.53474
16.38989
16.23973
16.08401
15.92245
15.75474
15.55586
15.34864
15.13266
14.90746
14.67254
14.42739
14.17147
Sweden.
population of
Female
16.820
17.719
18.344
18.780
18.927
19.045
19.131
19.162
19.151
19.109
19.041
18.952
18.840
18 707
18.568
18.424
18.290
18.151
18.013
17.872
17.725
17.573
17.414
17.252
17.087
16.915
16.751
16.588
16.427
16.261
16.104
15.941
15.787
15.629
15.465
15.278
15.070
14 854
14.629
14.401
14.185
13.994
13.798
13.596
Sweden.
population of
Male
16.503
17.355
17.935
18.328
18.503
18.622
18.693
18.725
18.715
18.674
18.600
18.491
18.378
18 246
18.105
17.958
17.803
17.643
17.492
17.335
17.192
17.042
16.887
16.742
16.592
16.436
16.274
16.105
15.930
15.751
15.575
15.395
15.208
15.014
14 812
14.601
14.382
14.154
13.916
13.668
13 426
13.196
12.984
12.763
France.
population of
General
11.61829
14.74285
16.51642
17.47403
17.96071
18.17773
18.23890
18.20941
18.12684
18.01340
17.88230
17.74155
17.59600
17.44876
17.30112
17.15438
17.00904
16.86524
16.72295
16.58201
16.44214
...
::
15.74696
...
•1•
15.02106
..
...
...
14.21380
:
...
13.28559
...
:::
Register.
the Northampton
According to
10.32794
13.46627
15.63361
16 46262
17.01059
17.24906
17 48276
17.61222
17.66320
17.62599
17.52522
17.39440
17.25244
17.10497
16.95169
16.79229
16.62651
16.46382
16.31112
16.16912
16.03543
15.91414
15.79966
15.68269
15.56300
15.44054
15.31518
15.18697
15 05549
14.92119
14.78350
14.64234
14.49771
14.34946
14.19742
14.04148
13.88147
13.71724
13.54860
13.37541
13.19743
13,01842
12.83904
12.65804
12.47260
hypothesis.
Mr. Demoivre's
According to
...
...
...
16.791
16.882
16.882
16.791
16.698
16.604
16.508
16.410
16.341
16.209
、 16.105
15.999
15.891
15.781
15 669
15 554
15 437
15.318
15197
150 73
14.946
14.816
14.684
14.549
14.411
14.270
14.126
13.979
13.829
13.676
13.519
13.359
13.196
13 028
12.858
12.683
12.504
1728 to 1737.
LondonRegister,
According to the
::::
16.2
16.3
16.4
16.4
16.4
16.4
16.3
16.2
16.
15.8
15.6
15.4
15.2
15.
14.8
14.7
14.5
14.3
14.1
14.
13.8
13.6
13.4
13.2
13.1
12.9
12.7
12.6
12.4
12.3
12 1
11.9
11.8
11.6
11.5
11.4
11.2
11.1
11.
137

1
years of Age.
Accomplished
1
45
46
47
48
aeroe toCA
49
50
52
53
54
56
57
58
NENO 8023 2282 28
59
60
61
62
63
64
66
67
68
69
70
71
72
***K DO** **** *** FAIN KIN
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
France.
of
State-Annuitants
13.90419
13 62495
13.35670
13.07649
12.80700
12.52560
12.25477
11.99530
11.72507
11.44340
11.17264
10.89082
10.59721
10.31409
10.01961
9.71298
9.39332
9.05966
8.73389
8.39417
8.03935
7.69096
7.35019
7.01869
6.69280
6.39375
6 08366
5.79394
5.50583
5.22183
4.94546
4.65225
4.36972
4.10520
3.83448
3.59617
3.36953
3.16395
2.93934
2.67866
2.42422
2.18465
1.97716
1 71050
1.44602
1.18743
0 94061
0.71191
0.48077
...
13.383
13.151
12.894
12.620
12.333
12.049
11.769
11.492
11.220
10.937
10 642
10.334
10.012
9.692
9.358
9.039
B
8 739
8.453
8.166
7.870
7.566
7.252
6.930
6.596
6.253
5.897
5.564
5.261
4.998
4.792
4.582
4.367
4.145
3.913
3.668
3.102
3.145
2.905
2.699
2.559
2.552
2.518
2.431
2.294
2.108
1.873
1.628
1.349
1.071
0.799
0.544
0.320
12.535
12.297
12.051
11 795
11.528
11.267
11.030
10.785
10 531
10.269
9 998
9.717
9.425
9.140
8.845
8540
8.241
7.950
7.669
7.382
7.090
6.792
6.489
6.201
5.933
5.670
5.418
5.180
4.9.10
4 724
4.487
4.253
4.021
3.768
3.512
3.260
3.017
2792
2.600
2.473
2.371
2.281
2.154
1.955
1.698
1.117
1 154
0.835
0.477
0.240
12.21762
U.
...
...
11.02642
10.76184
10.50375
10.24205
9.97514
9.70936
9.43912
9.16682
8.89286
8.61766
8.34165
8.06525
7.78882
7.51307
7.23816
6.96465
6.69298
6.42364
6.15723
5.89566
5.63459
5.37974
5.13002
4.88619
4.64912
4.41988
4 19975
3.99034
3.79371
3.61249
3.45006
3 31080
3.20026
3.12498
3.09187
*3.10564
*3.16162
*3.23293
*3.24895
3.08870
2.92992
2.77258
2.61678
2.46296
2.31109
2.16129
2.01390
12.28355
12 08921
11.88995
11.68556
11.47585
11.26490
11.05734
10 84966
10.63788
10.42197
10.20191
9.97771
9.74937
9.51692
9.28043
9.03998
8.79572
8.54781
8.29132
8.03096
7.76161
7.48817
7.21094
6.93037
6.64708
6.36185
6.07580
5.79038
5.50764
5.23040
4.96265
4.71022
4.45743
4.19787
3.92173
3.64386
3.37766
3.12192
2.88716
2.70810
2.54364
2.39339
2 25156
2 13157
hypothesis.
Mr. Demoivre's
According to
1.96769
1.75819
1.47388
1.17152 98
99
100
...
12.322
12.135
11.944
11.748
0.82703
0.53008 101
0.24038
102
103
104
11.548
11.344
11.135
10.921
10.702
10.478
10.248
10.014
9.773
9.527
9.275
9.017
8.753
8.482
8.205
7.921
7.631
7.333
7.027
6.714
6 394
6.065
5 728
5.383
5.029
4.666
4.293
3.912
3.520
3.111
2.707
2.284
Age
1.850
1.406
0.950
0.481
3.0
•
London Register,
1728 to 1737.
According to the {
108
10.7
10.5
10.4
102
10 1
99
9.8
9.6
9.4
9.3
9.1
8.9
87
86
84
8.2
8.1
7.9
7.7
7.5
7.3
1.86887
1.72697
1.58733
1.45149
1.31928
1.19073
1.06720
0.91615
976 TH
7 1
69
...
6.7
6.5
...
6.3
6.1
5.9
56
54
...
...
...
pad →
:::
...
Continued, for the
populat. of France
* This interruption of the decreasing values is accounted for in a note to the preceding page.
(N. B.) — When the rate of Interest is other than 4 per Cent. the difference in value is ascer-
tainable by the rule `given in Section XXXIV.
138

TABLE (No. IV.), of the VALUES OF ANNUITIES during the Joint continuance of two lives,
according to various systems of the Decrements and Probabilities of Life as stated in Table
No. I. those values expressed by quantities of Years' purchase; admitting as a Standard Rate
of Perpetual Interest that of four per centum, to accrue by full yearly instalments,—and the
Annuity to be payable at the expiration of each successive year, until the last accomplished
previously to the failure of either of the lives.
N. B.-The Values expressed in the column of French-Annuitants are deduced (by Dr. Price's method described
in Section XXXIV.) from Mr. Maseres's values at 4 per cent. The values expressed in the next three columns are
correctly ascertained according to first principles, as set forth in Section XXVII. The values by the Hypothesis have
been computed by Mr. Simpson's rule, and the London values are taken from Mr. Simpson himself. It will be remarked,
relatively to the general population of France, that the values of lives increase instead of decreasing from the 85th to
88th year, which is accounted for by the notes to the preceding TABLes.
FIRST PART:-Referable to Lives of similar ages.
Common ages.
1
* C10 land
12.252
13.583 12.27756
3 14.560 14.558 13.76966
15.070
2
4
157∞
8
15.396
6 15.630 15.820
DOES 290 7298 22** FONO DO 2700 5828 #*** 2079 2AAS
12
15
9 15.983 16.152 14.95325 14.396
10 16.010 16.141 11.79867 14.277
11 15.965 16.087 1.4.62981 14.133
15.846 15.982 14.45473 13.966
13 15.721 15.855
14 15.590 15.701
15.453 15.535
16 15.310 15.361
21
23
14.747
14.525
14.653 14 360
14.557
14.194
24 14.459 11.020
25 14.357
28
of France.
Annuitants
17
15.198 15.196 13.60044 13.019
18
19
20 14.838
15.082
14.962
15.023 13.44131 12.841
14.854 13.28623 12.679
11 682
13.13501
29
30
of Sweden.
Population
26 14.254 13.671
27 14.147 13.495
14.037 13.323
11.107
12.325
15.267 14.58898 13.185
15.577 14.99316 13.591
15.14883 14.005
15.790
16.003 15.15756 14.224
15.899 16.109 15.08019 14.399
37
39
31 13.685 12.795
32 13.559 12.624
40
33 13.429 12.456
34 13.295 12.286
42
12.860 11.683
38 12.665 11.452
35 13.149 12.109 10.95761
36 13.011 11.904
46
13.849 12.41793
13.923 13.148
13.806
48
41 12.019 10.732
49
50
of France.
Population
43 11.534 10.264
52
44 11.272 10.154
9.781928.252
11.445
11.313
12.965 11.71479
11.783 10.531
12.460 11.209
12.245 10.964 10.09527
45 10.996 9.954 9.10691
47 10.435 9.497
10.150 9.236
register.
Northampton
14.27858 13.789
14.10441 13.604
13 93166 13.411
13.76390 13.212
10.705 9.736
51 9.345 8.469 7.77629
12.409
12.293
12.179
12.062
11.944
11.822
11.699
11.573
12.535 12.341
11.179
11.042
10.307
10.149
9.986
9.820
9.654
9.491
9 326
9.160
hypothesis.
Mr. Demoivre's
9 885
8.966
8.266
9.605 8.707 8.00648 8.081
10.902
10.759
10.612 10.530
10.462
7.900
9.107 8.230 7.54368 7.723
..
13.342 13.0
12.860 12.3
11.182
8.990 9.063
8.815
8.637
8.453
1728 to 1737.
London register
::
11.786 10.5
8.235
:
11.3
:::
..
9.6
9.826 8.1
··
8.8
*::: :8
7.4
6.8
Common ages.
5828 5883 6868 80** **** **** **** **** *858 3433 522 2020
99
100
53 8.857
54
55
56
101
111
103
104
of France.
Annuitants
7.994
8.591 7.748
8.353 7.495
8.100 7.229
7.834
7.589
7.331
7.062
of Sweden.
Population
5.567
5.256
6.777 5.844
5.600
6.475
6.192
5.889
6.954
6.678
6.388
6.101
1.284
1.135
1.030
0.855
6.36023 6.805
6 12512 6.614
5.88600
6.421
5.65051 6.226
5.41669 6.030
5.18494 5.831
5.367 4.95563 5.626
5.128 4.72917 5.417
4.957 4.362 4.07024
4.671 4.130
3.85848
of France.
Population
7.30907
7.07298
6.83583
6 59808
0.703
0 548
0.407
0.298 0.326
3.448 2.935 2.87182 3.421
3.232
2.797
2.69107 3.211
3.029 2.648
2.51657 3.015
2.803 2.490 2.34904 2.833
0.949 1.61796
0.718
0.516
0.240 0.236
register.
Northampton
4.881 4.50590 5.201 5.277 5.0
4.626 4.28615
4.402
4.312
3.851
3.593
4.153
3 65115
3.44853
4.087 4.104
3.898 3.345 3.25091 3.862
3.671 3.128 3.05856 3.639
2.588 2.340
2.395 2.170
2.192 1.967
2.028 1.758
1.880 1.600
1.65279 1.869
1.761 1.472 1.55780
1.681
1.364 1.48862 1.510
1.452 1.276 1.45409
1.387
1.626
1.212 *1.46535
1.339
1.172*1.53006 1.195
1.127 *1.63718 1.124
1.071*1.71914 1.030
··
7.544
7.362
7.179
6.993
•
4.982
4.760
4.537
2.18912
2.656
2.03780 2.470
1 89647 2.271
1.76714 2.068
1.23084 0.365
0.190 1.13891 0.201
0.060
1.04907
0.96159
0.87629
0.79397
0.71358
0.63658
1.015
1.51856 0.922
1.42082 0.756
1.32477 0.583
0.56278
0.49213
0.42622
0.36289
hypothesis.
Mr. Demoivre's
7.332
::::
6.351 5.6
•
1728 to 1737.
London register
·
6.2
•
4.4
3.7
:
139

SECOND PART:---Referable to Lives of different ages.
VALUES when the youngest life is aged from 1 to 83 accomplished years, and the oldest exceeds
the preceding by 6, 12, 18, 24, 30, 36, and 42 years respectively; those values deduced from the
probabilities of life for the general population of Sweden. Rate of Interest 4 per cent. per annum.
life.
youngest
Age of the
1
2
3
4
5
6
7000=2
8
9
10
11
12
13
22000 222*** ****** 70**29 MODE=2* **29** 2378** 495988 588888 588RER
34
40
41
42
43
44
45
46
47
49
50
51
53
60
- 64
Excess of the
oldest life,
6 years.
13.989
14.780
15.323
15.65
15.817
15.887
15.914
15.888
15.824
15.729
15.617
15.477
15.327
15.164
15.001
14.832
14.665
14.491
14.320
14.144
13.976
13.807
13.635
13.455
13.284
13.108
12.935
12.763
12.586
12.390
12.192
11.988
11.779
11.568
11.361
11.156
10.953
10.741
10.519
10.286
10.049
9.813
9.581
9 351
9.129
8.897
8.658
8.402
8.139
7.874
7.613
7.351
7.083
6.814
6.555
6.299
6.045
5.788
5.519
5.249
4.984
4.729
4.482
4.231
3.982
3.750
3.527
3.340
3.147
2.916
2.752
2.558
Excess of the
oldest life,
12 years.
13.894
14.557
14.988
15.259
15.326
15.354
15.351
15.310
15.244
15.149
15.033
14.889
14.736
14.566
14.392
14.216
14.042
13.860
13.687
13.512
13.345
13.173
12.997
12.801
12.599
12.387
12.170
11.953
11.742
11.543
11.359
11.170
10.978
10.775
10.557
10.314
10.059
9.805
9.558
9.308
9.066
8.830
8.597
8.354
8.101
7.841
7.563
7.281
7.008
6.749
6.505
6.256
6.001
5.743
5.474
5.204
4.936
4.664
4.395
4.149
3.927
3.747
3.563
3.370
3.180
2.974
2.743
2.514
2.324
2.155
2.004
1.875
Excess of the
oldest life,
18 years.
13.389
14.008
14.417
14.671
14.725
14.740
14.727
14.673
14.590
14.484
14.357
14.202
14.045
13.874
13.700
13 520
13.340
13.141
12.934
12.720
12.505
12.286
12.073
11.873
11.683
11.485
11.284
11.072
10.847
10.606
10.365
10.128
9.905
9.679
9.452
9.207
8.951
8.683
8.404
8.124
7.839
7.559
7.318
7.075
6.836
6.586
6.323
6.048
5.764
5.487
5.221
4.953
4.694
4.455
4.231
4.043
3.844
3.637
3.430
3.210
2.974
2.744
2.557
2.396
2.252
2.123
2.010
1.910
1.798
1.661
1.464
1.189
Excess of the
oldest life,
24 years.
12.832
13.409
13.778
14.003
14.037
14.033
14.006
13.944
13.855
13.741
13.604
13.428
13.234
13.023
12.798
12.570
12.351
12.146
11.951
11.751
11.550
11.335
11.107
10.862
10.612
10.364
10.130
9.894
9.659
9.413
9.167
8.912
8.651
8.389
8.114
7.833
¿
7.561
7.296
7.033
6.763
6.492
6.225
5.957
5.689
5.426
5.153
4.884
4.633
4.398
4.205
4.008
3.803
3.605
3.389
3.150
2.909
2.710
2.539
2.385
2.248
2.135
2.037
1.926
1.790
1.585
1.290
1.017
0.764
0.617
0.514
0.441
Excess of the
oldest life,
30 years.
12.196
12.730
13.066
13.264
13.277
13.242
13.170
13.059
12.913
12.743
12 563
12.379
12.196
11.997
11.787
11.562
11.328
11.076
10.819
10.567
10.332
10.092
9.852
9.602
9.347
9.080
8.807
8.534
8.250
7.967
7.702
7.446
7.196
6.942
6.679
6.402
6.115
5.828
5.543
5.254
4.977
4.730
4.507
4.322
4.128
3.921
3.715
3.489
3.238
2.990
2.792
2.623
2.475
2.344
2.232
2.130
2.010
1.864
1.644
1.333
1.050
0.789
0.639
0.533
0.456
Excess of the
oldest life,
36 years.
11.465
11.913
12.164
12.284
12.242
12.185
12.112
12.004
11.865
11.694
11.493
11.259
11.011
10.759
10.514
10.264
10.018
9.761
9.500
9.228
8.953
8.675
8.385
8.097
7.823
7.557
7.297
7.032
6.761
6.481
6.197
5.917
5.642
5.364
5.093
4.840
4.603
4.405
4.195
3.975
3.762
3.539
3.295
3.052
2.854
2.684
2.533
2.396
2.277
2.171
2.050
1.901
1.681
1.366
1.078
0.810
0.655
0.546
0.464
life.
est
Young.
73 2.355
74 2.172
Excess of the
oldest life,
42 years.
B
10.546
10.946
11.168
11.260
11.183
11.064
10.915
10.743
10.560
10.357
10.140
9.898
9.644
9.371
9.087
8.799
8.503
8.208
7.928
7.658
7.396
7.127
6.851
6.566
6.275
5.986
5.702
5.415
5.136
4.881
4.646
4.453
4.251
4.040
3.833
3.605
3.352
3.098
2.889
2.710
2.553
2.418
2.305
2.203
2.083
1.933
.1.708
1.385
1.090
0.818
0.662
Excess of the oldest,
6
12 18 yrs.
0.551
0.468
1.768 | 0.937
1.692 0.708
75 2.017 1.605 0.575
761.877 1.497 0.481
77 ||1.756 1.339 0.421
781.639 1.097
79 1.524
0.863
80 1.416 0.638
81 1.320 0.511
82 1.225 0.427
83 1.094 0.379
140

By
By
Ages. Northampton Mr. Demoivre's
register. hypothesis.
5-10
15
20
25
30
35
40
45
50
158622
55
60
70
75
10-15
20
25
30
35
40
45
50
55
60
65
15 PLJAJ
70
75
80
15-20
25
30
35
40
45
50
AIZIABA
55
60
65
70
75
80
85
20-25
30
35
40
45
50
55
38322838
60
65
70
75
80
85
90
25-30
35
40
45
50
55
60
65
GUASNJAgugtêŭ GEASNJA8:
70
75
80
85
90
95
30-35
40
45
50
55
60
65
70
75
80
85
90
95
SECOND PART continued:---Lives of different ages,
According to various systems of the Decrements and Probabilities of life.
Rate of interest, four per cent. per annum.
13.933
13.479
12.993
12.633
12.220
11.732
11.150
10.500
9.742
8.931
8.011
6.963
5.768
4.557
13.841
13.355
12.998
12.586
12.098
11.513
10.851
10.085
9.256
8.314
7.236
6.008
4.725
3.517
12.961
12.630
12.246
11.787
11.234
10.607
9.872
9.077
8.170
7.127
5.933
4.695
3.492
2.462
12.229
11.873
11.445
10.924
10.330
9.630
8.869
7.995
6.986
5.826
4.619
3.443
2.431
1.704
11.618
11.217
10.725
10.160
9.488
8.754
7.906
6.920
5.780
4.589
3.425
2.421
1.699
0.236
10.948
10.490
9.959
9.321
8.619
7.802
6.844
5.729
4.557
3.406
2.411
1.694
1.236
13.093
12.808
12.480
12.102
11.665
11.156
10.564
9.871
9.059
8.105
6.980
5.652
12 593
12.281
11.921
11.501
11.013
10.440
9.767
8.975
8.041
6.934
5.623
12.051
11.711
11.314
10.847
10.297
9.648
8.879
7.967
6.882
5.590
11.468
11.093
10.655
10.131
9.509
8.766
7.880
6.826
5.551
•
10.838
10.428
9.936
9.345
8.634
7.779
6.748
5.505
By London
register,
1728 to 1737.
•
·
12.7
12.2
11.6
10.9
10.2
9.6
9.0
8.4
7.8
7.2
6.5
5.8
5.1
11.8
11.8
10.6
10.0
9.4
8.8
8.2
7.6
7.0
6.4
5.7
5.0
10.8
10.3
9.8
9.2
8.6
8.0
7.5
6.9
6.3
5.7
5.0
10.1
9.6
9.1
8.5
7.9
7.4
6.8
6.2
5.6
4.9
1820
•
•
9.2
8.8
8.3
7.8
7.3
6.7
6.1
5.5
4.9
•
By
By
Ages. Northampton Mr. Demoivre's
register. hypothesis.
35-40
45
50
55
383228333 78388223383 3883228383 883228383 8urroB83 3P2O2A3 23536
40-45
80
45—50
55
65
90
95
50-55
60
75
80
85
90
55-60
60-65
70
85
65-70
85
70-75
18383 2685 383
80
85
90
95
75-80
90
95
80-85
90
95
85-90
95
90-95
10.196
9.706
9.110
8.448
7.669
6.747
5.663
4.516
3.383
2.398
1.683
0.235
9.381
8.834
8.221
7.490
6.614
5.571
4.457
3.349
2.379
1.679
0.235
8.503
7.948
7.274
6.453
5.460
4.386
3.308
2.356
1.668
0.234
7.593
6.989
6.236
5.306
4.285
3.247
2.322
1.651
0.233
6.659
5.986
5.132
4.171
3.180
2.284
1.633
0.232
5.658
4 900
4.021
3.092
2.234
1.608
0.230
4.573
3.806
2.965
2.163
1.575
0.228
3.471
2.757
2.042
1.515
0.224
2.148
1.856
1.413
0.217
1.608
1.278
0.208
1.054
0.187
0.177
10.157
9.702
9.149
8.476
7.658
6.662
5.450
··
·
9.418
8.911
8.283
7.510
6.556
5.383
8 619
8 044
7.332
6.425
5.300
7.738
7.091
6.258
5.193
6.781
6.036
6.053
5.730
4.858
4.571
··
•
..
··
**
By London
register,
1728 to 1737.
8.5
8.1
7.6
7.1
6.5
6.0
5.4
4.8
7.8
7.4
6.9
6.4
5.9
5.4
4.8
7.1
6.7
6.3
5.8
5.3
4.7
•
:
6.5
6.1
5.7
5.2
4.6
··
..
5.5
5.9
6.1
4.5
5.3
4.9
4.4
4.6
4.2
4.0
··
··
··
··
··
..
..
་
•
Tables,
SOLVING THE QUESTIONS THAT DEPEND ON
Compound Interest,
ACCRUING EITHER ANNUALLY, HALF-YEARLY, or QUARTERLY.
>
RATE OF INTEREST: 3 per
-CORRESPONDING VALUE OF
Cent. per Ann.-

DESK COMME
Years.
མ་བསར2e2ele
={*={*}<
2
-«HEN-
♡
={{}<
**
ནི ནྟི བྷིསཾ པཉྩནྟི
10
-
PICKYLAN
7
∞
HHN:0
T
ܢܚܙܣܢܚܐ
10
P[05]=${
11
BAKAL BAGA
12
13
EL PICADILL
14
OM SA GARA
15
16
17
18
19
20
RATIO of INTEREST:
ANNUALLY, 0.03.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0300 000
1.0609 000
1.0927 270
....
1.1255 088
1.1592 741
1.2667 701
1.0000 000
2.0300 000
1.1940 523 6.4684 099
....
3.0909 000
4.1836 270
1.2298 739 7.6624 622
5.3091 358
....
1.3047 732 10.159 106
1.3439 164 11.463 879
8.8923 360
21
1.8602 946
22 1.9161 031
23
24
25
1.9735 865
2.0327 941
2.0937 779
1.3842 339 12.807 796
....
1.4257 609 14.192 030
....
1.4685 337 15.617 790
....
1.5125 897 | 17.086 324
1.5579 674 18.598 914
1.6047 064 20.156 881
1.6528 476
1.7024 331
1.7535 061
1.8061 112
21.761 588
23.414 435
25.116 868
26.870 374
28.676 486
30.536 780
32.452 884
34.426 470
36.459 264
0.9708 74
..
1.9134 70
....
2.828 612
....
3.717 098
4.579 707
···
5.417 191
6.230 283
7.019 692
cc..
7.786 109
……….
8.530 203
9.252 624
9.954 005
10.634 96
11.296 07
11.937 93
12.561 10
13.166 12
13.753 52
14.323 80
14.877 47
RATIO of INTEREST:
HALF-YEARLY, 0.015.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0150 000
1.0302 250
...
1.0456 781
……..
1.0613 636
...
1.0772 840 2.5761 335
1.0934 433
3.1147 755
1.1779 489
1.1956 182
0.5000 000
1.0075 000
1.1098 449
3.6614 971
1.1261 926 4.2164 196
...
1.5226 125
2.0454 517
1.1433 900 4.7796 658
1.1605 409 5.3513 627
....
1.2135 524 7.1184 148
1.2317 557 7.7251 910
5.9316 312
6.5206 057
1.2502 321
8.3410 689
1.2689 855 8.9661 849
1.4083 772
1.4295 028
1.2880 203 9.6006 777
1.3073 406 10.244 688
....
1.3269 507 10.898 358
1.3168 550
11.561 832
...
1.3670 578 12.235 261
1.3875 637 | 12.918 790
13.612 572
14.316 760
1.4509 454 15.031 512
1.4727 095 15.756 984
1.4948 002 16.493 339
1.5172 222 17.240 739
1.5399 805 17.999 350
1.5630 802 18.769 341
1.5865 264 | 19.550 881
1.6103 243 20.344 144
1.6589 964 | 21.966 546
1.7091 395 | 23.637 985
1.7607 983
1.8140 184
25.359 943
27.133 947
15.415 02 1.8688 471 | 28.961 571
15.936 92 1.9253 330
16.443 61 1.9835 262
30.844 434
32.784 207
34.782 610
16.935 51
2.0434 783
17.413 15
2.1052 424
36.841 414
0.1926 11
0.9779 42
1.4561 00
1.9271 92
..
2.3913 23
2.8485 94
3.2991 07
3.7429 63
4.1802 59
4.6110 93
5.0355 59
5.4537 53
5.8057 66
6.2716 91
6.6716 17
…………
7.0656 32
7.4538 25
7.8362 80
....
8.2130 84
8.5843 20
8.9500 69
...
9.3104 12
9.6654 31
10.015 20
10.359 81
10.699 32
11.033 81
11.363 36
11.688 04
..
12.007 92
12.323 07
12.633 57
13.240 86
13.830 34
14.402 53
14.957 92
RATIO of INTEREST:
QUARTERLY, 0.0075.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
1.0075 000
1.0150 563
1.0226 692
1.0303 392
1.0380 667
1.0458 522
1.0536 961
1.0615 988
1.0695 608
1.0775 825
1.0856 644
1.0938 069
1.1354 446
1.1439 604
1.1525 401
1.1611 841
1.2053 866
1.2144 270
1.2235 352
1.2327 117
1.1020 104
1.1102 755
3.4003 483
3.6758 509
1.1186 026 | 3.9534 198
1.1269 922
4.2330 721
1.2419 571
1.2512 718
1.2606 563
1.2701 112
1.3584 612
1.3686 497
1.3789 146
1.3892 564
1.1698 930
5.6631 008
1.1786 672 5.9555 740
1.1875 072
6.2502 408
1.1964 135 | 6.5471 176
1.3996 758
1.4101 734
1.4207 497
1.4314 053
1.4421 409
1.4529 570
1.4638 541
1.4748 330
1.4858 943
1.4970 385
1.5082 663
1.5195 783
0.2500 000
0.5018 750
0.7556 391
1.0113 064
1.5309 751
1.5424 574
1.5540 258
1.5656 810
1.2688 912
1.5284 078
1.7898 709
2.0532 949
1.5774 236
1.5892 543
1.6011 737
1.6131 825
2.3186 946
2.5860 848
1.2796 371 9.3212 353
1.2892 343 9.6411 446
1.2989 036 | 9.9634 532
1.3086 454 10.288 179
1.3184 602 | 10.615 340
1.3283 486 | 10.944 955
1.3383 113
1.3483 486
11.277 043
11.611 620
1.6621 252
1.7125 527
1.7645 102
1.8180 440
2.8554 805
3.1268 966
4.5148 185
4.7986 796
5.0846 697
5.3728 047
6.8462 210
7.1475 677
7.4511 744
7.7570 583
8.0652 362
8.3757 255
8.6885 434
9.0037 075
11.948 708
12.288 323
12.530 485
12.975 214
13.322 528
13.672 447
14.024 990
14.380 178
14.738 029
15.098 564
15.461 804
15.827 767
16.196 475
16.567 949
16.942 209
17.319 275
17.699 170
18.081 947
18.467 528
18.856 034
19.247 455
19.641 811
20.039 125
20.439 418
22.070 840
23.751 757
25.483 673
27.268 133
15.497 03
16.020 31
29.106 732
31.001 113
1.8732 020
1.9300 334
16.528 24 1.9885 891
17.021 28
32.952 969
34.964 041
2.0489 212
17.499 85
2.1110 839 37.036 128
1
0.2481 39
0.4944 31
0.7388 89
0.9815 28
1.2223 60
1.4613 99
1.6986 59
1.9341 53
2.1678 94
2.3998 95
2.6301 69
2.8587 28
3.0855 86
3.3107 56
3.5342 49
3.7560 78
3.9762 56
4.1947 95
4.4117 07
4.6270 05
4.8407 00
5.0528 04
5.2633 29
5.4722 86
5.6796 89
5.8855 47
6.0898 73
6.2926 78
6.4939 73
6.6937 70
6.8920 79
7.0889 13
7.2842 80
7.4781 94
7.6706 64
7.8617 01
8.0513 16
8.2395 20
8.4263 22
8.6117 35
8.7957 66
8.9784 28
9.1597 30
9.3396 83
9.5182 95
9.6955 78
9.8715 42
10.046 20
10.219 55
10.391 61
10.562 39
10.731 90
10.900 15
11.067 15
11.232 90
11.397 42
11.560 72
11.722 80
11.883 67
12.043 34
12.201 83
12.359 14
12.515 27
12.670 24
13.278 69
13.869 21
14.442 35
14.998 61
15.538 49
16.062 48
16.571 03
17.064 61
17.543 66
Page 2.
PERPETUAL STOCK. 3 p. C. 100; 3; p. C. 116,6667; 4 p. C. 133,3333; 5p. C. 166,6667.

Years,
Q2
Co
→*
**
-]
OD
CHANG
+++-454:34+*++
Halo tud
APKAL
ANAL
===22}+
❤
-~-~~
AK:N
CACCAL SA:DAY
BARCHA
12
*====*****
* * **02 20** 3
14
21
22
RATIO of INTEREST:
ANNUALLY, 0.03.
23
21
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
25
0.9708 737 1.0000 000
0.9425 9600.4926 108
10 0.7440 939
0.9151 417
0.8884 871
0.8626 088
0.8374 843
0.7891 093
0.7224 213
0.7013 798
0.6809 513
0.6611 178
··
0.8130 915 0.1305 063
0.1305 063 0.1605 063
0.6418 620
0.7664 167 0.0984 339
0.5375 493
0.5218 925
0.5066 918
0.4919 338
0.3235 303 0.3535 303
……….
0.2390 270 | 0.2690 270
0.1883 546 0.2183 546
0.1545 975 0.1845 975
··
0.1124 564
1.0300 000
0.5226 108
0.0780 774
0.0704 621
0.0872 305 | 0.1172 305
0.0585 263
0.1424 564
0.0459 525
0.0427 087
0.0398 139
0.0372 157
0.1284 339
0.0640 295 0.0940 295
0.4776 056 0.0274 279
0.1080 774
10 0.6231 669 0.0496 108 0.0796 108
17 0.6050 164
18 0.5873 946
19 0.5702 860
0.5536 758
0.0759 525
0.0727 087
0.0698 139
0.0672 157
0.1004 621
..
0.0537 666 0.0837 666
0.0885 263
0.0348 718
0.0648 718
0.0327 474
0.0327 474 | 0.0627 474
0.0308 139
0.0608 139
0.0290 474 | 0.0590 474
0.0274 279 0.0574 279
RATIO of INTEREST:
HALF-YEARLY, 0.015.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9852 217 2.0000.000
0.9706 618 | 0.9925 558
0.9563 170 | 0.6567 659
0.6567 659
0.9421 841
0.4888 895
0.9282 603
0.9145 421
....
…………
0.8489 332
0.8363 874
0.9010 267 0.2731 123
0.8877 111 0.2371 681
··
0.7998 515
0.7880 310
....
0.8745 922 0.2092 196
0.8616 672 | 0.1868.683
0.3881 786 0.4181 786
0.3210 504
0.3510 504
....
0.6892 058
0.6790 205
0.1685 877
0.1533 600
0.2092 196 0.2392 196
0.2168 683
0.1198 886
0.1115 302
0.7536 074 0.0917 569
|
0.7424 701 | 0.0864 915
0.0864 915
0.7314 979 0.0817 310
0.7206 876 | 0.0774 066 |
0.8240 270
0.1401 807 0.1704 807
0.8118 493 0.1294 466
0.1294 466 0.1594 466
0.7100 371 0.0734 615
0.6995 439 0.0698 482
0.0665 269
0.0634 639
....
0.7763 853 0.1041 593
0.1341 593
0.7649 116 0.0976 116 0.1276 116
0.6689 857 0.0606 305
0.6590 993 0.0580 022
2.0300 000
1.0225 558
0.6493 589 0.0555 576
0.6397 624 0.0532 784
0.6867 659
0.5188 895
0.0511 486
0.6303 078
0.6209 929 0.0491 542
0.6027 741
0.5850 897
0.5579 242
0.5512 623
0.0455 238
0.0423 048
0.0394 323
0.0368 542
•••
...
0.5350 893 0.0345 285
0.5193 907 0.0324 208
0.5041 527 0.0305 025
0.4893 617 0.0287 500
0.4750 047 0.0271 434
0.3031 123
0.2671 681
0.1985 877
0.1833 600
....
0.1498 886
0.1415 302
0.1217 569
0.1164 915
0.1117 3 310
0.1074 066
....
0.1034 615
0.0998 482
0.0965 269
0.0931 639
0.0906 305
0.0880 022
0.0855 576
0.0332 784
0.0811 486
0.0791 542
0.0755 238
0.0723 048
0.0694 323
0.0668 542
0.0645 285
0.0624 208
0.0605 025
0.0587 500
0.0571 434
RATIO of INTEREST:
QUARTERLY, 0.0075.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9925 558
0.9851 671
0.9778 333
0.9705 542
0.9633 292 | 0.7880 896
0.9561 580
0.6542 756
0.5586 995
0.4870 221
0.9490 402
0.9119 754
0.9349 632
0.9280 031
0.9210 949
0.9142 381
0.9074 324
0.9006 773
0.8939 726
0.8873 176 |
0.8807 123 0.2214 928
0.8741 561
0.2083 907
0.8676 487 | 0.1966 696
0.8611 899
0.1861 225
0.8051 808
0.7991 869
0.7932 376
0.7873 326
0.8547 790
0.1765 817
0.8481 159 | 0.1679 099
0.8421 001
0.8358 313
0.1599 938
0.1527 390
4.0000 000
1.9925 745
1.3233 831
0.9888 200
0.8296 093 | 0.1460 660
0.8234 336 | 0.1399 077
0.8173 038
0.8112 197
0.1342 071
0.1289 148
0.7584 605
0.7528 144
0.7472 103
0.7416 480
0.4312 771
0.3866 819
0.3502 038
0.3198 059
0.7144 511
0.7091 326
0.7038 537
0.6986 141
0.2940 875
0.2720 459
0.2529 456
0.2362 352
0.7814 715 | 0.1072 819
0.1037 221
0.7756 542
0.7698 801
0.7641 490
0.1003 668
0.0971 989
0.6934 135
0.6882 516
0.6831 282
0.6780 429
0.6531 785
0.6483 161
0.6434 899
0.6386 997
0.7361 270 | 0.0836 911
0.7306 472
0.7252 081
0.7198 095
0.0813 781
0.0791 735
0.0770 700
0.6339 451
0.6292 259
0.6245 418
0.6198 927
0.1239 889
0.1193 926
0.1150 941
0.1110 653
0.0942 033
0.0913 663
0.0886 758
0.0861 206
0.0750 608
0.0731 398
0.0713 013
0.0695 402
0.6729 954 | 0.0617 418
0.6679 855
0.6630 129
0.0603 575
0.0590 242
0.6580 773 | 0.0577 391
0.0678 517
0.0662 315
0.0645 755
0.0631 801
0.0564 998
0.0553 039
0.0541 491
0.0530 334
0.0519 549
0.0509 118
0.0499 024
0.0189 251
0.6016 39+
0.5839 236
0.0453 087
0.0421 021
0.5667 295 0.0392 408
0.5500 417 0.0366 728
0.5338 452 0.0343 563
0.5181 257 0.0322 569
0.5028 691 0.0303 463
0.4880 617 0.0280 008
0.4736 903 0.0270 007
4.0300 000
2.0225 745
1.3533 831
1.0188 200
0.8180 896
0.6842 756
0.5886 995
0.5170 221
0.4612 771
0.4166 849
0.3802 038
0.3498 059
0.3240 875
0.3020 459
0.2829 456
0.2662 352
0.2514 928
0.2383 907
0.2266 696
0.2161 225
0.2065 817
0.1979 099
0.1899 938
0.1827 390
0.1760 660
0.1699 077
0.1642 071
0.1589 148
0.1539 889
0.1493 926
0.1450 941
0.1410 653
0.1372 819
0.1337 221
0.1303 668
0.1271 989
0.1242 033
0.1213 663
0.1186 758
0.1161 206
0.1136 911
0.1113 781
0.1091 735
0.1070 700
0.1050 608
0.1031 398
0.1013 013
0.0995 402
0.0978 517
0.0962 315
0.0946 755
0.0931 801
0.0917 418
0.0903 575
0.0890 242
0.0877 391
0.086+ 998
0.0853 039
0.0841 491
0.0830 334
0.0819 519
0.0809 118
0.0799 024
0.0789 251
0.0753 087
0.0721 021
0.0692 408
0.0666 728
0.0643 563
0.0622 569
0.0603 463
0.0586 008
0.0570 007
Page 3.
莘
​RATE OF INTEREST: 3 per
CORRESPONDING VALUE OF
Cent. per Ann.-

}
•
Years.
26
2582* H**** 85829 79333 94928 F8AAR 85888 78878 CE882 78M40 27208 ***** 26398 48335 35888
29
30
47
52
54
57
59
60
61
62
4.5154 232
4.6508 859
53 4.7904 124
4.9341 248
55 5.0821 486
63
3.3598 989 78.663 297
3.4606 959 | 82.023 196
85.483 892
89.048 408
92.719 861
3.5645 168
3.6714 523
45 3.7815 958
6+
65
66
61
56 5.2340 130
5.3916 514
5.5534 010
5.7200 030
5.8916 031
68
69
70
71
72
75
77
79
81
RATIO of INTEREST:
ANNUALLY, 0.03.
83
1100
Improved Cumulated Value of
Principal. Annuity.
Annuity.
2.1565 913
2.2212 890
2.2879 277
2.3565 655
2.4272 625
2.5000 803
2.5750 827
2.6523 352
2.7319 053
2.8138 624
2.8982 783
2.9852 267
3.0747 835
3.1670 270
3.2620 378
3.8950 437
4.0118 950
4.1322 519
4.2562 194
4.3839 060
6.0683 512
6.2504 017
6.4379 138
6.6310 512
6.8299 827
7.0348 822
7.2459 286
7.4633 065
7.6872 057
7.9178 219
9.4542 934
9.7379 222
10.030 060
10.330 962
10.640 891
(
10.960 117
11.288 921
11.627 588
11.976 416
12.335 708
12.705 780
13.086 953
13.479 562
13.883 949
14.300 467
38.553 042
40.709 633
42.930 922
45.218 850
47.575 415
14.729 481
15.171 365
15.626 506
16.095 302
16.578 161
50.002 678
52.502 758
55.077 841
57.730 176
60.462 081
63.275 944
66.174 222
69.159 449
72.234 232
75.401 259
96.501 457
100.39 650
104.40 840
108.54 065
112.79 687
8.1553 565
238.51 188
8.4000 172 | 246.65 724
8.6520 177
8.9115 783
9.1789 256
255.06 726
263.71 928
272.63 085
117.18 077
121.69 620
120.34 708
131.13 749
136.07 162
141.15 377
146.38 838
151.78 003
157.33 343
163.05 344
168.94 504
175.01 339
181.26 379
187.70 171
194.33 276
201.16 274
208.19 762
215.44 355
222.90 686
230.59 406
281.80 978
291.26 407
301.00 200
311.03 206
321.36 302
332.00 391
342.96 402
354.25 295
365.88 053
377.85 695
390.19 266
402.89 844
415.98 539
429.46 495
443.34 890
457.64 937
472.37 885
487.55 021
503.17 672
519.27 202
17.075 506
535.85 018
17.587 771 552.92 569
18.115 404 570.51 346
18.658 866 588.62 886
19.218 632 607.28 773
Value of Perpetual Ann.
17.876 84
18.327 03
18.764 11
19.188 45
19.600 44
20.000 43
20.388 76
20.765 79
21.131 84
21.487 22
21.832 25
22.167 24
22.492 46
22.808 22
23.114 77
23.412 40
23.701 36
23.981 90
24.254 27
24.518 71
24.775 45
25.024 71
25.266 71
25.501 66
25.729 76
25.951 22
26.166 24
26.374 99
26.577 66
26.774 43
26.965 46
27.150 94
27.331 01
27.505 83
27.675 56
27.840 36
28.000 34
28.155 67
28.306 48
28.452 89
28.595 04
28.733 05
28.867 04
28.997 12
29.123 42
29.246 04
29.365 09
29.480 67
29.592 88
29.701 83
RATIO of INTEREST:
HALF-YEARLY, 0.015.
Improved Cumulated
Principal. Annuity.
30.932 48
31.002 41
2.1688 734
2.2344 275
2.3019 631
2.3715 400
2.4432 198
2.5170 661 50.568 870
2.5931 444 53.104 814
2.6715 922 55.717 407
2.7522 690 58.408 966
2.8354 563 | 61.181 876
2.9211 580 64.038 599
3.0094 500 | 66.981 665
3.1004 106 70.013 686
3.1941 205 | 73.137 350
3.2906 628 76.355 426
3.3901 231
79.670 769
3.4925 895 | 83.086 318
3.5981 531
3.7069 072
3.8189 485
86.605 102
90.230 241
93.961 950
3.9343 762 | 97.812 541
4.0532 927
4.1758 034
4.3020 171
4.4320 456
4.5660 041
4.7040 116
4.8461 904
4.9926 665
5.1435 698
5.2990 342
5.4591 975
5.6242 018
5.7941 933
5.9693 228
6.1497 455
6.3356 216
6.5271 158
6.7243 978
6.9276 428
7.1370 308
7.3527 475
7.5749 843
7.8039 382
8.0398 122
29.807 60
29.910 29
30.009 99 10.202 406
30.106 79
30.200 77
38.962 446
41.147 586
43.398 771
45.717 999
48.107 326
10.510 773
10.828 461
30.292 01
11.155 752
30.380 58 11.492 934
30.466 59
11.840 308
30.550 09 12.198 181
12.566 871
30.631 15
101.77 642
105.86 011
110.06 724
114.40 152
8.2828 156
8.5331 637
242.76 052
251.10 546
8.7910 785 | 259.70 262
9.0567 889
9.3305 303
268.55 963
277.68 434
31.070 30 15.025 180
31.136 21 15.479 316
118.86 680
123.46 705
128.20 635
133.08 888
138.11 899
143.30 114
148.63 992
154.14 006
159.80 611
165.64 409
9.6125 456
287.08 485
9.9030 848 | 296.76 949
306.74 685
317.02 578
327.61 538
31.381 22 17.437 335
31.438 08 17.964 378
31.493 28 18.507 351
$1.546 87 19.066 736 |
31.598 91
19.643 028
33.333 33
171.65 818
177.85 405
184.23 719
190.81 326
197.58 809
204.56 769
211.75 825
219.16 614
226.79 794
234.66 041
30.709 85
398.22 351
12.946 705
30.786 27 13.338 019 | 411.26 731
30.860 46 13.741 161
14.156 488
14.584 367
424.70 537
438.54 959
452.81 225
338.52 505
349.76 447
361.34 361
373.27 272
385.56 238
467.50 600
482.64 387
31.200 21 15.947 178 | 498.23 928
31.262 33 16.429 182 514.30 606
31.322 65 16.925 751 530.85 846
547.91 116
565.47 927
583.57 838
600.22 245
621.43 428
Value of
Annuity.
17.964 37
18.415 27
18.852 94
19.277 77
19.690 13
20.090 40
20.478 93
20.856 05
21.222 12
21.577 44
21.922 33
22.257 11
22.582 07
22.897 49
23.203 66
23.500 85
23.789 32
24.069 32
24.341 11
24.604 93
24.861 00
25.109 57
25.350 84
25.585 03
25.812 35
26.033 00
26.247 18
26.455 08
26.656 87
26.852 75
27.042 88
27.227 43
27.406 57
27.580 44
27.749 23
27.913 05
28.072 07
28.226 43
28.376 26
28.521 69
28.662 86
28.799 88
28.932 88
29.061 99
29.187 30
29.308 94
29.427 01
29.511 61
29.652 85
29.760 83
29.865 64
29.967 38
30.066 13
30.161 98
30.255 02
30.345 34
30.433 00
30.518 09
30.600 69
30.680 85
30.758 68
30.834 21
30.907 53
30.978 70
31.047 78
31.114 84
31.179 92
31.243 10
31.304 42
31.363 95
RATIO of INTEREST:
QUARTERLY, 0.0075.
Improved Cumulated
Principal. Annuity.
2.1751 324
2.2411 242
2.3091 181
2.3791 749
2.4513 571
2.5257 293
2.6023 579
2.6813 113
2.7626 601
2.8464 770
2.9328 368
3.0218 167
3.1134 962
3.2079 571
3.3052 839
3.4055 636
3.5088 856
3.6153 424
3.7250 289
3.8380 433
3.9544 864
4.0744 623
4.1980 782
4.3254 445
4.4566 750
4.5918 869
4.7312 010
4.8747 419
5.0226 376
5.1750 203
5.3320 263
5.4937 956
5.6604 729
5.8322 071
6.0091 516
6.1914 643
6.3793 084
6.5728 514
6.7722 664
6.9777 315
7.1894 302
7.4075 517
7.6322 908
7.8638 484
8.1024 312
8.3482 524
8.6015 316
8.8624 951
9.1313 760
9.4084 146
9.6938 583
9.9879 621
10.290 989
10.603 209
10.924 902
11.256 354
11.597 863
11.949 733
12.312 278
12.685 823
13.070 700
13.467 255
13.875 840
14.296 822
14.730 576
15.177 490
15.637 963
16.112 406
16.601 243
17.104 912
31.421 73
17.623 861
31.477 81 18.158 554
31.532 25 18.709 470
31.585 09 19.277 100
31.636 38
19.861 952
33.333 33
39.171 081
41.370 806
43.637 26y
45.972 495
48.378 570
50.857 643
53.411 929
56.043 710
58.755 337
61.549 233
64.427 893
67.393 889
70.449 872
73.598 570
76.842 798
80.185 452
83.629 520
87.178 079
90.834 297
94.601 443
98.482 880
102.48 208
106.60 261
110.84 815
115.22 250
119.72 956
124.37 337
129.15 806
134.08 792
139.16 734
144.40 876
149.79 319
155.34 910
161.07 357
166.97 172
173.04 881
179.31 028
185.76 171
192.40 888
199.25 772
206.31 434
213.58 506
221.07 636
228.79 495
236.74 771
244.94 175
253.38 439
262.08 317
271.04 537
280.28 049
289.79 528
299.59 874
309.69 963
320.10 697
330.83 006
341.87 848
353.26 211
364.99 110
377.07 594
389.52 742
402.35 668
415.57 516
429.19 468
443.22 741
457.68 587
472 58 300
487.93 209
503.74 686
520.04 144
536.83 039
554.12 869
571.95 185
590.31 567
609.23 668
628.73 173
Value of
Annuity.
18.008 60
18.459 85
18.897 81
19.322 87
19.735 42
20.135 82
20.524 44
20.901 61
21.267 67
21.622 95
21.967 77
22.302 44
22.627 25
22.942 51
23.248 47
23.545 43
23.833 64
24.113 37
24.384 86
24.648 35
24.904 09
25.152 30
25.393 19
25.627 00
25.853 92
26.074 15
26.287 91
26.495 36
26.696 72
26.892 14
27.081 80
27.265 88
27.444 54
27.617 94
27.786 24
27.949 58
28.108 11
28.261 97
28.411 30
28.556 23
28.596 90
28.833 42
28.965 93
29.094 52
29.219 34
29.340 48
29.458 05
29.572 17
29.682 93
29.790 41
29.894 73
29.995 98
30.094 26
30.189 63
30.282 20
30.372 04
30.459 24
30.543 87
30.626 00
30.705 73
30.783 10
30.858 20
30.931 08
31.001 81
31.070 47
31.137 10
31.201 77
31.264 53
31.325 45
31.384 58
31.441 96
31.497 65
31.551 70
31.604 16
31.655 08
33.333 33
Page 4.
PERPETUAL STOCK: 3p. C. 100; 3½ p. C. 116,6667; 4p. C. 133,3333; 5 p. C. 166,6667.

Years.
26
27
28
29
30
31
32
33
3-4
*3 85823 79333 95333 88888 85988 58838 BECOR FERER CEP28 J**** 85888 18846 85228
| 51
94
RATIO of INTEREST:
ANNUALLY, 0.03.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.4636 947
0.4501 891
0.4370 768
0.4243 464
0.4119 866
0 3999 871
0.3883 370
0.3770 262
0.3660 449
0 3553 834
0.3450 324
0.3349 829
0.3252 262
0.3157 535
0.3065 568
0.2976 280
0.2889 592
0.2805 429
0.2723 718
0.2644 386
0.2567 365
0.2492 588
0.2419 988
0.2319 503
0.2281 071
0.2214 632
0.2150 128
0.2087 503
0.2026 702
0.1967 672
0.1647 894
0.1599 897
0.1553 298
0.1508 057
0.1464 132
0.1421 488
0.1380 085
0.1339 889
0.1300 863
0.1262 973
0.1226 188
0.1190 474
0.1155 800
0.1122 135
0.1089 452
0.1057 720
0.1026 913
0.0997 003
0.0967 964
0.0939 771
0.1910 361 | 0.0070 845
0.1854 719
0.0068 311
0.0065 885
0.0063 559
0.0061 330
0.1800 698
0.1748 251
0.1697 331
0.0912 399
0.0885 824
0.0860 024
0.0834 974
0.0810 655
0.0787 043
0.0764 120
0.0741 864
0.0720 256
0.0699 278
0.0259 333
0.0245 642
0.0232 932
0.0221 147
0.0210 193
0.0678 911
0.0659 136
0.0639 938
0.0621 299
0.0603 203
0.0199 989
0.0199 466
0.0181 561
0.0173 219
0.0165 393
0.0158 038
0.0151 116
0.0144 593
0.0138 438
0.0132 624
0.0127 124
0.0121 917
0.0116 981
0.0112 298
0.0107 852
0.0559 383
0.0545 642
0.0532 932
0.0521 147
0.0510 193
0.0499 980
0.0490 466
0.0481 561
0.0473 219
0.0465 393
0.0103 625 0.0403 625
0.0099 605 | 0.0399 605
0.0095 778
0.0092 131
0.0088 655
0 0049 711
0.0018 031
0.0016 416
0.0044 862
0.0043 366
0.0458 038
0.0451 116
0.0444 593
0.0438 438
0.0432 624
0.0427 124
0.0421 917
0.0416 981
0.0412 298
0.0107 852
0.0085 338 0.0385 338
0.0082 172
0.0382 172
0.0079 147 0.0379 147
0.0076 256 | 0.0376 256
0.0073 491
0.0373 491
0.0035 485
0.0034 333
0.0033 222
0.0032 151
0.0031 117
0.0395 778
0.0392 131
0.0388 655
0.0059 191 | 0.0359 191
0.0057 138
0.0055 168
0.0053 276
0.0051 458
0.0357 138
0.0355 168
0.0353 276
0.0351 458
0.0370 845
0.0368 311
0.0365 885
0.0363 559
0.0361 330
0.0349 711
0.0348 031
0.0346 416
0.0314 862
0.0343 366
0.0041 927
0.0311 927
0.0040 540
0.0340 540
0.0039 205 0.0339 205
0.0037 919
0.0337 919
0.0036 680 | 0.0336 680
0.0030 120 0.0330 120
0.0029 158 0.0329 158
0.0028 228 0.0328 228
0.0027 331 0.0327 331
0.0026 465 | 0.0326 465
0.0025 628 | 0.0325 628
0.0024 820 0.0324 820
0.0024 039
0.0324 039
0.0023 285 0.0323 285
0.0022 556
0.0322 556
0.0021 851 0.0321 851
0.0021 169 0.0321 169
0 0020 511 0.0320 511
0.0019 874 | 0.0319 874
0.0019 258 0.0319 258
0.0585 634
0.0018 662
0.0018 086
0.0568 577
0.0552 016
0.0318 662
0.0318 086
0.0017 528 | 0.0317 598
0.0535 938 0.0016 989 0.0316 989
100 0.0520 328 0.0016 467 0.0316 457
Ratio of Perpetual Ann. 0.0300 000
RATIO of INTEREST:
HALF-YEARLY, 0.015.
•
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.0256 657
0.0256 657 | 0.0556 657
0.4610 689
0.4475 419
0.4344 118
0.4216 669
0.0243 028
0.0230 421
0.0218 732
0.0543 028
0.0530 421
0.0518 732
0.4092 960 | 0.0207 869 | 0.0507 869
0.3972 879 | 0.0197 750
0.3856 322
0.3743 185
0.3633 366
0.3526 769 |
0.0188 307
0.0179 477
0.0171 207
0.0163 417
0.3423 300
0.3322 866
0.3225 380
0.3130 752
0.3038 902
0.2949 746 | 0.0125 517
0.2863 205
0.2779 204
0.2697 666
0.2618 522
0.0120 357
0.0115 467
0.0110 828
0.0106 423
0.1040 307
0.0335 485
0.0334 333 0.1009 786
0.0333 222 0.0980 161
0.0332 151 0.0951 405
0.0331 117
0.0923 492
0.2541 699 | 0.0102 236
0.2457 130
0.0098 255
0.2394 748 0.0094 464
0.2324 491
0.0090 854
0.2256 295
0.0087 411
0.1887 136
0.1831 771
0.1778 030 |
0.1725 866
0.1675 232
0.1626 084
0.1578 377
0.1532 070
0.1487 122
0 1443 492
0.1401 143
0.1360 036
0.1320 135
0.1281 40-4
0.1243 810
0 2190 099
0.0084 128 | 0.0384 128
0.2125 845 0.0080 993 0.0380 993
0.2063 476 0.0077 999 0.0377 999
0.2002 938 0.0075 138 0.0375 138
0.194k 175 | 0.0072 401 0.0372 401
0.0156 156 | 0.0456 156
0.0419 295
0.0442 829
0.0136 729
0.0430 966
0.0149 295
0.0112 829
0.0136 729
0.0130 966
0.0896 399
0.0870 100
0.0844 573
0.0819 794
0.0795 743
0.0772 397
0.0749 736
0.0727 741
0.0706 390
0.0685 666
0.0665 549
0.0646 023
0.0627 070
0.0608 673
0.0590 816
0.1207 319
0.0041 193
0 1171 898 0.0039 824
0.1137 517 0.0038 506
0.1104 144 0.0037 236
0.1071 750 | 0.0036 012
0.0497 750
0.0488 307
0.0479 477
0.0471 207
0.0163 417
0.0048 884
0.0047 224
0.0015 627
0.0044 092
0.0042 615
0.0069 783 | 0.0369 783
0.0067 277 | 0.0367 277
0.0064 876 0.0361 876
0.0062 576 | 0.0362 576
0.0060 370 | 0.0360 370
0.0425 517
0.0420 357
0.0415 467
0.0410 828
0.0406 423
0.0058 255 0.0359 255
0.0056 226 0.0356 226
0.0054 278 0.0354 278
0.0052 407 0.0352 407
0.0050 610 | 0.0350 610
0.0402 236
0.0398 255
0.0394 464
0.0390 854
0.0387 411
0.0034 833
0.0033 696
0.0032 600
0.0343 881
0.0347 224
0.0345 627
0 0344 092
0.0342 615
0.0334 833
0.0333 696
0.0332 600
0.0031 543 0.0331 543
0.0030 524
0.0330 524
0.0341 193
0.0339 824
0.0338 506
0.0337 235
0.0336 012
C
0.0029 540
0.0028 591
0.0027 674
0.0026 790 0.0326 790
0.0329 540
0.0328 591
0.0327 674
0.0025 936
0.0325 936
0.0025 112 0.0325 112
0.0024 315 0.0324 315
0.0023 546 0.0323 546
0.0022 802 0.0322 802
0.0022 081 0.0322 084
0.0021 390 0.0321 390
0.0020 719 0.0320 719
0.0020 071 0.0320 071
0.0019 444 0.0319 444
0 0018 837 0.0318 837
0.0573 482 0.0018 251 0.0318 251
0.0556 657 0.0017 684
0.0317 684
0.0540 326 | 0.0017 136 0.0317 136
0.0524 474 0.0016 605 0.0316 605
0.0509 086 | 0.0016 092 | 0.0316 092
0.0300 000
RATIO of INTEREST:
QUARTERLY, 0.0075.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.4597 421
0.4462 042
0.4330 658
0.4203 138
0.4079 373
0.3959 253
0.3842 669
0.3729 518
0.3619 700 |
0.3513 115
0.3409 668
0.3309 268
0.3211 824
0.3117 248
0.3025 459
0.2528 773
0.2454 312
0.2382 042
0.2311 901
0.2243 825
0.2177 754
0.2113 628
0.2051 391
0.1990 986
0.1932 360
0.2936 372
0.0124 711
0.2849 908 | 0.0119 575
0.2765 990
0.2684 543
0.0114 708
0.0110 091
0.2605 494 | 0.0105 706
0.0105 706
0.0255 290
0.0241 715
0.0229 162
0.0217 521
0.0206 703
0.1197 856
0.1162 584
0.1128 350
0.1095125
0.1002 878
0.0196 627
0.0187 224
0.0178 432
0.0170 197
0.0162 472
0.1031 581
0.1001 205
0.0971 724
0.0943 111
0.0915 340
0.0155 212
0.0148 381
0.0141 945
0.0135 872
0.0130 136
0.1875 459 0.0069 252
0.1820 235 0.0066 759
0.1766 637 0.0064 371
0.1714 617 | 0.0062 083
0.1661 129 0.0059 890
|
0.0888 387
0.0862 228
0.0836 839
0.0812 197
0.0788 282
0.1615 127 | 0.0057 787
0.1567 568
0.0055 769
0.1521 410
0.0053 832
0.1476 610 | 0.0051 973
0.1433 130 0.0050 186
0.0765 070
0.0742 542
0.0720 677
0.0699 456
0.0678 860
0.0101 540
0.0097 578
0.0093 806
0.0090 213
0.0086 789
0.1390 931
0.1349 974
0.1310 223
0.1271 642
0.1234 198 | 0.0042 239
0.0048 470
0.0046 820
0.0045 233
0.0043 707
0.0083 522
0.0080 403
0.0077 425
0.0074 578
9.0071 856
0.0040 826
0.0039 466
0.0038 156
0.0036 894
0.0035 679
0.0034 507
0.0033 378
0.0032 289
0.0031 240
0.0030 227
0.0029 250
0.0028 308
0.0027 398
0.0026 520
0.0025 672
0.0024 854
0.0024 063
0.0023 299
0.0022 562
0.0021 849
0.0658 870
0.0021 160
0.0039 4690.0020 495
0 0620 640
0.0602 365
0.0584 627
0.0019 851
0.0019 229
0.0018 628
0.0567 413
0.0018 016
0.0550 705 0.0017 484
0.0531 489 | 0.0016 940
0.0518 750
0.0016 414
0.0503 475 0.0015 905
0.0555 290
0.0541 716
0.0529 162
0.0517 521
0.0506 703
0.0496 627
0.0487 224
0.0478 432
0.0470 197
0.0462 472
0.0455 212
0.0448 381
0.0441 945
0.0435 872
0.0430 136
0.0424 711
0.0419 575
0.0414 708
0.0410 091
0.0405 706
0.0401 540
0.0397 578
0.0393 806
0.0390 213
0.0386 789
0.0383 522
0.0380 403
0.0377 425
0.0374 578
0.0371 856
0.0369 252
0.0366 759
0.0364 371
0.0362 083
0.0359 890
0.0357 787
0.0355 769
0.0353 832
0.0351 973
0.0350 186
0.0348 470
0.0346 820
0.0345 233
0.0343 707
0.0342 239
0.0340 826
0.0339 466
0.0338 156
0.0336 894
0.0335 679
0.0334 507
0.0333 378
0.0332 289
0.0331 240
0.0330 227
0.0329 250
0.0328 308
0.0327 398
0.0326 520
0.0325 672
0.0324 854
0.0324 063
0.0323 299
0.0322 562
0.0321 819
0 0321 160
0.0320 495
0.0319 851
0.0319 229
0.0318 628
0.0318 046
0.0317 484
0 0316 940
0.0316 414
0.0315 905
0.0300 000
Page 5.
RATE OF INTEREST: 34 per Cent. per Ann.--CORRESPONDING VALUE of

Years.
**
Jak
PKNP
2
w
*}}}=}=
A/c:1
#
10
t
P1006/
=====
~{<={\{0}**
HGHAMT
8
HARDW
ස
AKONIAL DA Face CAKEHAN
11
12
GARLAL FAKDAL
14
15
& DOOR D2565 SAAN
16
18
19
20
RATIO of INTEREST:
ANNUALLY, 0.0325.
21
92
Improved Cumulated Value of
Principal. Annuity. Annuity.
25
1.0325 000
10 1.3768 943 11.596 748
1.0660 563 2.0325 000
1.0000 000
000
1.1007 031 3.0985 563
1.1364 759 4.1992 593
1.1734 114 5.3357 353
1.2115 473 6.5091 467
1.2509 226 | 7.7206 939
1.2915 775 8.9716 165
1.3335 538 | 10.263 194
1.4216 434
CAG.
1.5155 518
12.973 642
1.4678 468 14.395 285
1.6681 725 20.559 155
17 1.7223 881 22.227 327
1.7783 658 | 23.949 715
1.8361 626 | 25.728 081
1.8958 379 27.564 244
15.863 132
1.5648 072 17.378 684
1.6156 635 | 18.943 491
1.9574 527 29.460 082
2.0210 699 31.417 534
23 2.0867 546 33.438 604
2.1545 742 35.525 359
2.C245 978 37.679 933
24
0.9685 23
1.9065 60
2.8150 70
3.6949 83
4.5471 99
5.3725 90
6.1720 00
6.9462 47
7.6961 23
8.4223 95
9.1258 07
9.8070 77
10.466 90
11.105 96
11.724 90
RATIO of INTEREST:
HALF-YEARLY, 0.01625.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0162 500
1.0327 641
1.0495 465
1.0666 016
....
1.0839 339
1.1015 478
...
1.1194 480
..
1.1376 390
1.3152 536
1.3366 265
..
1.3583 466
1.3804 198
1.1561 256 4.8038 654
1.1749 127 5.3819 282
0.5000 000
1.0081 250
1.1940 050 5.9693 845
1.2134 076 6.5663 869
....
1.5245 070
2.0492 803
1.2331 254 7.1730 907
1.2531 637 7.7896 531
1.4488 147
1.4723 579
2.5825 811
1.2735 276 8.4162 353
1.2942 225 9.0529 991
1.4962 838
1.5205 984
....
3.1245 480
3.6753 219
4.2350 459
···
12.324 36
12.904 95 1.7298 920
13.467 26 1.7865 703
14.011 88 1.8451 056
14.539 35 1.9055 587
15.050 21 1.9679 926
15.545 00 2.0324 720
16.024 22 2.0990 641
16.428 34 2.1678 379
16.937 86 2.2388 651
1.4028 516 12.395 434
1.4256 479 13.096 859
9.7001 103
10.357 737
11.026 050
11.705 224
13.809 683
14.534 091
1.5453 081 16.778 711
1.5704 194 17.551 365
15.270 270
16.018 412
1.5959 387 18.336 575
1.6218 727 19.134 544
1.6482 281 19.945 480
1.6750 118 20.769 594
22.458 215
24.202 163
26.003 249
27.863 346
....
29.784 387
31.768 370
33.817 355
35.933 474
38.118 926
0.4920 05
0.9761 43
1.4525 39
1.9213 18
2.3826 00
2.8365 07
....
3.2831 56
3.7226 63
...
4.1551 42
...
4.5807 05
4.9994 66
5.4115 29
5.8170 03
6.2159 93
...
6.6086 04
6.9949 36
7.3750 91
7.7491 67
8.1172 61
....
8.4794 70
8.8358 87
9.1866 05
9.5317 14
....
9.8713 06
10.205 47
10.534 28
10.857 84
11.176 23
11.489 53
11.797 81
...
12.101 17
..
12.399 67
12.982 44
13.546 72
14.093 09
14.622 14
15.134 40
15.630 41
16.110 68
16.575 72
17.026 00
RATIO of INTEREST:
QUARTERLY, 0.008125.
Improved Cumulated
Principal. Annuity.
1.0081 950
1.0163 160
1.0245 736
1.0328 982
1.0412 905
1.0497 510
1.0582 803
1.0668 788
1.0755 472
1.0842 860
1.0930 958
1.1019 772
1.1109 308
1.1199 571
1.1290 567
1.1382 303
1.1474 785
1.1568 017
1.1662 007
1.1756 762
1.1852 285
1.1918 585
1.2045 667
1.2143 538
1.2242 204
1.2341 672
1.2441 948
1.2543 039
1.2644 951
1.2747 691
1.2851 266
1.2955 683
1.2060 948
1.3167 068
1.3274 051
1.3381 902
1.3710 743
1.3822 143
1.3934 448
1.4047 666
1.4161 803
1.4276 867
1.4392 867
1.4509 809
1.4627 701
1.4746 551
1.4866 367
1.4987 156
1.5108 927
1.5231 687
1.5355 444
1.5480 207
1.5605 984
1.5732 783
1.6382 398
1.6515 505
1.6649 693
1.6784 972
0.2500 000
0.5020 313
0.7561 103
1.0122 536
1.2704 782
1.5308 008
1.7932 386
2.0578 087
1.3490 630
10.740 400
1.3600 241 11.077 666
11.417 672
11.760 441
1.7907 530
1.8496 657
1.9105 161
2.3245 284
2.5934 152
2.8644 867
3.1377 606
1.9733 691
2.0382 895
2.1053 456
2.1746 078
2.2461 485
3.4132 549
3.6909 876
3.9709 769
4.2532 411
4.5377 987
4.8246 683
5.1138 687
5.4054 189
5.6993 379
5.9956 450
6.2943 596
6.5955 013
6.8990 898
7.2051 449
7.5136 867
7.8247 354
8.1383 114
8.4544 351
8.7731 274
9.0911 091
9.4183 012
9.7448 248
10.074 002
10.405 853
12.105 994
12.454 356
12.805 547
13.159 592
13.516 514
13.876 336
14.239 081
14.604 773
1.5850 612
1.5989 479
18.032 651
18.429 166
1.6119 394 18.828 903
1.6250 364 | 19.231 888
14.973 437
15.345 096
15.719 775
16.097 498
16.478 291
16.862 177
17.249 182
17.639 331
1.7337 168 | 22.575 901
24.330 862
26.143 560
28.015 890
19.638 147
20.047 707
20.160 595
20.876 837
29.949 818
31.947 368
34.010 634
36.141 778
38.343 032
Value of
Annuity.
0.2479 85
0.4939 72
0.7379 76
0.9800 13
1.2201 00
1.4582 51
1.6944 84
1.9288 12
2.1612 52
2.3918 18
2.6205 27
2.8473 92
3.0724 28
3.2956 51
3.5170 75
3.7367 14
3.9545 83
4.1706 96
4.3850 67
4.5977 11
4.8086 41
5.0178 71
5.2254 14
5.4312 85
5.6354 96
5.8380 62
6.0389 95
6.2383 09
6.4360 16
6.6321 30
6.8266 64
7.0196 29
7.2110 39
7.4009 07
7.5892 45
7.7760 64
7.9613 78
8.1451 98
8.3275 37
8.5084 06
8.6878 18
8.8657 83
9.0423 14
9.2174 23
9.3911 20
9.5634 17
9.7343 26
9.9038 57
10.072 02
10.238 83
10.404 30
10.568 43
10.731 24
10.892 73
11.052 93
11.211 83
11.369 46
11.525 81
11.680 90
11.834 74
11.987 35
12.138 72
12.288 87
12.437 81
13.021 68
13.586 94
14.134 21
14.664 04
15.177 00
15.673 62
16.154 42
16.619 91
17.070 57
Page 6.
|ė
PERPET. STOCK: 3 p. C. 92,3077; 3½ p. C. 107,6923; 4 p. C. 123,0769; 5 p. C. 153,8462.
Years.
***
1
-THRE
20
HTA:
~~:04
~*~~:04
10
===
PRU-AL
7
∞
6:0
FR-PL
❤
14~2:940
32]e#e [22l#S?
6 0.8253 908 0.1536 300
ܢܚܘܐܝܩܕܝܚ ܘܗܘܘܐܝܗ ܩܝܝ
PPONHAL
14
GARN-AL
RATIO of INTEREST:
ANNUALLY, 0.0325.
SA1:A
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
16
...
0.9685 230 1.0000 000
2002 20
17
18
19
0.9380 368 0.4920 049 0.5245 049
0.9085 102
0.8522 161
0.3227 310
0.8799 130 0.2381 373 0.2706 373
0.7742 470
0.1874 156
0.7994 100 0.1295 220
0.1114 627
0.7262 722 | 0.0862 311
1.0325 000
12 0.6812 700 0.0694 672 0.1019 672
|
15 0.6189 408 0.0527 886
0.3552 310
13 0.6598 257 | 0.0630 393
0.0630 393 | 0.0955 393
0.6390 564 0.0575 418
0.2199 156
0.7498 760 0.0974 356 0.1299 356
21
22
0.5108 680
0.4947 874
23 0.4792 130
24 0.4641 288
25
0.1861 300
0.1620 220
0.1439 627
0.7034 113 0.0770 794 0.1095 794
0.1187 311
0.0900 418
0.0852 886
0.5994 584 0.0486 401
0.0811 401
0.5805 892 0.0149 897 0.0774 897
0.5623 140 0.0417 542 0.0742 542
0.5446 141 0.0388 680 0.0713 680
0.5274 712 | 0.0362 789 0.0687 789
0.0339 442
0.0318 294
0.0299 056
0.0281 489
0.0664 442
0.0643 294
0.0624 056
0.0606 489
0.4495 194 0.0265 393 0.0590 393
RATIO of INTEREST:
HALF-YEARLY, 0.01625.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9840 102
0.9682 753 | 0.9919 405
0.9225 656
0.9078 135
0.8649 579
0.8511 271
2.0000.000
0.9527 925 0.6559 497
0.6884 497
0.9375 572 | 0.4879 762 | 0.5204 762
0.7852 205
0.7726 647
....
0.8932 975 0.2720 850
0.8790 134
0.2361 250
···
....
0.3200 463
0.7128 337
0.7014 355
;

0.3872 095 0.4197 095
0.3525 463
..
....
2.0325 000
1.0244 405
0.2000 213
0.8375 174 0.1675 213
0.8241 254 0.1522 906 | 0.1847 906
....
0.2081 657 0.2406 657
0.1858 070
0.2183 070
0.8109 475 0.1394 099 0.1719 099
0.7979 803 0.1283 753 0.1608 753
...
...
0.3045 850
0.2686 250
0.1030 915 0.1355 915
0.7603 097
0.7481 522 0.0965 461 | 0.1290 461
0.1188 179 0.1513 179
0.1104 606 | 0.1429 606
0.6265 905 0.0545 358
0.6165 712 | 0.0522 615
....
...
0.7361 891 0.0906 942
0.1231 942
0.7244 173 0.0854 319 0.1179 319
0.0384 567
0.0358 894
....
0.0806 748 | 0.1131 748
0.0763 541 0.1088 541
0.5081 320
0.4920 117
0.4764 028
0.4612 891
0.4466 549 0.0262 337
...
0.6902 194 0.0724 129 0.1049 129
0.6791 826 | 0.0688 037 0.1013 037
0.6683 224 0.0654 867 0.0979 867
0.6576 359 0.0624 281 0.0949 281
•
0.6471 202 0.0595 993 | 0.0920 993
0.6367 726 0.0569 756 0.0894 756
0.6067 121 0.0501 367 0.0826 367
0.5970 107
0.0481 473
0.0806 473
0.5780 708 0.0445 271
0.5597 317 0.0413 186
0.5419 744
0.5247 805
0.0770 271
0.0738 186
0.0709 567
0.0683 894
0.0870 358
0.0847 615
0.0335 746 0.0660 746
0.0314 779 | 0.0639 779
0.0295 706 0.0620 706
0.0278 292 | 0.0603 292
0.0587 337
RATIO of INTEREST:
QUARTERLY, 0.008125.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9919 405
0.9839 459
0.9760 158
0.9681 496
0.9603 468
0.9526 069
0.9449 294
0.9373 136
0.9297 593
0.9222 659
0.9148 329
0.9074 598
0.9001 461
0.8928 913
0.8856 951
0.8785 568
0.8714 760
0.8644 523
0.8574 853
0.8505 741
0.8437 192
0.8369 192
0.8301 741
0.8231 833
0.8168 464
0.8102 630
0.8037 327
0.7972 550
0.7908 295
0.7844 558
0.7781 334
0.7718 620
0.7656 412
0.7594 705
0.7533 495
0.7472 779
0.7412 552
0.7352 810
0.7293 551
0.7234 769
0.7176 459
0.7118 620
0.7061 248
0.7004 339
0.6726 593
0.6672 380
0.6618 604
0.6565 261
0.6512 349
0.6459 863
0.6407 799
0.6356 155
0.6304 927
0.6254 i12
0.6203 707
0.6153 708
4.0000 000
1.9919 078
1.3225 585
0.9878 947
0.7871 053
0.6532 528
0.5576 503
0.4859 538
0.5767 954
0.5584 243
0.5406 382
0.5234 187
0.4301 948
0.3855 919
0.4626 948
0.4180 919
0.3491 027
0.3816 027
0.3186 986 | 0.3511 986
0.2929 755
0.2709 302
0.2518 272
0.2351 148
0.1449 467
0.1387 898
0.1330 905
0.1277 999
0.2203 712 0.2528 712
0.2072 682 0.2397 682
0.1955 467 0.2280 467
0.1849 996 | 0.2174 996
0.1754 590 | 0.2079 590
0.1667 878
0.1588 724
0.1516 186
0.1992 878
0.1913 724
0.1841 186
0.1061 763
0.1026 186
0.0992 654
0.0960 998
0.6947 887
0.0739 836
0.6891 890 0.0720 652
0.6836 345
0.6781 246
0.0826 037
0.0802 932
0.0780 913
0.0759 902
4.0325 000
2.0244 078
1.3550 585
1.0203 917
0.1228 756 | 0.1553 756
0.1182 812 0.1507 812
0.1139 844
0.1464 844
0.1099 577 0.1424 577
0.8196 053
0.6857 528
0.5901 503
0.5184 538
0.3254 755
0.3034 302
0.2843 272
0.2676 148
0.0931 061
0.0902 718
0.0875 835
0.0850 309
0.0850 309 | 0.1175 309
0.1256 064
0.1227 718
0.1200 835
0.0554 550
0.0542 618
0.0531 098
0.0519 970
0.6104 112
0.6054 916
0.0509 213
0.0498 810
0.6006 116 0.0488 744
0.5957 710
0.0479 000
0.1774 467
0.1712 898
0.1655 905
0.1602 999
0.1064 836
0.1045 652
0.0702 293 0.1027 293
0.0684 708 | 0.1009 708
0.0442 950
0.0411 001
0.0382 503
0.0356 940
0.5067 476 0.0333 892
0.4906 075 0.0313 015
0.4749 814 0.0294 026
Ò.4598 530
0.0276 688
0.4452 065
0.0260 804
0.1386 763
0.1351 186
0.1317 654
0.1285 998
0.0667 849 0.0992 849
0.0651 674 | 0.0976 674
0.0636 141 0.0961 141
0.0621 215 | 0.0946 215
0.0606 859 0.0931 859
0.0593 043 0.0918 043
0.0579 738 | 0.0904 738
0.0566 915 0.0891 915
0.1151 037
0.1127 932
0.1105 913
0.1084 902
0.0879 550
0.0867 618
0.0856 098
0.0844 970
0.0834 213
0.0823 810
0.0813 744
0.0804 000
0.0767 950
0.0736 001
0.0707 503
0.0681 940
0.0658 892
0.0638 015
0.0619 026
0.0601 688
0.0585 804
Page 7.
RATE OF INTEREST: 34 per Cent, per Ann.-
CORRESPONDING VALUE OF

NIKE SOME A
Years.
8582 100** C**** ***33 85988 58846 85898 58876 86882 KRK** O*20%
26
27
2.2968 973
2.3715 464
2.4486 217
29 2.5282 019
30 2.6103 684
31
32
34
36
40
44
48
41
3.7110 130
42 3.8316 209
49
50
43 3.9561 486
4.0847 234
45 4.2174 769
51
52
53
46 4.3545 449
47 4.4960 676
55
56
57
60
61
63
64
66
67
71
72
73
74
75
RATIO of INTEREST:
ANNUALLY, 0.0325.
76
77
Improved Cumulated Value of
Principal. Annuity.
Annuity.
78
2.6952 054
2.7827 996
2.8732 406
2.9666 209
3.0630 361
3.1625 847
3.2653 687
3.3714 932
3.4810 668
3.5942 014
85838 683** 85882
100
4.6421 898
4.7930 610
4.9488 355
69
70 9.3821 900
5.1096 726
5.2757 370
5.4471 984
5.6242 324
5.8070 200
5.9957 481
6.1906 099
6.3918 047
6.5995 384
6.8140 234
7.0354 791
7.2641 322
7.7439 736
7.9956 527
8.2555 114
8.5238 155
8.8008 395
9.0868 668
79
80 12.918 284
39.904 531
42.201 428
44.572 975
47.021 596
49.549 798
185.70 705
192.74 253
7.5002 165 200.00 666
52.160 167
54.855 372
57.638 172
60.511 412
63.478 033
15.651 112
16.159 773
16.684 965
17.227 227
17.787 112
66.541 069
69.703 654
72.969 023
76.340 516
79.821 583
18.365 193
18.962 061
19.578 328
20.214 624
20.871 599
96 21.549 926
97 22.250 299
83.415 784
87.126 797
90.958 418
94.914 566
98.999 290
103.21 677
107.57 131
112.06 738
116.70 957
121.50 263
126.45 147
131.56 114
136.83 688
142.28 407
147.90 831
153.71 533
159.71 107
165.90 168
172.29 349
178.89 303
9.6871 112
10.001 942
267.29 573
276.98 284
10.327 005 286.98 478
10.662 633
11.009 169
297.31 179
307.97 442
.11.366 967 318.98 359
11.736 393
330.35 056
12.117 826 342.08 695
12.511 655 354.20 477
366.71 643
81
13.338 128
379.63 471
82
13.771 617 392.97 284
83
14.219 195 406.74 446
84
14.681 319 420.96 365
85 15.158 462
435.64 497
207.50 688
215.25 085
223.24 650
231.50 202
240.02 583
248.82 667
257.91 354
450.80 343
466.45 455
482.61 432
499.29 928
516.52 651
632.30 543
653.85 535
17.373 23
17.794 90
18.203 29
18.598 83
18.981 92
98 22.973 434 676.10 565
19.352 95
19.712 30
20.060 34
20.397 42
20.723 89
21.010 09
21.346 33
21.642 94
21.930 21
22.208 43
22.477 90
22.738 89
22.991 66
23.236 47
23.473 58
23.703 23
23.925 64
24.141 06
24.349 69
24.551 76
24.747 47
24.937 02
25.120 60
25.298 40
25.470 60
25.637 39
25.798 92
25.955 37
26.106 90
26.253 66
26.395 79
26.533 46
26.666 78
26.795 92
26.920 99
27.042 12
27.159 43
27.273 06
-
- 27.383 11
27.489 70
27.592 92
27.692 90
27.789 74
27.883 52
27.974 36
28.062 33
28.147 54
28.230 06
28.309 99
28.387 39
28.462 37
28.534 98
28.605 31
28.673 42
28.739 39
534.31 362 29.093 82
552.67 881
29.146 56
571.64 088 29.197 63
591.21 920 29.247 10
611.43 383 29.295 02
28.803 29
28.865 17
28.925 10
28.983 15
29.039 37
29.341 42
29.386 36
29.429 89
23.720 070 699.07 909 29.472 05
24.490 973 722.79 916 29.512 88
Value of Perpetual Ann. 30.769 23
RATIO of INTEREST:
HALF-YEARLY, 0.01625.
Improved Cumulated Value of
Principal. Annuity. Annuity.
2.3122 194 40.375 981
2.3879 771 42.706 988
2.4662 169 45.114 367
2.5470 202 47.600 622
2.630 710 | 50.168 337
2.7166 559
2.8056 646
2.8975 895
2.9925 263
3.0905 737
3.1918 334
3.2964 108
3.4044 146
3.5159 571
3.6311 541
3.7501 255 84.619 246
3.8729 948 88.399 841
3.9998 899 | 92.304 304
4.1309 425.96.336 693
4.2662 889
100.50 120
5.1767 473
5.3463 586
5.5215 270
5.7024 346
5.8892 696 |
4.4060 699 104.80 215
4.5504 306 109.24 402
4.6995 212 113.83 142
4.8534 966 | 118.56 913
5.0125 169 | 123.46 206
6.0822 261
6.2815 045
6.4873 121
6.6998 628
6.9193 775
7.1460 844
7.3802 192
7.6220 252
7.8717 537
8.1296 613
8.3960 251
8.6711 130
8.9552 139
9.2486 231
9.5516 456
9.8645 963
10.137 801
11.590 039
11.969 776 |
12.361 954
12.766 982
13.185 280
52.820 181
55.558 909
58.387 370
61.308 502
64.325 343
13.617 284
14.063 441
14.524 217
15.000 089
15.491 553
67.441 028
70.658 795
73.981 989
77.414 064
80.958 589
15.999 119
16.523 315
17.064 686
17.623 795
18.201 222
128.51 530
133.73 411
139.12 391
144.69 030
150.43 906
22.035 501
22.809 111
23.556 430
156.37 619
162.50 783
168.84 037
175.38 039
182.13 469
10.521 594 | 292.97 213
10.866 324
11.222 319
303.57 921
314.53 383
189.11 029
196.31 444
203.75 462
211.43 857
219.37 429
227.57 000
236.03 425
244.77 581
253.80 379
263.12 756
272.75 681
282.70 155
325.84 736
337.53 157
349.59 860
362.06 099
374.93 171
17.462 00
17.884 17
18.292 94
18.688 75
19.072 00
19.443 09
19.802 41
20.150 32
20.487 20
20.813 40
21.129 24
21.435 07
21.731 19
22.017 92
22.295 55
-
22.564 38
22.824 67
23.076 71
23.320 75
23.557 05
23.785 85
24.007 40
24.221 92
24.429 63
24.630 75
24.825 49
25.014 06
25.196 63
25.373 42
25.544 60
25.710 36
25.870 85
26.026 24
26.176 71
26.322 41
26.463 48
26.600 08
26.732 35
26.860 42
26.984 42
27.104 49
27.220 74
27.333 35
27.442 33
27.547 88
27.650 07
27.749 03
27.844 84
27.937 62
28.027 45 ||
28.114 43
28.198 65
28.280 20
28.359 17
28.435 63
388.22 412
28.509 66
401.95 204 28.581 34
416.12 975 28.650 75
430.77 197 28.717 96
445.89 394 28.783 03
461.51 136 28.846 05
477.64 047
28.907 06
494.29 804 28.966 13
511.50 138 29.023 34
529.26 836
29.078 73
18.797 568
19.413 452
20.049 516
547.61 747
29.132 36
566.56 777 29.184 29
586.13 896 29.234 57
20.706 420 606.35 137 29.283 26
21.384 846 | 627.22 603
29.330 40
648.78 463 29.376 04
671.04 958 29.420 24
694.04 401 29.463 04
24.328 235 717.79 184 29.504 48
25.125 327 742.31 774 29.544 60
30.769 23
RATIO of INTEREST:
QUARTERLY, 0.008125.
Improved Cumulated
Principal. Annuity.
2.3200 429
2.3963 682
2.4752 045
40.616 704
42.965 176
45.390 909
2.5566 344 | 47.896 443
2.6107 432
50.484 406
53.157 508
2.7276 190
2.8173 529
2.9100 388
55.918 550
58.770 426
3.0057 740 | 61.716 123
3.1046 587
64.758 729
3.2057 965
67.901 431
3.3122 945 | 71.147 522
74.500 405
3.4212 632
3.5338 167
3.6500 731
77.963 591
81.540 710
3.7701 541
3.8941 855
4.0222 974
4.1546 239
4.2913 037
4.4324 801
4.5783 009
4.7289 189
4.8844 921
5.0451 833
5.2111 610
5.3825 990
5.5596 771
5.7425 807
5.9315 015
6.1266 375
6.3281 931
6.5363 795
6.7514 150
6.9735 246
7.2029 414
7.4399 055
7.6846 653
7.9374 773
8.1986 064
11.705 042
12.090 117
12.487 861
12.898 689
13.323 033
85.235 509
89.051 862
92.993 765
97.065 350
101.27 088
105.61 477
110.10 157
114.73 597
119.52 283
124.46 718
16.178 876
16.711 133
17.260 900
17.828 753
18.415 288
129.57 418
134.84 920
140.29 776
145.92 556
151.73 851
8.4683 261
8.7469 192
9.0346 774
9.3319 025
229.79 465
238.36 674
247.22 084
256.36 623
9.6389 057 | 265.81 248
19.021 118
19.646 880
20.293 225
20.960 837
21.650 412
157.74 269
163.94 440
170.35 014
176.96 661
183.80 076
9.9560 087
10.283 544
10.621 854
10.971 295
275.56 950
285.64 751
296.05 706
306.80 907
11.332 231 | 317.91 480
190.85 973
198.15 091
205.68 201
213.46 084
221.49 558
13.761 338
14.214 062
392.65 655
406.58 652
14.681 679 | 420.97 475
15.164 681 435.83 634
15.663 572 | 451.18 684
329.38 590
341.23 437
353.47 263
366.11 352
379.17 026
467.04 235
483.41 948
500.33 538
554.49 595
573.75 015
593.63 770
614.17 960
635.39 728
Value of
Annuity.
22.362 672 | 657.31 299
23.098 305 679.94 968
23.858 260 | 703.25 416
24.643 155
727.48 170
752.42 683
25.453 872
17.506 88
17.929 29
18.338 24
18.734 18
19.117 50
19.488 61
19.847 83
20.195 75
20.532 52
20.858 56
21.174 22
21.479 83
21.775 70
22.062 15
22.339 47
22.607 96
22.867 90
23.119 56
23.363 21
23.599 09
23.827 46
24.048 56
24.262 62
24.469 86
24.670 50
24.864 74
25.052 80
25.234 87
25.411 14
25.581 80
25.747 02
25.906 98
26.061 85
26.211 78
26.356 93
26.497 47
26.633 53
26.765 25
26.892 78
27.016 24
27.135 78
27.251 50
27.363 55
27.472 02
27.577 03
27.678 71
27.777 14
27.872 44
27.964 70
28.054 03
28.140 51
28.224 23
28.305 29
28.383 77
28.459 75
28.867 41
28.927 98
28.986 62
517.80 779 29.043 40
535.85 501 29.098 37
28.533 31
28.604 52
28.673 47
28.740 23
28.804 84
29.151 59
29.203 11
29.252.99
29.301 28
29.348 04
29.393 30
29.437 13
29.479 55
29.520 63
29.500 40
30.769 23
Page 8.
PERPET. STOCK: 3p. C. 92,3077; 34 p. C. 107,6923; 4p. C. 123,0769; 5 p. C. 153,8462.

Years.
***RA 788*8 85889 29339 95998 88858 85888 58858 86888 PPPPP PER2: 58833 85888 58833 85883
62
63
64
65
66
67
69
70
72
73
74
75
78
79
82
86
RATIO of INTEREST:
ANNUALLY, 0.0325.
100
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.4353 699
0.4216 658
0.4083 930
0.3955 380
0.3830 877
0.3710 293
0.3593 504
0.3480 391
0.3370 838
0.3264 735
0.3161 971
0.3062 441
0.2966 045
0.2872 683
0.2782 259
0.2694 682
0.2609 862
0.2527 711
0.2448 146
0.2371 086
0.2296 451
0.2224 166
0.2154 156
0.2086 349
0.2020 677
0.1957 073
0.1895 470
0.1835 806
0.1778 020
0.1722 054
76 0.0879 742
0.0852 051
0.0825 231
0.0799 255
0.0774 097
0.1667 849
0.1615 350
0.1564 503
0.1515 257
0.1467 562
0.1421 367
0.1376 627
0.1333 295
0.1291 327
0.1250 680
0.1211 312
0.1173 184
0.1136 255
0.1100 489
0.1055 849
0.1032 300
0.0999 806
0.0968 335
0.0937 855
0.0908 334
0.0749 730
0.0726 131
0.0703 275
0.0681 138
0.0659 698
0.0638 932
0.0618 821
0.0599 342
0.0580 476
0.0562 205
0.0250 598 0.0575 598
0.0236 959 | 0.0561 959
0.0549 351
0.0224 351
0.0537 668
0.0212 668
0.0526 817
0.0201 817
0.0191 717 | 0.0516 717
0.0182 298 0.0507 298
0.0173 496 0.0498 496
0.0165 258 0.0490 258
0.0157 535 | 0.0482 535
0.0150 283 0.0475 283
0.0143 465 | 0.0468 465
0.0462 044
0.0137 044
0.0130 992
0.0455 992
0.0125 279 | 0.0450 279
0.0119 881
0.0114 775
0.0109 940
0.0105 358
0.0101 011
0.0096 884 0.0421 884
0.0092 962 | 0.0417 962
0.0089 231
0.0085 683
0.0082 303
0.0414 231
0.0410 683
0.0407 303
0.0079 082
0.0076 009
0.0073 080
0.0070 282
0.0067 609
0.0065 055
0.0062 613
0.0060 277
0.0058 041
0.0055 899
0.0053 819
0.0051 883
0.0049 998
0.0048 191
0.0046 457
0.0044 794
0.0043 196
0.0041 662
0.0040 189
0.0038 773
0.0037 412
0.0036 103
0.0034 845
0.0033 635
0.0032 470
0.0031 350
0.0030 271
0.0029 232
0.0028 232
0.0027 269
0.0026 341
0.0025 447
0.0024 585
0.0023 755
0.0022 954
0.0022 183
0.0021 438
0.0020 721
0.0020 028
0.0019 360
0.0444 881
0.0439 775
0.0134 940
0.0430 358
0.0426 011
0.0018 716
0.0018 094
0.0017 493
0.0016 914
0.0016 355
0.0404 082
0.0401 009
0.0398 080
0.0395 282
0.0392 609
0.0390 055
0.0387 613
0.0385 277
0.0383 041
0.0380 899
0.0378 849
0.0376 883
0.0374 998
0.0373 191
0.0371 457
0.0369 794
0.0368 196
0.0366 662
0.0365 189
0.0363 773
0.0362 412
0.0361 103
0.0359 845
0.0358 635
0.0357 470
0.0356 350
0.0355 271
0.0354 232
0.0353 232
0.0352 269
0.0351 341
0.0350 447
0.0349 585
0.0348 755
0.0347 954
0.0347 183
0.0346 438
0.0345 721
0.0345 028
0.0344 360
0.0343 716
0.0343 094
0.0342 493
0.0341 914
0.0341 355
0.0544 508
0.0527 369
0.0510 769
0.0494 691
0.0479 120
0.0464 039
0.0449 432
0.0435 285
0.0421 584
0.0408 314
0.0015 815
0.0015 294
0.0014 791
0.0014 304
0.0013 835
0.0340 815
0,0340 294
0.0339 791
0.0339 304
0.0338 835
Ratio of Perpetual Ann. 0.0325 000
RATIO of INTEREST:
HALF-YEARLY, 0.01625.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.4324 849 | 0.0247 672 | 0.0572 672
0.0234 154 | 0.0559 151
0.4187 644
0.0221 659 | 0.0546 659
0.0535 081
0.0210 081
0.0524 329
0.0199 329
0.4051 793
0.3926 156
0.3801 601
0.3680 996 0.0189 322
0.0179 991
0.3564 218
0.0171 270
0.3451 145
0.3341 658 | 0.0163 110
0.3235 645 | 0.0155 460
0.3132 996 | 0.0148 278 | 0.0473 278
0.0466 525
0.3033 603 0.0141 525
0.0135 168 0.0460 168
0.2037 362
0.0454 176
0.2844 176 0.0129 176
0.0148 520
0.2753 945 | 0.0123 520
0.2666 577
0.0118 176
0.0113 122
0.2581 981
0.2500 069 | 0.0108 337
0.0103 803
0.2420 755
0.2343 957 | 0.0099 501
0.2269 596 |
0.2197 594
0.2127 876
0.2060 370
0.1995 006
0.1931 715
0.1870 432
0.0077 812
0.0074 775 |
0.1811 093 | 0.0071 878
0.1753 637
0.1698 003
0.0069 113
0.0066 472
0.1644 135
0.1591 975
0.1541 470
0.1492 568
0.1445 217
0.1399 368
0.1354 973
0.1311 987
0.1270 365
0.1230 063
0.1191 040
0.1153 255
0.1116 668
0.1081 242
0.1046 940
0.0095 418 0.0420 418
0.0091 538 | 0.0416 538
0.0087 849
0.0084 339
0.0080 997
0.0412 849
0.0409 339
0.0405 997
0.0625 034
0.0605 205
0.0586 005
0.0567 415
0.0549 414
0.0052 879 |
0.0050 939
0.0049 079
0.0047 295
0.0015 584
0.0043 943
0.0042 367 |
0.0040 854
0.0039 401
0.0038 004
0.0862 810 0.0030 689
0.0835 437 0.0029 627
0.0028 604 |
0.0808 931
0.0783 270 | 0.0027 620
0.0758 421
0.0026 672
0.0152 786
0.0138 421
0.0424 513
0.0411 045
0.0398 005
0.0388 948
0.0063 948
0.0386 535
0.0061 535
0.0384 228
0.0059 228
0.0382 019
0.0057 019
0.0054 904 | 0.0379 904
0.0734 361 0.0025 758 |
0.0024 879
0.0711 064
0.0688 505 0.0024 031
0.0666 663 0.0023 214
0.0645 513 0.0022 427
0.0514 322
0.0504 991
0.0496 270
0.0488 110
0.0480 460
0.1013 726 | 0.0036 663 | 0.0361 663
0.0981 566 0.0035 373
0.0360 373
0.0359 133
0.0950 426 0.0034 133
0.0357 940
0.0032 940
0.0920 274
0.0891 079 | 0.0031 793 | 0.0356 793
0.0531 984
0.0018 261
0.0515 107 0.0017 650
0.0498 765 | 0.0017 061
0.0482 912
0.0467 621
0.0016 492
0.0015 943
0.0143 176
0.0438 122
0.0433 337
0.0428 803
0.0424 501
0.0015 413
0.0014 902
0.0014 408
0.0013 932
0.0013 471
0.0402 812
0.0399 775
0.0396 878
0.0394 113
0.0391 472
D
0.0377 879
0.0375 939
0.0374 079
0.0372 295
0.0370 584
0.0368 943
0.0367 367
0.0365 854
0.0364 401
0.0363 004
0.0021 668 | 0.0346 668
0.0020 936 0.0345 936
0.0345 231
0.0020 231
0.0344 550
0.0019 550
0.0018 894 0.0343 894
0.0355 689
0.0354 627
0.0353 604
0.0352 620
0.0351 672
0.0350 758
0.0349 879
0.0349 031
0.0348 214
0.0347 427
0.0313 261
0.0342 650
0.0342 061
0.0341 492
0.0340 943
0.0340 413
0.0339 902
0.0339 408
0.0338 932
0.0338 471
- 0.0325 000
RATIO of INTEREST:
QUARTERLY, 0.008125.
Discounted Annuity for Ratio of
Principal. Cumulation
Cumulation Annuity.
0.4310 265
0.4172 981
0.4040 070
0.3911 392
0.3786 812
0.3666 201
0.3549 431
0.3436 380
0.3326 930
0.3220 966
0.3118 377
0.3019 055
0.2922 897
0.2829 801
0.2739 671
0.2652 411
0.2567 931
0.2486 141
0.2406 957
0.2330 291
0.2256 073
0.2184 216
0.2114 648
0.2047 296
0.1982 088
0.1632 216
0.1580 230
0.1529 899
0.1481 171
0.1433 995
0.1388 322 |
0.1344 103
0.1301 293
0.1259 846
0.1219 719
0.1180 871
0.1143 259
0.1106 846
0.1071 593
0.1037 462
0.0854 333
0.0827 122
0.0800 778
0.0775 272
0.0750 580
0.1918 958 0.0077 176
0.0074 157
0.1857 838
0.1798 665 | 0.0071 277
0.1741 377 0.0068 528
0.1685 914 | 0.0065 903
0.0726 673
0.0703 529
0.0681 121
0.0659 427
0.0638 424
0.0618 090
0.0598 403
0.0579 3+
0.0560 892
0.0543 027
0.0246 204
0.0232 747
0.0220 308
0.0208 784
0.0198 081
0.0525 731
0.0508 987
0.0492 775
0.0477 080
0.0461 885
0.0188 120
0.0178 832
0.0170 154
0.0162 032
0.0154 419
0.0147 272
0.0140 553
0.0134 227
0.0128 265
0.0122 638
0.0419 142
0.0405 792
0.0392 867
0.0117 322
0.0112 294
0.0107 534
0.0094 684
0.0090 825
0.0087 157
0.0083 666
0.0080 342
0.0036 289
0.0035 008
0.1004 418
0.0972 427
0.0941 455 | 0.0033 777
0.0911 469
0.0882 439
0.0032 594
0.0031 455
0.0063 394
0.0060 996
0.0058 703
0.0056 508
0.0054 407
0.0103 023 | 0.0428 023
0.0423 745
0.0098 745
0.0052 395
0.0052 395
0.0050 467
0.0048 619
0.0046 847
0.0045 148
0.0043 517
0.0041 952
0.0040 450
0.0039 007
0.0037 621
0.0030 360
0.0029 305
0.0028 291
0.0027 314
0.0026 373
0.0025 468
0.0024 595
0.0023 754
0.0022 944
0.0022 164
0.0021 411
0.0020 686
0.0019 987
0.0019 312
0.0018 662
0.0018 034
0.0017 429
0.0016 845
0.0016 282
0.0015 738
0.0015 214
0.0447 174
0.0432 931 0.0014 707
0.0014 218
0.0013 746
0.0013 290
0.0571 204
0.0557 747
0.0545 308
0.0533 784
0.0523 081
0.0513 120
0.0503 832
0.0495 154
0.0487 032
0.0479 419
0.0472 272
0.0465 553
0.0459 227
0.0453 265
0.0447 638
0.0442 322
0.0437 294
0.0432 534
0.0419 684
0.0415 825
0.0412 157
0.0408 666
0.0405 342
0.0402 176
0.0399 157
0.0396 277
0.0393 528
0.0390 903
0.0388 394
0.0385 996
0.0383 703
0.0381 508
0.0379 407
0.0377 395
0.0375 467
0.0373 619
0.0371 847
0.0370 148
0.0368 517
0.0366 952
0.0365 450
0.0364 007
0.0362 621
0.0361 289
0.0360 008
0.0358 777
0.0357 594
0.0356 455
0.0355 360
0.0354 305
0.0353 291
0.0352 314
0.0351 373
0.0350 468
0.0319 595
0.0348 754
0.0317 944
0.0317 164
0.0346 411
0.0345 686
0.0344 987
0.0344 312
0.0343 662
0.0343 034
0.0342 429
0.0341 845
0.0341 282
0.0340 738
0.0340 214
0.0339 707
0.0339 218
0.0338 746
0.0338 290
0.0325 000
Page 9.
RATE OF INTEREST: 3½ per Cent. per Ann.——CORRESPONDING VALUE OF

Years.
AKNAR
Jak
-
PANLAL
2.
10
مستندات
110025-
**
2014
-~-~~-
======f22}=
PRICHAL
❤
−4-20
❤
10
Brace Fe
11
======]< ¢2
13
PPHQ-A
14
***** +000
15
16
17
19
20
RATIO of INTEREST:
ANNUALLY, 0.035.
Improved Cumulated Value of
Principal. Annuity. Annuity.
25
1.0350 000
1.0712 250
1.0000 000
1.1475 230
2.0350 000
1.1087 179 3.1062 250
12 1.5110 687 14.601 962
4.2149 429
1.1876 863 5.3624 659
1.2292 553 | 6.5501 522
..
1.2722 793 7.7794 075
1.3168 090 | 9.0516 868
1.3628 974 10.368 496
1.4105 988 11.731 393
1.4599 697 13.141 992
1.5639 561 16.113 030
1.7946 756 22.705 016
18 1.8574 892 24.499 691
1.9225 013 | 26.357 181
1.9897 889 28.279 682
1.6186 945 17.676 986
1.6753 488 19.295 681
1.7339 860 20.971 030
21
2.0594 315 30.269 471
22 2.1315 116 32.328 902
2.2061 145 34.460 414
2.2833 285 | 36.666 528
93
24
2.3632 450 38.949 857
0.9661 84
1.8996 94
2.801 636
3.673 080
4.515 052
5.328 553
6.114 544
6.873 956
7.607 686
8.316 605
9.001 552
9.663 333
10.30 274
10.92 052
11.51 741
RATIO of INTEREST:
HALF-YEARLY, 0.0175.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0175 000
1.0353 063
1.0534 241
...
1.0718 590
1.1291 221
1.1488 818
1.2529 895
1.2749 168
1.0906 166 2.5890 447
1.1097 024
1.3430 281
...
1.3665 311
1.3904 454
1.4147 782
1.4395 368
...
1.1689 872 4.8282 061
1.1894 445
5.4126 997
1.4647 287
1.2102 598 6.0074 220
1.2314 393
6.6125 519
1.4903 615
0.5000 000
1.0087 500
...
1.5164 428
..
1.5429 805
1.5699 827
1.5264 031
1.2972 279 | 8.4922 247
1.3199 294 9.1408 386
···
2.0531 152
12.09 411
12.65 132 1.8037 245
13.18 968 1.8674 073
13.70 984 1.9333 384
14.21 240 2.0015 973
....
14.69 797 2.0722 662
15.16 712 2.1454 302
15.62 041 2.2211 773
16.05 837 2.2995 987
16.48 151 2.3807 889
3.1343 530
3.6892 042
4.2537 652
....
7.2282 715
7.8547 663
9.8008 033
..
10.472 317
11.155 583
11.850 806
12.558 195
13.277 963
14.010 327
14.755 508
1.5974 574 17.070 211
1.6254 129 17.868 940
15.513 730
16.285 220
1.6538 576 18.681 646
1.6828 001 19.508 575
1.7122 491 20.349 975
1.7422 135 21.206 100
22.963 558
24.783 065
26.666 812
28.617 067
...
30.636 178
32.726 577
34.890 780
37.131 392
39.451 112
0.491 400
0.974 349
1.448 992
#
1.915 471
2.373 928
··
2.824 499
....
3.267 321
3.702 526
4.130 247
4.550 611
....
4.963 746
5.369 775
5.768 821
6.161 002
6.546 440
6.925 249
7.297 541
...
7.663 431
8.023 028
8.376 441
8.723 775
9.065 135
9.400 623
9.730 342
10.05 439
10.37 287
10.68 586
10.99 348
11.29 580
11.59 292
11.88 494
12.17 193
12.73 119
13.27 138
13.79 314
14.29 712
1
RATIO of INTEREST:
QUARTERLY, 0.00875.
Improved Cumulated Value of
Principal. Annuity.
Annuity. Annuity.
1.0087 500
1.0175 766
1.0264 804
1.0354 621
1.0445 224
1.0536 619
1.0628 815
1.0721 817
1.0815 633
1.0910 269
1.1005 734
1.1102 034
1.1199 178
1.1297 171
1.1396 021
1.1495 737
1.1596 324
1.1697 792
1.1800 148
1.1903 399
1.2007 554
1.2112 620
1.2218 605
1.2325 518
1.2433 366
1.2542 158
1.2651 902
1.2762 606
1.3330 828
1.3447 472
1.3565 138
1.3683 833
1.3803 566
1.3924 347
1.4046 186
1.4169 090
1.4293 069
1.4418 134
1.4544 292
1.4671 555
1.4799 931
1.4929 430
1.5060 063
1.5191 838
1.5324 767
1.5458 858
1.5594 123
1.5730 572
1.5868 215
1.6007 062
1.6147 123
1.6288 411
1.2874 279
1.2986 929
1.3100 565
8.2122 234
8.5340 804
8.8587 536
1.3215 1959.1862 677
1.7013 609
1.7162 478
1.7312 650
1.7464 136
0.2500 000
0.5021 875
0.7565 816
1.0132 017
1.8083 450
1.8724 726
1.9388 744
2.0076 308
1.2720 672
1.5331 978
1.7966 133
2.0623 337
14.78 390
2.0788 258
15.25 409 2.1525 452
15.70 824 2.2288 789
16.14 690
2.3079 195
16.57 060
2.3897 631
2.3303 791
2.6007 699
2.8735 267
3.1486 700
3.4262 209
3.7062 003
3.9886 296
4.2735 301
4.5609 236
4.8508 317
5.1432 765
5.4382 802
5.7358 651
6.0360 540
6.3388 695
6.6443 346
6.9524 726
7.2633 067
7.5768 607
7.8931 582
9.5166 176
9.8499 183
10.186 105
10.525 234
10.867 329
11.212 419
11.560 527
11.911 682
12.265 909
12.623 236
12.983 689
13.347 296
13.714 085
14.084 084
14.457 319
14.833 821
1.6430 934
18.374 095
1.6574 705 18.784 868
1.6719 734 19.199 236
1.6866 031
19.617 229
15.213 617
15.596 736
15.983 207
16.373 061
16.766 325
17.163-030
17.563 207
17.966 885
20.038 880
20.464 220
20.893 282
21.326 099
23.095 568
24.927 786
26.824 979
28.789 449
30.823 585
32.929 854
35.110 817.
37.369 121
39.707 509
0.217 831
0.493 513
0.737 064
0.978 502
1.217 846
1.455 114
1.690 323
1.923 493
2.154 640
2.383 782
2.610 936
2.836 120
3.059 350
3.280 645
3.500 020
3.717 492
3.933 077
4.146 793
4.358 655
4.568 679
4.776 881
4.983 277
5.187 883
5.390 715
5.591 786
5.791 114
5.988 713
6.184 598
6.378 783
6.571 285
6.762 116
6.951 292
7.138 828
7.321 736
7.509 032
7.691 729
7.872 842
8.052 384
8.230 368
8.406 808
8.581 719
8.755 111
8.927 000
9.097 398
9.266 317
9.433 772
9.599 771
9.761 336
9.927 471
10.08 919
10.24 951
10.40 843
10.56 598
10.72 216
10.87 699
11.03 047
11.18 262
11.33 346
11.48 298
11.63 121
11.77 815
11.92 381
12.06 822
12.21 137
12.77 166
13.31 277
13.83 534
14.34 002
14.82 740
15.29 810
15.75 268
16.19 170
16.61 567
Page 10.
PERPETUAL STOCK: 3 p. C. 85,7143; 34 p. C. 100; 4 p. C. 114,2857; 5 p. C. 142,8571.

Years.
1****
ARCHAŁ
Q2
09
Kako do ka
~~2.04
PSULAL
ໂດ
t
мала
роди
| ==| | |/
AKKAH
*UN-AL
~*~*
10
~~~:9}={
-NTO
~~~~~~
13
~~<~~~\}{0}~*~*~
14
****TO TODO 934 3
16
17
18
19
20
21
23
24
RATIO of INTEREST:
ANNUALLY, 0.035.
25
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9661 836
0.9335 107 0.4914 005
0.9019 427 | 0.3219 342
0.3219 342 0.3569 342
0.8714 422 0.2372 511
1.0000 000
0.8419 731 0.1861 814
0.8135 006 0.1526 682
0.7594 116
0.7859 910 | 0.1285 445
··
0.7089 188
0.7337 310 | 0.0964 460
0.6617 833
0.1104 766
0.6849 457 0.0760 920
0.5572 038
0.5383 611
0.0852 414
0.5201 557
0.5025 659
0.0684 839
0.5968 905 0.0518 251
0.5767 059 0.0476 848
0.4855 709 0.0330 366
92 0.4691 506 | 0.0309 321
0.4532 856 0.0290 188
0.4379 572 | 0.0272 728
0.0964 460 0.1314 460
1.0350 000
0.5264 005
0.0440 431
0.0408 168
0.0379 403
0.0353 611
0.2722 511
0.4231 470 | 0.0256 740
0.2214 814
0.1876 682
0.6394 042 | 0.0620 616 0.0970 616
0.1635 445
0.6177 818 0.0565 707 0.0915 707
0.1454 766
0.1202 414
0.1110 920
0.1034 839
0.0868 251
0.0826 848
0.0790 431
0.0758 168
0.0729 403
0.0703 611
RATIO of INTEREST:
HALF-YEARLY, 0.0175.
0.0256 740 0.0606 740
Discounted Annuity for] Ratio of
Principal. Cumulation Annuity.
0.9828 010 2.0000 000
0.9658 978 0.9913 259
0.9492 853
0.9329 585
0.9169 125 | 0.3862 428
0.9011 425 0.3190 451
0.8856 438
0.8704 116
0.8554 412 0.2071 162
0.8407 286 | 0.1847 507
....
0.6551 349 0.6901 349
0.4870 648
0.5220 648
0.7980 913
0.1383 458
0.7813 619 0.1273 112
0.2710 612
0.2350 858
....
0.8262 689 | 0.1661± 607
0.2014 607
0.8120 579 0.1512 276
0.1512 276 0.1862 276
...
0.7708 746 | 0.1177 548
0.6480 963
0.6369 497
....
0.7576 163 | 0.1093 991
...
0.7445 850 0.1020 325
0.7317 799 0.0954 899
0.6259 945
0.6152 283
...
...
....
0.5840 271
0.5739 824
0.5544 084
0.5355 018
0.5172 400
0.4996 010
0.0680 366 0.4825 635
0.0659 321 0.4661 070
0.0640 188 0.4502 117
0.0622 728 0.4348 585
0.4200 288
***
2.0350 000
1.0263 259
0.7191 940 0.0896 412 0.1246 412
0.7068 246 0.0843 824 0.1193 824
0.4212 428
0.3540 451
0.6946 679 0.0796 293 0.1146 293
0.6827 203 0.0753 128 0.1103 128
0.3060 612
0.2700 858
0.6709 782 0.0713 759 0.1063 759
|
0.6594 380 0.0677 713
0.1027 713
0.0585 816
0.0559 630
0.2121 162
0.2197 507
....
··
0.1733 458
0.1623 112
0.1527 548
0.1443 991
0.0374 998
0.0349 442
0.1370 325
0.1304 899
0.0644 590 0.0994 590
0.0614 054 | 0.0964 054
0.0964 054
..
0.6046 469 0.0535 285 0.0885 285
0.5942 476 0.0512 595 0.0862 595
|
0.0935 816
0.0909 630
0.0491 401
0.0841 401
0.0471 562 0.0821 562
0.0435 473 0.0785 473
0.0403 501 0.0753 501
0.0724 998
0.0699 412
0.0326 411
0.0305 562
0.0286 609
0.0269 314
0.0676 411
0.0655 562
0.0636 609
0.0619 314
0.0253 478 0.0603 478
RATIO of INTEREST:
QUARTERLY, 0.00875.
Ratio of
Discounted Annuity for
Principal. Cumulation Annuity.
0.9913 259
0.9827 271
0.9742 028
0.9657 525
0.9573 754 | 0.7861 220
0.9490 710
0.6522 316
0.5566 028
0.4848 876
0.9408 387
0.9326 777
0.9245 876 | 0.4291 148
0.9165 677
0.9086 172
0.9007 358
0.3845 015
0.3480 04t
0.3175 944
0.8929 227
0.8851 771
0.877+ 993
0.8698 878
0.8623 423
0.8548 622
0.8474 471
0.8400 962
0.8328 092
0.8255 853
0.8184 241
0.8113 250
0.8042 874
0.7973 110
0.7903 951
0.7835 391
0.7767 425
0.7700 050
0.7633 260
0.7507 018
4.0000 000
1.9912 881
1.3217 344
0.9869 702
0.2918 667
0.2698 181
0.2507 127
| 0.2339 986
0.2339 986
0.2192 539
0.2061 502
0.1944 286
0.1838 817
0.6301 907
0.6247 243
0.6193 054
0.6139 335
0.1743 416
0.1656 712
0.1577 568
0.1505 042
0.1438 337
0.1376 783
0.1319 808
0.1266 920
0.7501 410
0.7436 342
0.1050 790
0.1015 237
0.7371 839 | 0.0981 730
0.7307 895
0.0950 098
0.5877 618
0.5826 665
0.5776 121
0.5726 021
0.1217 697
0.1171 772
0.1128 827
0.1088 582
0.7244 505 0.0920 189
0.7181 666 | 0.0891 868
0.7119 371 0.0865 012
0.7057 617 | 0.0839 512
0.6996 399
0.0815 268
0.6935 711 0.0792 190
0.6875 550 0.0770 197
0.6815 911 | 0.0749 215
0.6756 789 | 0.0729 177
0.6698 180 0.0710 021
0.6640 079 0.0691 691
0.6582 482 | 0.0674 135
0.6525 386 0.0657 306
|
0.6468 784 0.0641 160
0.6412 673 | 0.0625 657
0.6357 049 | 0.0610 759
0.0596 434
0.0596 434
0.0582 648
0.0569 372
0.0556 579
0.6086 082 0.0544 245
0.6033 291
0.0532 343
0.5980 957
0.0520 854
0.5929 077 | 0.0509 756
0.0509 756
4.0350 000
2.0262 881
1.3567 3++
1.0219 702
0.5529 919 | 0.0432 983
0.5340 533
0.5157 632
0.1980 996 |
0.0401 159
0.0372 787
0.0317 349
0.0317 349
0.8211 220
0.6872 316
0.5916 028
0.5198 876
0.4641 1.18
0.4195 015
0.3830 041
0.3525 9++
0.3268 667
0.3048 181
0.2857 127
0.2689 986
0.2542 539
0.2411 502
0.2294 286
0.2188 817
0.2093 416
0.2006 712
0.1927 568
0.1855 042
0.1788 337
0.1726 783
0.1669 808
0.1616 920
0.1567 697
0.1521 772
0.1478 827
0.1438 582
0.1400 790
0.1365 237
0.1331 730
0.1300 098
0.1270 189
0.1241 868
0.1215 012
0.1189 512
0.1165 268
0.1142 190
0.1120 197
0.1099 215
0.1079 177
0.1060 021
0.1041 691
0.1024 135
0.1007 306
0.0991 160
0.0975 657
0.0960 759
0.0946 434
0.0932 643
0.0919 372
0.0906 579
0.0894 245
0.0882 343
0.0870 854
0.0859 756
0.0499 030 0.0849 030
0.0488 658 0.0838 658
0.0478 623 0.0828 623
0.0468 909
0.0818 909
0.0782 983
0.0751 159
0.0722 787
0.0697 349
0.0674 427
0.0653 676
0.0634 812
0.0617 601
0.4810 409
0.0324 427
0.4615 664 0.0303 676
0.4486 561 0.0284 812
0.4332 907 | 0.0267 601
0.4184 516 0.0251 842 | 0.0601 842
Page 11.
RATE OF INTEREST: 3½ per
-CORRESPONDING VALUE OF
Cent. per Ann.-

Years.
36
37
38
39
40
26
2.4459 586 41.313 102
27 2.5315 671 | 43.759 050
2.6201 720 46.290 627
29 2.7118 780 48.910 799
2.8067 937 51.622 677
28
30
41
31
2.9050 315
32 3.0067 076
33 3.1119 424
34
35
**** 95998 58848 85888 580J6 85882 NOR** ***28 58*** ***** *88*3 86882
94
95
96
RATIO of INTEREST:
ANNUALLY, 0.035.
100
Improved Cumulated Value of
Principal. Annuity.
Annuity.
54.429 471
57.331 503
60.341 210
3.2208 603 63.453 152
3.3335 905 | 66.674 013
3.6960 113
3.4502 661 | 70.007 603
3.5710 254 73.457 859
77.028 895
80.724 906
84.550 278
3.8253 717
3.9592 597
4.0978 338 88.509 538
4.2412 580 92.607 371
4.3897 020 96.848 629
4.5433 416 101.23 833
4.7023 586 | 105.78 167
4.8669 411
5.0372 840
5.2135 890
5.3960 616
5.5849 269
5.7803 993 136.58 284
5.9827 133 142.36 324
6.1921 083 148.34 595
6.4088 320 154.53 806
6.6331 412 | 160.94 689
6.8653 011
7.1055 866
7.3542 822
7.6116 820
7.8780
909
852
9.6841
10.023 132
10.373 941
10.737 029
11.112
825
241
8.1538
8.4392 079
8.7345 802
9.0402 905 | 229.72 259
9.3567 007 | 238.76 288
204.39 497
212.54 880
220.98 801
11.501 774
11.904 336
12.320 988
12.752 223
13.198
550
110.48 403
115.35 097
120.38 826
125.60 185
130.99 791
13.660 500
14.138 617
14.633 459
15.145 640
15.675 738
16.224 388
16.792 242
167.58 003
174.44 533
181.55 092
188.90 520
196.51 688
248.11 958
257.80 376
267.82 689
278.20 084
288.93 787
300.05 069
311.55 246
323.45 680
335.77 779
348.53 001
361.72 856
375.38 906
389.52 768
404.16 115
419.30 679
434.98 252
451.20 691
17.379 970 | 467.99 916
17.988 269 485.37 913
18.617 859 | 503.36 740
19.269
484
521.98 525
19.943 916 | 541.25 474
20.641 953
21.364 421
561.19 861
581.84 061
22.112 175 | 603.20 503
22.886
102 625.31 720
23.687 116 | 648.20 331
24.516 165 | 671.89 042
25.374 231 | 696.40 659
26.262 329 721.78 082
16.89 035
17.28 537
17.66 702
18.03 577
18.39 204
18.73 628
19.06 887
19.39 021
19.70 068
20.00 066
20.29 050
20.57 053
20.84 109
21.10 250
21.35 507
21.59 910
21.83 488
22.06 269
22.28 279
22.49 545
22.70 092
22.89 912
23.09 125
23.27 657
23.45 561
23.62 862
23.79 577
23.95 726
24.11 330
24.26 405
24.40 971
24.55 044
24.68 643
24.81 779
24.94 474
25.06 737
25.18 587
25.30 036
25.41 097
25.51 785
25.62 111
25.72 088
25.81 728
25.91 041
26.00 040
26.08 734
26.17 135
26.25 250
26.33 092
26.40 669
26.47 989
26.55 062
26.61 896
26.68 498
26.74 877
26.81 041
26.86 996
26.92 750
26.98 309
27.03 680
27.32 301
27.36 522
27.40 602
27.44 543
27..48 351
27.181 510| 748.04 315
28.132 853775.22 466
29.117 513 803.35 752
30.136 626 | 832.47 503
31.191 408 862.61 166
27.52 029
27.55 584
27.59 018
27.62 337
27.65 543
Value of Perpetual Ann. 28.57 143
RATIO of INTEREST:
HALF-YEARLY, 0.0175.
Improved Cumulated
Principal. Annuity.
2.4648 457
2.5518 700
2.6419 670
2.7352 450
2.8318 164
2.9317 972
3.0353 079
3.1424 732
3.2534 221
3.3682 882
3.4872 098
3.6103 302
3.7377 974
3.8697 652
4.0063 921
4.1478 428
4.2942 874
4.4459 026
4.6028 708
4.7653 808
4.9336 291
5.1078 165
5.2881 543
5.4748 594
5.6681 561
5.8682 775
6.0754 639
6.2899 661
6.5120 411
6.7419 571
6.9799 907
7.2264 275
7.4815 653
7.7457 111
8.0191 839
8.3023 109
8.5954 349
8.8989 071
9.2130 938
9.5383 738
9.8751 382
10.223 792
10.584 756
10.958 463
11.345 366
11.745 929
12.160 633
12.589 981
13.034 485
13.494 684
13.971 130
14.464 399
14.975 083
15.503 795
16.051 177
27.08 870
27.13 884
27.18 728
19.766 050
20.463 917
21.186 418
27.23 409 21.934 434
27.27 932 22.708 856
16.617 883
17.204 599
17.812 029
18.440 904
19.091 984
23.510 621
24.340 691
25.200 071
26.089 790
27.010 922
41.852 742
44.339 142
46.913 348
49.578 427
52.337 617
55.194 206
58.151 653
61.213 518
64.383 502
67.665 381
71.063 139
74.580 868
78.222 787
81.993 292
85.896 921
89.938 361
94.122 500
98.454 368
102.93 917
107.58 231
112.38 940
117.36 376
122.51 869
127.85 312
133.37 589
139.09 365
145.01 324
151.14 190
157.48 690
164.05 592
170.85 688
177.89 793
185.18 757
192.73 461
200.54 192
208.63 745
217.01 245
225.68 305
234.65 982
243.95 354
253.57 545
263.53 694
273.85 019
284.52 751
295.58 190
307.02 654
318.87 477
331.14 229
343.84 239
356.99 098
370.60 374
384.69 711
399.28 812
414.39 410
430.03 361
446.22 521
462.98 850
480.31 365
498.31 159
516.91 390
536.17 289
556.11 194
576.75 481
599.12 669
620.25 306
64.316 062
666.87 692
691.43 060
716.85 121
743.16 917
27.964 577
770.41 650
28.951 900 798.62 576
29.974 085
31.032 357
32.127 993
827.83 100
858.06 733
889.37 122
Value of
Annuity.
16.97 986
17.37 516
17.75 698
18.12 577
18.48 199
18.82 606
19.15 841
19.47 941
19.78 947
20.08 895
20.37 822
20.65 763
20.92 751
21.18 818
21.43 997
21.68 317
21.91 807
22.14 497
22.36 412
22.57 580
22.78 027
22.97 776
23.16 852
23.35 277
23.53 074
23.70 264
23.86 867
24.02 905
24.18 395
24.33 357
24.47 810
24.61 769
24.75 252
24.88 275
25.00 855
25.13 005
25.24 741
25.36 076
25.47 025
25.57 601
25.67 816
25.77 683
25.87 213
25.96 418
26.05 309
26.13 898
26.22 193
26.52 610
26.59 614
26.66 350
26.72 856
26.79 141
26.85 211
26.91 074
26.96 738
27.02 208
27.07 492
27.12 595
27.17 524
27.22 285
27.26 885
27.31 327
27.35 617
27.39 761
27.43 764
27.47 631
27.51 365
RATIO of INTEREST:
QUARTERLY, 0.00875.
27.54 973
27.58 457
97.61 823
27.65 073
27.68 213
28.57 143
Improved Cumulated
Principal. Annuity.
2.4745 085 42.128 813
2.5522 598 44.635 991
2.6531 227
47.232 078
2.7472 079 | 49.920 225
2.8445 296 | 52.703 704
2.9455 060 55.585 887
58.570 296
61.650 505
64.860 318
68.173 589
3.0499 598
3.1581 177
3.2701 110
3.3860 758
3.5061 530
71.604 373
3.6304 886 | 75.156 818
78.835 227
82.644 088
86.588 022
3.7592 331
3.8925 431
4.0305 803
4.1735 136
90.671 817
4.3215 148 | 94.900 430
4.4747 649
4.6334 491
4.7977 610
99.279 000
103.81 285
108.50 746
11.872 100
12.293 109
26.30 205 12.729 048
26.37 944
26.45 420
13.180 446
13.647 852
4.9678 994
5.1440 712
5.3264 905
5.5153 790
5.7109 654 |
5.9134 881
6.1231 928
6.3403 333
6.5651 747
6.7979 894
7.0390 604
7.2886 798
7.5471 515
7.8147 890
8.0919 177
8.3788 735
8.6760 072
8.9836 750
9.3022 544
9.6321 311
113.36 858
118.40 204
123.61 403
129.01 093
134.59 900
140.38 538
146.37 695
152.58 096
159.00 501
165.65 683
266.49 701
276.96 070
287.79 552
11.465 509 | 299.01 455
9.9737 067 | 256.39 162
10.327 395
10.693 625
11.072 843
172.54 457
179.67 653
187.06 147
194.70 825
202.62 622
23.834 791
24.680 022
25.555 226
26,461 468
27.399 845
210.82 496
219.31 451
228.10 501
237.20 727
246.63 233
28.371 502
29.377 613
30.419 403
31.498 138
32.615 128
14.131 833
375.19 521
14.632 977 | 389.51 359
15.151 892
15.689 209
16.245 582
404.33 976
419.69 166
435.58 802
310.63 141
322.66 026
335.11 570
348.01 273
361.36 723
16.821 683
17.418 214
18.035 900
18.675 .490
452.04 806
469.09 186
486.74 000
505.01 394
19.337 762 | 523.93 605
20.023 520 543.52 916
20.733 596 | 563.81 698
21.468 851
22.230 179
23.018 508
584.82 432
606.57 659
629.10 025
652.42 258
676.57 208
701.57 789
727.47 052
754.28 128
782.04 288
810.78 900
849.55 437
871.37 540
903.28 940
Value of
Annuity.
17.02 512
17.42 056
17.80 245
18.17 126
18.52 744
18.87 142
19.20 363
19.52 445
19.83 429
20.13 351
20.42 249
20.70 157
20.97 110
21.23 139
21.48 276
21.72 554
21.96 000
22.18 642
22.40 509
22.61 627
22.82 022
23.01 718
23.20 740
23.39 111
23.56 852
23.73 986
23.90 533
24.06 513
24.21 946
24.36 851
24.51 245
21.65 146
24.78 571
24.91 536
25.04 057
25.16 149
25.27 828
25.39 106
25.49 998
25.60 517
25.70 675
25.80 486
25.89 961
25.99 111
26.07 918
26.16 483
26.24 725
26.32 684
26.40 371
26.47 796
26.54 965
26.61 889
26.68 576
26.75 034
26.81 271
26.87 294
26.93 111
26.98 729
27.04 154
27.09 393
27.14 454
27.19 340
27.24 050
27.28 617
27.33 019
27.37 270
27.41 375
27.45 340
27.49 169
27.52 867
27.56 438
27.59 887
27.63 218
£7.66 434
27.69 541
28.57 143
Page 12.
PERPETUAL STOCK: 3 p. C. 85,7143; 34 p. C. 100; 4p. C. 114,2857; 5 p. C. 142,8571.

Years.
O7828 78*** 85882 79333 95998 588*3 85888 78838 86882 *22*2 LE*** ***** 85888 53313 85333
77
100
RATIO of INTEREST:
ANNUALLY, 0.035.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.4088 377
0.3950 122
0.3816 543
0.3687 482
0.3562 784
0.3442 304 0.0183 724
0.3325 897
0.0174 415
0.3213 427 0.0165 724
0.3104 761 0.0157 597
0.2999 769 | 0.0149 983
0.0242 054 | 0.0592 054
0.0228 524
0.0216 026
0.0204 454
0.0193 713
0.0578 524
0.0566 026
0.0554 454
0.0543 713
0.2898 327
0.2800 316 | 0.0136 132
0.2705 620 | 0.0129 821
0.0123 878
0.0118 273 |
0.2614 125
0.2525 725
0.2440 314
0.2357 791
0.2278 059
0.2201 023
0.2126 592
0.0142 842 | 0.0492 842
0.0486 132
0.0479 821
0.0473 878
0.0468 273
0.0112 982
0.0107 983
0.0103 254
0.0098 777
0.0094 534
0.2054 679
0.0090 511
0.0440 511
0.1985 196 | 0.0086 692 | 0.0436 692
0.1918 065
0.1853 202
0.1790 534 | 0.0076 337 | 0.0426 337
0.0083 065
0.0079 617
0.0433 065
0.0429 617
0.1729 984 0.0073 216 | 0.0423 216
0.1671 482
0.1614 959
0.1560 347
0.1507 581
0.0420 243
0.0417 410
0.0414 709
0.0412 132
0.0059 673
0.0057 325
0.1456 600
0.1407 343
0.1359 752 | 0.0055 081
0.1313 770 0.0052 937
0.1269 343 | 0.0050 886
0.0732 038
0.0707 283
0.0683 365
0.0660 256
0.0637 929
0.0616 356
0.0595 513
0.0575 375
0.0555 918
0.0537 119
0.0070 243
0.0067 410
0.0064 709
0.0062 132
0.1226 418
0.1184 945
0.0048 925
0.0047 048
0.0045 251
0.1144 875
0.1106 159 | 0.0043 531
0.1068 753
0.0041 883
0.0533 724
0.0524 415
0.0515 724
0.0507 597
0.0499 983
|
0.0462 982
0.0457 983
0.0453 254
0.0448 777
0.0444 534
0.1032 611
0.0997 692
0.0963 954
0.0931 356
0.0040 303 0.0390 303
0.0388 789
0.0038 789
0.0387 338
0.0037 338
0.0035 945 0.0385 945
0.0899 861 | 0.0034 610 | 0.0384 610
0.0409 673
0.0407 325
0.0405 081
0.0402 937
0.0400 886
0.0869 431 0.0033 328
0.0840 030 0.0032 097
0.0030 916
0.0383 328
0.0382 097
0.0380 916
0.0811 623
0.0784 177 | 0.0029 782 0.0379 782
0.0757 659 0.0028 692 0.0378 692
|
0.0398 925
0.0397 018
0.0395 251
0.0393 531
0.0391 883
0.0518 955
0.0501 406
0.0184 450
0.0019 158
0.0018 476
0.0017 819
0.0468 068 | 0.0017 187
0.0452 240
0.0027 645 | 0.0377 645
0.0026 639 0.0376 639
0.0025 672 | 0.0375 672
0.0024 743 0.0374 743
0.0023 849 | 0.0373 849
0.0022 989 | 0.0372 989
0.0372 163
0.0022 163
0.0021 368 0.0371 368
0.0020 602 0.0370 602
0.0019 866 | 0.0369 866
0.0369 158
0.0368 476
0.0367 819
0.0367 187
0.0016 578 | 0.0366 578
0.0365 992
0.0365 427
0.0364 883
0.0364 359
0.0363 855
RATIO of INTEREST:
HALF-YEARLY, 0.0175.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.4057 049
0.3918 695
0.3785 058
0.3655 980
0.3531 303
0.3410 877
0.3294 559
0.3182 207
0.3073 687
0.2968 867
0.2867 622
0.2769 830
0.2675 372
0.2584 136
0.2496 011
0.2410 892
0.2328 675
0.2249 262
0.2172 557
0.2098 468
0.2026 906
0.1957 781
0.1891 019
0.1826 531
0.1764 242
0.1704 078
0.1645 965
0.1589 834
0.1535 617
0.1483 249
0.1432 667
0.1383 810
0.1336 619
0.1291 037
0.1247 010
0.1204 484
0.1163 408
0.1123 733
0.1085 412
0.1048 397
0.1012 644
0.0978 111
0.0944 755
0.0912 537
0.0881 417
0.0851 359
0.0822 326
0.0794 282
0.0767 196
0.0741 033
0.0715 762
0.0691 353
0.0667 776
0.0645 003
0.0623 007
0.0238 933 0.0588 933
0.0225 534 0.0575 534
0.0213 159 | 0.0563 159
0.0201 701 0.0551 701
0.0191 067 0.0541 067
0.0136 946 | 0.0015 992
0.0422 170 | 0.0015 427
0.0407 894 | 0.0014 883
0.0394 101 0.0014 359
0.0380 774 | 0.0013 855
0.0013 368 0.0363 368
0.0357 595
0.0355 456 | 0.0012 899
0.0362 899
0.0345 400
0.0012 418
0.0362 448 0.0333 622
0.0362 012
0.0012 012
0.0322 244
0.0320 601 0.0011 593 | 0.0361 593 0.0311 255
0 0367 897
0.0343 436
0.0331 822
Ratio of Perpetual Ann. 0.0350 000
0.0181 178
0.0171 964
0.0163 363
0.0155 319
0.0147 786
0.0425 340
0.0410 835
0.0396 824
0.0383 292
0.0370 221
0.0140 720
0.0490 720
0.0134 083 0.0484 083
0.0127 840 0.0477 840
0.0121 961 | 0.0471 961
0.0116 419 | 0.0466 419
0.0111 187
0.0106 245
0.0101 570
0.0097 145
0.0092 952
0.0088 976
0.0088 976 0.0438 976
0.0085 203 0.0435 203
0.0081 620 | 0.0431 620
0.0078 215 0.0428 215
0.0074 976 | 0.0421 976
0.0058 528
0.0056 212
0.0053 999
0.0051 885
0.0049 863
0.0071 891
0.0421 894
0.0068 959 | 0.0418 959
0.0066 163 0.0416 163
0.0063 497 | 0.0413 497
0.0060 955 | 0.0410 955
0.0032 570
0.0031 360
0.0030 198
0.0029 083
0.0028 012
0.0601 761
0.0022 410
0.0581 240 0.0021 599
0.0561 418 0.0020 818
0.0542 273 0.0020 068
0.0523 780
0.0019 346
0.0531 178
0.0521 964
0.0513 363
0.0505 319
0.0497 786
0.0047 930
0.0046 080
0.0044 310
0.0042 615
0.0397 930
0.0396 080
0.0394 310
0.0392 615
0.0040 991 | 0.0390 991
0.0026 983
0.0025 994
0.0025 045
0 0024 132
0.0023 254
0.0505 918 0.0018 651
0.0488 665 0.0017 982
0.0472 000 0.0017 338
0.0155 901 0.0016 719
0.0440 357 0.0016 122
0.0039 436
0.0039 436 | 0.0389 436
0.0037 945
0.0036 516
0.0035 146
0.0033 832
0.0387 945
0.0386 516
0.0385 146
0.0383 832
0.0-161 187
0.0456 245
0.0451 570
0.0447 145
0.0142 952
0.0012 980
0.0012 522
0.0012 090
0.0011 654
0.0011 244
0.0408 528
0.0406 212
0.0403 999
0.0401 885
0.0399 863
E
0.0382 570
0.0381 360
0.0380 198
0.0379 083
0.0378 012
0.0376 983
0.0375 994
0.0375 045
0.0374 132
0.0373 254
0.0372 410
0.0371 599
0.0370 818
0.0370 068
0.0369 346
0.0015 548 | 0.0365 548
0.0014 995 0.0364 995
1
0.0014 463 0.0364 463
0.0013 950 0.0363 950
0.0013 456 | 0.0363 456
RATIO of INTEREST:
QUARTERLY, 0.00875.
0.0362 980
0.0362 522
0.0362 080
0.0361 654
0.0361 244
- 0.0350 000
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.4041 207 | 0.0237 367
0.3902 805 0.0224 034
0.3769 143 0.0211 721
0.3610 059 0.0200 320
0.3515 396 0.0189 740
0.3395 002 | 0.0179 902
0.0179 902
0.5278 732 | 0.0170 735
0.0162 178
0.3166 443
0.3058 000 0.0154 177
0.2953 271 0.0146 684
0.2852 129
0.2751 450
0.2660 117 | 0.0126 847
0.2569 015 0.0121 001
0.2481 032 0.0115 489
0.2396 063
0.2314 003
0.2234 754
0.0110 288
0.0105 374
0.0100 726
0.2158 219 | 0.0096 327
0.2084 306
0.0092 160
0.2012 923
0.1943 985
0.1877 409
0.1813 112
0.1751 018
0.1691 049
0.1633 135
0.1577 204
0.1523 189
0.1471 023
0.1002 636 |
0.0968 299
0.0935 137
0.0903 110
0.0872 181
0.1420 644
0.1371 991
0.1325 003
0.1279 625
0.0057 956
0.0057 956
0.0055 656
0.0053 458
0.0051 359
0.1235 801 | 0.0049 352
0.0139 656
0.0133 055
0.1193 478 | 0.0017 433
0.1152 604 0.0045 597
0.1113 130
0.0043 839
0.0042 157
0.1075 008
0.1038 192 0.0040 546
0.0707 622
0.0683 388
0.0659 983
0.0637 381
0.0615 552
0.0594 471
0.0574 112
0.0554 450
0.0535 461
0.0517 123
0.0088 208
0.0084 458
0.0080 897
0.0077 513
0.0074 295
0.0842 311
0.0813 464
0.0032 192
0.0030 992
0.0785 605 | 0.0029 840
0.0758 700
0.0732 716
0.0028 735
|
0.0027 673
0.0071 232
0.0068 317
0.0065 539
0.0062 891
0.0060 366
|
0.0391 309
0.0377 908
0.0361 966
0.0352 466
0.0340 395
0.0328 738
0.0317 479
0.0306 606
0.0039 003
0.0037 524
0.0036 106
0.0034 747
0.0033 443
0.0499 413
0.0368 651
0.0367 982
0.0018 398
0.0482 309 | 0.0017 736
0.0367 338 || 0 0165 791 0.0017 099
0.0366 719
0.0016 486
0.0366 122
0.0015 896
0.0149 839
0.0134 433
0.0026 653
0.0026 653
0.0025 673
0.0024 732
0.0023 827
0.0022 957
0.0022 122
0.0021 318
0.0020 545
0.0019 801
0.0019 086
0.0419 555 | 0.0015 327
0.0405 186 | 0.0014 780
0.0014 254
0.0013 746
0.0013 258
0.0012 787
0.0012 33-4
0,0011 897
0.0011 476
0.0011 071
·
0.0587 367
0.0574 034
0.0561 721
0.0550 320
0.0539 740
0.0529 902
0.0520 735
0.0512 178
0.0504 177
0.0496 684
0.0489 656
0.0483 055
0.0476 847
0.0471 001
0.0465 489
0.0460 288
0.0455 374
0.0450 726
0.0446 327
0.0442 160
0.0438 208
0.0434 458
0.0430 897
0.0427 513
0.0424 295
0.0421 232
0.0418 317
0.0415 539
0.0412 891
0.0410 366
0.0407 956
0.0405 656
0.0403 458
0.0401 359
0.0399 352
0.0397 433
0.0395 597
0.0393 839
0.0392 157
0.0390 546
0.0389 003
0.0387 524
0.0386 106
0.0384 747
0.0383 443
0.0382 192
0.0380 992
0.0379 840
0.0378 735
0.0377 673
0.0376 653
0.0375 673
0.0374 732
0.0373 827
0.0372 957
0.0372 122
0.0371 318
0.0370 545
0.0369 801
0.0369 086
0.0368 398
0.0367 736
0 0367 099
0.0366 486
0.0365 896
0.0365 327
0.0361 780
0.0364 251
0.0363 746
0.0363 258
0.0362 787
0.0362 334
0.0361 897
0.0361 476
0.0361 071
0.0350 000
Page 13.
RATE OF INTEREST: 33 per Cent. per Ann.--CORRESPONDING VALUE OF

Years.
mihis?://
20
Co
स
#KKHP}=
=={}~~0{x#
ex
*BA
14124
OO
4124
PICS=AL
HAHORR
Fax-A
SAKOKAL
10
HTHA:OTH
CPAL BAC
A POUKAH
& DAUN 2565 5 Gala
17
18
20
RATIO of INTEREST:
ANNUALLY, 0.0375.
Improved Cumulated Value of
Principal. Annuity. Annuity.
23
....
25
1.0375 000 1.0000 000
1.0764 063 | 2.0375 000
....
1.1167 715 3.1139 063
1.1586 504 | 4.2306 777
1.2020 998 | 5.3893 281
1.2471 785 6.5914 280
1.2939 4777.8386 065
12 1.5554 543 14.812 116
1.3424 708 9.1325 542
13 1.6137 839 16.367 570
& Z
1.3928 134 | 10.475 025
………
1.4450 439 11.867 838
1.4992 331 13.312 882
15 1.7370 870 | 19.655 654
1.6743 008 17.981 354
22
21 2.1664 577 | 31.105 539
2.2476 999 | 33.271 996
2.3319 886 35.519 696
37.851 685
2.5101 671 40.271 123
24 2.4194 382
1.8022 278 | 21.392 742
1.8698 113 23.194 969
1.9399 293 25.064 781
27.004 710
29.017 387
2.0126 766
2.0881 520
0.9638 55
1.8928 73
2.7883 11
3.6513 84
4.4832 62
5.2850 72
6.0579 00
6.8027 96
....
7.5207 67
8.2127 87
8.8797 95
9.5226 94
10.142 36
..
10.739 62
11.315 30
11.870 17
12.404 98
12.920 46
13.417 31
13.896 20
RATIO of INTEREST:
HALF-YEARLY, 0.01875.
14.357 79
14.802 69
15.231 51
15.644 82
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0187 500 0.5000 000
1.0378 516 1.0093 750
1.0573 113 1.5283 008
1.0771 359 2.0569 564
...
1.0973 322 2.5955 244
1.1179 071 3.1441 904
1.1388 679 3.7031 440
1.1602 217 4.2725 780
1.1819 758 4.8526 888
1.2041 379 5.4436 767
1.2267 155 6.0457 456
1.2497 164 | 6.6591 034
1.2731 486 7.2839 616
1.2970 201 7.9205 358
1.3213 392 8.5690 459
1.3461 143 9.2297 155
1.3713 540
1.3970 669
1.4232 619 11.286 983
1.4499 480 11.998 614
...
9.9027 727
10.588 450
1.4771 346 12.723 588
1.5048 308 13.462 155
1.5330 464 14.214 571
1.5617 910 | 14.981 094
1.5910 746 15.761 989
1.6209 073 | 16.557 527
1.6512 993 | 17.367 980
1.6822 611 | 18.193 630
1.7138 035 19.034 761
1.7459 373 19.891 662
2.1819 265
2.2545 158
2.3502 313
2.4391 912
16.043 20 2.5315 184
1.7786 737
20.764 631
1.8120 238
21.653 968
1.8806 117 | 23.482 979
1.9517 958
2.0256 743
2.1023 493
25.381 222
27.351 316
29.395 981
31.518 040
33.720 422
36.006 167
38.378 432
40.840 491
0.4907 98
0.9725 62
1.4454 60
1.9096 54
2.3653 04
2.8125 69
3.2516 01,
3.6825 53
4.1055 74
4.5208 09
4.9284 01
5.3284 92
5.7212 19
6.1067 18
6.4851 22
6.8565 61
7.2211 65
7.5790 57
7.9303 63
8.2752 03
8.6136 96
8.9459 59
9.2721 07
...
9.5922 53
9.9065 06
10.214 97
10.517 77
...
10.814 99
11.106 73
11.393 11
11.674 22
11.950 16
12.486 88
13.004 03
13.502 32
13.982 41
14.445 05
14.890 79
15.320 26
15.734 08
16.132 80
RATIO of INTEREST:
QUARTERLY, 0.009375.
Improved Cumulated
Principal. Annuity.
0.2500 000
0.5023 438
1.0093 750
1.0188 379
1.0283 895 | 0.7570 532
1.0380 306 1.0141 506
1.0477 622
1.0575 850
1.0674 998
1.0775 076
1.0876 093
1.0978 056
1.1080 975
1.1184 859
1.1289 717
1.1395 559
1.1502 392
1.1610 227
1.1719 073
1.1828 939
1.1939 835
1.2051 771
1.2164 757
1.2278 801
1.2393 915
1.2510 108
1.3107 618
1.3230 502
1.3606 109
1.3733 666
1.3862 420
1.3992 380
1.4123 558
1.4255 967
1.4389 616
1.4524 519
1.2627 390
1.2745 772
1.2865 264 | 7.6407 029
1.2985 875
7.9623 345
1.4798 130
1.4936 863
1.5076 896
8.2869 814
8.6146 718
1.3354 538| 8.9454 344
1.3479 737
9.2792 978
1.5797 001
1.5945 098
1.6094 584
1.6245 470 |
1.2736 583
1.5355 988
1.6397 772
1.6551 501
1.6706 671
1.6863 296
1.7999 950
2.0668 700
1.7021 389
1.7180 965
1.7342 037
1.7501 618
2.3362 469
2.6081 492
2.8826 006
3.1596 250
1.7668 724
1.7834 368
1.8001 565
1.8170 330
3.4392 465
3.7214 894
4.0063 784
4.2939 382
4.5841 938
4.8771 707
5.1728 941
5.4713 900
2.1898 466
2.2731 279
2.3595 764
2.4493 126
2.5424 616
5.7726 843
6.0768 032
6.3837 733
6.6936 211
1.4660 686 | 12.428 497
12.795 014
13.164 968
13.538 389
7.0063 738
7.3220 586
1.5218 242 | 13.915 311
1.5360 913
1.5504 921
1.5650 280
14.295 768
14.679 790
15.067 413
9.6162 912
9.9564 439
10.299 786
10.646 346
10.996 156
11.349 245
11.705 644
12.065 384
15.458 670
15.853 595
16.252 223
16.654 587
17.060 724
17.470 669
17.884 456
18.302 123
18.723 705
19.149 240
19.578 764
20.012 315
20.449 930
20.891 649
21.337 508
21.787 547
1.8861 359 | 23.630 292
1.9578 669 25.543 118
2.0323 259 | 27.528 690
2.1096 165
29.589 774
31.729 243
33.950 077
36.255 371
38.648 337
41.132 309
Value of
Annuity.
0.2476 78
0.4930 56
0.7361 54
0.9769 95
1.2155 99
1.4519 86
1.6361 78
1.9181 95
2.1480 57
2.3757 84
2.6013 96
2.8249 13
3.0463 53
3.2657 37
3.4830 83
3.6984 10
3.9117 38
4.1230 84
4.3324 67
4.5399 05
4.7454 17
4.9490 20
5.1507 32
5.3505 70
5.5485 53
5.74-16 96
5.9390 18
6.1315 35
6.3222 63
6.5112 21
6.6984 23
6.8838 87
7.0676 28
7.2496 62
7.4300 06
7.6086 74
7.7856 84
7.9610 49
8.1347 85
8.3069 08
8.4774 32
8.6463 72
8.8137 43
8.9795 60
9.1438 37
9.3065 87
9.4678 26
9.6275 68
9.7858 26
9.9426 14
10.097 95
10.251 83
10.404 29
10.555 34
10.704 98
10.853 23
11.000 10
11.145 61
11.289 77
11.432 59
11.574 08
11.714 26
11.853 14
11.990 73
12.528 41
13.046 40
13.545 41
14.026 14
14.489 25
14.935 40
15.365 20
15.779 26
16.178 14
Page 14.
PERPETUAL STOCK: 3 p. C. 80; 34 p. C.93,3333; 4 p. C. 106,6667; 5 p. C. 133,3333.

Years.
ཡཛྲ་ཡན1212]མས
+
PIKOHAL
HTHARL
3
سته است
HS
N
20
HONHAL
14234
24~24
∞
HAHAHA
❤
TAKCHAL QANLAH BERKHAL CARE-AL CANAL
14
Zapiral Sur
17
18
19
20
RATIO of INTEREST:
ANNUALLY, 0.0375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
25
0.9638 554
0.9290 173 0.4907 976 0.5282 976
0.8954 384
0.8630 731
0.8318 777
0.8018 098
1.0000 000 1.0375 000
0.3211 400
0.6920 205
0.2363 687 | 0.2738 687
0.1855 519 0.2230 519
0.1517 122
0.7728 287 0.1275 737 | 0.1650 737
0.3586 400
0.7448 952 0.1094 984 0.1469 984
0.7179 713 0.0954 652
0.0954 652 0.1329 652
21
0.4615 829
22
0.4448 993
23 0.4288 186
24
0.4133 191
0.1892 122
..
0.6670 077 0.0751 152
0.0842 614 0.1217 614
0.6428 990 0.0675 123
13 0.6196 616 0.0610 964 0.0985 964
....
0.0751 152 | 0.1126 152
0.1050 123
0.5972 642 | 0.0556 132 0.0931 132
0.5756 764 0.0508 759 0.0883 759
0.5548 688 | 0.0167 448
0.0842 448
0.5348 133 0.0431 128 0.0806 128
0.5154 827 0.0398 966 | 0.0773 966
0.0370 306 0.0745 306
0.0341 621 | 0.0719 621
0.4968 508
0.1788 923
0.0321 486 0.0696 486
0.0300 553 0.0675 553
0.0281 534 0.0656 534
0.0264 189 0.0639 189
0.3983 799 0.0248 317 0.0623 317
RATIO of INTEREST:
HALF-YEARLY, 0.01875.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9815 951 2.0000 000
0.9635 291 0.9907 121
0.9457 953 | 0.6543 215 | 0.6918 215
0.9283 880
0.4861 547
0.5236 547
0.9113 011
0.8945 286
0.8780 650
0.8619 013
0.7854 543
0.7709 981
0.3852 786 0.4227 786
0.3180 4690.3555 469
0.2700 408
0.2340 508
0.8160 410 0.2060 713 0.2435 713
0.8304 697 0.1836 994 | 0.2211 994
0.7026 114
0.6896 799
0.8151 850 0.1654 056 | 0.2029 056
0.8001 816 0.1501 704 0.1876 704
0.1372 879
0.1262 541
0.7568 079 0.1166 991
0.7428 789 0.1083 457
···
2.0375 000
1.0282 121
0.5834 974
0.5727 582
0.6522 960 | 0.0703 503
0.6402 905 0.0667 508
0.3075 408
0.2715 508
0.7292 063 | 0.1009 818 0.1384 818
0.7157 853 | 0.0944 425 0.1319 425
...
……….
....
....
0.0885 976 | 0.1260 976
0.0833 430 0.1208 430
0.6769 864 0.0785 942 0.1160 942
|
0.6645 265 0.0742 823 0.1117 823
0.1747 879
0.1637 541
0.1541 991
0.1458 457
...
...
0.6285 060 | 0.0634 438 | 0.1009 438
0.6169 385 0.0603 955 | 0.0978 955
0.1078 503
0.1042 508
....
0.6055 837 0.0575 772 0.0950 772
0.5944 381 0.0549 643
0.0549 643 | 0.0924 643
0.0525 355 | 0.0900 355
0.0502 723 0.0877 723
0.5622 167 | 0.0481 588 0.0856 588
0.5518 691 0.0461 809 | 0.0336 809
0.5317 419 0.0425 840 0.0800 840
0.5123 487 0.0393 992 | 0.0768 992
0.4936 628 | 0.0365 613 | 0.0740 613
0.4756 584 0.0340 183 0.0715 183
0.4583 106 0.0317 279 | 0.0692 279
0.4415 955 | 0.0296 556 | 0.0671 556
0.4254 900 0.0277 730 | 0.0652 730
0.4099 720 | 0.0260 563 | 0.0635 563
0.3950 198 | 0.0244 855 0.0619 855
RATIO of INTEREST:
QUARTERLY, 0.009375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9907 121 4.0000 000
0.9815 104 1.9906 687
0.9723 942 1.3209 111
0.9633 627 | 0.9860 469
0.9544 151
0.9455 505
0.9367 683
0.9280 677
0.8533 099
0.8453 844
0.8375 325
0.8297 535
0.8857 617 | 0.2907 614
0.8775 319
0.8693 844
0.8613 097
0.2687 096
0.2496 020
0.2328 865
0.8220 469 |
0.8144 118
0.8068 475
0.7993 536
0.919 479
0.9109 081
0.9024 477 | 0.3169 090
0.4280 369
0.3834 137
0.4655 369.
0.4209 137
0.3844 090
0.8940 658 | 0.3164 932 | 0.3539 932
0.7919 293
0.7845 739
0.7772 868
0.7700 675
0.7629 151
0.7558 292
0.7488 092
0.7418 542
0.6820 963
0.6757 610
0.6694 846
0.6632 665
0.7851 598
0.6512 118
0.6571 061
0.6510 030
0.6449 565
0.6389 662
0.5555 571
0.4838 234
0.6330 315
0.6271 520
0.6213 271
0.6155 562
0.7349 640 | 0.1039 902
0.7281 376 0.1004 375
0.7213 748 0.0970 894
0.7146 747 | 0.0939 289
0.5301 845
0.5107 599
0.4920 471
0.4740 198
0.1427 272
0.1365 736
0.1308 780
0.1255 913
0.4566 529
0.4399 225
0.4238 049
0.4082 778
0.2181 409
0.2556 409
0.2050 369 | 0.2425 369
0.1933 154 0.2308 154
0.1827 689 0.2202 689
0.1206 712
0.1160 810
0.1117 889
0.1077 668
0.1732 296 | 0.2107 296
0.1645 602 0.2020 602
0.1566 472 | 0.1941 472
0.1493 960
0.1493 960 | 0.1868 960
4.0375 000
2.0281 687
1.3584 111
1.0235 469
0.7080 369 0.0909 409
0.7014 607 0.0881 116
0.6949 455 0.0854 289 | 0.1229 289
0.6884 910 | 0.0828 817
0.1203 817
0.0804 603
0.0781 554
0.0759 592
0.0738 640
0.8226 598
0.6887 118
0.5930 571
0.5213 234
0.6098 390 0.0586 141
0.6041 748
0.0572 388
0.5985 633 | 0.0559 145
0.5930 039 0.0546 385
0.0646 886
0.0630 772
0.0615 300
0.0600 435
0.3282 614
0.3062 096
0.2871 020
0.2703 865
0.5659 718
0.5607 151
0.0488 999
0.0478 660
0.5555 072 0.0468 658
0.5503 477
0.0458 978
0.1802 272
0.1740 736
0.1683 780
0.1630 913
0.1581 712
0.1535 810
0.1492 889
0.1452 668
0.1414 902
0.1379 375
0.1345 894
0.1314 289
0.0718 633
0.0699 508
0.0681 209
0.1093 633
0.1074 508
0.1056 209
0.0663 684 | 0.1038 684
0.1284 409
0.1256 116
0.1179 603
0.1156 554
0.1134 592
0.1113 610
0.0534 082
0.5874 961
0.0909 082
0.5820 395
0.0522 214 0.0897 214
0.5766 335 0.0510 757 0.0885 757
0.5712 778 | 0.0499 692 | 0.0874 692
0.1021 886
0.1005 772
0.0990 300
0.0975 435
0.0961 141
0.0947 388
0.0934 145
0.0921 385
0.0863 999
0.0853 660
0.0843 658
0.0833 978
0.0423 186 | 0.0798 186
0.0391 495 | 0.0766 495
0.0363 257 0.0738 257
0.0337 955 | 0.0712 955
0.0315 167 0.0690 167
0.0294 550 | 0.0669 550
0.0275 821 0.0650 821
0.0258 743 0.0633 743
0.3933 196 0.0243 118 | 0.0618 118
Page 15.
RATE OF INTEREST: 33 per Cent.
CORRESPONDING VALUE OF
per
Ann.

Years.
26
27
28
29
30
ZZZZZ 18838 **200 gyugur erercneer CARRA ZA*** *☺☺☺o co co co co co
33
34
31 3.1306 266
32 3.2480 251
3.3698 260
3.4961 945
36
37
40
42
43
44
46
48
49
50
41 4.5238 930
4.6935 389
4 8695 467
5.0521 547
45 5.2416 105
55
56
62
63
70
71
73
74
75
76
77
78
79
80
81
82
83
13 85338 38836 85888
96
RATIO of INTEREST:
ANNUALLY, 0.0375.
99
100
Improved Cumulated Value of
Principal. Annuity.
Annuity.
2.6042 984 | 42.781 290
2.7019 596 45.385 588
2.8032 830 48.087 548
50.890 831
3.0174 714 | 53.799 237
2.9084 062
3.6273 018 | 70.061 381
3.7633 256 73.688 682
3.9044 503
4.0508 672
4.2027 747
4.3603 788
5.4381 709
5.6421 023
5.8536 811
6.0731 941
6.3009 389
6.5372 241
6.7823 700
7.0367 089
7.3005 855
7.5743 574
7.8583 958
8.1530 857
8.4588 264
8.7760 324
9.1051 336
56.816 709
59.947 335
63.195 360
66.565 186
13.650 718
14.162 620
14.693 718
15.244 732
15.816 410
16.409 525
17.024 882
17.663 315
18.325 690
19.012 903
77.452 008
81.356 458
85 407 326
89.610 100
19.725 887
20.465 608
21.233 068
22.029 308
22.855 407
93.970 479
98 494 372
103.18 791
108.05 746
113.10 961
118.35 122
123.78 939
129.43 150
135.28 518
141.35 837
9.4465 761 225.24 203
9.8008 227
234.68 861
10.168 354 244.48 943
10.549 667
254.65 778
10.945 279 265.20 745
147.65 931
154.19 653
160.97 890
168.01 561
175.31 620
11.355 727 276.15 273
11.781 567
12.223 376
12.681 752
13.157 318
287.50 846
299.29 002
311.51 340
324.19 515
182.89 056
190.74 895
198.90 204
207.36 086
216.13 690
337.35 247
351.00 319
365.16 581
379.85 952
395.10 426
410.92 067
427.33 019
444.35 507
462.01 839
480.34 408
499.35 698
519.08 287
539.54 848
560.78 154
582.81 085
23.712 485 605.66 626
24.601 703 629.37 874
25.524 267 653.98 045
26.481 427 679.50 471
27.474 480 705.98 614
28.504 773 733.46 062
29.573 702 761.96 539
30.682 716 | 791.53 909
31.833 318
33.027 067
16.427 18
16.797 29
17.154 01
17.497 84
17.829 24
34.265 582
35.550 542
36.883 687
38.266 825 993.78 201
39.701 831
1032.0 488
18.148 67
18.456 55
18.753 30
19.039 33
19.315 01
19.580 73
19.836 85
20.083 71
20.321 65
20.550 99
20.772 04
20.985 10
21.190 46
21.388 39
21.579 17
21.763 06
21.940 30
22.111 13
22.275 79
22.434 49
22.587 46
22.734 90
22.877 02
23.013 99
23.146 02
23.273 27
23.395 92
23.514 14
23.628 09
23.737 92
23.843 77
23.945 81
24.044 15
24.138 94
24.230 30
24.318 37
24.403 24
24.485 05
24.563 91
24.639 91
24.713 17
24.783 78
24.851 83
24.917 43
24.980 65
25.041 59
25.100 33
25.156 95
25.211 51
25.264 11
25.314 81
25.363 67
25.410 76
25.456 16
25.499 91
25.542 08
25.582 73
25.621 91
25.659 67
25.696 07
25.731 15
25.764 96
25.797 56
822.22 181 25.828 97
854.05 513 25.859 25
25.888 43
25.916 56
956.89 832 25.943 67
887.08 220
921.34 778
25.969 81
25.994 99
Value of Perpetual Ann. 26.666 67
RATIO of INTEREST:
HALF-YEARLY, 0.01875.
Improved Cumulated Value of
Principal. Annuity. Annuity.
2.6273 403 43.395 742
2.7267 893-46.047 714
2.8300 025 48.800 066
2.9371 225 | 51.656 600
3.0482 972 54.621 258
3.5367 005
3.6705 701
57.698 133
60.891 473
3.4077 132 | 64.205 685
3.1636 800
3.2834 302
67.645 346
71.215 203
4.5871 716
4.7608 032
4 9410 070
5.1280 318
5.3221 359
3 8095 069 74.920 184
3.9537 027 78.765 405
4.1033 565 82.756 174
4.2586 750 86.898 000
4.4198 725 | 91.196 600
6.6511 604
6.9029 172
7.1642 034
7.4353 797
7.7168 204
8.0089 141
8.3120 640
8.6266 886
8.9532 223
9.2921 157
5.5235 870 | 120.62 899
5.7326 634
5.9496 537
6.1748 574
6.4085 854
126.20 436
131.99 077
137.99 620
144.22 894
9.6438 368
10.008 871
10.387 723
10.780 914
11.188 988
11.612 509
12.052 061
12.508 250
12.981 707
13.473 085
13.983 062 |
14.512 343
16.837 535
17.474 862
18.136 313
18.822 800
19.535 273
24.413 550
25.337 641
26.296 711
27.292 082
28.325 131
29.397 281
30.510 014
31.664 865
32.863 430
34.107 362
95.657 908
100.28 808
105.09 352
110 08 085
115.25 696
346.21 499
360.32 914
15.061 658 | 374.97 754
15.631 765 390.18 040
16.223 452 | 405.95 871
39.572 101
41.069 967
150.69 761
157.41 112
164.37 876
171.61 012
179.11 521
186.90 438
194.98 837
203.37 836
212.08 593
221.12 309
230.50 232
240.23 656
250.33 927
260.82 437
271.70 636
283.00 025
294.72 162
306.88 667
319.51 219
332.61 560
20.274 713 | 513.99 236
21.042 143
21.838 621
22.665 247
23.523 162
422.33 426
439.32 965
456.96 834
475.27 468
494.27 394
534.45 715
555.69 656
577.73 992
600.61 765
624.36 134
649.00 377
674.57 895
701.12 219
728.67 014
757.26 082
786.93 370
817.72 974
849.69 147
882.86 299
35.398 379
917.29 011
36.738 263 | 953.02 035
38.128 864
990.10 303
1028.5 894
1068.5 324
16.516 98
16.867 16
17.243 82
17.587 49
17.918 61
18.237 66
18.545 08
18.841 28
19.126 68
19.401 67
19.666 64
19.921 94
20.167 92
20.404 94
20.633 31
20.853 35
21.065 37
21.269 66
21.466 49
21.656 15
21.838 89
22.014 96
22.184 61
22.348 08
22.505 58
22.657 34
22.803 56
22.944 46
23.080 21
23.211 01
23 337 04
23.458 48
23.575 48
23.688 22
23.796 85
23.901 52
24.002 36
24.099 53
24.193 16
24.283 37
24.370 29
24.454 04
24.534 74
24.612 49
24.687 41
24.759 60
24.829 15
24.896 17
24.960 74
25.022 95
25.082 90
25.140 67
25.196 32
25.249 94
25.301 62
25.351 40
25.399 37
25.445 59
25.490 12
25.533 04
25.574 38
25.614 21
25.652 60
25.689 58
25.725 22
25.759 55
25.792 64
25.824 52
25.855 23
25.884 82
25.913 34
25.940 81
25.967 28
25.992
26.017 37
79
26.666 67
RATIO of INTEREST:
QUARTERLY, 0.009375.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
2.6391 531 | 43.710 748
2.7395 218 46.387 247
2.8437 075 | 49.165 534
2.9518 556 52.049 482
3.0611 166 55.043 108
3.1806 469
3.3016 090
3.4271 713
3.5575 088
3.6928 032
3.8332 429 | 75.553 143
3.9790 236
4.1303 484
4.2874 283
79.440 629
83.475 958
87.664 753
4.4504 819 | 92.012 851
4.6197 366
4.7954 282
4.9778 015
5.1671 105
5.3636 190
5.5676 009
5.7793 401
5.9991 324
6.2272 833
6.4641 110
6.7099 453
6.9651 288
7.2300 172
7.5049 794
7.7903 987
8.0866 726
8.3942 140
8.7134 513
9.0448 295
9.3888 103
58.150 584
61.376 239
64.724 567
68.200 235
71.808 085
17.059 778
17.708 573
18.382 041
19.081 122
19.806 789
96.526 310
101.21 142
106.07 471
111.12 295
116.36 317
20.560 054
21.341 967
22.153 615
22.996 132
23.870 690
121.80 269
127.44 908
133.31 020
139.39 422
145.70 963
152.26 521
159.07 010
166.13 379
173.46 612
181.07 730
9.7458 728
10.116 515
10.501 252
10.900 622
11.315 179275.07 145
29.862 490
30.998 180
32:177 061
33.400 775
34.671 028
188.97 794
197.17 904
205.69 204
214.52 879
223.70 161
11.745 503 286.54 675
12.192 192 298.45 845
12.655 869
310.82 317
13.137 180
323.65 813
13.636 795336.98 121
233.22 327
243.10 706
253.36 673
264.01 658
14.155 412
14.693 751
350.81 097
365.16 669
15.252 564380.06 837
15.832 629 395.53 677
16.434 754 | 411.59 344
428.26 075
415.56 194
463.52 109
482.16 325
501.51 438
521.60 145
542.45 244
564.09 641
586.56 351
609.88 505
24.778 507
634.09 353
25.720 850 | 659 22 267
685.30 748
26.699 031
27.714 412
28.768 409
712.38 432
740.49 091
769.66 640
799.95 147
831.38 829
864.02 068
897.89 409
35.989 590933.05 574
37.358 297 969.55 460
38.779 058 1007.4 415
40.253 850 1016.7 693
41.784 730 | 1087.5 928
-
16.562 41
16.932 61
17.289 24
17.632 80
17.963 78
18.282 63
18.589 80
18.885 71
19.170 78
19.445 41
19.709 98
19.964 86
20.210 39
20.446 93
20.674 81
20.894 33
21.105 82
21.309 55
21.505 82
21.694 90
21.877 05
22.052 51
22.221 58
22.384 44
22.541 33
22.692 46
22.838 07
22.978 34
23.113 47
23.243 65
23.369 06
23.489 88
23.606 27
23.718 39
23.826 41
23.930 47
24.030 71
24.127 29
24.220 32
24.309 95
24.396 30
24.479 47
24.559 61
24.636 80
24.711 17
24.782 82
24.851 84
24.918 33
24 982 38
25.044.09
25.103 54
25.160 80
25.215 98
25.269 12
25.320 33
25.369 65
25.417 17
25.462 95
25.507 05
25.549 54
25.590 47
25.629 89
25.667 88
25.704 47
25.739 73
25.773 68
25.806 40
25.837 92
25.868 28
25.897 53
25.925 71
25.952 86
25.979 01
26.004 20
26 028 48
26.666-67
Page 16.
PERPETUAL STOCK: 3p. C. 80; 3 p. C. 93,3333; 4p. C. 106,6667; 5 p. C. 133,3333.

Years.
O7:28 M**** 85889 #3333 95998 E88ËR AKA88 58888 85882 F2P2P 2E028 ***** ***** 48833 85838
46
50
56
60
64
66
68
69
70
72
73
58 0.1182 197
0.1139 467
0.1098 282
7.4
RATIO of INTEREST:
ANNUALLY, 0.0375.
83
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
100
0.0233 747
0.3839 806
0.0608 747
0.3701 017 | D.0220 334 0.0595 334
0.0207 954 0.0582 954
0.0196 499 0.0571 499
0.0185 876 | 0.0560 876
0.3567 246
0.3438 309
0.3314 033
0.3194 249
0.3078 794
0.2967 512
0.2860 253
0.2756 870
0.2657 224
0.2561 182
0.2468 607
0.2379 380
0.2293 379
0.2210 486
0.2130 589
0.2053 579
0.1979 354
0.1907 811
0.1838 854
0.1772 389
0.1708 327
0.1646 580
0.1587 065
0.1529 701
0.1474 411
0.1421 119
0.1369 753
0.1320 244
0.1272 524
0.1226 529
0.1058 581
0.1020 322
0.0983 443
0.0947 897
0.0913 636
0.0820 613
0.0848 784
0.0818 105
0.0788 535
0.0760 033
75 0.0632 255
0.0176 005
0.0166 813
0.0158 239
0.0150 229
0.0142 732
0.0609 402
0.0587 376
0.0566 145
0.0545 682
0.0525 959
0.0135 706 0.0510 706
0.0129 112 0.0504 112
0.0122 916 0.0497 916
0.0117 086
0.0492 086
0.0111 595 | 0.0486 595
0.0106 416
0.0101 529
0.0096 911
0.0092 543
0.0088 410
0.0084 494
0.0080 782
0.0077 261
0.0073 918
0.0070 742
0.0067 723
0.0064 852
0.0050 276
0.0048 225
0.0046 267
0.0551 005
0.0541 813
0.0533 239
0.0525 229
0.0517 732
0.0442 723
0.0139 852
0.0062 120 | 0.0437 120
0.0059 518
0.0057 040
0.0434 518
0.0432 040
0.0014 397
0.0042 610
0.0040 902
0.0039 268
0.0037 706
0.0054 678 | 0.0429 678
0.0052 425
0.0427 425
0.0425 276
0.0423 225
0.0421 267
0.0036 212
0.0034 782
0.0033 412
0.0032 101
0.0030 846
0.0481 416
0.0476 529
0.0471 911
0.0467 543
0.0463 410
0.0459 494
0.0455 782
0.0452 261
0.0448 918
0.0445 742
0.0732 562
0.0706 084
0.0680 563
0.0029 643
0 0028 490
0.0027 385
0.0655 964 | 0.0026 326 0.0401 326
0.0404 643
0.0403 490
0.0402 385
0.0025 310 0.0400 310
0.0419 397
0.0417 610
0.0415 902
0.0414 268
0.0412 706
0.0421 719
0.0406 476
0.0391 784
0.0377 623
0.0363 974 | 0.0014 165
0.0411 212
0.0409 782
0.0408 412
0.0407 101
0.0405 846
0.0024 336 0.0399 336
0.0023 401 0.0398 401
0.0022 505 0.0397 505
0.0021 644 0.0396 644
0.0020 818 0.0395 818
0.0395 026
0.0506 948 0.0020 026
0.0488 625 | 0.0019 265 0.0394 265
0.0470 964 0.0018 534
0.0453 941 | 0.0017 832
0.0437 533
0.0393 534
0.0392 832
0.0017 158 0.0392 158
0.0016 511
0.0391 511
0.0015 889 0.0390 889
0.0015 291 0.0390 291
0.0014 717 0.0389 717
0.0389 165
0.0350 819
0.0338 138
0.0013 634
0.0013 124
0.0388 634
0.0388 124
0.0325 916 | 0.0012 634 0.0387 634
0.0314 136
0.0302 782
0.0012 162 0.0387 162
0.0011 709 0.0386 709
0.0291 838 | 0.0011 273
0.0281 290
0.0271 123
0.0010 854
0.0010 450
0.0386 273
0.0385 854
0.0385 450
0.0385 063
0.0384 689
0.0261 323 0.0010 063
0.0251 878 | 0.0009 689
Ratio of Perpetual Ann. 0.0375 000
RATIO of INTEREST:
HALF-YEARLY, 0.01875.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3806 130
0.3667 317
0.3533 566
0.3404 693
0.3280 520
0.3160 876 0.0173 316
0.3045 595 | 0.0164 227
0.2934 519 0.0155 749
0.2827 494 0.0147 830
0.2724 372 0.0140 419
0.2625 011
0.2529 275
0.2437 029
0.2348 148
0.2262 509
0.1810 418
0.1744 390
0.1680 770
0.1619 471
0.1560 407
0.2179 993 0.0104 539
0.2100 486
0.0099 713
0.2023 879
0.0095 153
0.1950 066 | 0.0090 842
0.1878 945
0.0086 763
0.1248 609
0.1203 071
0.1503 497
0.0066 358
0.1448 663 0.0063 5£8
0.1395 829 | 0.0060 835
0.1344 921
0.1295 871
0.0058 272
0.0055 830
0.0230 438
0.0217 166
0.0204 918
0.0193 586
0.0183 079
0.0053 503
0.0051 285
0.1159 193 | 0.0049 169
0.1116 916
0.1076 181
0.0047 151
0.0015 22-4
0.0861 140
0.0829 734
0.0799 472
0.0770 315
0.0742 220
0.0715 151
0.0689 069
0.0663 938
0.0639 723
0.0616 392
0.0133 475
0.0126 959
0.0120 837
0.0115 077
0.0109 653
0.1036 932
0.0043 384
0.0999 114 0.0041 626
0.0962 675 | 0.0039 946
0.0927 565 0.0038 340
0.0893 736 | 0.0036 804
0.0593 911
0.0572 251
0.0551 380
0.0531 271
0.0511 895
0.0493 225
0.0475 237
0.0457 904
0.0441 204
0.0425 113
0.0082 899
0.0079 237
0.0075 763
0.0072 466
0.0069 334
0.0409 609
0.0394 670
0.0380 276
0 0366 407
0.0353 043
0.0035 336
0.0033 930
0.0032 585
0.0031 298
0.0030 065
0.0028 884
0.0027 752
0.0026 668
0.0025 629
0.0024 633
0.0023 678
0.0022 762
0.0021 883
0.0021 040
0.0020 232
0.0019 456
0.0019 456
0.0018 711
0.0017 995
0.0017 309
0.0016 650
0.0016 016
0.0015 408
0.0014 824
0.0014 263
0.0013 724
0.0340 168 0.0013 205
0.0327 761 | 0.0012 708
0.0315 807 0.0012 229
0.0304 290 | 0.0011 769
0.0293 192 0.0011 327
0.0282 499
0 0010 902
0.0272 196 0.0010 493
0.0262 268 0.0010 100
0.0252 703 0.0009 722
0.0243 487 0.0009 359
0.0605 438
0.0592 166
0.0579 918
0.0568 586
0.0558 079
0.0548 316
0.0539 227
0.0530 749
0.0522 830
0.0515 419
0.0508 475
0.0501 959
0.0495 837
0.0490 077
0.0184 653
0.0479 539
0.0474 713
0.0470 153
0.0465 842
0.0461 763
0.0457 899
0.0454 237
0.0450 763
0.0447 466
0.0144 334
0.0441 358
0.0438 528
0.0435 835
0.0433 272
0.0430 830
0.0428 503
0.0426 285
0.0424 169
0.0422 151
0.0420 224
0.0418 384
0.0416 626
0.0414 946
0.0413 340
0.0411 804
0.0410 336
0.0408 930
0.0407 585
0.0406 298
0 0405 065
0.0403 884
0.0402 752
0.0401 668
0.0400 629
0.0399 633
0.0398 678
0.0397 762
0.0396 883
0.0396 040
0.0395 232
0.0394 456
0.0393 711
0.0392 995
0.0392 309
0.0391 650
0.0391 016
0.0390 408
0.0389 824
0.0389 263
0.0388 724
RATIO of INTEREST:
QUARTERLY, 0.009375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3789 094
0.3650 272
0.3516 536
0.3387 700
0.3263 584
0.3144 015
0.3028 826
0.2917 858
0.2810 956
0.2707 970
0.2608 757
0.2513 179
0.2421 103
0.2332 401
0.2246 948
0.2164 625
0.2085 320
0.2008 919
0.1935 318
0.1864 413
0.1490 325
0.1435 724
0.1383 123
0.1332 449
0.1283 631
0.1236 603
0.1191 297
0.1147 651
0.1105 604
0.1065 098
0.1026 075
0.0988 483
0.0952 267
0.0917 379
0 0883 769
0.1796 106
0.1730 301
0.0082 100
0.0078 463
0.1666 908 | 0.0075 013
0.1605 837
0.1547 003
0.0851 390
0.0820 197
0.0790 147
0.0761 198
0.0733 310
0.0706 444
0.0680 561
0.0655 628
0.0631 607
0.0608 467
0.0586 174
0.0564 698
0.0544 000
0.0524 078
0.0504 877
0.0486 380
0.0468 560
0.0451 394
0.0434 856
0.0418 924
0.0388 205
0.0334 868 |
0.0387 708 0.0322 600
0.0387 229 0.0310 780
0.0380 769
0.0386 327
0.0228 777
0.0215 577
0.0203 395
0.0192 125
0.0181 676
0.0299 394
0.0283 425
0.0171 967
0.0162 929
0.0154 501
0.0146 627
0.0139 259
0.0385 902
0.0277 858
0.0385 493 0.0267 678
0.0385 100 0.0257 871
0.0384 722 0.0248 42 3
0.0384 359
0.0239 322
· 0.0375 000
0.0132 357
0.0125 880
0.0119 795
0.0114 071
0.0108 680
0.0103 599
0.0098 803
0.0094 273
0.0089 990
0.0085 938
0.0071 739
0.0068 630
0.0065 675
0.0062 865
0.0060 192
0.0057 648
0.0055 225
0.0042 877
0.0041 134
0.0039 468
0.0037 876
0.0036 354
0.0103 576 | 0.0015 771
0.0388 790
0.0015 169
0.0374 515
0.0360 823.
0.0347 603
0.0603 777
0.0590 577
0.0578 395
0.0567 125
0.0556 676
0.0546 967
0.0537 929
0.0529 501
0.0521 627
0.0514 259
0.0019 172
0.0018 435
0.0017 727
0.0017 048
0.0016 397.|
0.0507 357
0.0500 880
0.0494 795
0.0489 071
0.0183 680
0.0052 916 0.0427 916
0.0050 715 | 0.0425 715
0.0048 616
0.0046 614
0.0044 702
0.0423 616
0.0421 614
0.0419 702
0.0478 599
0.0473 803
0.0469 273
0.0464 990
0.0460 938
0.0457 100
0.01453 463
0.0450 013
0.0146 739
0.0443 630
0.0012 993
0.0012 501
0.0012 028
0.0011 574
0.0011 137
0.0410 675
0.0437 865
0.0435 192
0.0432 648
0.0430 225
0.0034 898
0.0033 506
0.0032 173
0.0030 897
0.0409 898
0.0408 506
0.0407 173
0.0405 897
0.0029 675 | 0.0404 675
0.0028 505
0.0027 385
0.0026 311
0.0025 282
0.0403 505
0.0402 385
0.0401 311
0.0400 282
0.0024 296 | 0.0399 296
0.0417 877
0.0116 134
0.0414 468
0.0412 876
0.0411 354
0.0023 350 0.0398 350
0.0022 444 0.0397 444
0.0021 574 | 0.0396 574
0.0020 740 0.0395 740
0.0019 940 | 0.0394 940
0.0390 771
0.0390 169
0.0014 592 | 0.0389 592
0.0014 037 | 0.0389 037
0.0013 505
0.0388 505
0.0394 172
0.0393 435
0.0392 727
0.0392 018
0.0391 397
0.0387 993
0.0387 501
0.0387 028
0.0380 574
0.0386 137
0.0010 717
0.0010 314
0.0385 717
0.0385 314
.0.0009 925 | 0.0384 926
0.0009 553
0.0009 195
0.0384 553
0.0384 195
0.0375 000
Page 17.
RATE OF INTEREST: 4 per Cent. per Ann.
CORRESPONDING VALUE OF
Years.
03
p
CO
FINAN***
20
**
10
مراه سنتان
PUN-AL
∞
ANDT
ཝ-ve}༢ཏི
PWCHAP
مستتر
=<>:{**
PONHALL
14541710714211
用​T区 ​的​SAS 18
12
13
CPIN-A-
15
* 2 **** 225
19
हूँ

20
21
22
23
24
RATIO of INTEREST:
ANNUALLY, 0.04.
Improved Cumulated Value of
Principal. Annuity. Annuity.
...
1.0400 000
1.0816 000
1.1248 640 3.1216 000
1.1698 586 4.2464 610
1.2166 529 5.4163 226
1.2653 190 6.6329 755
1.3685 691
1.0000 000
1.3159 318 7.8982 945
1.4233 118
2.0400 000
1.4802 443
1.5394 541
1.6010 322
1.8009 435
9.2142 263
14 1.7316 764 18.291 911
………
10.582 795
12.006 107
13.486 351
15.025 805
1.6650 735 16.626 838
1.8729 812 21.824 531
18
17 1.9479 005 23.697 512
2.0258 165 25.645 413
2.1068 492 27.671 229
2.1911 231 29.778 079
20.023 588
2.2787 68131.969 202
2.3699 188 34.247 970
2.4647 155 36.617 889
2.5633 042 39.082 604
2.6658 363 41.645 908
0.9615 38
1.8860 95
2.7750 91
3.6298 95
...
4.4518 22
5.2421 37
6.0020 55
....
...
6.7327 45
7.4353 32
8.1108 96
8.7604 77
9.3850 74
9.9856 48
10.563 12
11.118 39
RATIO of INTEREST:
HALF-YEARLY, 0.02.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0200 000 0.5000 000
1.0404 000
1.0100 000
1.5302 000
1.0612 080
1.0824 322 | 2.0608 040
..
1.1040 808
1.1261 624
…………
1.1486 857
3.7171°417
1.1716 594 4.2914 845
1.1950 926 4.8773 142
1.2189 944 5.4748 605
1.2433 743
1.2682 418
...
1.2936 066 7.3401 658
1.3194 788
7.9869 691
1.3458 683
....
1.3727 857
1.4002 414
...
1.4282 462
1.4568 112
1.4859 474
1.5156 663
1.5459 797
...
2.6020 201
3.1540 605
1.5768 993
1.6084 372
6.0843 577
6.7060 449
8.6167 085
9.3196 426
12.165 67 1.9606 760
12.659 30 2.0398 873
13.133 94 2.1222 988
13.590 33 2.2080 397
...
14.029 16
2.2972 445
14.451 12 2.3000 531
14.856 84 2.4866 113
15.246 96
15.622 08
10.006 035
10.706 156
•
11.420 279
12.148 685
12.891 659
13.649 492
14.422 482
1.6406 062 | 16.015 150
1.6734 181 16.835 453
··
..
15.210 931
1.7068 865 17.672 162
1.7410 242
18.525 605
1.7758 447 19.396 117
1.8113 616 | 20.284 040
...
1.8475 888| 21.189 720
11.652 30 1.8845 406 22.113 515
24.016 901
25.997 184
28.057 470
30.200 992
32.431 112
34.751 327
37.165 282
2.5870 704 | 39.676 760
2.6915 880 42.289 701
0.4901 96
0.9707 80
1.4419 42
1.9038 64
2.3567 30
2.8007 15
3.2359 95
3.6627 41
4.0811 18
4.4912 92
4.8934 24
5.2876 71
5.6741 87
6.0531 24
6.4246 32
6.7888 55
7.1459 36
7.4960 16
7.8392 31
··
8.1757 17
8.5056 05
8.8290 24
9.1461 02
9.4569 63
9.7617 28
10.060 52
10.353 45
10.640 64
....
10.922 19
....
11.198 23
11.468 85
11.734 17
12.249 30
12.744 42
13.220 32
13.677 74
14.117 39
14.539 98
14.946 16
15.336 56
15.711 80
RATIO of INTEREST:
QUARTERLY, 0.01.
Improved Cumulated, Value of
Principal. Annuity.
Annuity.
1.0100 000
1.0201 000
1.0303 010
1.0406 040
1.0510 101
1.0615 202
1.0721 354
1.0828 567
1.0936 853
1.1046 221
1.1380 933
1.1494 742
1.1609 690
1.1725 786
1.1843 044
1.1961 475
1.2081 090
1.2201 900
1.1156 683 | 2.8917 087
1.1268 250
3.1706 258
1.4450 765
1.4595 272
1.4741 225
1.4888 637
0.2500 000
0.5025 000
0.7575 250
1.0151 003
1.2323 919 | 5.8097 985
1.2147 159
1.2571 630
1.2697 346
6.1178 964
6.4290 755
6.7433 662
1.5037 524
1.5187 899
1.5339 778
1.5493 176
1.2752 513
1.5380 038
1.8033 838
2.0714 176
1.2821 320 | 7.0607 998
1.2952 563
7.3814 079
1.3082 089
1.3212 910
1.5648 107
1.5804 589
1.5962 631
1.6122 261
2.3421 318
2.6155 531
1.6283 483
1.6446 318
1.6610 781
1.6776 889
1.3345 039
1.3478 489
8.3625 969
8.6962 229
1.3613 274 | 9.0331 851
1.3749 407 9.3735 170
1.6944 658
1.7114 105
1.7285 246
1.7458 098
3.4523 320
3.7368 553
4.0242 239
4.3144 661
1.3886 901 | 9.7172 521
1.4025 770
10.064 425
1.4166 028 10.415 069
1.4307 688 | 10.769 220
4.6076 108
4.9036 869
5.2027 238
5.5047 510
1.9672 222
2.0470 993
2.1302 198
2.2167 152
7.7052 219
8.0322 742
11.126 912
11.488 181
11.853 063
12.221 593
12.593 809
12.969 747
13.349 445
13.732 939
14.120 269
14.511 471
14.906 586
15.305 652
15.708 708
16.115 796
16.526 953
16.942 223
17.361 645
17.785 262
18.213 114
18.615 245
1.7632 679 | 19.081 698
1.7809 006 19.522 515
1.7987 09619.967 740
1.8166 967 | 20.417 417
1.8348 637 | 20.871 592
1.8532 123 21.330 308
1.8717 444 | 21.793 611
1.8904 619 22.261 547
24.180 555
26.177 483
28.255 494
30.417 880
2.3067 227
2.4003 849
32.668 069
35,009 623
2.4978 502 37.446 255
2.5992 729 | 39.981 823
2.7048 138
42.620 346
0.2475 25
0.4925 99
0.7352 46
0.9751 91
1.2133 58
1.4488 69
1.6820 49
1.9129 19
2.1415 04
2.3678 26
2.5919 07
2.8137 69
3.0334 35
3.2509 26
3.4662 63
/3.6794 68
3.8905 63
4.0995 67
4.3065 02
4.5113 88
4.7142 46
4.9150 95
5.1139 55
5.3108 47
5.5057 89
5.6988 01
5.8899 02
6.0791 11
6.2664 46
6.4519 27
6.6355 71
6.8173 97
6.9974 23
7.1756 66
7.3521 45
7.5268 76
7.6998 77
7.8711 66
8.0107 58
8.2086 71
8.3749 22
8.5395 27
8.7025 02
8.8638 63
9.0236 27
9.1818 09
9.3384 25
9.4934 90
9.6470 20
9.7990 29
9.9495 34
10.098 55
10.246 09
10.392 17
10.536 80
10.680 00
10.821 78
10.962 16
11.101 15
11.238 76
11.375 01
11.509 91
11.643 48
11.775 72
12.291 72
12.787 60
13.264 12
13.722 05
14.162 11
14.585 00
14.991 39
15.381 93
15.757 22
Page 18.
PERPETUAL STOCK: 3 p. C.75; 3 p. C.87,5; 4 p. C. 100; 5 p. C. 125.

Years.
명예심​을 ​기
​1
PKEUP
20
♡
241254
**
2
AKC-PY
L
PICISTALL
--- ***
7
K
CC
*N**
H42:04 O
HAT
****
COPROLPH
HENN
GARE-PL
15
T** 700 28** 3
17
18
19
20
RATIO of INTEREST :
ANNUALLY, 0.04.
21
22
23
24
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
25
··
0.9615 384 1.0000 000
0.9245 562 0.4901 961
0.8889 963
0.8518 042
0.8219 271
0.4901 961 0.5301 961
12 0.6245 971 | 0.0665 522
0.3203 485
13 0.6005 741 0.0601 437
0.1846 271
0.7903 145 | 0.1507 619
0.2354 900 0.2754 900
0.7306 902 | 0.1085 278
0.7025 867 | 0.0944 930
0.1507 619 0.1907 619
0.7599 178 0.1266 096 0.1666 096
0.6755 642 | 0.0832 909
0.6495 810 0.0741 490
0.5339 082
0.5133 733
0.4936 281
0.4746 424
0.4563 870
0.4388 336
0.4219 554
0.4057 254
0.3901 215
0.3751 168
1.0400 000
0.5552 645 | 0.0499 411
0.3603 485
14 0.5774 751 0.0546 690 0.0946 690
0.2246 271
0.0312 801
0.0291 988
0.0273 090
0.0255 868
0.0240 120
0.1485 278
0.1344 930
0.1232 909
0.1141 490
0.1065 522
0.1001 437
0.0899 411
0.0458 200
0.0858 200
0.0421 985 0.0821 985
0.0389 933 0.0789 933
0.0361 386 0.0761 386
0.0335 818 0.0735 818
RATIO of INTEREST:
HALF-YEARLY, 0.02.
0.0712 801
0.0691 988
0.0673 090
0.0655 868
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9803 921 2.0000 000
0.9611 688 | 0.9900 990
0.9423 223 0.6535 094 0.6935 094
0.9238 453 0.4852 475
0.5252 475
...
0.9057 308
0.3843 168
0.4243 168
0.8879 714 0.3170 516 0.3570‍516
0.8367 553
0.8203 483
0.8705 602 | 0.2690 239 | 0.3090 239
0.8534 904 0.2330 196 | 0.2730 196
0.8042 630
0.1643 559
0.7884 932 0.1491 192
0.7430 147
0.7284 458
...
0.2050 309
0.1826 531
0.7730 326 | 0.1362 367 | 0.1762 367
0.7578 751 0.1252 040 0.1652 040
0.7141 626 | 0.0999 397
0.7001 594 0.0934 042
....
0.0875 635
0.6864 307
0.6729 713 0.0823 134
....
D
2.0400 000
1.0300 990
30
...
...
....
0.4353 041
0.4184 007
0.4021 537
0.3865 376
0.0640 120 0.3715 279
0.1156 509 0.1556 509
0.1073 003
0.1473 003
....
0.2450 309
0.2226 531
..
0.2043 559
0.1891 192
....
··
0.1399 397
0.1334 042
•
0.6597 758 0.0775 695 0.1175 695
0.6468 390 0.0732 628
0.1132 628
0.1275 635
0.6341 559 0.0693 362 0.1093 362
0.1057 422
..
0.6217 215 0.0657 422
...
0.1223 134
0.6095 309 | 0.0624 409
0.1024 409
0.5975 793 0.0593 985 0.0993 985
....
0.5858 621 0.0565 862 | 0.0965 862
0.5743 745 0.0539 793 | 0.0939 793
0.5631 123 0.0515 567 0.0915 567
0.5520 709 0.0492 998 | 0.0892 998
0.5412 460 0.0471 927 0.0871 927
0.5306 333
0.0452 212
0.0852 212
0.5100 282
0.4902 232
0.0416 373
0.0384 657
0.4711 872 | 0.0356 411
0.4528 904
0.0816 373
0.0784 657
0.0756 411
0.0731 115
0.0331 115
0.0308 346 0.0708 346
0.0287 759 | 0.0687 759
0.0269 068 | 0.0669 068
0.0252 037 | 0.0652 037
0.0236 464 | 0.0636 461
RATIO of INTEREST:
QUARTERLY, 0.01.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9900 990
0.9802 960
0.9705 901
0.9609 804
0.9514 657
0.9420 453
0.9327 181
0.9234 832
0.9143 398
0.9052 870
0.8963 237
0.8871 492
0.8786 625
0.8699 629
0.8613 491
0.8528 212
0.8443 775
0.8360 173
0.8277 399
0.8195 444
0.8114 301
0.8033 962
0.7954 418
0.7875 662
0.7797 684
0.7720 480
0.7641 039
0.7568 356
0.7493 422
0.7419 229
0.7345 771
0.7273 041
0.7201 031
0.7129 733
0.7059 143
0.6989 249
0.6920 049
0.6851 533
0.6783 697
0.6716 531
0.6650 031
0.6584 189
0.6518 999
0.6154 454
0.6390 550
0.6327 277
0.626+ 630
0.6202 604
0.6141 192
0.6080 388
0.6020 187
0.5960 581
0.5901 565
0.5843 134
0.5785 271
0.5728 001
0.5671 288
0.5615 136
0.5559 541
0.5504 496
0.5449 996
0.5396 036
0.5342 692
0.5289 713
0.5083 310
0.4884 961
0.4694 351
0.4511 179
0.4335 155
0.4165 998
0.4003 412
0.3847 229
0,3697 112
4.0000 000
1.9900 497
1.3200 885
0.9851 244
0.7841 592 0.8241 592
0.6501 935 | 0.6901 935
0.5945 131
0.5227 611
0.5545 131
0.4827 611
0.4269 614 | 0.1659 614
0.4223 283
0.3823 283
0.3458 163
0.3153 951
0.3858 163
0.3553 951
0.2896 593
0.2676 047
0.2484 951
0.2317 784 | 0.2717 784
0.1721 230
0.1634 549
0.1555 43.4
0.1482 939
0.2170 322 0.2570 322
0.2039 282 | 0.2439 282
0.1922 070 0.2322 070
0.1816 613 | 0.2216 613
0.1416 270
0.1354 755
0.1297 821
0.1244 977
0.1029 098
0.0993 599
0.0960 147
0.0928 572
4.0400 000
2.0300 497
1.3600 885
1.0251 244
0.0794 041
0.0771 025
0.1195 801 0.1595 801
0.1149 925 | 0.1549 925
0.1107 029
0.1066 835
0.1507 029
0.1466 835
0.3296 593
0.3076 047
0.2884 951
0.0708 202
0.0689 110
0.0670 799
0.0653 353
0.0898 722 | 0.1298 722
0.0898 722
0.0870 460
0.0843 664
0.0818 224
0.1270 460
0.1243 664
0.1218 224
0.0575 982
0.0562 263
0.0519 055
0.0536 330
0.2121 230
0.2034 549
0.1955 434
0.1882 939
0.1816 270
0.1754 755
0.1697 821
0.1641 977
0.1194 041
0.1171 025
0.0749 095 | 0.1149 095
0.0728 176
0.1128 176
0.0479 120
0.0468 817
0.0158 850
0.0149 205
0.0413 555
0.0382 008
0.0353 913
0.0328 754
0.1429 098
0.1393 599
0.1360 147
0.1328 572
0.0636 590 | 0.1036 590
0.0620 509
0.0605 072
0.0590 241
0.1020 509
0.1005 072
0.0990 241
0.1108 202
0.1089 110
0.1070 799
0.1053 353
0.0524 062 | 0.0924 062
0.0512 229 0.0912 229
0.0500 808 0-0900 808
0.0189 778 | 0.0889 778
0.0975 982
0.0962 263
0.0949 055
0.0936 330
0.0879 120
0.0868 817
0.0858 850
0.0849 205
0.0813 555
0.0782 008
0.0753 913
0.0728 754
0.0306 109 0.0706 109
0.0285 636 | 0.0685 636
0.0267 049 0.0667 049
0.0250 114 0.0650 114
0.0234 630 0.0634 630
Page 19.
RATE OF INTEREST. 4
Ann.-CORRESPONDING VALUE OF
per
Cent. per

Years.
2018 2029 203
26
27
2.7724 698
2.8833 686
2.9987 033
3.1186 515
30 3.2433 975
29
34
35
31 3.3731 334
32 3.5080 587
33 3.6483 811
3.7943 163
3.9460 890
Oma 3333 95838 583
37
38
36 4.1039 326
4.2620 899
4.4388 135
4.6163 660
4.8010 206
39
40
41
45
4.9930 615
42 5.1927 839
43 5.4004 953
44 5.6165 151
5.8411 757
47
RATIO of INTEREST:
ANNUALLY, 0.01.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
46 6.0748 227
6.3178 156
49
48 6.5705 232
6.8333 404
50 7.1066 833
85888 58858 85888 F2CJR DER28 78*** ***** **346 85888
51 7.3909 507
52 7.6865 887
7.9940 523
5+ 8.3138 143
8.6463 669
55
91
92
100
8.9922 216
9.3519 105
9.7259 869
10.115 026
10.519 627
10.940 413
11.378 029
11.833 150
12.306 476
12.798 735
13.310 685
13.843 112
14.396 836
14.972 710
15.571 618
16.194 483
16.842 262
17.515 953
18.216 591
18.945 255
19.703 065
20.491 187
21.310 835
22.163 268
23.049 799
23.971 791
24.930 653
25.927 889
26.965 005
28.043 605
35.484 107
36.903 471
38.379 610
93
94 39.914 794
41.511 386
29.165 349
30.331 963
31.545 242
32.807 051
34.119 333
96 43.171 841
44.898 715
46.694 664
48.562 450
44.311 745
47.084 214
49.967 583
52.966 £86
56.084 938
59.323 335
62.701 469
66.209 527
69.857 909
73.652 225
77.598 314
81.702 246
85.970 336
90.409 150
95.025 516
99.820 536
104.81 960
110.01 238
115.41 289
121.02 939
126.87 057
132.94 539
139.26 321
145.83 373
152.66 708
159.77 377
167.16 472
174.85 131
182.84 536
191.15 917
199.80 554
208.79 776
218.14 967
227.87 566
237.99 069
248.51 031
259.45 073
270.82 875
282.66 190
294.96 838
307.76 712
321.07 780
334.92 091
349.31 775
364.29 046
379.86 208
396.05 656
412.89 882
430.41 478
448.63 137
467.57 662
487.27 969
507.77 087
529.08 171
551.24 498
574.29 478
598.26 657
623.19 723
649.12 512
676.09 012
704.13 373
733.29 908
763.63 104
795.17 628
827.98 333
15.982 77
16.329 59
16.663 06
16.983 71
17.292 03
17.588 49
17.873 55
18.147 65
18.411 20
18.661 61
18.908 28
19.142 58
19.367 86
19.584 48
19.792 77
19.993 05
20.185 63
20.370 79
20.548 8+
20.720 04
20.384 65
21.042 94
21.195 13
21.341 47
21.482 18
21.617 49
21.747 58
21.872 67
21.992 96
22.108 61
22.219 82
22.326 75
22.429 57
22.528 43
22.623 49
29.714 89
22.802 78
22.887 29
22.968 55
23.046 68
23.121 81
23.194 05
23.263 51
23.330 30
23.391 51
23.456 26
23.515 64
23.572 73
23.627 62
23.680 41
23.731 16
23.779 96
23.826 89
23.872 01
23.915 39
23.957 11
23.997 £2
24.035 79
24.072 87
24.108 53
24.142 82
24.175 79
24.207 49
24.237 97
24.267 28
862.10 267
24.295 46
897.58 677 24.322 56
934.49 024 24.348 61
972.86 985 24.373 67
1012.7 846
24.397 76
1054.2 960
1097.4 679
1142.3 666
1189.0 613
50.504 948 1237.6 237
24.420 92
24.443 19
24.454 61
24.485 20
24.505 00
Value of Perpetual Ann. 25.000 00
RATIO of INTEREST:
HALF-YEARLY, 0.02.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
2.8003 282
2.9134 614
3.0311 653
3.1536 244
3.2810 308
3.4135 844
3.5514 932
3.6949 736 |
3.8442 505
3.9995 582
5.0724 069
5.2773 321
5.4905 364
5.7123 540
5.9431 331
6.1832 357
6.4330 384
6.6929 332
6.9633 277
7.2446 461
4.1611 404 79.028 509
4.3292 504 83.231 261
4.5041 522 87.603 804
4.6861 199 92.152 998
4.8754 392 | 96.885 979
7.5373 298
7.8418 379
8.1586 484
8.4882 578
8.8311 834
9.1879 632
9.5591 569
9.9453 469
10.347 139
10.765 163
11.200 076
11.652 559
12.123 322
12.613 105
13.122 674
13.652 830
14.204 404
14.778 262
15.375 304
15.996 466
16.642 724
17.315 090
18.014 619
18.742 410
19.499 603
20.287 387
21.106 998
21.959 720
22.846 893
23.769 908
24.730 212
25.729 312
26.768 777
27.850 235
28.975 385
30.145 990
31.363 888
32.630 989
33.949 281
35.320 832
45.008 205
47.836 536
50.779 132
53.840 609
57.025 770
44.795 356
46.605 088
48.487 934
50.446 846
52.484 899
60.339 611
63.787 331
67.374 339
71.105 263
74.988 956
101.81 017
106.93 330
112.26 341
117.80 885
123.57 833
129.58 089
135.82 596
142.32 333
149.08 319
156.11 615
163.43 325
171.04 595
178.96 621
187.20 644
195.77 959
204.69 908
213.97 892
223.63 367
233.67 847
244.12 908
255.00 190
266.31 397
278.08 306
290.32 762
303.06 685
316.32 075
330.11 011
344.45 656
359.38 260
374.91 166
391.06 809
407.87 724
425.36 548
443.56 025
462.49 008
482.18 468
502.67 494
523.99 301
546.17 233
569.24 769
593.25 530
618.23 281
644.21 941
671.25 588
699.38 462
728.64 976
759.09 721
790.77 473
823.73 203
858.02 081
36.747 794
38.232 405
893.69 485
930.81 012
39.776 994 969.42 485
41.383 984 1009.5 996
43.055 897 1051.3 974
1094.8 839
1140.1 272
1187.1 983
1236.1 712
1287.1 225
16.072 47
16.419 14
16.752 35
17.072 61
17.380 44
17.676 32
17.960 71
18.234 05
18.496 78
18.749 31
18.992 03
19.225 33
19.449 57
19.665 10
19.872 26
20.071 37
20.262 76
20.446 71
20.623 52
20.793 46
20.956 81
21.113 81
21.264 72
21.409 77
21.549 18
21.683 18
21.811 97
21.935 77
22.054 75
22.169 12
22.279 05
22.384 71
22.486 26
22.583 87
22.677 70
22.767 87
22.854 55
22.937 86
23.017 94
23.094 90
23.168 88
23.239 98
23.308 32
23.374 02
23.437 15
£3.497 84
23.556 17
23.612 24
23.666 13
23.717 92
23.767 71
23.815 56
23.861 56
23.905 76
23.948 25
23.989 09
24.028 35
24.066 08
24.102 34
24.137 20
24.170 70
24.202 91
24.233 86
24.263 61
21.292 20
24.319 68
24.346 10
24.371 50
24.395 90
24.419 36
24.441 91
24.463 58
24.484 41
24.504 43
24.523 67
25.000 00
RATIO of INTEREST:
QUARTERLY, 0.01.
Improved Cumulated
Principal. Annuity.
2.8146 401
2.9289 258
3.0478 519
3.1716 069
3.3003 869
4.1906 156
4.3607 714
4.5378 362
4.7220 905
4.9138 264
3.4343 958
3.5738 461
3.7189 586
3.8699 632
67.973 964
71.749 080
4.0270 992 | 75.677 480
5.1133 474
5.3209 698
5.5370 225
5.7618 478
5.9958 020
6.2392 556
6.4925 944
6.7562 197
7.0305 491
7.3160 178
7.6130 775
7.9221 990
8.2438 720
8.5786 063
8.9269 321
9.2894 014
9.6665 883
10.059 091
10.467 530
10.892 554
13.830 653
14.392 233
14.976 615
15.584 726
16.217 528
16.876 025
17.561 259
18.274 317
19.016 327
19.788 466
20.591 957
21.428 073
22.298 139
23.203 533
24.145 689
45.366 003
48.223 145
51.196 298
54.290 173
57.509 672
30.658 618
31.903 481
33.198 891
34.546 899
35.949 641
60.859 896
64.346 152
37.409 341
38.928 310
40.508 956
42.153 782
43.865 394
79.765 390
84.019 285
11.334 835
258.37 088
11.795 075 | 269.87 687
12.274 002 281.85 005
12.772 376
294.30 939
13.290 985 | 307.27 464
45.645 505
47.499 936 |
49.428 624
51.435 625
53.524 117
88.445 905
93.052 264
97.845 659
102.83 369
108.02 425
113.42 556
119.04 520
124.89 505
130.98 139
137.31 486
143.90 549
150.76 373
157.90 045
165.32 694
173.05 497
181.09 680
189.46 516
198.17 330
207.23 503
216.66 471
226.47 726
236.68 825
247.31 384
320.76 632
334.80 582
349.41 538
364.61 814
380.43 820
25.126 101
26.146 322
603.15 253
628.65 804
27.207 967 | 655.19 918
28.312 720
29.462 330
682.81 799
711.55 824
396.90 062
414.03 148
431.85 791
450.40 818
469.71 166
489.79 893
510.70 183
532.45 348
555.08 832
578.64 224
741.46 546
772.58 703
| 804.97 227
838.67 247
873.74 103
910.23 352
948.20 775
987.72 389
1028.8 415
1071.6 349
1116.1 626
1162.4 984
1210.7 156
1260.8 906
1313.1 029
Value of
Annuity.
16.117 87
15.464 45
16.797 50
17.117 56
17.425 13
17.720 70
18.004 74
18.277 69
18.539 99
18.792 06
19.034 29
19.267 07
19.490 77
19.705 73
19.912 31
20.110 83
20.301 61
20.484 94
20.661 12
20.830 40
20.993 11
21.149 46
21.299 70
21.444 09
21.582 84
21.716 18
21.844 31
21.967 44
22.085 77
22.199 49
22.308 77
22.413 77
22.514 69
22.611 66
22.704 85
22.794 41
22.880 47
22.963 17
23.042 65
23.119 03
23.192 42
23.262 95
23.330 73
23.395 87
23.458 46
23.518 61
23.576 41
23.631 96
23.685 34
23.736 64
23.785 93
23.833 31
23.878 83
23.922 58
23.964 62
24.005 02
24.043 81
24.081 15
24.117 00
24.151 46
24.184 57
24.216 39
24.246 90
24.276 35
24.304 58
24.331 72
24.357 79
24.382 85
24.406 93
24.430 08
24.452 31
24.473 68
24.494 22
24.513 95
24.532 92
25.000 00
Page 20.
PERPETUAL STOCK: 3 p. C. 75; 34 p. C. 87,5; 4p. C. 100; 5 p. C. 125.

Years.
*7*2* H**** 85883 29949.95996 588ZR AKAAO J8838 86882 E8PER DE223 ***** ****8 58833 85838
60
62
64
100
RATIO of INTEREST:
ANNUALLY, 0.04.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3606 892 0.0225 674
0.3468 166 0.0212 385
0.3334 775 0.0200 130
0.3206 514 0.0188 799
0.3083 187 | 0.0178 301
0.2964 602 0.0168 554 0.0568 554
0.2850 579 | 0.0159 486 0.0559 486
0.2740 942 0.0151 036 0.0551 036
0.2635 521 0.0143 148 0.0543 148
0.2534 154 0.0135 773 | 0.0535 773
0.0128 869 0.0528 869
0.0122 396 | 0.0522 396
0.0116 319
0.0516 319
0.0110 608 0.0510 608
0.0105 235 | 0.0505 235
0.2436 687
0.2342 968
0.2252 851
0.2166 206
0.2082 890
0.2002 779
0.1925 749
0.0100 174 | 0.0500 174
0.0095 402 0.0495 402
0.0090 899 | 0.0490 899
0.1851 682
0.1780 464 | 0.0086 645 | 0.0486 645
0.0082 625 | 0.0482 625
0.1711 984
0.0078 820
0.1646 139
0.1582 826 | 0.0075 219 |
0.0071 806
0.1463 411 0.0068 571
0.1407 126 | 0.0065 502
0.1521 948
0.0914 042
0.0878 887
63 0.0845 084
0.0812 580
0.0781 327
0.1112 072
0.1069 300
0.1028 173
0.0988 628
0.0950 604
0.1353 006 0.0062 588 | 0.0462 588
0.1300 967 | 0.0059 821
0.1250 930 | 0.0057 191
0.1202 817 0.0054 691
0.1156 555 0.0052 312
0.0625 674
0.0612 385
0.0600 130
0.0588 799
0.0578 301
0.0050 019
0.0047 893
0.0045 841
0.0013 884
0.0042 018
0.0751 276 0.0032 492
0.0722 381 0.0031 145
0.0694 597 0.0029 858
0.0667 882 0.0028 627
0.0642 194 | 0.0027 451
0.0507 535
0.0488 015
0.0469 245
0.0451 197
0.0433 843
0.0478 820
0.0475 219
0.0471 806
0.0468 571
0.0465 502
816
0.0281
0.0270 977
0.0260 555
0.0250 534
0.0240
0.0021 387
0.0020 522
0.0019 694
0.0018 901
0.0018 142
0.0459 821
0.0457 191
0.0454 691
0.0452 312
0.0040 240 0.0440 240
0.0038 543 0.0438 543
0.0036 924 | 0.0436 924
0.0035 378
0.0033 902
0.0435 378
0.0433 902
0.0450 049
0.0447 893
0.0445 841
0.0443 884
0.0442 018
0.0617 494 0.0026 325 | 0.0426 325
0.0593 744 0.0025 249 0.0425 249
0.0024 219 0.0424 219
0.0023 233 0.0423 233
0.0022 290 0.0422 290
0.0570 908
0.0548 950
0.0527 837
0.0432 492
0.0431 145
0.0429 858
0.0428 627
0.0427 451
0.0421 387
0.0420 522
0.0419 694
0.0418 901
0.0418 142
0.0417 157
0.0401 112
0.0017 413 | 0.0417 413
0.0016 715 0.0416 715
0.0385 685 | 0.0016 046 | 0.0416 046
0.0015 405
0.0014 791
0.0415 405
0.0414 791
0.0370 851
0.0356 588
0.0342 873
0.0329 685
0.0317 005
0.0304 813
0.0293 089
0.0014 202 | 0.0414 202
-0.0013 637
0.0013 637 0.0413 637
0.0013 095 0.0413 095
0.0012 576 | 0.0412 576
0.0012 078 | 0.0412 078
0.0011 600 | 0.0411 600
0.0011 141 0.0411 141
0.0010 701 0.0410 701
0.0010 279 0.0410 279
898 0.0009 874 | 0.0409 874
0.0231
0.0222 723
0.0214 157
0.0205 920
0.0198 000
Ratio of Perpetual Ann. 0.0400 000
632 | 0.0009 485 | 0.0409 485
0.0009 112 0.0409 112
0.0008 754 | 0.0408 754
0.0008 410 0.0408 410
0.0408 080
0.0008 080
RATIO of INTEREST:
HALF-YEARLY, 0.02.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3571 010
0.3432 343
0.3299 061
0.3170 955
0.3047 823
0.2929 472
0.2815 717
0.2706 379
0.2601 287
0.2500 276
0.2403 187
0.2309 869
0.2220 174
0.2133 962
0.2051 097
0.1617 276
0.1554 476
0.1494 113
0.1436 094
0.1380 330
0.1971 450 | 0.0098 222
0.1894 897 0.0093 516
0.1821 316 0.0089 076
0.1750 592 0.0084 883
0.1682 614 0.0080 920
0.1088 381
0.1046 117
0.1005 495
0.0966 451
0.0928 922
0.0222 182
0.0209 045
0.0196 931
0.0185 733
0.0175 359
0.0892 851
0.0858 181
0.0824 856
0.0792 826
0.0762 040
0.0165 729
0.0156 771
0.0148 424
0.0140 635
0.0133 353
0.0126 537
0.0120 147
0.0114 150
0.0108 515
0.0103 214
0.1326 730 | 0.0061 187
0.1275 211 0.0058 464
0.1225 693 0.0055 876
0.1178 098
0.0053 417
0.1132 351 0.0051 078
0.0077 172
0.0073 624
0.0070 263
0.0067 077
0.0064 055
0.0048 852
0.0046 734
0.0044 716
0.0042 794
0.0040 962
0.0306 457
0.0294 557
0.0283 119
0.0039 215
0.0037 550
0.0035 960
0.0034 444
0.0032 996
0.0732 449
0.0031 613
0.0704 007
0.0030 293
0.0676 670 | 0.0029 031
0.0650 394 | 0.0027 825
0.0625 138 0.0026 673
0.0600 863 | 0.0025 571
0.0577 531 0.0024 517
0.0555 105 0.0023 509
0.0533 549 0.0022 545
0.0512 831 0.0021 622
0.0492 917
0.0473 777
0.0455 379
0.0020 739
0.0019 894
0.0019 084
0.0437 696 | 0.0018 309
0.0420 700 | 0.0017 567
0.0017 567
0.0404 364
0.0388 662
0.0016 856
0.0016 175
0.0373 570 | 0.0015 523
0.0359 063
0.0345 121
0.0014 897
0.0014 298
0.0331 719 0.0013 724
0.0318 838 0.0013 174
0.0012 646
0.0012 140
0.0011 655
0.0272 125 0.0011 190
0.0261 558 0.0010 743
0.0251 402 0.0010 315
0.0241 639 | 0.0009 905
0.0232 256 | 0.0009 511
0.0223 237 0.0009 133
0.0214 569 0.0008 771
0.0206 237 0.0008 423
0.0198 228 0.0008 089
0.0190 531 0.0007 769
G
·
0.0622 182
0.0609 045
0.0596 931
0.0585 733
0.0575 359
0.0565 729
0.0556 771
0.0548 424
0.0540 635
0.0533 353
0.0526 537
0.0520 147
0.0514 150
0.0508 515
0.0503 214
0.0498 222
0.0493 516
0.0489 076
0.0484 883
0.0180 920
0.0477 172
0.0473 624
0.0470 263
0.0467 077
0.0464 055
0.0461 187
0.0458 464
0.0455 876
0.0453 417
0.0451 078
0.0448 852
0.0446 734
0.0444 716
0.0442 794
0.0440 962
0.0439 215
0.0437 550
0.0435 960
0,0434 444
0.0432 996
0.0431 613
0.0430 293
0.0429 031
0.0427 825
0.0426 673
0.0425 571
0.0424 517
0.0423 509
0.0422 545
0.0121 622
0.0420 739
0.0419 894
0.0419 084
0.0118 309
0.0417 567
0.0416 856
0.0416 175
0.0415 523
0.0114 897
0.0414 298
0.0413 724
0.0113 174
0.0412 646
0.0412 140
0.0411 655
0.0411 190
0.0410 743
0.0410 315
0.0109 905
0.0409 511
0.0409 133
0.0108 771
0.0408 423
0.0408 089
0.0407 769
0.0400 000
RATIO of INTEREST:
QUARTERLY, 0.01.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3552 852
0.3414 221
0.0620 429
0.0607 369
0.0220 429
0.0207 369
0.3280 999 | 0.0195 326 | 0.0595 326
0.0584 195
0.0573 884
0.3152 976
0.3029 918
0.0184 195
0.0173 884
0.2911 720
0.2798 106
0.2688 925
0.2584 004
0.2483 177
0.2386 28+
0.0125 368
0.2293 172 | 0.0119 020
0.2203 693 0.0113 063
0.2117 706 | 0.0107 466
0.2035 074 0.0102 202
0.1602 755
0.1540 216
0.1480 118
0.1422 364
0.1366 864
0.1313 529.
0.1262 276
0.1213 022
0.1165 690
0.1120 206
0.1076 496
0.1034 491
0.0994 126
0.0955 335
0.0918 058
0.1955 666 | 0.0097 244
0.0497 214
0.1879 357 | 0.0092 572 | 0.0492 572
0.1806 025 0.0088 164
0.1735 555 0.0084 001
0.1667 834 | 0.0080 067
0.0488 164
0.0484 001
0.0480 067
0.0882 236
0.0847 812
0.0814 730
0.0782 940
0.0752.390
0.0723 032
0.0694 819
0.0667 708
0.0641 654
0.0616 617
0.0592 557
0.0569 435
0.0547 216
0.0525 864
0.0505 345
0.0485 626
0.0466 678
0.0448 468
0.0430 969
0.0114 153
0.0397 992
0.0382 463
0.0367 539
0.0353 198
0.0339 416
0.0326 173
0.0313 445
0.0301 215
0.0289 462
0.0278 167
0.0164 312
0.0155 409
0.0147 115
0.0267 313
0.0256 882
0.0246 859
0.0237 227
0.0227 970
0.0139 375
0.0132 140
0.0219 075
0.0210 527
0.0202 312
0.0191 418
0.0186 832
0.0076 347 0.0476 347
0.0072 825 0.0472 825
0.0069 490 0.0469 490
0.0066 329 | 0.0466 329
0.0063 331 0.0463 331
0.0048 254
0.0046 154
0.0044 155
0.0042 250
0.0040 434
0.0050 486 | 0.0460 486
0.0057 785 0.0457 785
0.0055 219 0.0455 219
0.0052 780 0.0452 780
0.0050 461 0.0450 461
0.0038 704
0.0037 051
0.0035 480
0.0033 978
0.0032 544
0.0031 175
0.0029 868
0.0028 619
0.0027 426
0.0026 285
0.0025 195
0.0024 153
0.0023 156
0.0022 202
0.0021 290
0.0020 417
0.0019 581
0.0018 781
0.0018 015
0.0017 282
0.0564 312
0.0555 409
0.0547 115
0.0539 375
0.0532 140
0.0016 580
0.0015 907
0.0015 263
0.0014 645
0.0014 054
0.0525 368
0.0519 020
0.0513 063
0.0507 466
0.0502 202
0.0013 487
0.0012 914
0.0012 423
0.0009 720
0.0009 332
0.0448 25+
0.0446 154
0.0444 155
0.0442 250
0.0440 434
0.0438 704
0.0437 054
0.0435 480
0.0433 978
0.0432 544
0.0431 175
0.0129 868
0.0128 619
0.0427 426
0.0426 285
0.0125 195
0.0424 153
0.0423 156
0.0122 202
0.0121 290
0.0120 417
0.0419 581
0.0118 781
0.0118 015
0.0117 282
0.0416 580
0.0415 907
0.0415 263
0.0414 645
0.0414 054
0.0113 487
0.0412 944
0.0412 423
0.0011 924 0.0411 924
0.0011 445 0.0111 445
0.0010 986 0.0410 986
0.0010 546 0.0110 546
0.0010 124 0.0410 124
0.0409 720
0.0409 332
0.0408 602
0.0008 959 | 0.0108 959
0.0008 602
0.0008 260
0.0007 931
0.0408 260
0.0407 931
0.0007 616 | 0.0407 616
0.0100 000
Page 21.
RATE OF INTEREST: 44 per
CORRESPONDING VALUE OF
Cent. per
Ann.-

Years.
TH
Hay
Co
-H
~{C}{0}<
ໄດ້
هم
===||26|<
t-
∞
Ma
سراها انهار
PPNA
ܗ
==2<
+++{Q}#{***
ܢܚ ܛܠܪܢ ܐܘܘܘܐܝ
10
ܝܚܘܓܞܝܝܐܟܬܘܟܐܝܒܝ ܝܚ
11
ܢܚܙܛܠܪܢܚ ܦ02ܐܙ ܓ
{* ¥832 v¢¥c{#! «#*
13
2 2000 200 2
15
18
19
21
22
23
24
25
RATIO of INTEREST:
ANNUALLY, 0.0425.
Improved Cumulated
Principal. Annuity.
1.0425 000
1.0868 063
1.1811 478
1.1329 955 3.1293 063
...
1.2313 466
1.3382 352
1.4544 024
16
1.9463 324
17 2.0290 516
2.1152 862
1.2836 788 6.6747 962
1.5162 145
1.5806 536
1.0000 000
1.6478 314
2.0425 000
1.3951 102 9.2967 102
4.2623 018
5.4434 496
2.2051 859
2.2989 063
7.9584 750
10.691 820
12.146 223
13.662 437
1.7178 642 16.890 922
15.243 091
1.7908 734 18.608 786
1.8669 855 20.399 660
22.266 645
24.212 978
26.242 029
28.357 316
30.562 501
2.3966 098
32.861 408
2.4984 657 35.258 018
2.6046 505 37.756 483
2.7153 482
40.361 134
2.8307 505 43.076 482
Value of
Annuity.
0.9592 33
1.8793 60
2.7619 76
3.6086 10
4.4207 29
5.1997 40
5.9469 93
6.6637 82
7.3513 50
8.0108 87
8.6435 37
9.2503 95
9.8325 13
10.390 90
10.926 52
11.440 31
11.933 15
12.405 90
12.859 38
13.294 37
13.711 62
14.111 87
14.495 80
14.864 08
15.217 34
RATIO of INTEREST:
HALF-YEARLY, 0.02125.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0212 500 0.5000 000
1.0429 516 1.0106 250
1.0651 143
1.0877 480
1.1108 626
2.6085 319
1.1344 684 3.1639 632
1.2083 385
1.2340 157
1.1585 759 3.7311 374
1.1831 956 4.3104 854
1.5321 008
2.0646 579
1.2602 386 6.1232 603
1.2870 186 | 6.7533 796
1.4297 009
1.4600 820
1.3143 6787.3968 889
1.3422 981 8.0540 728
1.4911 088
1.5227 948
1.3708 219 | 8.7252 219
1.3999 519 9.4106 328
1.5551 542
1.5882 012
4.9020 832
5.5062 525
1.8400 517
1.8791 528
10.110 609
10.825 459
……..
11.555 500
12.301 055
13.062 452
13.840 029
1.6219 505 14.634 130
1.6564 170 | 15.445 105.
...
1.6916 158 16.273 313
1.7275 627 | 17.119 121
··
1.7642 734 17.982 903
1.8017 642 18.865 039
19.765 921
20.685 947
1.9190 848
21.625 524
1.9598 653 | 22.585 066
2.1318 395
2.2234 053
2.3189 041
2.0440 446 | 24.565 755
26.631 517
28.786 008
31.033 037
2.4185 046 | 33.376 579
2.5223 832 35.820 780
2.6307 235 38.369 964
2.7437 172 41.028 639
2.8615 641 43.801 508
0.4895 96
0.9690 05
...
1.4384 38
1.8981 03
2.3482 04
2.7889 39
3.2205 03
3.6430 88
4.0568 79
4.4620 60
4.8588 10
5.2473 05
5.6277 16
6.0002 12
6.3649 56
6.7221 12
..
7.0718 35
7.4142 82
7.7496 03
8.0779 46
8.3994 58
8.7142 79
9.0225 50
9.3244 06
9.6199 82
9.9094 07
10.192 81
10.470 32
10.742 05
11.008 12
11.268 67
11.523 78
12.018 21
12.492 27
12.946 81
13.382 63
RATIO of INTEREST:
QUARTERLY, 0.010625. .
14.585 33
14.953 67
15.306 84
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0106 250 0.2500 000
1.0213 629 | 0.5026 563
1.0322 149 | 0.7579 970
1.0431 822
1.0160 507
1.5404 127
1.0542 660 | 1.2768 462
1.0654 675
1.0767 881 1.8067 796
1.0882 290 | 2.0759 766
1.0997 914 2.3480 339
1.1114 767 | 2.6229 818
1.1232 862 | 2.9008 509
1.1352 211 3.1816 725
1.1472 828
1.1594 727
1.1717 921
1.1842 424
1.1968 249 | 4.6311 752
1.2095 412 4.9303 815
1.2223 926 | 5.2327 668
1.2353 805 5.5383 649
1.2485 064 | 5.8472 100
1.2617 718 | 6.1593 366
1.2751 781 6.4747 796
1.2887 269 6.7935 741
1.3024 196
1.3162 578
1.3302 431
1.3443 769 | 8.1029 860
7.1157 559
7.4413 608
7.7704 252
1.3586 609
8.4390 802
1.3730 967 | 8.7787 454
1.3876 858
1.4024 300
9.1220 196
9.4089 411
3.4654 777
3.7522 984
4.0421 666
4.3351 146
1.4173 308
9.8195 486
1.4323 900 | 10.173 881
1.4476 091
1.4629 899
10.531 979
10.893 881
1.4785 342
11.259 629
1.4942 436 | 11.629 262
1.5101 200
1.5261 650
12.002 823
12.380 353
1.5423 805
1.5587 683
1.5753 302
1.5920 681
1.6089 838
1.6260 793
1.6433 564
1.6608 170
1.6784 632
1.6962 969
1.7143 200
1.7325 347
12.761 894
13.147 489
13.537 181
13.931 014
2.0517 410
2.1403 396
2.2327 641
14.329 031
14.731 277
15.137 797
15.548 636
1.7509 429 17.669 244
1.7695 466
18.106 980
1.7883 481 18.549 366
1.8073 493 | 18.996 453
13.800 50
2.4297 586
14.201 17 2.5346 808
2.6441 338
2.7583 132
2.8774 231
15.963 840
16.383 456
16.807 530
17.236 110
1.8265 523 19.448 291
1.8459 595 | 19.904 929
1.8655 728 20.366 418
1.8853 945 20.832 812
1.9054 26821.304 160
1.9256 720 21.780 517
1.9461 322
22.261 935
1.9668 099 22.748 468
24.746 847
26.831 519
29.006 213
2.3291 796 | 31.274 814
33.641 379
36.110 137
38.685 501
41.372 075
44.174 661
0.2473 72
0.4921 43
0.7343 40
0.9739 92
1.2111 23
1.4457 62
1.6779 34
1.9076 65
2.1349 81
2.3599 07
2.5824 68
2.8026 90
3.0205 96
3.2362 11
3.4495 60
3.6006 65
3.8695 51
4.0762 41
4.2807 58
4.4831 25
4.6833 64
4.8814 98
5.0775 49
5.2715 39
5.4634 90
5.6531 22
5.8413 58
6.0273 17
6.2113 22
6.3933 92
6.5735 48
6.7518 10
6.9281 98
7.1027 32
7.2754 30
7.4463 13
7.6154 00
7.7827 08
7.9482 58
8.1120 67
8.2741 54
8.4345 37
8.5932 34
8.7502 63
8.9056 40
9.0593 84
9.2115 12
9.3620 40
9.5109 86
9.6583 66
9.8041 97
9.9184 91
10.091 27
10.232 55
10.372 35
10.510 67
10.647 54
10.782 97
10.916 98
11.049 58
11.180 78
11.310 61
11.439 07
11.566 18
12.061 39
12.536 10
12.991 17
13.427 40
13.845 56
14.246 42
14.630 69
14.999 05
15.352 16
Page 22.
PERPETUAL STOCK: 3p. C. 70,5882; 34 p. C.82,3529; 4 p. C. 94,1176; 5 p.C. 117,6471.

Years.
PREH
ACOOLAK
20
[NC]$
Со
They h
In
*****NS[*
##
H*2**
~~*~*~*~*
AIKKAL
∞
*****
A CLAY
10
COLDAL (OPBKLA
12
-*-*-*
14
10 10 10 10 CHE SARA G
17
18
19
21
22
23
24
RATIO of INTEREST:
ANNUALLY, 0.0425.
25
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9592 326 1.0000 000 1.0425 000
0.9201 272 0.4895 961 0.5320 961
|
0.8826 161 0.3195 597 0.3620 597
0.8166 341
0.8121 190
0.7790 111
0.7472 528
0.6595 373
0.7167 893 0.1075 649 0.1500 619
0.6068 582
20 0.4349 894
0.2316 150 0.2771 150
0.1837 070 | 0.2262 070
0.6875 676 0.0935 291
0.1498 173 | 0.1923 173
0.5356 228
0.1256 522 0.1681 522
0.0935 291 0.1360 294
0.6326 497 0.0731 934 0.1156 934
|
0.4172 561
0.4002 456
0.3839 287
0.3682 769
0.3532.632
0.0823 301
0.5821 182 0.0592 034
0.5583 868 0.0537 381
0.0656 035
0.5137 869 0.0149 104
0.4928 411
0.4727 493
0.4534 765
0.0413 002
0.0381 068
0.0352 643
0.0327 198
0.1248 301
0.1081 035
0.1017 034
0.0962 381
0.0490 204 | 0.0915 204
RATIO of INTEREST:
HALF-YEARLY, 0.02125.
0.0304 308 0.0729 308
0.0283 623
0.0708 623
0.0264 855 0.0689 855
0.0247 763 0.0672 763
|
0.0232 145 0.0657 145
Discounted Aunuity for Ratio of
Principal. Cumulation Annuity.
0.9791 921
2.0000 000
2.0425 000
0.9588 173 0.989± 867
0.9894 867 1.0319 867
0.9388 661 0.6526 985 0.6951 985
0.9193 306
0.4813 417
0.5268 417
0.9002 013
0.8814 701
0.8631 286 0.2680 105
0.8451 688 | 0.2319 924
+4
0.8275 827 | 0.2039 949
0.8103 624 0.1816 117
0.7608 221
0.7449 910
0.3833 574
0.3160 593
0.7294 894 | 0.1146 103
0.7143 104 0.1062 628
....
0.2058 117
0.7935 005 | 0.1633 117
0.7769 895 0.1480 740 0.1905 740
0.6991 470 0.0989 060
0.6848 930 | 0.0923 748
0.6430 231
0.6296 431
0.6706 419 | 0.0865 389
0.6566 873 0.0812 938
0.5434 630
0.5321 547
0.1351 920 0.1776 920
0.1241 608
0.1666 608
....
0.0765 553
0.0722 542
0.6165 416 | 0.0683 334
0.6037 127 | 0.0647 455
...
0.5911 508 0.0014 503
0.5788 502 0.0584 142
0.5668 056 0.0555 084
0.5550 116
0.0530 081
0.0865 389 0.1290 389
0.1237 938
0.0505 921
0.0483 420
0.0874 104
0.5210 817 | 0.0462 417
0.5102 392 0.0442 770
0.0838 002
0.4892 261
0.0806 068 0.4690 785
0.0777 643 0.4497 606
0.0752 198 0.4312 382
0.4258 574
0.3585 593
……….
0.0407 071
0.0375 495
0.0347 391
0.0322 237
0.3105 105
0.2714 924
0.4134 786 | 0.0299 611
0.3964 505 0.0279 168
0.3801 236 | 0.0260 621
0.3644 691
0.3494 592
0.2464 949
0.2241 117
0.1571 103
0.1487 628
0.1414 060
0.1348 748
0.1190 553
0.1147 542
0.1108 334
0.1072 455
0.1039 503
0.1009 142
0.0981 084
0.0955 081
0.0930 921
0.0908 120
0.0887 417
0.0867 770
0.0332 071
0.0800 195
0.0772 391
0.0747 237
0.0724 611
0.0704 168
0.0685 621
0.0243 732 | 0.0668 732
0.0228 303 | 0.0653 303
RATIO of INTEREST:
QUARTERLY, 0.010625.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9891 867
0.9790 840
0.9687 906
0.9586 053
0.9485 273
0.9385 551
0.9286 878
0.9189 242
0.8716 217
0.8624 610
0.8533 938
0.8111 217
0.9092 637 | 0.4258 883
0.8997 040 0.3812 455
0.8902 451 | 0.3447 264
0.8808 857 0.3143 001
4.0000 000
1.9891 933
1.3192 665
0.9842 028
0.7360 188
0.7282 808
0.7206 242
0.7130 481
0.8355 +41
0.8267 597
0.8180 678 0.1911 035
0.8094 672 | 0.1805 587
0.7831 797 0.8256 797
0.6491 767 | 0.6916 767
0.5534 710 0.5959 710
0.5242 010
0.4817 010
0.6763 456
0.6692 349
0.6621 991
0.6552 372
0.2885 605
0.2665 043
0.2473 921
0.2306 744
0.2159 279
0.2028 240
0.7678 017 | 0.1405 332
0.7597 296 0.1343 840
0.1286 931
0.1234 113
0.7517 423
0.7.138 390
0.5957 831
0.5895 195 |
0.5833 230
0.5771 890
0.1710 217
0.1623 552
0.1514 454
0.8009 570
0.7925 363
0.7842 042
0.7759 596 | 0.1471 983 | 0.1896 983
0.2135 217
0.2048 552
0.1969 454
0.1018 377
0.0982 909
0.7055 515
0.6981 339
0.6907 942 0.0949 489
0.6835 317 | 0.0917 947
0.5711 208
0.5651 165
0.1184 963
0.1139 115
0.1096 248
0.1056 081
0.6483 485 0.0783 583
0.6415 322 0.0760 601
0.6347 876 0.0738 706
0.6281 139 | 0.0717 823
0.0888 129
0.0859 900
0.0833 137
0.0807 732
0.6215 103 0.0697 881
0.6149 762 | 0.0678 828
0.6085 108
0.0660 598
0.6021 133
0.0643 143
4.0425 000
2.0319 933
1.3617 665
1.0267 028
0.4683 883
0.4237 455
0.3872 264
0.3568 001
0.3310 605
0.3090 043
0.2898 921
0.2731 744
0.2584 279
0.2453 240
0.2336 035
0.2230 587
0.1830 332
0.1768 840
0.1711 931
0.1659 113
0.1609 963
0.1564 115
0.1521 248
0.1481 084
0.1443 377
0.1407 909
0.1374 489
0.1342 947
0.1313 129
0.1284 900
0.1258 137
0.1232 732
0.1208 583
0.1185 601
0.1163 706
0.1142 823
0.1122 884
0.1103 828
0.1085 598
0.1068 143
0.0626 416
0.0610 372
0.0594 971
0.0580 177 | 0.1005 177
0.1051 416
0.1035 372
0.1019 971
0.0565 955 | 0.0990 955
0.0552 273 | 0.0977 273
0.5591 753 | 0.0539 102 0.0964 102
0.5532 965 0.0526 414 | 0.0951 414
0.5474 795 | 0.0514 184 | 0.0939 184
0.5417 237 0.0502 388 | 0.0927 388
0.5360 281 | 0.0491 004 0.0916 004
0.5303 930 0.0180 012 | 0.0905 012
0.5248 168 | 0.0469 392 | 0.0891 392
0.5192 992 | 0.0459 126 0.0884 126
0.5138 397 0.0449 197 0.0874 197
0.5084 375 | 0.0439 590 | 0.0861 590
0.4873 910 | 0.0404 092 | 0.0829 092
0.4672 156 | 0.0372 696 | 0.0797 696
0.0344 754 0.0769 754
0.4478 753
0.4293 357 | 0.0319 746 | 0.0744 746
0.4115 635 0.0297 253
0.3945 270 | 0.0276 931
0.3781 957 0.0258 495
0.3625 404 0.0241 709
0.0722 253
0.0701 931
0.0683 495
0.0666 709
0.3475 332 0.0226 374 | 0.0651 374
Page 23.
RATE OF INTEREST: 44 per Cent.
-CORRESPONDING VALUE OF
per Ann.

Years.
***9* H**** 85800 73333 95938 58888 95978 588** 86882 NARZR 2F228 588*83 85888 58833 8582g
41
RATIO of INTEREST:
ANNUALLY, 0.0425.
96
Improved Cumulated Value of
Principal. Annuity. Annuity.
99
100
2.9510 574 | 45.907 233
3.0764 773 48.858 290
3.2072 276 51.934 767
3.3435 348 55.141 995
3.4856 350 58.485 530
3.6337 745 | 61.971 105
3.7882 099
3.9492 088
4.1170 502
4.2920 248
65.604 939
69.393 149
73.342 358
77.459 408
4.4744 359
4.6645 994
4.8628 449
5.0695 158
5.2849 702
8.3537 071
8.7087 397
53 9.0788 611
9.4047 127
9.8669 630
5.5095 815
5.7437 387
106.10 780
111.61 738
5.9878 476 | 117.36 112
6.2423 311 123.34 897
6.5076 302 | 129.59 130
6.7842 045
7.0725 331
7.3731 158
7.6864 732
8.0131 483
10.286 309
10.723 477
11.179 225
11.654 342
12.149
12.666 012
13.204 317
13.765 501
14.350 531
14.960 432
81.751 433
86.225 869
90.890 468
95.753 313
100.82 283
218.50 139
228.78 770
239.51 117
250.69 040
651 | 262.34 474
136.09 893
142.88 313
149.95 567
157.32 878
165.01 525
173.02 840
181.38 211
190.09 085
199.16 971
208.63 442
15.596 250
343.44 119
16.259 091
359.03 744
16.950 102 375.29 653
17.670 482 392.24 663
18.421 477 | 409.91 711
47.981 511
50.020 725
95 52.146 606
274.49 439
287.16 040
300.36 472
314.13 022
328.48 075
19.204 390 428.33 859
20.020 577 | 447.54 298
20.871 451 467.56 356
21.758 488 488.43 501
22.683 224 510.19 350
44.149 094
46.025 430
23.647 261 532.87 672
24.652 269 556.52 398
25.699 991
581.17 625
26.792 240 606.87 624
27.930 910 | 633.66 848
29.117 974 661.59 939
30.355 488 690.71 736
31.645 596| 721.07 285
32.990 534 752.71 845
34.392 632 785.70 898
35.854 319
820.10 161
855.95 593
37.378 127
38.966 698 893.33 406
932.30 076
972.92 354
40.622 782
42.349 250
1015.2 728
1059.4 219
1105.4 473
1153.4 288
1203.4 496
54.362 837 1255.5 962
56.673 257 | 1309.9 590
98 59.081 871 1366.6 323
61.592 850 1425.7 141
64.210 546 | 1487.3 070
15.556 20
15.881 25
16.193 04
16.492 13
16.779 02
17.054 21
17.318 19
17.571 41
17.814 30
18.047 29
18.270 78
18.485 16
18.590 80
18.888 06
19.077 28
19.258 78
19.432 88
19.599 88
19.760 08
19.913 75
20.061 15
20.202 54
20.338 17
20.468 27
20.593 06
20.712 77
20.827 60
20.937 74
21.043 40
21.144 74
21.241 96
21.335 22
21.424 67
21.510 47
21.592 78
21.071 73
21.747 46
21.820 10
21.889 79
21.956 64
22.020 75
22.082 25
22.141 25
22.197 85
22.252 13
22.304 20
22.354 15
22.402 06
22.448 02
22.492 11
£2.534 40
22.574 96
22.613 87
22.651 19
22.687 00
22.721 34
22.754 28
22.785 88
22.816 20
22.845 27
22.873 16
22.899 92
22.925 58
22.950 20
22.973 81
23.096 59
23.114 24
23.131 16
23.147 40
23.162 97
Value of Perpetual Ann. 23.529 41
RATIO of INTEREST:
HALF-YEARLY, 0.02125.
2.9841 727
3.1126 605
3.2463 541
3.3857 901
3.5312 151
Improved Cumulated Value of
Principal. Annuity. Annuity.
3.6828 863
3.8410 720
4.0050 521
4.1781 183
4.3575 750
4.5447 396
4.7399 433
4.9435 313
5.1558 637
5.3773 161
5.6082 802
5.8491 646
6.1003 953
6.3624 168
6.6356 926
6.9207 060
7.2179 611
7.5279 838
7.8513 225
8.1885 490
8.5402 600
8.9070 775
9.2896 504
9.6886 554
10.104 798
10.538 815
10.991 474
11.463 575
11.955 953
12.469 480
13.005 064
13.563 652
14.146 232
14.753 834
15.387 535
16.048 453
16.737 759
17.456 672
18.206 464
18.988 460
24.438 501
25.488 173
26.582 930
27.724 708
28.915 528
22.996 46
23.018 19
45.923 677
47.896 171
23,039 03 49.953 386
23.059 02
23.078 20
46.693 476
49.709 659
52.855 391
56.136 238
59.558 002
63.126 737
66.848 753
70.730 637
74.779 253
79.001 764
52.098 962
54.336 694
83.405 638
87.998 666
92.788 971
97.785 027
102.99 567
108.43 012
114.09 799
120.00 930
126.17 451
132.60 453
139.31 073
146.30 497
153.59 962
161.20 759
169.14 233
177.41 788
186.04 888
195.05 060
204.43 895
214.23 055
224.44 271
235.09 350
246 20 176
257.78 713
269.87 012
19.804 044 | 442.44 809
20.651 658
21.541 808
22.467 062
23.432 058
462.40 255
483.33 666
505.10 735
527.81 313
282.17 209
295.61 533
309.32 310
323.61 963
338.53 022
354.08 125
370.30 022
387.21 582
404.85 797
423.25 787
30.157 495
686.05 871
31.452 807 | 716.53 663
32.803 754
34.212 726
35.682 216
748.32 362
781.47 591
816.05 215
551.49 415
576.19 231
601.95 130
628.81 667
656.83 595
37.214 823 852.11 349
38.813 258 889.72 372
40.480 348 928.94 937
42.219 042
44.082 416
969.85 982
1012.5 274
1057.0 277
1103.4 393
1151.8 444
1202.3 285
1254.9 810
56.670 540 1309.8 951
59.104 628 | 1367.1 677
61.643 264
64.290 939
67.052 335
1426.9 003
1489.1 986
1554.1 726
15.645 47
15.970 16
16.281 46
16.579 95
16.866 15
17.140 56
17.403 67
17.655 94
17.897 82
18.129 75
18.352 13
18.565 35
18.769 78
18.965 79
19.153 73
19.333 93
19.506 72
19.672 38
19.831 22
19.983 53
20.129 56
20.269 57
20.403 82
20.532 54
20.655 95
20.774 30
20.887 76
20.996 55
21.100 86
21.200 87
21.296 77
21.388 71
21.476 87
21.561 40
21.642 45
21.720 16||
21.794 67
21.866 11
21.934 61
22.000 29
22.063 26
22.123 64
22.181 54
22.237 05
22.290 27
22.311 30
22.390 23
22.437 15
22.482 13
22.525 26
22.566 61
22.606 26
22.644 28
22.680 73
22.715 68
22.749 19
22.781 32
22.812 13
22.841 67
29.869 99
22.897 15
22.923 19
22.948 15
22.972 09
22.995 01
23.017 05
23.038 15
23.058 38
23.077 78
23.096 38
23.114 21
23.131 31
23.147 70
23.163 43
23.178 50
23.529 41
RATIO of INTEREST:
QUARTERLY, 0.010625.
Improved Cumulated Value of
Principal. Aunuity.
Annuity.
3.0016 764
3.1312 953
3.2665 114
3.4075 664
3.5547 124
3.7082 125
3.8683 412
4.0353 845
4.2096 411
4.3914 224
4.5810 535
4.7788 733
4.9852 353
5.2005 085
5.4250 777
5.6593 442
5.9037 269
6.1586 625
6.4246 068
6.7020 352
6.9914 435
7.2933 491
7.6082 916
7.9368 341
8.2795 637
8.6370 931
9.0100 613
9.3991 352
9.8050 101
10.228 412
13.181 629
13.750 840
14.344 631
14.964 063
15.610 214
47.098 269
50.148 124
53.329 679
56.648 620
60.110 880
24.852 571
25.925 759
27.045 289
28.213 163
29.431 468
63.722 648
67.490 380
71.420 811
75.520 966
79.798 175
84.260 082
88.914 665
93.770 242
98.835 494
104.11.947
10.670 096
11.130 854
11.611 508
227.53 168
238.37 304
249.68 255
12.112 918 261.48 043
12.635 980 | 273.78 777
109.63 163
115.38 181
121.38 029
127.63 781
134.16 553
46.856 901
48.880 283
50.991 039
53.192 941
55.489 927
140.97 514
148.07 880
155.48-922
163.21 963
171.28 385
57.886 102
60.385 748
62.993 335
65.713 523
68.551 174
179.69 631
188.47 203
197.62 671
207.17 671
217.13 910
16.284 328
16.987 520
17.721 078
18.486 312
411.44 323
19.284 591 | 430.22 567
286.62 657
300.01 977
313.99 132
328.56 619
343.77 044
359.63 124
376.17 694
393.43 712
20.117 341
20.986 051
21.892 274 | 491.58 292
22.837 630
23.823 808
513.82 658
537.03 077
449.81 979
470.26 003
30.702 382
32.028 177
33.411 223
34.853 991
698.87 957
730.07 475
762.61 700
796.56 450
36.359 062 | 831.97 792
561.23 697
586.48 811
612.83 032
640.30 971
668.99 570
37.929 124 868.92 057
39.566 985 907.45 848
41.275 573 | 947.66 054
43.057 941
44.917 276
989.59 862
1033.3 477
1078.9 859
1126.5 949
1176.2 597
1228.0 692
1282.1 159
1338.4 965
1397.3 117
1458.6 667
1522.6 711
1589.4 394
15.690 65
16.015 14
16.326 19
16.624 37
16.910 20
17.184 19
17.446 85
17.698 61
17.940 00
18.171 38
18.393 17
18.605 78
18.809 59
19.004 97
19.192 26
19.371 79
19.543 89
19.708 87
19.867 02
20.018 63
20.163 95
20.303 27
20.436 81
20.564 83
20.687 55
20.805 18
20.917 95
21.026 05
21.129 68
21.229 01
21.324 24
21.415 52
21.503 02
21.586 91
21.667 32
21.744 40
21.818 29
21.889 12
21.957 02
22.022 11
22.081 50
22.144 31
22.201 65
22.256 61
22.309 30
22.359 80
22.408 22
22.454 63
22.499 12
22.541 77
22.582 65
22.621 84
22.659 41
22.695 33
22.729 94
22.763 04
22.794 77
22.825 18
22.854 33
22.882 27
22.909 06
22.934 74
22.959 35
22.982 95
23.005 57
23.027 26
23.048 04
23.067 97
23.087 07
23.105 38
23.122 93
23.139 76
23.155 89
23.171 35
23.186 17
23.529 41
Page 24.
PERPETUAL STOCK: 3 p. C. 70,5882; 31 p. C. 82,3529; 4p. C. 94,1176; 5 p. C. 117,6471.

Years.
9**** H***8 AKA83 #9933 95998 ESKAK AK8A8 58888 86882 PRRZR RFR28 58833 85838 63832 85338
56
7.4
76
78
87
89
92
95
RATIO of INTEREST:
ANNUALLY, 0.0425.
96
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3388 616
0.3250 471
0.3117 958
0.2990 847
0.2868 918
0.2751 959
0.2639 769
0.2532 153
0.2428 923
0.2329 903
0.2234 919
0.2143 807
0.2056 409
0.1972 575
0.1892 158
0.1815 020
0.1741 026
0.1670 049
0.1601 966
0.1536 658
0.1474 012
0.1413 921
0.1356 279
0.1300 987
0.1247 949
0.0972 166
0.0932 533
0.0894 517
0.0858 049
0.0823 069
0.0520 714
0.0199 486
0.0479 123
0.0459 591
0.0440 855
0.0217 831
0.0642 831
0.0204 674 0.0629 674
0.0192 549 0.0617 549
0.0181 350 0.0606 350
0.0170 982 0.0595 982
0.1197 073 0.0057 794
0.1148 272 0.0055 132
0.1101 460 | 0.0052 606
0.1056 556
0.0050 208
0.0047 931
0.1013 483
0.0161 365 | 0.0586 365
0.0152 428 0.0577 428
0.0144 106 0.0569 106
0.0136 347
0.0561 317
0.0129 100 0.0554 100
0.0122 322
0.0115 974
0.0110 023
0.0104 435
0.0099 184
0.0789 515 | 0.0036 431
0.0757 328
0.0034 824
0.0033 293
0.0031 834
0.0030 443
0 0726 454
0.0696 838
0.0668 430
0.0226 505
0.0217 271
0.0208 414
0.0199 917
0.0191 767
0.0094 244 0.0519 244
0.0089 592 0.0514 592
0.0085 207 0.0510 207
0.0081 071
0.0506 071
0.0077 166 | 0.0502 166
0.0073 476
0.0069 987
0.0066 686
0.0063 561
0.0060 600
0.0029 117
0.0027 852
0.0641 180
0.0615 041
0.0589 967 | 0.0026 646
0.0565 916
0.0025 494
0.0024 395
0.0542 845
0.0045 766
0.0043 709
0.0041 732
0.0039 890 |
0.0038 118
0.0183 949
0.0176 450
98 0.0169 257
0.0162 357
0.0343 430
0.0329 430
0.0015 115
0.0014 478
0.0316 000
0.0013 868
0.0303 117
0.0013 285
0.0290 760 0.0012 727
0.0023 346
0.0022 344
0.0021 387
0.0020 474
0.0019 600
0.0547 322
0.0540 974
0.0535 023
0.0529 435
0 0524 184
0.0007 964
0.0007 634
0.0007 317
0.0007 014
100 0.0155 738 0.0006 724
0.0498 476
0.0494 987
0.0491 686
0.0488 561
0.0185 600
0.0482 794
0.0480 132
0.0477 606
0.0475 208
0.0472 931
0.0422 882
0.0018766 0.0443 766
0.0405 642 0.0017 969 | 0.0442 969
0.0389 105 0.0017 206 0.0142 206
0.0373 242
0.0441 478
0.0358 026
0.0440 781
0.0016 478
0.0015 781
0.0470 766
0.0468 709
0.0466 752
0.0464 890
0.0463 118
0.0461 431
0.0459 824
0 0458 293
0.0456 834
0.0455 443
0.0454 117
0.0452 852
0.0451 646
0.0150 494
0.0449 395
0.0278 906 0.0012 194
0.0267 536 | 0.0011 683
0.0437 194
0.0136 683
0.0436 194
0.0010 726 | 0.0435 726
0.0010 278
0.0256 629
0.0246 167
0.0236 132
0.0011 194
0.0435 278
0.0448 346
0.0147 344
0.0446 387
0.0443 474
0.0444 600
0.0440 115
0.0439 478
0.0438 868
0.0438 285
0.0437 727
0.0432 964
0.0432 634
6.0432 317
0.0432 014
0.0431 724
Ratio of Perpetual Ann. 0.0425 000
0.0009 850
0.0009 439
0.0009 046
0.0434 850
0.0434 439
0.0434 046
0.0008 670 | 0.0433 670
0.0008 309 0.0433 309
RATIO of INTEREST:
HALF-YEARLY, 0.02125.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
76
0.3350 676 | 0.0214 163 | 0.0639 163
0.3212 686 0.0201 168 0.0626 168
0.3080 379 | 0.0189 195 0.0614 195
0.2953 520] 0.0178 138 0.0603 138
0.2831 887 0.0167 903 0.0592 903
0.2715 262
0.2603 440
0.2496 223
0.2393 422
0.2294 854
0.2200 346 0.0119 896
0.2109 730| 0.0113 638
0.2022 846 0.0107 771
0.1939 539 0.0102 264
0.1859 664
0.0097 091
0.1783 078 0.0092 225
0.1709 646 0.0087 614
0.1639 238 0.0083 327
0.1571 730 | 0.0079 255
0.1507 002 0.0075 412
0.1170 925
0.1122 703
0.1076 467
0.1032 135
0.0989 629 |
0.0768 931
0.0737 264
0.0706 902
0.0677 790
0.0649 877
0.0158 411
0.0149 591
0.0141 381
0.0133 727
0.0126 579
0.1444 939 | 0.0071 782 | 0.0496 782
0.1385 433
0.0493 350
0.1328 377
0.0490 104
0.1273 671
0.0487 032
0.1221 218
0.0484 122
0.0068 350
0.0065 104
0.0062 032
0.0059 122
0.0948 873 0.0044 555
0.0909 796 0.0042 536
0.0042 536 |
0.0872 328 0.0040 617
0.0836 403 0.0038 792
0.0801 958 | 0.0037 055
0.0623 113
0.0597 452
0.0572 847
0.0549 255
0.0526 636
0.0504 947
0.0484 152
0.0464 214
0.0445 096
0.0426 766
0.0331 593
0.0317 937
0.0304 843
0.0292 289
0.0280 252
0.0268 710
0.0257 644
0.0247 033
0.0236 860
0.0227 105
0.0035 402
0.0033 828
0.0032 329
0.0030 900
0.0029 539
0.0056 364 0.0481 364
0.0053 749 0.0478 749
0.0051 269 | 0.0476 269
0.0048 914 0.0473 914
0.0046 679 | 0.0471 679
0.0028 242
0.0027 005
0.0025 825
0.0024 700
0.0023 626
0.0409 190
0.0392 339
0.0376 181
0.0018 133
0.0017 355
0.0016 612
0.0360 689 | 0.0015 903
0.0345 835
0.0015 224
0.0022 602
0.0021 623
0.0020 689
0.0019 798
| 0.0018 916
0.0018 946
0.0014 576
0.0013 956
0.0013 363
0.0012 796
0.0012 254
0.0011 736
0.0011 239
0.0010 765
0.0010 311
0.0009 876
0.0217 753 0.0009 460
0.0208 785 0.0009 062
0.0200 187 0.0008 682
0.0191 942
0.0184 038
0.0008 317
0.0007 968
0.0583 411
0.0574 591
0.0566 381
0.0558 727
0.0551 579
0.0544 896
0.0538 638
0.0532 771
0.0527 264
0.0522 091
**
0.0517 225
0.0512 644
0.0508 327
0.0504 255
0.0500 412
H
0.0469 555
0.0467 536
0.0465 617
0 0463 792
0.0462 055
0.0460 402
0.0458 828
0.0457 329
0.0455 900
0.0454 539
0.0453 242
0.0452 005
0.0450 825
0.0449 700
0.0448 626
0.0447 602
0.0446 623
0.0445 689
0.0444 798
0.0443 946
0.0443 133
0.0442 355
0.0441 612
0.0440 903
0.0440 224
0.0439 576
0.0438 956
0.0438 363
0.0437 795
0.0437 254
0.0436 736
0.0436 239
0.0435 765
0.0435 311
0.0434 876
0.0176 459
0.0007 634 0.0432 634
0.0169 192 0.0007 314 | 0.0432 314
0.0162 224 0.0007 008 | 0.0432 008
0.0155 543 0.0006 715 0.0431 715
0.0119 137
0.0431 434
0.0006 434
- 0.0425 000
0.0434 460
0.0434 062
0.0433 682
0.0433 317
0.0432 968
RATIO of INTEREST:
QUARTERLY, 0.010625.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3331 472
0.3193 567
0.3061 370
0.2934 646
0.2813 167
0.2696 717
0.2585 088
0.2478 079
0.2375 499
0.2277 166
0.2182 904
0.2092 543
0.2005 923
0.1922 889
0.1843 292
0.1766 989
0.1693 845
0.1623 729
0.1556 516
0.1492 084
0.1430 320
0.1371 112
0.1314 355
0.1259 948
0.1207 793
0.1157 797
0.1109 870
0.1063 928
0.1019 887
0.0977 669
0.0937 199
0.0898 404
0.0861 215
0.0825 565
0.0791 391
0.0758 632
0.0727 228
0.0697 125
0.0668 268
0.0640 605
0.0614 087
0.0588 667
0.0564 300
0.0540 941
0.0518 549
0.0497 084
0.0476 507
0.0456 782
0.0437 874
0.0419 748
0.0402 373
0.0395 717
0.0369 750
0.0354 444
0.0339 772
0.0325 708
0.0312 225
0.0299 301
0.0286 911
0.0275 035
0.0263 650
0.0252 736
0.0242 274
0.0232 245
0.0222 631
0.0213 416
0.0204 531
0.0196 113
0.0187 995
0.0180 213
0.0172 753
0.0165 602
0,0158 747
0.0152 176
0.0145 876
0.0212 322
0.0199 409
0.0187 513
0.0176 527
0.0166 359
0.0156 930
0.0148 169
0.0140 015
0.0132 414
0.0125 316
0.0118 680
0.0112 467
0.0106 644
0.0101 178
0.0096 044
0.0078 347
0.0074 535
0.0070 934
0.0067 532
0.0064 313
0.0061 267
0.0058 383
0,0055 649
0.0053 058
0.0050 600
0.0048 268
0.0046 058
0.0516 215
0.0511 669
0.0091 215
0.0086 669
0.0082 386 | 0.0507 386
0.0503 347
0.0499 535
0.0043 950
0.0041 951
0.0040 051
0.0038 241
0.0036 525
0.0034 889
0.0033 331
0.0031 818
0.0030 435
0.0029 089
0.0027 806
0.0026 583
0.0025 417
0.0024 305
0.0023 244
0.0022 231
0.0021 265
0 0020 342
0.0019 462
0.0018 621
0.0017 818
0.0017 051
0.0016 318
0.0015 618
0.0014 918
0.0014 309
0.0013 697
0.0013 113
0.0012 554
0.0012 020
0.0637 322
0.0624 409
0.0612 513
0.0601 527
0.0591 359
0.0581 930
0.0573 169
0.0565 015
0.0557 414
0.0550 316
0.0007 471
0.0007 157
0.0006 856
0.0006 567
0.0006 292
0.0543 680
0.0537 467
0.0531 644
0.0526 178
0.0521 044
0.0495 934
0.0492 532
0.0489 313
0.0186 267
0.0483 383
0.0480 649
0 0178 058
0.0475 600
0.0473 268
0.0471 053
0.0468 950
0.0466 951
0.0465 051
0.0463 244
0.0461 525
0.0459 889
0.0458 331
0.0456 848
0.0455 435
0.0454 089
0.0452 806
0.0451 583
0.0450 417
0.0449 305
0.0448 244
0.0447 231
0.0446 265
0.0445 312
0.0444 462
0.0443 621
0.0442 818
0.0442 051
0.0441 318
0.0440 618
0.0439 948
0.0439 309
0.0438 697
0.0438 113
0.0437 554
0.0437 020
0.0011 509
0.0011 020
0.0436 509
0.0436 020
0.0010 552 | 0.0435 552
0.0010 105
0.0009 677
0.0435 105
0.0434 677
0.0009 268 0.0434 268
0.0008 876 | 0.0433 876
0.0008 502 0.0133 502
0.0008 143
0.0007 800
0.0433 143
0.0432 800
0.0432 471
0.0432 157
0.0431 855
0.0431 567
0.0431 292
0.0425 000
Page 25.
RATE OF INTEREST: 44 per CentAnn.--CORRESPONDING VALUE OF
Cent. .
per Ann.-

Years.
d
={*}}}<
P101SL1
Q2
w
#{2}-A-
+
OCASA
S
Halako
-triazol
t-
∞
FECHAL
#={0080}£}
CASCHI SA
10
11
CANAL BAKAL
12
GARNA FARLAN
15
~*~2:KO
7998 72728
17
18
19
20
21
RATIO of INTEREST:
ANNUALLY, 0.045.
23
24
25
Improved Cumulated Value of
Principal. Annuity. Anuuity.
1.0450 000
1.0920 250 2.0450 000
1.1411 661
1.1925 186
13 1.7721 961
14 1.8519 449
1.2461 819 | 5.4707 097
1.3022 601 6.7168 917
1.4860 951
1.3608 618 8.0191 518
1.5529 694
1.4221 006 | 9.3800 136
1.6228 530
1.6958 814
1.0000 000
3.1370 250
4.2781 911
··
2.1133 768
2.2084 788
2.3078 603
2.4117 140
2.5202 412
2.6336 520
2.7521 663
2.8760 138
3.0054 345
··
2903
10.802 114
12.288 209
13.841 179
15.464 032
17.1599 13
18.932 109
2.0223 702 22.719 337
24.741 707
26.855 084
29.063 562
31.371 423
0.9569 38
33.783 137
36.303 278
38.937 030
41.689 196
44.565 210
1.8726 68
2.7489 64
....
3.5875 26
4.3899 77
5.1578 72
5.8927 01
6.5958 86
...
7.2687 90
1.9352 824 20.784 054 10.739 55
7.9127 18
8.5289 17
9.1185 81
9.6828 52
10.222 83
13 404 72
13.784 42
14.147 77
14.495 48
14.828 21
RATIO of INTEREST:
HALF-YEARLY, 0.0225.
Improved Cumulated Value of
Principal. Aunuity. Annuity.
1.0225 000
1.0455 063
....
1.0690 301
1.0930 833
....
1.1176 777
....
....
2.6150 599
1.1428 254 3.1738 987
1.1685 390
1.1948 311
1.2773 105
1.3060 500
....
1.3354 361
....
1.3654 834
1.2217 148 4.9269 965
1.2492 034 | 5.5378 539
1.3962 068
1.4276 215
1.4597 429
1.4925 872
....
0.5000 000
1.0112 500
1.5340 031
2.0685 182
1.7441 463
1.7833 896
....
....
...
3.7453 114
4.3295 809
..
6.1624 556
6.8011 109
7.4541 359
8.1218 539
1.5261 704 | 11.692 675
1.5605 092
12.455 760
8.8045 957
9.5026 991
1.5956 207 13.236 015
1.6315 221
10.216 510
10.946 381
1.6682 314 14.849 586
1.7057 666
15.683 702
....
14.033 825
16.536 585
17.408 658
1.8235 159 18.300 353
1.8645 450 | 19.212 111
....
1.9064 973 | 20.144 383
1.9493 934 21.097 632
....
1.9932 548
2.0381 030 23.068 956
22.072 329
11.234 01
11.707 19 2.1308 495 25.129 988
12.159 99 2.2278 164 27.284 809
12.593 29 2.3291 960 29.537 689
13.007 94 2.4351 890 | 31.893 088
2.5460 053 34.355 673
2.6618 614 | 36.930 321
2.7829 959 39.622 131
2.9096 396 42.436 436
3.0420 464 45.378 809
0.4889 98
...
0.9672 35
1.4349 48
1.8923 70
2.3397 26
2.7772 38
3.2051 23
3.6235 92
4.0328 53
4.4331 08
4.8245 56
....
5.2073 89
5.5817 99
5.9479 70
....
6.3060 83
6.6563 16
6.9988 42
7.3338 31
7.6614 48
7.9818 56
8.2952 14
8.6016 76
8.9013 95
9.1945 18
9.4811 91
9.7615 56
10.035 75
10.303 91
10.566 17.
10.822 66
11.073 51
11.318 84
11.793 41
12.247 33
12.681 50
13.096 76
13.493 95
13.873 85
14.237 22
14.584 77
14.917 20
RATIO of INTEREST:
QUARTERLY, 0.01125.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0112 500
1.0226 266
1.0341 311
1.0457 651
1.0575 299
1.0694 272
1.0814 582
1.0936 246
1.1309 512
1.1436 744
1.1059 279 | 2.3539 532
1.1183 696
2.6304 351
1.1565 408 3.1786 840
1.1695 519 | 3.7678 192
4.0602 071
1.1827 093
1.1960 148 4.3558 844
1.2091 700 4.6548 881
1.2230 765 | 4.9572 556
1.2368 361
1.2507 505
5.2630 248
5.5722 338
1.3832 399
1.3988 013
1.4145 379
1.4304 514
1.2648 215
1.2790 507
1.2931 400
1.3079 912 | 6.8442 495
5.8849 214
6.2011 268
6.5208 895
1.4465 440
1.4628 176
1.4792 743
1.4959 161
1.3227 061 7.1712 473
1.3375 866 | 7.5019 238
1.3526 344
1.3678 516
7.8363 205
8.1744 791
1.5127 452
1.5297 636
1.5469 734
1.5043 769
0.2500 000
0.5028 125
0.7584 691
1.0170 019
1.5819 761
1.5997 733
1.6177 708
1.6359 707
1.2784 432
1.5428 257
1.8101 825
2.0805 470
1.6543 754
1.6729 871
1.6918 082
1.7108 410
1.7300 880
1.7495 515
1.7692 340
1.7891 378
2.9100 275
3.1927 653
1.9786 574 |
2.0009 173
2.0234 277
2.0461 912
2.1398 353
2.2377 651
2.3401 766|
2.4472 750
8.5164 420
8.8622 519
9.2119 523
9.5655 867
9.9231 996
10.284 836
10.650 540
11.020 359
11.394 338
11.772 524
12.154 965
12.541 708
12.932 802
13.328 296
13.728 240
14.132 682
14.541 675
14.955 269
15.373 516
15.796 468
1.8092 656 | 17.983 681
1.8296 199
1.8502 031
1.8710 179
18.435 997
18.893 402
19.355 953
16.224 178
16.656 700
17.094 088
17.536 396
1.8920 668
19.823 707
1.9133 526 | 20.296 724
1.9348 778
1.9566 452
20.775 062
21.258 782
21.747 943
22.242 607
22.742 837
23.248 694
25.329 674
27.505 891
29.781 702
32.161 666
2.5592 747
2.6764 002
2.7988 858
2.9269 771
34.650 550
37.253 337
39.975 241
42.821 713
3.0609 305 45.798 454
0.2472 19
0.4916 87
0.7334 36
0.9724 96
1.2088 96
1.4425 66
1.6738 35
1.9024 32
2.1281 87
2.3520 27
2.5730 80
2.7916 73
3.0078 35
3.2215 92
3.4329 71
3.6419 99
3.8487 01
4.0531 03
4.2552 32
4.4551 12
4.6527 68
4.8482 26
5.0415 09
5.2326 42
5.4216 48
5.6085 52
5.7933 76
5.9761 45
6.1568 80
6.3356 04
6.5123 41
6.6871 11
6.8599 36
7.0308 39
7.1998 41
7.3669 63
7.5322 25
7.6956 49
7.8572 55
8.0170 63
8.1750 93
8.3313 66
8.4858 99
8.6387 14
8.7898 28
8.9392 64
9.0870 32
9.2321 59
9.3776 61
9.5205 54
9.6618 58
9.8015 90
9.9397 68
10.076 41
10.211 53
10.345 15
10.477 28
10.607 94
10.737 14
10.864 91
10.991 26
11.116 21
11.239 76
11.361 94
11.837 21
12.291 68
12.726 £6
13.141 83
13.539 21
13.919 20
14.282 56
14.630 01
14.962 27
Page 26.
PERPETUAL STOCK: 3 p. C. 66,6667; 3½ p. C.77,7778; 4 p.C. 88,8889; 5 p.C. 111,1111.

1
Years.
ཨྠ་པཝཾqtce=
HEANNA
20
++~~~
یں
+24
-H
~4~*~4~:~]*
5
PINHAL
C
HEANDE
7
~~:~
∞
10-6:00
→
0:CM平​老
​HANPH
IHHH CANAL GAA AAA BAA
13
14
19
20
2*7* 2
21
22
23
24
RATIO of INTEREST:
ANNUALLY, 0.045.
25
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9569 378 1.0000 000
0.9157 300 0.4889 976 0.5339 976
|
0.8762 966 0.3187 734 0.3637 734
|
0.8385 614
0.8024 510
...
0.7678 957
0.7348 281
0.7031 851
0.6729 044
0.6439 277
0.4944 693
17 0.4731 764
18
0.4528 004
0.4333 018
0.4146 429
0.5642 716
0.5399 728
0.5167 204
1.0450 000
0.2337 437 0.2787 437
0.3967 874
0.3797 009
0.3633 501
0.3477 035
0.3327 306
0.1827 919 | 0.2277 919
0.6161 988 0.0722 482
0.1488 784 0.1938 784
0.1247 015 0.1697 015
0.1066 097 0.1516 097
0.0925 745 0.1375 745
0.0813 788
0.5896 639 0.0646 662 0.1096 662
0.1263 788
0.0722 482 0.1172 482
0.0582 754 0.1082 754
0.0528 203 0.0978 203
0.0481 138 0.0931 138
0.0440 154 0.0890 154
0.0404 176
0.0404 176 | 0.0854 176
0.0372 369 0.0822 369
0.0344 073 0.0794 073
0.0318 761 0.0768 761
0.0296 006 0.0746 006
0.0275 456 0.0725 456
0.0256 825 0.0706 825
0.0239 870 0.0689 870
0.0224 390 | 0.0674 390
RATIO of INTEREST:
HALF-YEARLY, 0.0225.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9779 950 2.0000 000
0.9564 744 | 0.9888 752
0.9354 273
0.9148 434
0.8947 123
0.8750 243
0.7488 191
0.7323 413
……….
0.8185 216 | 0.2029 647
0.8005 101 0.1805 754
...
0.6518 891
0.4834 379
....
0.3120 005
0.8557 694 0.2670 005
0.8369 384 0.2309 692 0.2759 692
0.7828 950 0.1622 730
0.7656 675 | 0.1470 348
..
....
0.6552 348
0.6408 165
....
0.3823 996 0.4273 996
0.3150 699
0.3600 699
....
0.7162 263
0.1135 771
0.7004 659 | 0.1052 333
...
0.1341 537
0.1231 246
0.6850 521 0.0978 808
0.6699 776 0.0913 544
....
…………
..
0.6267 153 0.0755 514
0.6129 246 0.0712 564
0.0855 236
0.0802 841
0.5994 372 0.0673 419
0.5862 467 | 0.0637 605
0.5733 464 | 0.0604 720
0.5607 300 0.0574 427
2.0450 000
1.0338 752
0.6968 891
0.5281 379
0.5016 920 | 0.0453 056
0.4906 523 0.0433 483
0.4692 964 0.0397 931
0.4488 700 | 0.0366 504
0.4293 327
0.4106 458
0.0338 551
0.0313 548
....
0.3927 721 0.0291 073
0.3756 765 | 0.0270 780
0.3593 250 0.0252 384
0.3436 852 0.0235 647
0.2479 647
0.2255 754
0.2072 730
....
0.1920 348
0.1791 537
0.1681 246
0.1585 771
0.1502 333
0.1128 808
0.1363 544
0.1305 236
0.1252 841
....
0.1205 514
0.1162 56+
0.1123 419
0.1087 605
0.5483 912 0.0546 438
0.0546 438 0.0996 438
0.5363 239
0.0520 505
0.0970 505
0.1054 720
··
0.1024 427
0.5245 221 0.0496 416
0.0496 416 0.0946 416
0.5129 801 0.0473 987 0.0923 987
0.0903 056
0.0883 483
0.0847 931
0.0816 504
0.0788 551
0.0763 548
0.0741 073
0.0720 780
0.0702 384
0.0685 647
0.3287 261 | 0.0220 367
0.0220 367 0.0670 367
RATIO of INTEREST:
QUARTERLY, 0.01125.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9888 752 4.0000 000
0.9778 741 1.9888 130
0.9669 953 1.3184 452
0.9562 377 | 0.9832 823
0.9455 997
0.9350 800
0.9246 774
0.7822 013
0.6481 614
0.5524 305
0.9143 905 | 0.4806 428
0.9042 181
0.8941 588
0.8842 114
0.8743 747
0.8646 474 | 0.2874 650
0.8550 283
0.8455 163
0.8361 101
0.2054 055
0.2462 929
0.2295 745
0.8268 084
0.8176 104
0.8085 145
0.7995 200
0.7906 254
0.7818 298
0.7731 321
0.7645 311
0.7392 980
0.7310 736
0.7229 404
0.7148 978
0.7069 448
0 6990 800
0.7560 259 | 0.1394 457
0.7476 152
0.6610 499
0.6536 958
0.6464 235
0.6392 322
0.6321 208
0.6250 885
0.6181 345
0.6112 579
0.4248 173
0.3801 653
0.3436 394
0.3132 078
0.5780 053
0.5715 751
0.5652 164
0.5589 284
0.6913 029 | 0.1007 740
0.6836 122 0.0972 304
0.6760 072 | 0.0938 920
0.6684 867
0.0907 412
0.5527 104
0.5465 616
0.5404 812
0.5344 684
0.5285 226
0.5226 428
0.5168 285
0.5110 789
0.1699 258
0.1612 610
0 1533 533
0.1461 081
0.4673 257 |
0.4468 744
0.4273 182
0.4086 178
0.1332 991
0.1276 109
0.1223 319
0.2148 279 | 0.2598 279
0.2467 215
0.2017 245
0.1900 048
0.1794 612
0.1174 199
0.1128 382
0.1085 546
0.1045 414
0.6044 577 0.0687 679
0.5977 332 0.0668 661
0.5910 835 0.0650 469
0.5845 078 | 0.0633 053
0.0877 629
0.0849 436
0.0822 709
0.0797 338
0.0773 228
0.0750 283
0.0728 425
0.0707 580
0.0616 361
0.0600 355
0.0584 998
0.0570 243
4.0450 000
2.0338 130
1.3634 452
1.0282 823
0.8272 013
0.6931 614
0.5974 305
0.5256 428
0.4698 173
0.4251 653
0.3886 394
0.3582 078
0.3324 650
0.3104 055
0.2912 929
0.2745 745
0.0394 794
0.0363 559
0.0335 777
0.0310 929
0.2350 048
0.2244 612
0.2149 258
0.2002 610
0.1983 533
0.1911 081
0.1844 457
0.1782 991
0.1726 109
0.1673 319
0.1624 199
0.1578 382
0.1535 546
0.1495 414
0.1457 740
0.1422 304
0.1388 920
0.1357 412
0.1327 629
0.1299 436
0.1272 709
0.1247 338
0.1223 228
0.1200 283
0.1178 425
0.1157 580
0.1137 679
0.1118 661
0.1100 469
0.1083 053
0.1066 36+
0.1050 355
0.1034 998
0.1020 243
0.0556 060
0.0542 417
0.1006 060
0.0992 417
0.0529 285 0.0979 285
0.0516 637 | 0.0966 637
0.0504 446 | 0.0954 446
0.0192 690
0.0481 344
0.0470 394
0.0942 690
0.0931 314
0.0920 394
|
0.5053 932 | 0.0459 814
0.0909 814
0.4997 708 0.0449 588 0.0899 588
0.4942 109 | 0.0439 699 | 0.0889 699
0.4887 129
0.0430 132
0.0880 132
0.0844 794
0.0813 559
0.0785 777
0.0760 929
0.0288 596
0.3907 357
0.0738 596
0.3736 362 | 0.0268 432 0.0718 432
0.3572 850 0.0250 155
0 0700 155
0.3416 494
0.0233 526 | 0.0683 526
0.3266 981
0.0218 348
0.0668 348
Page 27.
RATE OF INTEREST: 4½ per Cent.
-CORRESPONDING VALUE OF
per
Ann'.

Years.
26
27
28
29
30
31
32
33
34
A ☺ ☺o co co ess
36
37
38
39
35 4.6673 478
40
41
**2*3 94938 KONAR OHOOO EXMZN 25982 H2MEN OFC0%
43
46
6.0781 009
42 6.3516 155
6.6374 382
44 6.9361 229
45 7.2482 484
50
51
52
53
7.5744 196
47 7.9152 685
54
55
56
61
65
66
67
68
11.762 842
57 12.292 170|
12.845 318
59 13.423 357
58
60
14.027 408
69
70
14.658 641
63
62 15.318 280
16.007 603
64 16.727 945
17.480 702
71
73
74
75
76
77
78
RATIO of INTEREST:
ANNUALLY, 0.045.
79
Improved Cumulated Value of
Principal. Annuity.
Annuity.
22.764 422
72 23.788 821
80
3.1406 790
3.2820 096
3.4297 000
3.5840 365
3.7453 181
81
82
83
84
3.9138 575
4.0899 810
4.2740 302
4.4663 615
* 85888 18848 26888
100
86.163 966
91.041 344
5.3262 192 | 96.138 205
4.8773 785
5.0968 605
5.5658 991
5.8163 645
101.46 442
107.03 032
8.2714 556
8.6436 711
9.0326 363
9.4391 049
9.8638 646
10.307 739
10.771 587
11.256 308
24.859 318
25.977 987
27.146 996
47.570 645
50.711 324
53.993 333
57.423 033
61.007 070
64.752 388
68.666 245
72.756 226
77.030 256
81.496 618
44.055 586
46.038 087
112.84 669
118.92 479
125.27 640
131.91 384
138.84 997
146.09 821
153.67 263
161.58 790
169.85 936
178.50 303
18.267 334
383.71 853
19.089 364 | 401.98 587
19.948 385 421.07 523
20.846 063 | 441.02 362
21.784 136 | 461.86 968
187.53 566
196.97 477
206.83 863
217.14 637
227.91 795
68.416 977
71.495 741
74.713 050
239.17 427
250.93 711
263.22 928
276.07 460
289.49 795
303.52 536
318.18 400
333.50 228
349.50 989
366.23 783
28.368 611
29.645 199
608.19 136
636.55 997
30.979 233 666.20 517
32.373 298 697.18 440
33.830 096 729.55 770
483.65 382
506.41 824
530.20 706
555 06 638
581.04 436
35.352 451 763.38 779
36.943 311 798.74 025
38 605 760 | 835.68 356
40.343 019
42.158 455
874.28 932
914.63 234
956.79 079
1000 8 464
1046.8 845
50 274 742 | 1094.9 943
52.537 105 1145.2 690
48.109 801
54.901 275
57.371 832
1197.8 061
1252.7 074
59.953 565 1310.0 792
62.651 475 | 1370.0 328
65.470 792 1432.6 843
15.146 61
15.451 30
15.742 87
16.021 89
16.288 89
16.544 39
16.788 89
17.022 86
17.246 76
17.461 01
17.666 04
17.862 24
18.049 99
18.229 66
18.401 58
18.566 11
18.723 55
18.874 21
19.018 38
19.156 35
19.288 37
19.414 71
19.535 61
19.651 30
19.762 01
19.867 95
19.969 33
20.066 34
20.159 18
20.248 02
20.333 03
20.414 39
20.492 24
20.566 73
20.638 02
20.706 24
20.771 52
20.833 99
20.893 77
20.950 98
21.005 72
21.058 11
21.108 24
21.156 21
21.202 11
21.246 04
21.288 08
21.328 31
21.366 80
21.403 63
21.438 88
21.472 62
21.504 90
21.535 79
21.565 35
21.717 81
21.739 53
21.760 32
21.780 21
21.799 24
21.817 46
21.834 89
21.851 57
21.867 53
21.882 80
1498.1 551
1566.5 720
21.897 42
21.911 41
1638.0 678 21.924 79
78.075 137 1712.7 808 21.937 60
81.588 518 1790.8 560
21.949 85
Value of Perpetual Ann. 22.222 22
3.1804 785
3.3252 102
3.4765 280
3.6347 318
3.8001 348
RATIO of INTEREST:
HALF-YEARLY, 0.0225.
Improved Cumulated
Principal. Annuity.
3.9730 647
4.1538 639
4.3428 907
4 5405 194
4.7471 414
6.2000 040
6.4821 429
6.7771 209
7.0855 223
7.4079 578
7.7450 662
8.0975 151
8.4660 027
8.8512 587
9.2540 463
9.6751 632
10.115 444
10.575 760
11.057 023
11.560 186
4.9631 660
88.070 355
5.1890 211 | 93.089 357
5.4251 540
5.6720 324
5.9301 453
98.336 755
103.82 294
109.55 878
12.086 247
12.636 247
13.211 275
13.812 471
14.441 024
15.098 181
15.785 243
16.503 570
17.254 586
18.039 777
18.860 700
19.718 979
20.616 316
21.554 488
22.535 351
21.593 63
36.766 924
21.620 70
38.440 049
21.646 61 40.189 312
21.671 39 42.018 177
21.695 11 48.930 266
23.560 851
24.633 017
25.753 973
26.925 940
28.151 238
29.432 296
30.771 649
32.171 951
33 635 976
35.166 623
45.929 368
48.019 441
50.204 626
48.455 078
51.671 337
55.033 956
58.549 595
62.225 217
71.673 201
74.934 780
78.344 781
81.909 958
85.637 373
66.068 104
70.085 865
74.286 460
78.678 209
83.269 809
115.55 564
121.82 540
128.38 015
135 23 383
142.39 906
149.89 036
157.72 256
165.91 117
174.47 242
183.42 325
192.78 141
202.56 541
212.79 466
223.48 939
234.67 081
246.36 105
258.58 326
271.36 167
284.72 157
298.68 944
313.29 292
328.56 095
344.52 378
361.21 301
378.66 171
396.90 444
415.97 732
435.91 814
456.76 639
478.56 337
501.35 224
525.17 815
550 08 829
576.13 199
603.36 085
631.82 879
661.59 220
692.71 003
725.24 391
759.25 829
794.82 054
832.00 109
870.87 359
911.51 504
954.00 592
998.43 040
1044.8 765
1093.4 361
52.489 250 -1144.2 056
54.877 839 1197.2 853
57.575 124 1252.7 805
59.986 051 1310.8 011
62.715 791 1371.4 620
65.569 751 1434.8 834
68.553 585
1501.1 908
1570.5 136
1642.9 951
1718.7 729
1797-9 991
1880.8 305
Value of
Annuity.
15.235 15
15.539 27
15.830 15
16.108 37
10.374 48
16.629 00
16.872 49
17.105 30
17.328 02
17.541 04
17.744 79
17.939 68
18.126 08
18.304 36
18.474 89
18.637 99
18.794 00
18.943 22
19.085 94
19.222 44
19.353 01
19.477 90
19.597 34
19.711 59
19.820 87
19.925 39
20.025 36
20.120 98
20.212 44
20.299 92
20.383 59
20.463 61
20.540 16
20.613 37
20.683, 40
20.750 37
20.814 44
20.875 71
20.934 32
20.990 38
21.043 99
21.095 27
21.144 33
21.191 24
21.236 12
21.279 04
21.320 09
21.359 36
21.396 91
21.432 83
21.467 19
21.500 05
21.531 48
21.561 55
21.590 31
21.617 82
21.644 12
21.669 29
21.693 35
21.716 37
21.738 39
21.759 45
21.779 59
21.798 85
21.817 28
21.834 91
21.851 77
21.867 89
21.883 31
21.898 06
21.912 17
21.925 67
21.938 58
21.950 93
21.962 73
22.222 22
RATIO of INTEREST:
QUARTERLY, 0.01125.
Improved Cumulated Value of
Principal. Annuity. Annuity.
3.2010 142 | 48.911 427
3.3475 089
52.166 864
3.5007 079
3.6609 181
3.8284 604
4.0036 702
4.1868 985
4.3785 123
4.5788 952
4.7884 488
6.2632 490
6.5498 871
6.8496 433
7.1631 178
7.4909 385
5.0075 926
89.057 612
5.2367 655 | 94.150 344
5.4764 265 | 99.476 144
105.04 568
5.7270 556
5.9891 548
110.87 011
7.8337 619
8.1922 747
8.5671 949
8.9592 733
151.86 138
159.82 833
168.15 989
176.87 274
9.3692 052 | 185.98 434
15.327 893
16.029 375
16.762 961
17.530 119
18.332 386
9.7980 813195.51 293
10.246 492
10.715 423
11.205 816
11.718 651
205.47 760
215.89 830
226.79 591
238.19 224
19.171 370
20.048 749
20 966 282
21.925 805
22.929 242
55.571 287
59.131 514
62.854 675
12.254 956 | 250.11 013
12.815 805
13.402 322
14.015 680
14.657 109
262.57 345
275.60 714
289.23 733
303.49 130
66.748 226
70.819 966
75.078 050
79.531 006
84.187 751
29.991 247
31.363 799
32.799 166
34.300 223
35.869 975
116.96 109
123.33 083
129.99 207
136.95 817
144.21 308
42.901 021
44.864 390
46.917 613
49.064 802
51.310 257
53.658 475
56.114 160
23.978 601
510.63 557
25.075 983
535.02 185
560 52 417
26.223 588
27.423 713 587.19 361
28.678 761 | 615.08 358
58.682 229
61.367 826
64.175 330
67.113 365
70.184 814
318.39 761
333.98 611
350.28 801
367 33 598
385.16 414
403.80 821
423.30 553
443.69 515
465.01 790
487.51 648
37.511 568
39.228 288
811.36 818
849.51 751
41.023 574 | 889.41 276
931.13 381
974.76 423
644.24 993
674 75 109
706 64 813
740.00 495
774.88 834
1020.3 914
1068.1 067
1118.0 057
1170.1 883
1224.7 591
1281.8 273
1341.5 073
1403.9 184
1469.1 859
1537.4 403
73.396 828
76.755 841
80.268 577
83.942 076
1608.8 184
1683.4 631
1761.5 239
1843.1 572
87.783 692 | 1928.5 265
15.279 98
15.533 79
15.874 30
16.152 10
16.417 74
16.671 76
16.914 66
17.146 93
17.369 04
17.581 42
17.784 51
17.978 72
18.164 43
18.342 00
18.511 81
18.674 19
18.829 47
18.977 94
19.119 91
19.255 67
19.385 50
19.509 61
19.628 35
19.741 86
19.850 41
19.954 20
20.053 46
20.148 37
20.239 12
20.325 91
20.408 90
20.488 25
20.564 14
20.636 70
20.706 08
20.772 43
20.835 88
20.896 55
20.954 56
21.010 04
21.063 09
21.113 81
21.162 32
21,208 70
21.253 06
21.295 47
21.336 03
21.374 81
21.411 89
21.447 35
21.481 27
21.513 69
21 544 70
21.574 35
21.602 70
21.629 82
21.655 74
21.680 53
21.704 23
21.726 90
21.748 58
21.769 31
21.789 13
21.808 08
21.826 20
21.843 54
21.860 11
21.875 95
21.891 11
21.905 60
21.919 46
21.932 70
21.945 37
21.957 49
21.969 07
22.222 22
Page 28.
PERPETUAL STOCK: 3 p. C. 66,6667; 3 p. C. 77,7778; 4p. C. 88,8889; 5 p. C. 111,1111.

Years.
Onda. H**** Okage 2333 95398 488*3 85838 58838 85832 ENC** DE22: ***** 8*898 CXKER 85338
¿
100
RATIO of INTEREST:
ANNUALLY, 0.045.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.3184 025
0.3046 914
0.2915 707
0.2790 150
0.2670 000
0.2555 024
0.2444 999
0.2339 712
0.2238 959
0.2142 544
0.2050 282
0.0116 058
0.1961 992 0.0109 840
0.1877 504 0.0104 017
0.1796 655 | 0.0098 557
0.1719 287 0.0093.431
0.1645 251
0.1574 403
0.1506 605
0.0088 616
0.0084 087
0.0079 823
0.0075 807
0.1441 728
0.1379 644 | 0.0072 020
0.1059 422
0.1013 801
0.0970 145
0.0928 368
0.0888 391
0.1320 233
0.0068 445
0.1253 381 0.0065 073
0.1208 977
0.0061 886
0.1156 916 | 0.0058 872
0.1107 096 0.0056 021
0.0850 135
0.0813 526
0.0778 494
0.0744 970
0.0712 890
0.0210 214 0.0660 214
0.0197 195 | 0.0647 195
0.0185 208 0.0635 208
0.0174 145
0.0624 145
0.0163 915 | 0.0613 915
0.0682 192
0.0652 815
0.0154 434 0.0604 434
0.0145 632 | 0.0595 632
0.0137 445 0.0587 445
0.0129 819 0.0579 819
0.0122 704 0.0572 704 ||
0.0547 425
0.0523 852
0.0501 294
0.0479 707
0.0459 050
0.0053 323
0.0050 768
0.0048 347
0.0046 052
0.0043 876
0.0041 811
0.0039 851
0.0037 990
0.0036 222
0.0034 543
0.0624 703
0.0597 802 0.0028 611
0.0572 059 | 0.0027 305
0.0226 986
0.0217 211
0.0207 858
0.0198 907
0.0190 342
0.0439 282
0.0420 365
0.0402 264
0.0384 941
75 0.0368 365 | 0.0017 210
0.0020 676
0.0019 747
0.0018 861
0.0018 016
0.0352 502
0.0016 442
0.0337 323
0.0015 709
0.0322 797
0.0015 010
0.0308 897
0.0014 343
0.0295 595 | 0.0013 707
0.0032 946 | 0.0482 946
0.0031 428 0.0481 428
0.0029 985 | 0.0479 985
0.0478 611
0.0177 305
0.0282 866
0.0270 685 | 0.0012 520
0.0259 029
0.0247 874
0.0237 200
0.0011 966
0.0011 438
0.0010 933
0.0566 058
0.0559 840
0.0554 017
0.0548 557
0.0543 431
0.0538 616
0.0534 087
0.0529 823
0.0525 807
0.0522 020
0.0026 061 0.0176 061
0.0024 877 0.0474 877
0.0023 749 | 0.0473 749
0.0022 675 | 0.0472 675
0.0021 651 0.0471 651
0.0518 445
0.0515 073
0.0511 886
0.0508 872
0.0506 021
0.0010 452
0.0009 992
0.0009 552
0.0009 133
0.0008 732
0.0503 323
0.0500 768
0.0498 347
0.0496 052
0.0493 876
0.0491 811
0.0489 851
0.0487 990
0.0486 222
0.0184 543
0.0013 100 | 0.0463 100
0.0462 520
0.0461 966
0.0461 438
0.0460 933
RATIO of INTEREST:
HALF-YEARLY, 0.0225.
Discounted Annuity for Ratio of
Principal. [Cumulation Annuity.
0.3144 181
0.3007 329
0.2876 433
0.2751 235
0.2631 486
0.0182 145
0.0174 302
0.0159 613
0.0008 349 0.0458 349
0.0007 983 0.0457 983
0.0166 796 | 0.0007 633 0.0457 633
0.0007 299 0.0457 299
0.0152 740 | 0.0006 980
0.0456 980
0.0146 163 0.0006 675 0.0456 675
0.0139 868 0.0006 383 0.0456 383
0.0006 105
0.0133 845
0.0456 105
0.0128 082
0.0005 838 0.0455 838
0.0122 566 0.0005 584 0.0455 584
Ratio of Perpetual Ann. 0.0450 000
0.2516 949
0.2407 397
0.2302 614
0.2202 391
0.2106 531
0.2014 843
0.1927 146
0.1843 266
0.1763 037
0.1686 299
0.1612 902
0.1542 700
0.1475 553
0.1411 329
0.1349 900
0.1291 145
0.1234 947
0.1181 195
0.1129 783
0.1080 608
0.1033 574
0.0988 587
0.0945 559
0.0904 403
0.0865 038
0.0827 387
0.0791 374
0.0756 929
0.0723 983
0.0692 472
0.0662 331
0.0533 503
0.0605 929
0.0579 555
0.0554 331
0.0170 676 0.0424 433
0.0469 747 0.0405 959
0.0468 861 0.0388 290
0.0468 016 0.0371 389
0.0467 210 0.0355 224
0.0530 203
0.0507 126
0.0485 053
0.0463 911
0.0443 747
0.0466 442
0.0339 763
0.0165 709 0.0324 974
0.0465 010 0.0310 830
0.0464 343 0.0297 301
0.0463 707
0.0284 361
0.0271 984
0.0260 145
0.0248 822
0.0237 992
0.0227 633
0.0460 452
0.0459 992
0.0217 726
0.0208 249
0.0459 552 0.0199 185
0.0459 133 0.0190 515
0.0458 732 0.0182 223
0.0206 377 | 0.0656 377
0.0193 531 0.0643 531
0.0181 7060.0631 705
0.0170 795 0.0620 795
0.0160 707 0.0610 707
0.0151 359 | 0.0601 359
0.0142 682 0.0592 682
0.0134 614 0.0584 614
0.0127 100 0.0577 100
0.0120 092 | 0.0570 092
0.0113 546 | 0.0563 545
0.0107 424 0.0557 424
0.0101 691 0.0551 691
0.0096 318 0.0546 318
0.0091 275 0.0541 275
0.0086 538
0.0082 084
0.0077 893
0.0073 946
0.0070 225
0.0066 715
0.0063 402
0.0060 273
0.0057 316
0.0051 519
0.0051 872
0.0049 367
0.0046 994
0.0044 745
0.0042 613
0.0040 591
0.0038 672
0 0036 851
0.0035 122
0.0033 480
0.0031 919
0.0030 436
0.0029 026
0.0027 684
0.0026 409
0.0019 946
0.0019 041
0.0018 179
0.0017 357 |
0.0016 574
0.0015 827
0.0015 115
0.0014 436|
0.0013 788
0.0013 171
0.0012 581
0.0012 019
0.0011 483
0.0010 971
0.0010 482
0.0536 538
0.0532 084
0.0527 893
0.0523 946
0.0520 225
0.0025 195
0.0024 040
0.0022 940
0.0021 893
0.0475 195
0.0474 040
0.0472 940
0.0471 893
0.0020 896 | 0.0470 895
0.0516 715
0.0513 402
0.0510 273
0.0507 316
0.0504 519
0.0501 872
0.0499 367
0.0496 994
0.0494 745
0.0492 613
I
0.0490 591
0.0488 672
0.0486 851
0.0485 122
0.0483 480
-
0.0481 919
0.0480 436
0.0479 026
0.0477 684
0.0476 409
0.0469 946
0.0469 041
0.0468 179
0.0467 357
0.0466 574
0.0.165 827
0.0465 115
0.0464 436
0.0463 788
0.0463 171
0.0010 016
0.0010 016 | 0.0460 016
0.0009 571
0.0009 145
0.0008 740
0.0008 352
0.0459 571
0.0459 145
0.0458 740
0.0458 352
0.0462 581
0.0462 019
0.0461 483
0.0460 971
0.0460 482
0.0174 292 0.0007 982 | 0.0457 982
0.0166 705 0.0007 629 | 0.0457 629
0.0159 449 0.0007 291 0.0457 291
0.0152 509 0.0006 969 | 0.0456 969
0.0006 661 | 0.0456 661
0.0145 871
0.0006 367 | 0.0456 367
0.0139 522
0.0133 449 0.0006 086 0.0456 086
0.0127 641 0.0005 818 0.0455 818
0.0122 085 0.0005 562 | 0.0455 562
0.0116 771 0.0005 317 | 0.0455 317
0.0450 000
RATIO of INTEREST:
QUARTERLY, 0.01125.
Discounted Annuity for
for Ratio of
Principal. Cumulation Annuity.
0.3124 010 0.0204 451 0.0654 451
0.2987 296 0.0191 693 | 0.0641 693
0.2856 565 | 0.0179 949 | 0.0629 949
0.2731 555 | 0.0169 114 | 0.0619 114
0.2612 016 0.0159 097 0.0609 097
|
0.2388 403
0.2283 881
0.2183 933
0.2088 359
0.2497 708 | 0.0149 817
0.0141 203
0.0133 195
0.0125 737
0.0118 782
0.0112 287
0.1996 968
0.1909 576 | 0.0106 213
0.1826 008
0.1746 098
0.1669 685
0.1596 616
0.1526 744
0.1459 930
0.1396 040
0.1334 946
0.1020 608
0.0975 944
0.0933 234
0.0892 394
0.0853 341
0.0815 997
0.0780 287
0.0746 139
0.0713 487
0.0682 263
0.0652 405
0.0623 855
0.0596 553
0.0570 447
0.0545 483
0.0417 039
0.0398 788
0.0381 336
0.0364 648
0.0348 690
0.0515 850
0.1276 526 0.0065 850
0.1220 662 | 0.0062 567 | 0.0512 567
0.1167 243 | 0.0059 467 | 0.0509 467
0.1116 162 0.0056 538 0.0506 538
0.1067 316 | 0.0053 768 | 0.0503 768
0.0333 431
0.0318 839
0.0304 885
0.0291 543
0.0278 785
0.0266 584
0.0254 918
0.0243 762
0.0233 095
0.0222 894
0.0213 140
0.0203 812
0.0194 893
0.0186 364
0.0178 208
0.0100 526
0.0095 197
0.0090 196
0.0170 409
0.0162 952
0.0155 821
0.0149 002
0.0142 481
0.0085 499
0.0081 083
0.0521 611 | 0.0024 761
0.0498 784 0.0023 624
0.0476 956 | 0.0022 538
0.0456 084
0.0436 124
0.0021 505
0.0020 521
0.0136 246
0.0130 283
0.0124 582
0.0119 130
0.0113 916
0.0076 928
0.0073 015
0.0069 327
0.0051 148
0.0048 667
0.0046 318
0.0044 093
0.0041 983
0.0031 407
0.0029 941
0.0028 548
0.0027 223
0.0025 963
0.0019 583
0.0018 691
0.0017 840
0.0017 030
0.0016 258
0:0039 982 | 0.0489 982
0.0038 085
0.0036 284
0.0034 574
0.0032 950
0.0488 085
0.0486 284
0.0484 574
0.0482 950
0.0015 522
0.0014 820
0.0014 151
0.0013 513
0.0012 905
0.0012 325
0.0011 771
0.0011 243
0.0010 740
0.0010 259
| 0.0599 817
0.0591 203
0.0583 195
0.0575 737
0.0568 782
0.0562 287
0.0556 213
0.0550 526
0.0545 197
0.0540 196
0.0007 801
0.0007 45+
0.0007 123
0.0006 806.
0.0006 504
0.0535 499
0.0531 083
0.0526 928
0.0523 015
0.0519 327
0.0006 216
0.0005 940
0.0005 677
0.0005 425
0.0005 185
0.0501 148
0.0498 667
0.0496 318
0.0494 093
0.0491 983
0.0481 407
0.0479 941
0.0478 548
0.0477 223
0.0475 963
0.0474 764
0.0473 624
0.0472 538
0.0471 505
0.0470 521
0.0469 583
0.0468 691
0.0467 840
0.0467 030
0.0466 258
0.0465 522
0.0464 820
0.0464 151
0.0163 513
0.0162 905
0.0009 800
0.0459 800
0.0009 362 | 0.0459 362
0.0008 944
0.0008 546
0.0008 165
0.0462 325
0.0461 771
0.0461 243
0.0460 740
0.0460 259
0.0458 944
0.0458 546
0.0458 165
0.0457 801
0.0457 454
0.0157 123
0.0156 806
0.0456 504
0.0456 216
0.0455 940
0.0455 677
0.0455 425
0.0455 185
0.0450 000
Page 29.
RATE OF INTEREST: 43 per Cent. per Ann.--CORRESPONDING VALUE OF

15
A
7
Years,
cr
CO
AKHAL
---
-T-2:-
9
POKLAH
POOPAL
P1006}~~
ات انتحار
FICHA
~+~2:40 +-~~T =
10
11
~~
12
Hedamlaco HNŽ
13
14
JARNYAL
rer {{c{©
17
18
19
20
21
22
23
24
25
RATIO of INTEREST:
ANNUALLY, 0.0475.
Improved Cumulated Value of
Principal. Annuity. Annuity.
....
1.0475 000
1.0972 563 2.0475 000
1.1493 759 3.1447 563
1.2039 713 4.2941 322
1.2611 599 5.4981 035
1.3210 650 6.7592 634
1.3838 156
....
1.0000 000
1.4495 468 9.4641 440
1.5184 003
1.5905 243
1.6660 742
1.7452 128
•
1.8281 104
8.0803 284
10.913 691
…….
12.432 091
14.022 615
15.688 690
...
15 2.0059 05521.176 958
17.433 902
1.9149 456 19.262 013
....
....
2.1011 860 23.182 864
2.2009 924 | 25.284 050
2.3055 395 | 27.485 042
2.4150 526 | 29.790 582
2.5297 676 | 32.205 635
2.6499 316
34.735 402
2.7758 034 37.385 334
2.9076 540
40.161 137
3.0457 676 43.068 791
3.1914 415 46.114 559
0.9546 54
••
1.8660 18
2.7360 55
3.5666 40
4.3595 61
..
5.1165 26
5.8391 65
6.5290 36
7.1876 21
....
7.8163 48
8.4165 61
8.9895 57
9.5365 70
10.058 78
....
10.557 31
11.033 23
11.487 57
11.921 31
12.335 38
12.730 67
13.108 04
13.468 29
13.812 21
14.140 54
14.453 97
RATIO of INTEREST:
HALF-YEARLY, 0.02375.
Improved Cumulated
Principal. Annuity.
...
1.0237 500
1.0480 641
1.0729 556 1.5359 070
1.0984 383
2.0723 848
1.1245 262
1.1512 337
....
1.2352 226
1.2645 591
....
1.1785 755 3.7591 839
1.2065 667
...
...
...
1.2945 924
6.2019 458
1.3253 390 6.8492 421
1.3890 402
....
1.4220 299
1.4558 031
1.3568 1587.5119 116
8.1903 195
...
1.4903 784
1.5257 749
....
1.5620 121
1.5991 098
1.6370 887
1.6759 696
0.5000 000
1.0118 750
1.7157 738
...
1.7982 409
1.8409 491
2.6216 040
3.1838 671
··
4.3487 716
2.0701 955
2.1193 626
2.2212 278
2.3279 890
2.4398 817
2.5571 523
4.9520 550
5.5636 663
2.6800 594
2.8088 739
8.8848 395
9.5958 545
10.323 756
11.068 945
15.068 923
...
1.7565 235 15.926 810
··
11.831 833
12.612 839
..
...
13.412 394
14.230 938
16.805 071
17.704 192
1.8846 717 18.624 666
1.9294 326 19.567 002
...
1.9752 566
20.531 719
2.0221 690 21.519 347
22.530 431
23.565 529
25.710 059
27.957 664
30.313 298
32.782 153
35 369 672
38.081 557
2.9438 798 40.923 786
3.0853 747
43.902 624
3.2336 703 47.024 638
Value of
Annuity.
0.4884 00
0.9654 71
....
1.4314 73
…………
1.8866 65
....
2.3312 96
2.7656 13
3.1898 54
3.6042 53
4.0090 38
4.4044 33
4.7906 55
5.1679 17
..
5.5364 27
5.8963 88
6.2479 98
6.5914 51
6.9269 36
7.2546 38
7.5747 38
7.8874 12
8.1928 32
8.4911 67
8.7825 81
9.0672 34
9.3452 84
9.6168 83
9.8821 81
10.141 32
10.394 46
10.641 72
10.883 24
11.119 16
11.574 71
12.009 36
12.424 09
12.819 79
13.197 35
13.557 59
13.901 31
14.229 27
14.542 19
RATIO of INTEREST:
QUARTERLY, 0.011875.
Improved Cumulated
Principal. Annuity.
1.0118 750
1.0238 910
1.0360 497
1.0483 528
1.0608 020
1.2800 421
1.0733 990 | 1.5452 426
1.0861 456 1.8135 924
1.0990 436 | 2.0851 288
1.1120 948
1.1253 009
1.1386 638
1.1521 855
1.1658 677
1.1797 123
1.1937 214
1.2078 969
1.2222 407
1.2367 548
1.2514 412
1.2663 021
1.2813 394
1.2965 553
1.3119 519
1.3275 314
1.3432 958
1.3592 474
1.3753 885
1.3917 212
1.4082 479
1.4249 709
1.4418 924
1.4590 149
1.4763 407
1.4938 722
1.5116 119
1.5295 623
1.5477 259
1.5661 051
1.5847 026
1.6035 210
1.6225 628
1.6418 307
1.6613 275
1.6810 557
1.7010 183
1.7212 178
1.7416 573
1.7623 395
1.7832 673
1.8044 436
1.8258 713
1.8475 536
1.8694 933
1.8916 935
1.9141 574
1.9368 880
1.9598 885
1.9831 622
2.0067 122
2.0305 419
2.0546 546
2.0790 537
2.1037 424
2.1287 241
0.2500 000
0.5029 688
0.7589 415
1.0179 539
2.2316 542
2.3395 609
2.4526 853
2.5712 795
2.3598 897
26379 134
2.9192 386
3.2039 046
3.4919 509
3.7831 179
4.0783 460
4.3767 763
4.6787 505
4.9813 107
5.2931 994
5.6063 597
5.9229 352
6.2432 701
6.5674 089
6.8953 969
7.2272 797
7.5631 037
7.9029 155
8.2467 626
8.5946 929
8.9167 549
9.3029 976
9.6634 707
10.028 224
10.397 310
10.770 778
11.148 681
11.531 071
11.918 003
12.309 529
12.705 705
13.106 585
13.512 226
13.922 683
14.338 015
14.758 279
15.183 534
15.613 838
16.049 .252
16.489 837
16.935 654
17.386 765
17.843 233
18.305 121
18.772 495
19.245 418
19.723 957
20.208 179
20.698 151
21.193 942
£1.695 620
22.203 255
22.716 919
23.236 683
23,762 618
25.929 561
28.201 283
30.582 848
33.079 568
2.6956 021
2.8259 483
2.9625 909
3.1058 405
35.697 013
38.441 017
41.317 703
44.333 484
3.2560 166 47.495 086
Value of
Annuity.
0.2470 66
0.4912 33
0.7325 34
0.9710 03
1.2066 74
1.4395 79
1.6097 51
1.8972 21
2.1220 22
2.3441 85
2.5637 41
2.7807 20
2.9951 52
3.2070 68
3.4164 97
3.6234 69
3.8280 11
4.0301 53
4.2299 23
4.4273 48
4.6224 56
4.8152 75
5.0058 31
5.1911 50
5.3802 59
5.5641 85
5.7459 52
5.9255 85
6.1031 11
6.2785 53
6.4519 36
6.6232 85
6.7926 22
6.9599 73
7.1253 59
7.2888 04
7.4503 32
7.6099 63
7.7677 22
7.9236 29
8.0777 06
8.2299 75
8.3804 57
8.5291 73
8.6761 44
8.8213 90
8.9649 31
9.1067 88
9.2469 80
9.3855 27
9.5224 48
9.6577 62
9.7914 88
9.9236 45
10.054 25
10.183 32
10.310 88
10.436 94
10.561 53
10.684 65
10.806 32
10.926 57
11.045 40
11.162 84
11.618 99
12.054 09
12.469 13
12.865 02
13.242 66
13.602 88
13.946 48
14.274 23
14.586 87
Page 30.
PERPETUAL STOCK: 3 p. C. 63,1579; 3½ p. C.73,6842; 4 p. C. 84,2105; 5 p.C. 105,2632.
Years.
yo
B
~}<
PORTAT
20
*{0.25)-4-
M
PRO:NHAIL
子
​FRANK
Wi
LO
**
PLAK
PKK-AN
Co
PROKAH
*~*~*~* =
-*-*-∞
12
ܤܚܘܠܕܝܝcdchܪܐܠ
13
ཨ¢ $ ¢2v72] ཨཱ ཨ 42790298
14
16
2002 22** 2
17
18
19
21
23
24
2.5
RATIO of INTEREST:
ANNUALLY, 0.0475.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9546 540
0.9113 641
0.8700 374
0.8305 846
0.7929 207
0.7569 650
0.6898 708
0.6287 235
0.6002 133
……..
0.5470 128
0.5222 081
0.4985 280
1.0000 000 1.0475 000
··
0.7226 396 | 0.1237 574
0.4759 217
0.4543 405
014337 380
0.4140 696
0.3952 932
···
··
0.3439 199
0.3283 245
0.3134 362
0.4884 005 0.5359 005
...
0.6585 879 0.0916 280
0.3179 897 0.3654 897
0.1818 809
0.2328 759 | 0.2803 759
0.5729 960 0.0637 402
I

0.1479 451 0.1954 451
0.1056 620
..
0.1237 574 0.1712 574
****
0.2293 809
0.1531 620
0.0801 370 | 0.1279 370
0.1391 280
...
0.3773 682
0.0287 891
0.3602 561 | 0.0267 485
0.0248 997
0.0232 187
0.0216 851
0.0713 134 0.1188 134
0.1112 402
..
0.0573 595 | 0.1048 595
0.0519 157 | 0.0994 157
0.0472211 0.0947 211
0.0131 353 0.0906 353
0.0395 506 0.0870 506
0.0363 834 0.0838 834
0.0810 677
0.0335 677
0.0310 505 0.0785 505
0.0762 891
0.0742 485
0.0723 997
0.0707 187
0.0691 851
RATIO of INTEREST:
HALF-YEARLY, 0.02375.
Ratio of
Discounted Annuity for
Principal. Cumulation Annuity.
0.9768 009
0.9541 401
..
0.6510 811
0.9320 050
0.9103 834 | 0.4825 359
...
0.8892 634
0.8686 334
...
...
0.8484 819 0.2659 940
0.8287 980 | 0.2299 500
2.0000 000
0.9882 641
0.8095 707 | 0.2019 368
0.7724 438
....
..
0.3814 459
0.3140 835
0.7907 894 0.1795 440
0.7370 197
0.1612 397
0.7545 239 0.1460 016
0.7032 201
0.6869 061
...
3.
...
...
....
....
0.1125 513
0.1331 219
0.1806 219
...
0.7199 216 | 0.1220 954
0.1220 954 0.1695 954
....
0.1042 117
0.6709 705 0.0968 640
0.6551 047
0.0903 428
...
0.6401 999 | 0.0845 178
0.6253 478 | 0.0792 843
.………
0.5828 274 0.0663 617
...
0.5693 064 | 0.0627 872
0.5560 990 | 0.0595 058
0.5431 981 0.0564 838
...
2.0475 000
1.0357 644
0.5985 811
0.5300 359
0.5062 633 0.0487 051
0.4945 185 0.0461 698
0.0461 698
0.4289 459
0.3615 835
0.4830 461 0.0443 844
0.4718 399 0.0424 319
0.0424 319
0.4502 015 0.0388 953
0.4295 553 | 0.0357 681
0.4098 560 | 0.0329 888
0.3910 600
0.0305 044
0.3134 940
0.2774 500
0.2494 368
0.2270 440
0.2087 397
0.1935 016
0.6108 404 0.0745 579
0.1220 579
0.5966 696 0.0702 694 0.1177 694
0.1600 513
0.1517 117
0.1443 640
…………
0.1378 428
0.1320 178
0.1267 843
...
0.1138 617
0.1102 872
0.1070 058
....
..
0.5305 964 0.0536 922 0.1011 922
0.5182 871 | 0.0511 065
0.0511 065 0.0986 065
0.1039 838
0.0962 051
0.0939 698
0.0918 844
0.0899 319
0.0363 953
0.0832 684
0.0804 888
0.0780 041
0.3731 261
0.3560 145
0.3396 878
0.3241 098
0.0282 598
0.0262 594
0.0214 357
0.0227 777
0.0757 598
0.0737 594
0.0719 357
0.0702 777
0.5092 461 0.0212 651 0.0687 654
RATIO of INTEREST:
QUARTERLY, 0.011875.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9282 641
0.9766 655
0.9652 047
0.9538 774 |
0.9426 830
0.9316 200
0.9206 869
0.9098 820 | 0.4795 867
0.7812 243
0.6471 476
0.5513 918
0.8577 303
0.8476 643
0.8377 163
0.8278 853
0.8992 039 | 0.4237 486
0.3790 875
0.3425 551
0.3121 191
0.8886 512
0.8782 223
0.8679 158
0.8181 694
0.8085 677
0.7990 787
0.7897 011
0.7804 333
0.7712 744
0.7622 231
0.7532 779
0.7444 377
0.7357 012
0.7270 673
0.7185 347
0.7101 023
0.7017 687
0.6935 330
0.6853 9.10
L
4.0000 000
1.9881 951
1.3176 247
0.9823 627 | 1.0298 627
0.9823 627
4.0475 000
2.0356 951
1.3651 247
0.2863 729
0.2643 113
0.2451 974
0.2284 787
0.5349 043
0.5286 269
0.5224 230
0.5162 921
0.2137 323
0.2006 295
0.1889 110
0.1783 689
0.1688 352
0.1601 725
0.1522 671
0.1450 243
0.6773 505 0.0997 185
0.6694 013 0.0961 787
0.6615 454 0.0928 431
0.6537 818 | 0.0896 967
0.1383 646
0.1322 209
0.1265 356
0.1212 597
0.6461 092 0.0867 222
0.6385 267 0.0839 067
0.6310 332 0.0812 379
0.6236 276 | 0.0787 048
0.4480 981
0.4274 306
0.4077 164
0.3889 115
0.1163 509
0.1117 724
0.1074 922
0.1034 825
0.6163 090 | 0.0762 975
0.6090 762 0.0740 071
0.6019 283 0.0718 252
0.5948 642 | 0.0697 447
0.3709 738
0.3538 635
0.3375 424
0.3219 741
0.3071 237
0.5878 832
0.5809 840
0.5741 658
0.5674 276 | 0.0623 082
0.0677 586
0.0658 608
0.0640 458
0.5607 684 | 0.0606 434
0.5541 875
0.0590 470
0.5476 837 0.0575 150
0.5412 563 | 0.0560 438
0.5102 331 0.0494 849
0.5042 452 0.0483 135
0.4983 276 | 0.0471 833
0.4924 793 0.0460 923
0.8287 243
0.69-16 476
0.5988 918
0.5270 867
0.0450 384
0.0440 201
0.0430 354
0.4712 486
0.4265 875
0.3900 551
0.3596 191
0.3338 729
0.3.118 113
0.2926 974
0.2759 787
0.2612 323
0.2481 295
0.2364 110
0.2258 689
0.2163 352
0.2076 725
0.1997 671
0.1925 243
0.1858 646
0.1797 209
0.1740 356
0.1687 597
0.1638 509
0.1592 724
0.1549 922
0.1509 825
0.1472 185
0.1436 787
0.1403 434
0.1371 967
0.1342 222
0.1314 067
0.1287 379
0.1262 048
0.1237 975
0.1215 071
0.1193 252
0.1172 447
0.1152 586
0.1133 608
0.1115 458
0.1098 082
0.0546 295 0.1021 295
0.0532 694 0.1007 694
0.0519 604 | 0.0994 604
0.0506 998 | 0.0981 998
0.1081 434
0.1065 470
0.1050 150
0.1035 438
0.0925 384
0.0915 201
0.0905 354
0.1697 619 | 0.0420 829 | 0.0895 829
0.4866 999
0.4809 881
0.4753 433
0.0969 849
0.0958 135
0.0946 833
0.0935 923
0.0385 660 | 0.0860 660
0.0354 591
0.0326 981
0.0302 301
0.0829 594
0.0801 981
0.0777 301
0.0280 135 0.0755 135
0.0260 139 | 0.0735 139
0.0242 027 0.0717 027
0.0225 563 | 0.0700 563
0.0210 548
0.0685 548
Page 31,
RATE OF INTEREST. 43 per
CORRESPONDING VALUE OF
Cent. per Ann.

Years.
26
OSIEN EINE 18298 A***2 36848 600OG CABAA ***** ** to co co co co co co co co 10 20 20 10
33
34
43
46
48
49
50
51
76
77
81
82
44 7.7050 843
8.0710 758
45
*** 8*8*8 58818 85888
96
RATIO of INTEREST:
ANNUALLY, 0.0475.
100
Improved Cumulated Value of
Principal. Annuity.
Annuity.
3.3419 875 49.305 000
3.5007 319 | 52.646 988
3.6670 167
56.147 720
59.814 736
63.655 936
3.8412 000
4.0236 570
4.2147 807
4.4149 828
67.679 593
71.894 374
4.6246 944 | 76.309 357
4.8443 674
5.0744 749
80.934 051
85.778 419
5.3155 124
5.5679 993
5.8324 792
6.1095 220
6.3997 243
6.7037 112 | 120.07 813
7.0221 375 | 126.78 184
7.3556 890 133.80 398
141.15 967
148.86 475
13.447 011
57 14.085 744
14.754 817
15.455 671
16.189 815
8.4544 519
8.8560 383
9.2767 001
156.93 583
165.39 028
174.24 632
9-7173 434 183.52 302
10.178 917193.24 036
10.662 416
11.168 881
11.699 402
12.255 124
12.837 242
90.852 894
96.168 406
101.73 641
107.56 888
113.67 841
203.41 928
214.08 170
225.25 058
236.94 998
249.20 510
262.04 234
275.48 936
289.57 510
304.32 992
319.78 559
16.958 832
335.97 540
17.764 376 | 352.93 424
18.608 184 370.69 861
19.492 073
389.30 680
20.417 946 408.79 887
21.387 799 429.21 681
22.403 719 450.60 461
23.467 895 473.00 833
24.582 621 496.47 623
25.750 295 | 521.05 885
26.973 434 546.80 914
28.254 673 | 573.78 258
29.596 769
31.002 616
32.475 240
602.03 725
631.63 402
662.63 664
34.017 814 695.11 188
35.633 660 | 729.12 969
37.326 259 | 764.76 335
39.099 257
40.956 471
802.08 961
841.18 887
68.236 521
71.477 756
74.872 950
78.429 415
82.154 812
42.901 904 882.14 534
44.939 744 925.04 724
47.074 382 969.98 699
49.310 415 1017.0 614
51.652 660 1066.3 718
54.106 161 1118.0 244
56.676 204 1172.1 306
59.368 323 1228.8 068
62.188 319 1288.1 751
65.142 264 1350.3 635
86.057 165
90.144 881
94.426 763
98.912 034
103.61 036 2160.2 180
14.753 20
15.038 85
15.311 55
15.571 89
15.820 42
1790.6 772
1876.7 343
1966.8 792
2061.3 060
16.057 68
16.284 18
16.500 41
16.700 84
16.903 90
17.092 03
17.271 62
17.443 08
17.606 76
17.763 02
17.912 19
18.054 59
18.190 54
18.320 33
18.444 23
18.562 51
18.675 42
18.783 22
18.886 13
18.984 37
19.078 16
19.167 69
19.253 17
19.334 77
19.412 67
19.487 04
19.558 03
19.625 80
19.690 50
19.752 27
19.811 23
19.867 53
19.921 27
19.972 57
20.021 55
20.068 30
20.112 94
20.155 55
20.196 23
20.235 06
20.272 14
20.307 53
20.341 32
20.373 57
20.404 36
20.433 76
20.461 82
20 488 61
20.514 18
20.538 60
20.561 91
20.584 16
20.605 40
20.625 68
20.645 04
1415.5 057
1483.7 422
1555.2 200
20.744 10
20.758 09
20.771 45
1630.0 929 20.784 20
1708.5 224 20.796 37
20.663 53
20.681 17
20.698 01
20.714 10
20.729 44
Value of Perpetual Ann. 21.052 63
RATIO of INTEREST:
HALF-YEARLY, 0.02375.
Improved Cumulated
Principal. Annuity.
3.3890 936
3.5519 872
3.7227 102
3.9016 388
4.0891 674
4.2857 094 69.172 829
4.4916 980 73.509 431
4.7075 872 78.054 468
4.9338 530 82.817 996
5.1709 940 | 87.810 400
5.4195 330
5.6800 177
5.9530 225
6.2391 489
6.5390 278
6.8533 200
7.1827 184
7.5279 490
7.8897 728
8.2689 874
8.6664 285
9.0829 723
9.5195 368
9.9770 844
10.456 624
10.959 211
11.485 956
12.038 017
12.616 613
13.223 019
15.954 454
16.721 290
17.524 983
18.367 305
19.250 112
20.175 351
21.145 060
28.024 373
29.371 538
30.783 044
32.262 602
33.813 273
50.296 708
53.726 047
57.320 214
61.087 132
65.035 102
35.438 477
37.141 794
38.926 979
40.797 968
13.858 571
14.524 670
15.222 785 | 299.42 705
314.83 061
330.97 452
93.042 800
98.526 689
104.27 416
110.29 787
116.61 111
44.814 050
46.967 995 |
49.225 468
51.591 444
51.071 138
123.22 779
130.16 249
137.43 051
145.04 785
153.03 131
161.39 819
170.16 784
179.35 867
188.99 125
199.08 681
20.807 99 90.621 582
20.819 08 94.977 223
20.829 68 99.542 214
20.839 78 104.32 662
209.66 761
220.75 696
232.37 931
244.56 028
257.32 672
22.161 377 | 445.50 268
23.226 543
467.92 723
24.342 905 491.42 958
25 512 924
516.06 156
26.739 179 | 541.87 745
270.70 676
284.72 990
347.89 438
365.62 747
384.21 288
403.69 159
424.10 652
725.02 056
760.87 987
798.46 272
837.85 196
42.758 884 | 879.13 440
568.93 416
597.29 132
627.01 144
658.16 003
690.80 575
922.40 105
967.74 727
1015.2 730
1065.0 830
1117.2 871,
56.670 017
1172.0 004
59.393 808 1229.3 433
62 248 516 1289.4 424
65.240 432 1352.4 301
63.376 152 1418.4 453
71.662 588 1487.6 $34
75.106 983 1560.1 470
78.716 930 1636.1 459
82.500 385 1715.7 976
86.465 689 1799.2 777
1886.7 701
1978.4 678
2074.5 729
2175.2 972
20.849 43 109.34 098 | 2280.8 627
-
Value of
Annuity.
14.840 75
15.125 63
15.397 45
15.656 79
15.904 24
16.140 35
16.365 62
16.580 57
16.785 66
16.981 34
17.168 05
17.346 19
17.516 17
17.678 35
17.833 10
17.980 74
18.121 62
18.256 03
18.384 29
18.506 66
18.623 41
18.734 82
18.841 11
18.942 53
19.039 30
19.131 63
19.219 73
19.303 79
19.383 99
19 460 51
19.533 53
19.603 19
19.669 66
19.733 09
19.793 60
19.851 34
19.906 43
19.959 00
20.009 15
20.057 00
20.102 66
20.146 23
20.187 80
20.227 46
20.265 30
20.301 41
20.335 86
20.368 72
20.400 09
20.430 02
20.458 57
20.485 81
20.511 81
20.536 61
20.560 28
20.582 85
20.604 40
20.624 95
20.644 57
20.663 28
20.681 14
20.698 17
20.714 43
20.729 94
20.744 73
20.758 86
20.772 33
20.785 19
20.797 45
20.809 15
20.820 32
20.830 97
20.841 14
20.850 84
20.860 09
21.052 63
RATIO of INTEREST:
QUARTERLY, 0.011875.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
3.4134 542
3.5785 043
3.7515 350
3.9329 323
4.1231 006
4.3224 641
4.5314 674
4.7505 766
4.9802 803
5.2210 909
5.4735 453
5.7382 066
6.0156 650
6.3065 394
6.6114 783
6.9311 618
7.2663 030
7.6176.492
7.9859 840
8.3721 287
8.7769 447
9.2013 347
9.6462 451
10.112 668
10.601 644
11.114 263
11.651 669
12.215 060
12.805 693
13.424 884
14.074 015
14.754 533
15.467 956
15.215 875
16.999 959
17.821 954
18.683 696
19.587 105
20.534 197
21.527 083
22.567 978
23.659 203
24.803 192
26.002 496
27.259 790
28.577 878
29.959 698
31.408 334
32.927 015
34.519 129
36.188 226
37.938 028
39.772 439
41.695 548
43.711 645
45.825 226
43.041 005
50.363 922
52.799 159
55.352 147
50.809 561
54.284 300
57.927 053
61.745 943
65.749 486
69.946 613
74.346 682
93.050 044
97.549 275
102.26 606
107.21 091
112.39 486
78.959 507
83.795 375
88.865 071
94.179 901
99.751 718
105.59 295
111.71 662
118.13 638
124.86 657
131.92 217
139.31 893
147.07 335
155.20 271
163.72 515
172.65 968
182.02 621
191 84 564
202.13 988
212.93 186
224.24 567
236.10 653
248.54 090
261.57 651
275.24 242
289.56 912
304.58 855
320.33 422
336.84 123
354.14 641
372.28 834
391.30 748
411.24 625
432.14 912
454.05 269
477.03 986
501.11 984
526.36 834
552.83 769
580.58 689
609 67 786
640.17 545
672.14 769
705.66 587
740.80 476
777.64 270
816.26 187
856.74 838
899.19 253
943.68 897
990.33 694
1039.2 405
1090.5 086
1144.2 557
58.028 579
60.834 424
63.775 940
1200.6 017
1259.6 721
1321.5 987
66.859 6861386.5 197
70.092 540
1454,5 798
73.481 711
77.034 758
80.759 605
1525.9 308
1600.7 318
1679.1 496
84.661 559 | 1761.3 591
88.758 329
1847.5 438
1937.8 957
2032.6 163
2131.9 170
2236.0 191
2345.1 549
14.885 09
15.169 55
15.440 89
15.699 72
15.946 61
16.182 12
16.405 76
16.621 04
16.825 43
17.020 40
17.206 38
17.383 78
17.553 00
17.714 41
17.868 38
18.015 24
18.155 33
18.288 96
18.416 43
18.538.02
18.654 00
18.764 63
18.870 16
18.970 82
19.066 84
19.158 43
19.245 80
19.329 13
19.408 62
19.484 45
19.556 78
19.625 77
19.591 58
19.754 36
19.814 24
19.871 35
19.925 84
19.977 81
20.027 38
20.074 67
20.119 78
20.162 80
20.203 84
20.242 99
20.280 33
20.315 95
20.349 93
20.382 34
20.413 26
20.442 75
20.470 87
20.497 70
20.523 30
20.547 71
20.571 01
20.593 22
20.614 40
20.634 62
20.653 90
20.672 29
20.689 83
20.706 56
20.722 52
20.737 75
20.752 26
20.766 13
20.779 34
20.791 95
20.803 97
20.815 44
20.826 38
20.836 81
20.846 76
20.856 26
20.865 32
21.052 63
Page 32.
PERPETUAL STOCK: 3 p. C. 63,1579; 3½ p. C. 73,6842; 4p. C. 84,2105; 5 p. C. 105,2632.

Years.
***2* H***8 85822 79333 959A8 5888K AKA88 88836 86882 *°*** C*228 ***** 85898 XXX*3 85888
26
27
64
70
72.
75
77
82.
85
86
96
RATIO of INTEREST:
ANNUALLY, 0.0475.
100
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2992 231
0.2856 546
0.2727 012
0.2603 352
0.2485 301
0.2372 603
0.2265 014
0.0147 755
0.0139 093
0.2162 305 0.0131 046
0.2054 253 0.0123 557
0.1970 647 | 0.0116 579
0.1182 809
0.1129 173
0.1077 970
0.1029 088
0.0982 423
0.0937 874
0.0895 345
0.0854 745
0.0815 986
0.0778 984
0.1881 286 0.0110 068 0.0585 068
0.1795 977 | 0.0103 984 | 0.0578 984
0.0573 293
0.0567 964
0.0562 967
0.1714 537 0.0098 293
0.1636 789 | 0.0092 964
0.0087 967 |
0.1562 567
0.1491 711
0.1424 068
0.0083 279
0.0078 876
0.1359 492 0.0074 736 |
0.1297 844 0.0070 842
0.1238 992 | 0.0067 175
78. 0.0267 908
0.0255 759
0.0244 162
0.0467 556
0.0446 354
0.0426 114
0.0406 791
0.0388 345
0.0202 819
0.0189 944
0.0189 944
0.0178 101
0.0167 183
0.0157 095
\?
0.0743 660 0.0038 162
0.0709 938 | 0.0036 299
0.0677 745
0.0647 012
0.0617 672
0.0034 533
0.0032 859
0.0031 271
0.0293 964
0.0280 634
0.0589 663
0.0029 764
0.0562 924
0.0028 334
0.0537 398
0.0026 977
0.0513 029 0.0025 687
0.0489 765 0.0024 462 |
0.0233 090
0.0222 520
0.0212 430
0.0202 797
0.0193 601
0.0049 160
0.0046 711
0.0184 822
0.0176 441
0.0168 440
0.0160 802
0.0153 510
0.0063 720 0.0538 720
0.0060 463 0.0535 463
0.0057 390
0.0532 390
0.0054 489 0.0529 489
0.0051 749 | 0.0526 749
0.0370 735 0.0018 288
0.0353 924 | 0.0017 428
0.0337 875 0.0016 610
0.0322 553 0.0015 832
0.0307 927
0.0015 091
3
0.0044 395 | 0.0519 395
0.0042 203 0.0517 203
0.0040 128 0.0515 128
0.0677 819
0.0664 944
0.0653 101
0.0642 183
0.0632 095
0.0622 755
0.0614 093
0.0606 016
0.0598 557
0.0591 579
0.0014 386
0.0013 715
0.0013 076
0.0012 467
0.0011 888
0.0558 279
0.0553 876
0.0549 736
0.0545 842
0.0542 175
0.0146 549
0.0139 904
0.0133 560 | 0.0006 430
0.0127 503
0.0121 721
0.0006 135
0.0005 853
0.0023 298 0.0498 298
0.0022 192 | 0.0497 192
0.0021 141 0.0496 141
0.0020 142 0.0495 142
0.0019 192 | 0.0494 192
0.0007 065
0.0006 740
0.0513 162
0.0511 299
0.0509 533
0.0507 859
0.0506 271
0.0504 764
0.0503 334
0.0501 977
0.0500 687
0.0499 462
0.0493 288
0.0492 428
0.0491 610
0.0490 832
0.0490 091
0.0010 810
0.0010 309
0.0009 832
0.0009 378
0.0011 336 | 0.0486 336
0.0485 810
0.0485 309
0.0484 832
0.0484 378
0.0489 386
0.0488 715
0.0488 076
0.0487 467
0.0486 888
0.0008 944 0.0483 944
0.0008 531 0.0483 531
0.0008 138 0.0483 138
0.0007 763 | 0.0482 763
0.0007 405 0.0482 405
0.0524 160 0.0912 474
0.0521 711 || 0.0870 628
RATIO of INTEREST:
HALF-YEARLY, 0.02375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.0116 202
0.0005 584
0.0110 933 0.0005 328
0.0105 902 | 0.0005 084
0.0101 100 0.0004 851
0.0096 515 | 0.0004 629 | 0.0479 629
0.0480 584
0.0480 328
0.0480 084
0.0479 851
Ratio of Perpetual Ann. 0.0475 000
0.2950 641 | 0.0198 820 0.0673 820
0.2815 325 0.0186 129 || 0.0661 129
0.2686 215 0.0174 458 | 0.0649 458
0.2563 025 0.0163 701 0.0638 701
0.2445 486 0.0153 763 0.0628 763
0.2333 336 0.0144 565 | 0.0619 565
0.2226 329
0.0136 037 0.0611 037
0.2124 230 0.0128 116 | 0.0603 116
0.2026 814 0.0120 747 0.0595 747
0.1933 861 | 0.0113 882 0.0588 882
0.1845 177 | 0.0107 477
0.1760 558 0.0101 495
0.1679 819 | 0.0095 901
0.1602 783 | 0.0090 664
0.1529 279 0.0085 755 |
0.1459 147
0.1392 231
0.1328 383
0.1267 464
0.1209 338
0.1153 878
0.1100 961
0.1050 471
0.1002 297
0.0956 332
0.0830 702
0.0792 606
0.0756 257
0.0570 614
0.0544 446
0.0519 478
0.0495 654
0.0472 924
0.0451 235
0.0130 542
0.0410 797
0.0391 958
0.0373 983
0.0356 832
0.0340 468
0.0324 854
0.0309 956
0.0295 742
0.0035 121
0.0721 575 | 0.0036 940 | 0.0511 940
0.0688 484
0.0656 910
0.0626 784
0.0598 040
0.0081 150 0.0556 150
0.0076 827 0.0551 827
0.0072 764 | 0.0547 764
0.0068 943 0.0543 943
0.0065 346 | 0.0540 346
0.0061 958 0.0536 958
0.0058 766 0.0533 766
0.0055 754 0.0530 754
0.0052 912 | 0.0527 912
0.0050 229 | 0.0525 229
0.0047 695 0.0522 695
0.0045 299 0.0520 299
0.0043 033 0.0518 033
0.0040 890 | 0.0515 890
0.0038 861 0.0513 861
0.0482 065 0.0139 543
0.0481 740 0.0133 143
0.0127 037
0.0481 430
0.0481 135
0.0180 853
0.0582 477
0.0576 495
0.0570 901
0.0565 664
0.0560 755
0.0510 121
0.0033 397 0.0508 397
0.0031 763 | 0.0506 763
0.0030 214 0.0505 214
0.0028 744 | 0.0503 744
0.0027 350 | 0.0502 350
0.0026 027 | 0.0501 027
0.0024 771
0.0023 579
0.0499 771
0.0498 579
0.0022 447
0.0021 371
0.0497 447
0.0496 371
0.0020 349 | 0.0495 349
0.0494 378
0.0493 454
0.0019 378
0.0018 454
0.0017 577 | 0.0492 577
0.0016 742 | 0.0491 742
0.0015 949
0.0015 194
0.0014 476
0.0282 179
0.0269 238
0.0256 891
0.0013 793
0.0013 143
0.0012 524
0.0245 110 0.0011 935
0.0233 870 0.0011 375 | 0.0486 375
0.0488 793
0.0488 143
0.0487 524
0.0486 935
0.0176 460 | 0.0008 532
0.0168 368 | 0.0008 134
0.0160 646 | 0.0007 755
0.0153 279 | 0.0007 394
0.0146 250
0.0007 050
0.0223 144 0.0010 841 | 0.0485-841
0.0212 911 0.0010 333 0.0485 333
0.0203 147 0.0009 850 0.0484 850
0.0193 831 0.0009 389 0.0484 389
0.0184 942 0.0008 950 0.0483 950
0.0006 722
0.0006 410
0.0006 112
0.0121 212 | 0.0005 828
0.0115 653
0.0005 558
0.0490 949
0.0490 194
0.0489 476
K
0.0483 532
0.0483 134
0.0482 755
0.0482 394
0.0432 050
0.0481 722
0.0481 410
0.0481 112
0.0480 828
0.0180 558
0.0110 349 0.0005 300 0.0480 300
0.0105 288 0.0005 054 0.0480 054
0.0100 460 | 0.0004 820 0.0479 820
0.0095 853 0.0004 597 0.0479 597
0.0091 457 | 0.0004 384 | 0.0179 384
0.0475 000
RATIO of INTEREST:
QUARTERLY, 0.011875.
Ratio of
Discounted Annuity for
Principal. Cumulation Annuity.
0.2929 584
0.2794 464
0.2665 576
0.2542 632
0.2425 359
0.2313 495
0.2206 791
0.2105 008
0.2007 919
0.1915 309
0.1826 969
0.1742 705
0.1662 327
0.1585 656
0.1512 521
0.1442 760
0.1376 216
0.1312 741
0.1252 194
0.1194 439
0.1139 348
0.1085 799
0.1036 673
0.0988 859
0.0943 250
0.0899 745
0.0858 246
0.0818 661
0.0780 903
0.0744 885
0.0710 529
0.0677 758
0.0646 498
0.0616 680
0.0588 237
0.0561 106
0.0535 226
0.0510 540
0.0486 992
0.0464 531
0.0443 106
0.0422 669
0.0403 174
0.0384 578
0.0366 841
0.0349 921
0.0333 782
0.0318 387
0.0303 702
0.0289 691
0.0276 333
0.0263 588
0.0251 430
0.0239 834
0.0228 772
0.0218 220
0.0208 155
0.0198 555
0.0189 397
0.0180 661
0.0172 329
0.0164 381
0.0156 799
0.0149 567
0.0142 669
0.0136 088
0.0129 812
0.0123 824
0.0118 113
0.0112 665
0.0107 469
0.0102 512
0.0097 784
0.0093 274
0.0088 972
0.0196 813
0.0184 215
0.0172 631
0.0161 954
0.0152 092 | 0.0627 092
0.0671 813
0.0659 215
0.0647 631
0.0636 954
0.0142 966
0.0134 505
0.0126 647
0.0119 338
0.0594 338
0.0112 530 | 0.0587 530
0.0106 180
0.0100 249
0.0094 703
0.0089 512
0.0084 648
0.0080 085
0.0075 802
0.0071 778
0.0067 993
0.0064 432
0.0061 078
0.0057 917
0.0051 937
0.0052 125
0.0049 471
0.0046 963
0.0044 594
0.0042 354
0.0040 235
0.0038 230
0.0036 332
0.0034 534
0.0032 831
0.0031 217
0.0029 688
0.0028 237
0.0026 861
0.0025 555
0.0024 316
0.0023 140
0.0022 023
0.0020 963
0.0019 955
0.0018 998
0.0018 088
0.0017 224
0.0016 402
0.0015 621
0.0014 878
0.0014 171
|
0.0013 499
0.0012 859
0.0012 251
0.0011 672
0.0011 121
0.0617 966
0.0609 505
0.0601 647
0.0581 180
0.0575 249
0.0569 703
0.0564 512
0.0559 648
0.0555 085
0.0550 802
0.0546 778
0.0542 993
0.0539 432
0.0536 078
0.0532 917
0.0529 937
0.0527 125
0.0524 471
0.0521 963
0.0519 594
0.0517 354
0.0515 235
0.0513 230
0.0511 332
0.0509 534
0.0507 831
0.0506 217
0.0504 688
0.0503 237
0.0501 861
0.0500 555
0.0499 316
0.0498 140
0.0497 023
0.0495 963
0.0494 955
0.0493 998
0.0493 088
0.0492 224
0.0491 402
0.0490 621
0.0489 878
0.0489 171
0.0488 499
0.0487 859
0.0487 251
0.0486 672
0.0486 121
0.0485 597
0.0485 098
0.0010 597
0.0010 098
0.0009 622 | 0.0484 622
0.0009 170 0.0484 170
0.0008 739 | 0.0183 739
0.0008 329 0.0483 329
0.0007 939
0.0482 939
0.0007 567 | 0.0482 567
0.0007 212 0.0482 212
0.0006 875
0.0181 875
0.0181 553
0.0481 247
0.0480 955
0.0005 677 | 0.0480 677
0.0005 413
0.0006 553
0.0006 247
0.0005 955
0.0480 413
0.0005 160 0.0480 160
0.0004 920 0.0479 920
0.0001 691 | 0.0479 691
0.0004 472 0.0479 472
0.0004 264 | 0.0479 264
0.0475 000
Page 33.
RATE OF INTEREST: 5
5 per
CORRESPONDING VALUE of
Cent.
per Ann.-

Years.
Food
~<~~~~}<
૭
co
P10:21P-
+
5
[3.85/bplat
CO
ه دار
~}{*}<
PICAL
عطورات انتر
HAN
سه سرانه دار
OPINHAL
ܚܩܝܚܝܘܗܢܫܓ step:ܘܩܝܺ toesn-eh
12
13
& 2222 222 O CANAL BHV-P
14
16
19
20
21
23
24
25
Improved Cumulated Value of
Principal. Annuity. Annuity.
RATIO of INTEREST:
ANNUALLY, 0.05.
1.0500 000
....
1.1025 000
1.1576 250
1.2155 063
1.2762 816
1.4071 004
....
...
··
....
1.0000 000
10 1.6288 946| 12.577 693
2.0500 000
1.3400 956 6.8019 128
3.1525 000
....
4.3101 250
•
··
5.5256 313
1.4774 554 9.5491 089
8.1420 085
...
1.5513 282 11.026 564
..
2.7859 626
2.9252 607
3.0715 238
3.2250 999
3.3863 549
1.7103 394 14.206 787
1.7958 563 15.917 127
1.8856 491 17.712 983
1.9799 316 19.598 632
17
2.2920 183
25.840 366
18 2.4066 19228.132 385
...
2.1828 746 23.657 492
2.5269 502 30.539 004
2.6532 977 33.065 954
0.9523 81
35.719 252
38.505 214
41.430 475
44.501 999
47.727 099
1.8594 10
2.7232 48
3.5459 51
4.3294 77
5.0756 92
5.7863 73
6.4632 13
DAO
....
2.0789 282 21.578 564 10.379 66
7.1078 22
7.7217 35
8.3064 14
8.8632 52
9.3935 73
9.8986 41
10.837 77
11.274 07
11.689 59
12.085 32
12.462 21
13.488 57
13.798 64
14.093 94
RATIO of INTEREST:
HALF-YEARLY, 0.025.
Improved Cumulated Value of
Principal. Annuity. Annuity,
1.0250 000
1.0506 250
....
....
1.0768 906
1.1038 129 2.0762 578
0.5000 000
....
....
1.0125 000
1.1314 082
2.6281 643
1.1596 934 3.1938 684
....
1.1886 858
3.7737 151
1 2184 029 4.3680 580
1.5378 125
1.2488 630 4.9772 594
1.2800 845
5.6016 909
··
1.3120 867 6.2417 332
1.3448 888 6.8977 765
1.6795 819
1.7215 714
1.3785 1107.5702 209
1.4129 738 8.2594 764
1.7646 107
1.8087 259
1.4482 982 8.9659 633
1.4845 056 9.6901 124
1.5216 183 10.432 365
..
1 5596 587| 11.193 174
1.5986 502 11.973 004
1.6386 164 12.772 329
12.821 15
2.8209 952
13.163 00 2.9638 081
13.591 637
14.431 428
15.292 214
16.174 519
1.8539 441 17.078 882
1.9002 927 | 18.005 854
1.9478 000 18.956 000
1.9964 950 19.929 900
2.0464 074 20.928 148
2.0975 676 21.951 352
....
2.1500 068 | 23.000 135
2.2037 569 | 24.075 139
2.3153 221
2.4325 353
2.5556 824
26.306 443
28.650 706
31.113 648
2.6850 638 | 33.701 277
36.419 904
39.276 162
42.277 017
45.429 791
3.1138 509
3.2714 896
3.4371 087 48.742 174
0.4878 05
....
0.9637 12
1.4280 12
1.8809 87
2.3229 14
2.7540 63
3.1746 95
...
3.5850 69
3.9854 33
4.3760 32
4.7571'04
5.1288 82
...
5.4915 92
5.8454 56
6.1906 89
6.5275 01
6.8560 99
7.1766 82
7.4894 46
7.7945 81
8.0922 74
8.3827 07
8.6660 55
8.9424 93
..
9.2121 88
....
9.4753 06
9.7320 05
....
9.9824 44
10.226 77
10.465 15
...
10.697 70
10.924 59
11.361 89
11.778 13
12.174 30
12.551 39
12.910 30
13.251 92
13.577 08
13.886 58
14.181 16
RATIO of INTEREST:
QUARTERLY, 0.0125.
Improved Cumulated
Principal. Annuity.
1.0125 000
1.0251 563
1.0379 707
1.0509 453
1.0640 822
1.0773 832
1.0908 505
1.1044 861
1.1182 922
1.1322 708
1.1464 242
1.1607 545
1.1752 639
1.1899 547
1.2048 292
1.2198 895
1.2351 382
1.2505 774
1.2662 096
1.2820 372
1.2980 627
1.3142 885
1.3307 171
1.3473 511
1.3641 929
1.3812 454
1.3985 109
1.4159 923
1.4336 922
1.4516 134
1.4697 585
1.4881 305
1.5067 321
1.5255 663
1.5446 359
1.5639 438
1.5834 931
1.6032 868
1.6233 279
1.6436 195
1.6641 647
1.6849 668
1.7060 289
1.7273 542
1.7489 461
1.7708 080
1.7929 431
1.8153 549
1.8380 468
1.8610 224
1.8842 852
1.9078 387
1.9316 867
1.9558 328
1.9802 807
2.0050 342
2.0300 971
2.0554 733
2.0811 668
2.1071 813
2.3273 525
2.4459 203
2.5705 285
2.7014 849
0.2500 000
0.5031 250
0.7594 141
1.0189 067
2.8391 130
2.9837 526
3.1357 608
3.2955 132
3.4634 043
1.2816 431
1.5476 636
1.8170 094
2.0897 220
2.3658 435
2.6454 166
2.9284 843
3.2150 904
3.5052 790
3.7990 950
4.0965 837
4.3977 910
4.7027 633
5.0115 479
5.3241 922
5.6407 446
5.9612 539
6.2857 696
6.6143 417
6.9470 210
7.2838 588
7.6249 070
7.9702 183
8.3198 461
8.6738 441
9.0322 672
9.3951 705
9.7626 102
10.134 643
10.511 326
10.892 717
11.278 876
11.669 862
12.065 736
12.466 557
12.872 389
13.283 294
13.699 335
14.120 577
14.547 084
14.978 923
15.416 159
15.858 861
16.307 097
16.760 936
17.220 447
17.685 703
18.156 774
18.633 734
19.116 656
19.605 614
20.100 684
2.1335 211
22.670 422
2.1601 901 23.203 803
2.1871 925
2.2145 324
20.601 943
21.109 467
21.623 335
22.143 627
23.743 850
24.290 648
26.547 050
28.918 405
31.410 570
34.029 699
36.782 260
39.675 051
42.715 217
45.910 265
49.268 085
Value of
Annuity.
0.2469 14
0.4907 79
0.7316 33
0.9695 14
1.2014 59
1.4365 02
1.6656 81
1.8920 31
2.1155 86
2.3363 81
2 5544 51
2.7698 28
2.9825 46
3.1926 38
3.4001 36
3.6050 73
3.8074 80
4.0073 87
4.2048 27
4.3998 27
4.5924 22
4.7826 39
4.9705 08
5.1560 57
5.3393 15
5.5203 11
5.6990 73
5.8756 28
6.0500 03
6.2222 25
6.3923 21
6.5603 17
6.7262 39
6.8901 12
7.0519 63
7.2118 15
7.3696 94
7.5256 24
7.6796 28
7.8317 32
7.9819 57
8.1303 28
8.2768 67
8.4215 97
8.5645 40
8.7057 19
8.8451 54
8.9828 68
9.1188 82
9.2532 17
9.3858 93
9.5169 32
9.6463 52
9.7741 75
9.9004 20
10.025 11
10.148 25
10.269 88
10.390 00
10.508 65
10.625 82
10.741 55
10.855 85
10.968 74
11.406 54
11.823 12
12.219 50
12.596 67
12.955 55
13.297 03
13.621 96
13.931 14
14.225 33
Page 34.
PERPETUAL STOCK: 3 p. Cent. 60; 3½ p. Cent. 70; 4 p. Cent. 80; 5 p. Cent. 100.

Years.
funk
*)
سانات
3
1
10.0
C
=====|5\22/1
5:9/1
-
patched pa
nhân tố khác nhau nên thanh
14-244
AN-
2.0
-*-**-9
Beste Pap-A-
ཨབྷཱ[ཨqཨཊིཀྑ¥ 2{ཡ# བ ཞི4eཨནཎ[ཀྑཎ:c]ཨནྟཱཿཝཾ པཱཨདྡྷིཝཾ
13
ནི}ff2
16
17
18
02 727* *
RATIO of INTEREST:
ANNUALLY, 0.05.
22
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
25
0.9523 810
....
0.9070 295 0.4878 049
0.4878 049 0.5378 049
0.8638 376 0.3172 085
0.8227 025
12 0.5568 375
0.7462 154
0.7835 262 0.1809 748
0.7106 813
0.6139 132
11 0.5846 793 0.0703 889
15 0.4810 171
1.0000 000
0.6768 394 | 0.1047 218
0.5303 214
0.2320 118
0.6446 089 0.0906 901
0.4581 115
0.4362 967
0.4155 207
0.3957 340
0.3768 895
0.1228 198
0.3589 423
0.3418 499
0 3255 713
0.3100 680
0.2953 028
....
..
0.1470 175 0.1970 175
..
0.0795 050
1.0500 000
....
0.3672 085
0.2820 118
..
0.2309 748
··
0.0279 961
0.0259 705
6.0241 368
0.0224 709
0.0209 525
0.1728 198
0.1547 218
0.1406 901
0.0628 254 0.1128 254
0.5050 679 0.0510 240 0.1010 240
0.1295 050
0.0564 558 0.1064 558
0.1203 889
0.0463 423 0.0963 423
0.0422 699 0.0922 699
0.0386 991
0.0355 462
0.0327 450
0.0302 426
0.0886 991
0.0855 462
0.0827 450
0.0802 426
0.0779 961
0.0759 705
0.0741 368
0.0724 709
0.0709 525
RATIO of INTEREST:
HALF-YEARLY, 0.025.
Discounted Annuity for Ratio of
Principal. Cumulation Aunuity.
0.9756 098 2.0000 000 2.0500 000
0.9876 543 1.0376 543
0.9518 145
..
0.9285 994
0.9059 507
0.5502 743 | 0.7002 743
0.4816 358 0.5316 358
0.8838 543 0.3804 938
0.8622 9690.3131 000
...
0.7254 204
0.7077 272
....
0.8412 652 | 0.2649 908 | 0.3149 908
0.8207 466 | 0.2289 347
0.2789 347
....
0.8007 284 0.2009 138 0.2509 138
0.7811 984 0.1785 175 | 0.2285 175
0.7621 447 | 0.1609 514 | 0.2109 514
0.7435 559 0.1449 742 | 0.1949 742
..
0.1320 965 0.1820 965
0.1710 731
……….
0.1210 731
0.4304 938
0.3631 000
0.6904 656 0.1115 329 | 0.1615 329
0.6736 249 0.1031 980
0.1531 980
...
0.6571 950 0.0958 555
0.1458 555
0.6411 659 0.0893 402 0.1393 402
0.6255 277 0.0835 212
0.6102 710 0.0782 943
0.5953 863 0.0735 747 | 0.1235 747
0.5808 647 | 0.0692 932 | 0.1192 932
0.5393 906 0.0585 518
0.5262 347 0.0555 375
•
0.1335 212
0.1282 943
0.5666 973 | 0.0653 928 | 0.1153 928
0.5528 753 | 0.0618 256 0.1118 256
....
...
0.5133 997 0.0527 537 Q.1027 537
0.5008 778 | 0.0501 759
0.0501 759 0.1001 759
0.1085 518
0.1055 375
0.4886 612 0.0477 825
0.0977 825
0.4767 427 0.0455 553 | 0.0955 553
··
0.02
0.0205 161
...
0.4651 148
0.0434 780
0.0934 780
....
0.4537 705 0.0415 366
0.0415 366 0.0915 366
0.4319 054
0.4110 937
0.0380 135
0.0349 032
0.3912 849 0.0321 402
0.0880 135
0.0849 032
0.0821 402
0.3724 306 0.0296 725 | 0.0796 725
|
0.3544 848
0.3374 038
0.3211 458
0.0274 575 | 0.0774 575
0.0254 607 0.0754 607
0.0236 535 0.0736 535
0.3056 712 0.0220 120 | 0.0720 120
0.2909 422
0.0705 161
RATIO of INTEREST:
QUARTERLY, 0.0125.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9876 543
0.9754 611
0.9634 183
0.9515 243
0.9397 771
0.9281 749
0.9167 159
0.9053 984
0.8942 207
0.8831 810
0.8722 775
0.8615 084
0.8508 727
0.8403 681
0.8299 932
0.8197 464
0.8096 260
0.7996 306
0.7897 587
0.7800 014
0.7703 788
0.7608 680
0.7514 746
0.7421 970
0.7330 342
0.7239 843
0.7150 463
0.7062 185
0.6974 998
0.6288 886
0.6803 839
0.6719 841
0.6636 880
0.6554 943
0.6474 018
0.6394 092
0.5315 152
0.6237 187
0.6160 185
0.6084 134
0.6009 020
0.5934 835
0.5861 565
0.5789 200
0.5717 729
0 5647 140
0.5577 422
0.5508 565
0.5140 558
0.5373 390
0.5307 052
0.5241 533
0.5176 823
0.5112 912
0.5049 789
0.1987 416
0.4687 087
0.46.9 222
0.4572 028
0.4515 626
0.4296 728
0.4088 441
0.3890 251
0.3701 668
4.0000 000 4.0500 000
1.9875 776 | 2.0375 776
1.3168 048 1.3668 048
0.9814 441 1.0314 441
0.3522 227
0.3351 484
0.3189 019
0.3034 429
0.2887 332
0.7802 485 0.8302 485
0.6461 353 | 0.6961 353
0.5503 549 | 0.6003 549
0.4785 325 0.5285 325
0.4226 822 | 0.4726 822
0.3780 123 0.4280 123
0.3414 736 | 0.3914 736
0.3110 332 0.3610 332
0.2852 840
0.2632 206
0.2441 058
0.2273 869
0.2126 409
0.2626 409
0.2495 392
0.1995 392
0.1878 219 | 0.2378 219
0.1772 816 | 0.2272 816
0.1677 500
0.1590 895
0.1511 866
0.1439 466
0.1372 899 0.1872 899
0.1311 491
0.1811 491
0.1254 671 | 0.1754 671
0.1201 945 0.1701 945
0.3352 840
0.3132 206
0.2941 058
0.2773 869
0.1152 891
0.1107 142
0.1064 377
0.1652 891
0.1607 142
0.1564 377
0.1024 316 | 0.1524 316
0.0986 715
0.1486 715
0.0951 355 0.1451 355
0.0918 045 0.1418 045
0.0886 613 | 0.1386 613
0.2177 500
0.2090 895
0.2011 866
0.1939 466
0.0856 908 | 0.1356 908
0.0828 793 | 0.1328 793
0.0802 146 | 0.1302 146
0.0776 857 0.1276 857
|
0.4925 873 0.0485 391
0.4865 060 | 0.0473 721
0.4804 997
0.4745 076
0.0462 463
0.0151 597
0.0752 826 | 0.1252 826
0.0729 962 | 0.1229 962
0.0708 186 0.1208 186
0.0687 423
0.1187 423
0.0596 625
0.0580 705
0.0565 428
0.0550 759
0.0667 605 0.1167 605
|
0.0648 670 0.1148 670
0.0630 562 0.1130 562
0.0613 230 0.1113 230
0.0536 661
0.0523 104
0.0510 058
0.0197 495
0.0441 103
0.0430 954
0.0421 162
0.0411 681
0.1096 625
0.1080 705
0.1065 428
0.1050 759
0.1036 661
0.1023 104
0.1010 058
0.0997 495
0.0985 391
0.0973 721
0.0962 463
0.0951 597
0.0941 10.3
0.0930 964
0.0921 162
0.0911 681
0.0376 690 0.0876 690
0.0345 801 0.0845 801
0.0318 364 0.0818 364
0.0293 861 | 0.0793 861
0.0271 871 0.0771 871
0.0252 048 | 0.0752 048
0.0234 109 0.0734 109
0.0217 816 | 0.0717 816
0.0202 971
0.0702 971
Page 35,
ANGELES
Years.
31
128** ***** ***** 95938 58886 97888 58OJU 85882 *°*** OF22: ***** 8*888 ***** 85282
26
3.5556 727
27 3.7334 563
28 3.9201 291
29
4.1161 356
4.3219 424
30
34
35
RATE OF INTEREST: 5 per
RATIO of INTEREST:
ANNUALLY, 0.05.
100
Improved Cumulated Value of
Principal. Annuity.
Annuity.
4.5380 395
4.7649 415
5.0031 885
5.2533 480
5.5160 154
5.7918 161
6.0314 069
6.3854 773
6.7047 512
7.0399 887
7.3919 881
7.7615 876
8.1496 669
8.5571 503
8.9850 078
10.921 333
11.167 400
15.367 413
16.135 783
16.942 572
17.789 701
18.679 186
9.4342 582
168.68 516
178.11 942
9.9059 711
10.401 270 | 188.02 539
198.42 666
209.34 800
19.613 145
20.593 803
21.623 493
22.704 667
23.839 901
896
25.031
26.283 491
12.040 770
12.642 808
13.274 949
13.938 696 | 258.77 392
272.71 262
245.49 893
14.635
631
51.113 454
54.669 126
58.402 583
62.322 712
66.438 848
70.760 790
75.298 829
80.063 771
85.066 959
90.320 307
40.774 321
42.813 037
44.953 688
47.201 373
49.561
441
95.836 323
101.62 814
107.70 955
114.09 502
120.79 977
127.83 976
135.23 175
142.99 334
151.14 301
159.70 016
84.766 884
89.005 228
93.455 490
98.128 264
103.03 468
96
108.18 641
97 113.59 573
119.27 552
125.23 930
131.50 126
480.63 792
505.66 981
27.597 665 | 531.95 330
28.977 548
30.426
559.55 097
588.52 852
426
220.81 540
232.85 617
287.34 825
302.71 506
318.85 145
335.79 402
353.58 372
747
31.947
33.545 134
35.222 391
36.983 511 719.67 021
686 756.65 373
618.95 494
650.90 269
684.44 782
38.832
372.26 291
391.87 605
412.46 986
434.09 335
456.79 802
52.039 514
54.641 489
57.373 564
60.242 242
63.254 354
66.417 072
69.737 925
73.224 822
1308.3 414
1374.7 585
1414.4 964
76.886 063 1517.7 213
80.730 366
1594.6 073
795.48 641
836.26 073
879.07 $77
924.02 746
971.22 883
1020.7 903
1072.8 298
1127.4 713
1184.8 448
1245.0 871
: 5 per Cent.
Cent. per
1675.3 377
1760.1 046
1849.1 098
1942.5 653
2040.6 935
2143.7 282
2251.9 147
2365.5 104
2484.7 859
2610.0 252
14.375 18
14.643 03
14.898 13
15.141 07
15.372 45
15.592 81
15.802 68
16.002 55
16.192 90
16.374 19
16.546 85
16.711 29
16.867 89
17.017 04
17.159 09
17.294 37
17.423 21
17.545 91
17.662 77
17.774 07
17.830 07
17.981 01
18.077 16
18.168 72
18.255 92
18.338 98
18.418 07
18.493 40
18.565 14
18.633 47
18.698 54
18.760 52
18.819 54
18.875 75
18.929 29
18.980 27
19.028 83
19.075 08
19.119 12
19.161 07
19.201 02
19.239 07
19.275 30
19.309 81
19.342 68
19.373 98
19.403 79
19.432 18
19.459 22
19.484 97
19.509 49
19.532 85
19.555 10
19.576 28
19.596 46
19.615 68
19.633 98
19.651 41
19.668 01
19.683 82
19.698 87
19.713 21
19.726 87
19.739 87
19.752 26
Value of Perpetual Ann. 20.000 00
RATIO of INTEREST:
HALF-YEARLY, 0.025.
Improved Cumulated
Principal. Annuity.
3.6111 123 52.222 247
3.7939 249 55.878 498
3.9859 924 59.719 847
4.1877 832 63.755 664
4.3997 897 | 67.995 795
4.6225 291
4.8565 446
5.1024 072
5.3607 166
5.6321 029
5.9172 281
6.2167 877
6.5315 126
7.5745 522
7.9580 139
8.3608 883
8.7841 583
9.2288 563
Ann.—
6.8621 704 | 117.24 341
7.2095 678
124.19 136
15.888 135
16.692 472
17.537 528
18.425 366
19.358 150
20.338 156
21.357 775
22.449 519
23.586 026
24.780 068
9.6960 672 | 173.92 134
10.186 931 183.73 861
10.702 644 | 194.05 288
11.244 465 204.88 931
11.813 716 216.27 433
26.034 559
27.352 559
28.737 282
30.192 107
31.720 583
72.450 582
77.130 893
82.048 144
87.214 332
92.642 057
12.411 786 | 228.23 571
13.040 132
13.700 289
240.80 265
254.00 578
14.393 866 | 207.87 732
15.122 555
282.45 111
98.344 561
104.33 575
110.63 025
131.49 104
139.16 028
147.21 777
155.68 317
164.57 713
69.904 454
73.443 367
77.161 437
81.067 735
85.171 789 |
19.764 06 89.483 611
19.775 29 94.013 719
19.785 99 98.773 163
19.796 18 103.77 355
19.805 89 109.02 709 |
297.76 270
313.84 944
330.75 057
348.50 731
367.16 300
386.76 312
407.35 551
428.99 038
451.72 052
475.60 137
33.326 437
646.52 874
35.013 588
680.27 176
36.786 151 | 715.72 302
38.648 450 | 752.96 899
40.605 027 | 792.10 054
500.69 119
527.05 118
554.74 564
583.84 214
614.41 165
42.660 657 833.21 314
44.820 353
47.089 383
49.473 283
876.40 706
921.78 766
969.46 566
51.977 868 | 1019.5 574
54.609 248 1072.1 850
57.373 841 1127.4 768
60.278 392 1185.5 678
63.329 985 1246.5 997
66.536 066 | 1310.7 213
1378.0 891
1448.8 673
1523.2 287
1601.3 547
1683.4 358
1769.6 722
1860.2 744
1955.4 633
2055.4 711
2160.5 418
19.815 13 114.54 659 2270.9 317
19.823 94 120.34 £51 2386.9 102
19.832 32 126.43 800 2508.7 600
19.840 30 132.83 892 | 2636.7 785
19.847 91 139.56 389
2771.2 779
-
-CORRESPONDING VALUE OF

I
Value of
Annuity.
14.461 54
14.728 42
14.982 43
15.224 20
15.454 33
15.673 37
15.881 85
16.080 28
16.269 15
16.448 93
16.620 04
16.782 91
16.937 92
17.985 47
17.225 91
17.359 58
17.486 81
17.607 91
17.723 17
17.832 88
17.937 31
18.036 70
18.131 30
18.221 35
18.307 05
18.388 63
18.466 27
18.540 17
18.610 52
18.677 47
18.741 20
18.801 86
18.859 59
18.914 54
18.966 85
19.016 63
19.064 01
19.109 11
19.152 03
19.192 90
19.231 79
19.268 81
19.304 04
19.337 57
19.369 49
19.399 88
19.428 79
19.456 32
19.482 51
19.507 45
19.531 18
19.553 78
19.575 28
19.595 74
19.615 22
19.633 76
19.651 41
19.668 20
19.684 19
19.699 41
19.713 89
19.727 68
19.740 80
19.753 29
19.765 18
19.776 50
19.787 26
19.797 52
19.807 27
19.816 56
19.825 39
19.833 81
19.841 82
19.849 44
19.856 70
20.000 00
RATIO of INTEREST:
QUARTERLY, 0.0125.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
3.6398 486
3.8252 819
4.0201 622
4.2249 707
4.4402 132
4.6664 214
4.9041 538
5.1539 976
5.4165 697
5.6925 187
5.9825 260
6.2873 078
6.6076 168
6.9442 440
7.2980 209
7.6698 210
8.0605 626
8.4712 107
8.9027 794
9.3563 345
9.8329 961
10.333 941
10.860 408
11.413 695
11.995 169
12.606 267
13.248 498
13.923 447
14.632 781
15.378 253
16.161 704
16.985 067
17.850 377
18.759 771
19.715 494
20.719 906
21.775 489
22.884 848
24.050 724
25.275 997
26.563 691
27.916 987
29.339 227
30.833 924
32.404 769
34.055 641
35.790 617
37.613 982
39.530 239
41.544 120
43.660 599
45.884 903
48.222 525
50.679 238
53.261 109
55.974 514
58.826 154
61.823 073
64.972 670
68.282 725
71.761 411
75.417 320
79.259 481
83.297 382
87.540 995
92.000 801
96.687 812
101.61 361
106.79 034
112.23 082
117.94 845
123.95 738
130.27 243
136.90 920
143.88 408
52.796 971
56.505 638
60.403 243
64.499 413
68.804 265
73.328 428
78.083 076
83.079 951
88.331 394
93.850 374
99.650 519
105.74 616
112.15 23+
118.88 488
125.96 042
133.39 642
141.21 125
149.42 421
158.05 559
167.12 669
176.65 992
186.67 883
197.20 815
208.27 389
219.90 338
232.12 534
244.96 995
258.46 894
272.65 563
287.56 507
303.23 408
319.70 134
337.00 754
355.19 541
374.30 987
394.39 812
415.50 977
437.69 696
461.01 449
485.51 993
511.27 382
538.33 974
566.78 455
596.67 848
628.09 538
661.11 281
695.81 234
732.27 964
770.60 478
810.88 240
853.21 199.
897.69 807
944.45 050
993.58 476
1045.2 222
1099.4 903
1156.5 231
1216.4 615
1279.4 534
1345.6 545
1415.2 282
1488.3 464
1565.1 896
1645.9 476
1730.8 199
1820.0 160
1913.7 562
2012.2 721
2115.8 069
2224.6 163
2338.9 690
2459.1 475
2585.4 485
2718.1 840
2857.8 817
14.505 26
14.771 63
15.025 08
15.266 24
15.495 71
15.714 06
15.921 82
16.119 52
16.307 63
16.486 62
16.656 93
16.818 99
16.973 19
17.119 92
17.259 54
17.392 38
17.518 79
17.639 06
17.753 51
17.862 41
17.966 03
18.064 63
18.158 45
18.247 72
18.332 66
18.413 49
18.490 40
18.503 57
18.633 21
18.699 46
18.762 51
18.822 50
18.879 58
18.933 89
18.985 57
19.034 75
19.081 54
19.126 06
19.168 42
19.208 74
19.247 09
19.283 59
19.318 32
19.351 36
19.382 81
19.412 73
19.441 19
19.468 28
19.494 06
19.518 59
19.541 92
19.564 13
19.585 26
19.605 36
19.624 49
19.642 69
19.660 02
19.676 50
19.692 18
19.707 10
19.721 30
19.734 81
19.747 67
19.759 90
19.771 54
19.782 61
19.793 15
19.803 18
19.812 72
19 821 80
19.830 44
19.838 65
19.846 47
19.853 92
16.861 02
20.000 00
Page 36.
PERPETUAL STOCK: 3 p. Cent. 60; 34 p. Cent. 70; 4p. Cent. 80; 5 p. Cent. 100.

Years.
***** ****3 85923 32332 95993 58886 95898 78888 86882 *22*2 LE228 ****3 35838 28813 85888
54
57
66
74
76
78
92
100
RATIO of INTEREST:
ANNUALLY, 0.05.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2812 407
0.2678 483
0.2550 936
0.2429 463
0.2313 775
0.2203 595
0.2098 662
0.1998 725
0.1903 548
0.1812 903
0.1726 574
0.1644 356
0.1566 054
0.1491 480
0.1420 457
0.1352 816
0.1288 396
0.1227 044
0.1168 613
0.1112 965
0.0362 349
69 0.0345 095
0.0328 662
70
0.0195 643
0.0182 919
0.0171 225
0.0160 455
0.0150 514
0.0141 321
0.0132 804
0.0124 900
0.0117 554
0.0110 717
0.0104 345
0.0098 398
0.0092 842
0.0087 646
0.0082 782
0.1059 967
0.1009 492
0.0059 282
0.0056 142
0.0961 421 0.0053 184
0.0915 639 | 0.0050 396
0.0872 037
0.0047 767
0.0399 490
0.0380 467
0.0078 223
0.0073 947
0.0069 933
0.0066 163
0.0062 617
0.0830 512
0.0045 287
0.0790 964 | 0.0042 945
0.0040 733
0.0038 644
0.0683 264 | 0.0036 669
0.0036 669
0.0753 299
0.0717 427
0.0650 728
0.0034 801
0.0619 741 0.0033 034
0.0590 229 | 0.0031 363
0.0562 123 | 0.0029 780
0.0535 355 0.0028 282
0.0509 862 | 0.0026 863
0.0485 583 0.0025 518
0.0462 460 0.0024 244
0.0440 438 0.0023 037
0.0419 465
0.0021 892
0.0245 252 | 0.0012 571
0.0233 574
0.0222 451
0.0211 858
0.0201 770
0.0011 958
0.0011 376
0.0010 822
0.0010 296
0.0192 162
0.0183 011
0.0009 796
0.0009 321
0.0174 296 0.0008 869
0.0165 996 0.0008 440
0.0158 092 0.0008 032
0.0136 566
0.0130 063
0.0123 869
0.0150 564 0.0007 643
0.0143 394 0.0007 274
0.0006 923
0.0006 589
0.0006 271
0.0695 643
0.0682 919
0.0671 225
0.0660 455
0.0650 514
0.0005 969
0.0005 681
0.0005 408
0.0005 148
0.0004 900
0.0641 321
0.0632 804
0.0624 900
0.0617 554
0.0610 717
0.0117 971
0.0112 353
0.0107 003
0.0101 907
0.0097 055
0.0092 433
0.0088 031
0.0083 840
0.0004 665
0.0004 441
0.0001 227
0.0079 817 0.0004 024
0.0076 045 0.0003 831
0.0604 345
0.0598 398
0.0592 842
0.0587 616
0.0582 782
0.0578 223
0.0573 947
0.0569 933
0.0566 163
0.0562 617
0.0020 806
0.0520 806
0.0019 776 | 0.0519 776
0.0018 799
0.0017 871
0.0016 992
0.0518 799
0.0517 871
0.0516 992
0.0559 282
0.0556 142
0.0553 184
0.0550 396
0.0547 767
0.0283 910
0.0270 391
0.0257 515
0.0313 011 0.0016 156 0.0516 156
0.0298 106 | 0.0015 363 | 0.0515 363
0.0514 610
0.0513 895
0.0513 216
0.0014 610
0.0013 895
0.0013 216
0.0545 287
0.0542 945
0.0540 733
0.0538 644
0.0536 669
0.0534 801
0.0533 034
0.0531 363
0.0529 780
0.0528 282
0.0526 863
0.0525 518
0.0524 244
0.0523 037
0.0521 892
0.0512 571
0.0511 958
0.0511 376
0.0510 822
0.0510 296
0.0509 796
0.0509 321
0.0508 869
0.0508 440
0.0508 032
0.0505 969
0.0505 681
0.0505 408
0.0505 148
0.0504 900
RATIO of INTEREST:
HALF-YEARLY, 0.025.
0.0504 665
0.0504 441
0.0504 227
0.0504 024
0.0503 831
Ratio of Perpetual Ann. 0.0500 000
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2769 230
0.2635 793
0.2508 786
0.2387 898
0.2272 836
0.2163 318
0.2059 077
0.1959 859
0.1865 422
0.1775 536
0.1689 980
0.0101 683
0.1608 548 | 0.0095 844
0.1531 039 0.0090 391
0.1457 265 0.0085 293
0.1337 046 | 0.0080 521 |
0.1031 346 |
0.0981 650
0.0934 349
0.0889 326
0.0846 474
0.0805 686
0.0766 861
0.0729 912 |
0.0694 740
0.0661 264 |
0.1320 210 0.0076 051 0.0576 051
0.1256 595 | 0.0071 860 | 0.0571 860
0.1196 045 0.0067 927
0.0567 927
0.1138 413 0.0064 233 0.0564 233
0.0060 762 | 0.0560 762
0.1083 558
0.0629 401
0.0599 072
0.0570 206
0.0542 730
0.0516 578 |
0.0491 687
0.0467 994
0.0445 444
0.0423 980
0.0403 550
0.0384 105
0.0365 597
0.0347 980 |
0.0331 212
0.0315 253
0.0300 062
0.0285 603
0.0271 841
0.0258 743
0.0246 275
0.0191 489 0.0691 489
0.0178 960 | 0.0678 960
0.0167 449 | 0.0667 449
0.0156 849 | 0.0656 849
0.0147 068 0.0647 068
0.0234 408
0.0223 113
0.0212 362
0.0202 129
0.0192 390 |
0.0138 025 0.0638 025
0.0129 650 | 0.0629 650
0.0121 880 | 0.0621 880
0.0114 660 0.0614 660
0.0107 942 | 0.0607 912
0.0183 119 |
0.0174 295
0.0165 897
0.0601 683
0.0595 844
0.0590 391
0.0585 293
0.0580 521
0.0057 497 | 0.0557 497
0.0054 425 0.0554 425
0.0051 532 0.0551 532
0.0048 807 0.0548 807
0.0046 238 0.0546 238
0.0043 814 0.0543 814
0.0041 528 0.0541 528
0.0039 369 | 0.0539 369
0.0037 331 0.0537 331
0.0035 404 | 0.0535 401
0.0033 584 0.0533 584
0.0031 862 | 0.0531 862
0.0030 234 0.0530 234
0.0028 694 | 0.0528 69+
0.0027 236 | 0.0527 236
0.0025 856 | 0.0525 856
0.0024 549
0.0023 311
0.0022 138
0.0021 026
0.0524 549
0.0523 311
0.0522 138
0.0521 026
0.0019 972 | 0.0519 972
0.0018 973
0.0018_026
0.0017 128 0.0517 128
0.0016 276 | 0.0516 276
0.0015 467 | 0.0515 467
0.0014 700 0.0514 700
0.0013 972 0.0513 972
0.0013 281 0.0513 281
0.0012 625 | 0.0512 625
0.0012 002 0.0512 002
0.0011 410 0.0511 410
0.0010 848 0.0510 848
0.0010 315 0.0510 315
0.0009 808 | 0.0509 808
0.0009 327 | 0.0509 327
0.0008 869
0.0008 435
0.0157 903 0.0008 022
0.0150 294 0.0007 629 | 0.0507 629
0.0508 869
0.0508 435
0.0508 022
0.0507 643 0.0143 052 | 0.0007 256 | 0.0507 256
0.0507 274 0.0136 159 | 0.0006 902 | 0.0506 902
0.0506 923 0.0129 598 0.0006 565 | 0.0506 565
0.0506 589 0.0123 354 0.0006 245 0.0506 245
0.0506 271 0.0117 410 0.0005 940 | 0.0505 910
0.0087 301 0.0004 403
0.0083 094 0.0004 190
0.0079 090 0.0003 986
0.0075 279 0.0003 793
0.0071 652 | 0.0003 608
0.0111 752 | 0.0005 651 0.0505 651
0.0106 367 0.0005 376 | 0.0505 376
0.0101 242 | 0.0005 114 0.0505 114
0.0096 364 | 0.0004 865 | 0.0504 865
0.0091 720 | 0.0004 628 0.0504 628
L
RATIO of INTEREST:
QUARTERLY, 0.0125.
0.0504 403
0.0504 190
0.0503 986
0.0503 793
0.0503 608
0.0500 000
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2747 367
0.2614 186
0.2487 462
0.2366 880
0.2252 144
0.2142 970
0.2039 088
0.1940 241
0.1846 187
0.1756 692
0.1303 811
0.1240 608
0.1180 469
0.1123 245
0.1068 795
0.0518 973 || 0.0358 205
0.0518 026
0.0340 841
0.0324 318
0.0308 597
0.1671 535
0.1590 506
0.1513 405
0.1440 042
0.1370 234 | 0.0079 391
0.1016 934
0.0967 685
0.0920 776
0.0876 140
0.0833 669
0.0618 747
0.0588 752
0.0560 212
0.0533 056
0.0507 215
0.0189 405
0.0176 973
0.0176 973
0.0165 554
0.0165 554
0.0155 040
0.0145 340
0.0136 373
0.0128 069
0.0120 366
0.0113 210
0.0106 553
0.0793 256 | 0.0043 080
0.0754 803
0.0718 213
0.0683 397
0.0650 269
0.0100 351
0.0094 566
0.0089 164
0.0084 115
0.0139 351
0.0132 596
0.0126 168
0.0120 052
0.0114 232
0.0574 965
0.0074 965
0.0074 965
0.0070 816 0.0570 816
0.0066 924 0.0566 924
0.0063 269 0.0563 269
0.0059 835 | 0.0559 835
0.0040 821
0.0038 689
0.0036 676
0.0034 775
0.0025 355
0.0024 067
0.0482 628
0.0459 232
0.0436 970 | 0.0022 817
0.0415 788 0.0021 691
0.0395 632
0.0084 783
0.0080 673
0.0076 762
0.0056 606 | 0.0556 606
0.0053 568 | 0.0553 568
0.0050 708
0.0550 708
0.0048 014
0.0045 475
0.0548 014
0.0545 475
0.0689 405
0.0676 973
0.0665 554
0.0655 040
0.0645 340
0.0636 373
0.0628 069
0.0620 366
0.0613 210
0.0606 553
0.0293 637 | 0.0015 126
0.0279 403 | 0.0014 372
0.0265 859 0.0013 656
0.0252 971
0.0012 977
0.0240 708
0.0012 332
0.0600 351
0.0594 566
0.0589 164
0.0584 115
0.0579 391
0.0032 978 | 0.0532 978
0.0031 279
0.0029 673
0.0028 154
0.0026 715
0.0531 279
0.0529 673
0.0528 154
0.0526 716
0.0376 454 | 0.0019 559 | 0.0519 559
0.0018 576
0.0017 643
0.0016 759
0.0015 921
0.0518 576
0.0517 643
0.0516 759
0.0515 921
0.0525 355
0.05:24 067
0.0522 847
0.0521 691
0.0020 596 | 0.0520 596
0.0543 080
0.0540 821
0.0538 689
0.0536 676
0.0534 775
0.0229 039 0.0011 720| 0.0511 720
0.0217 937 0.0011 140 | 0.0511 140
0.0207 372 0.0010 588 0.0510 588
0.0197 319
0.0010 065 0.0510 065
0.0187 754 | 0.0009 567 | 0.0509 567
0.0178 653 | 0.0009 095 | 0.0509 095
0.0169 992
0.0009 647
0.0008 221
0.0161 752
0,0153 911
0.0146 450
0.0508 647
0.0508 221
0.0007 816 | 0.0507 816
0.0007 431 0.0507 431
0.0001 275
0.0001 066
0.0003 868
0.0073 041 0.0003 679
0.0069 500 | 0.0003 499
0.0515 126
0.0514 372
0.0513 656
0.0512 977
0.0512 332
0.0007 066 | 0.0507 066
0.0006 719
0.0506 719
0.0006 389
0.0506 389
0.0006 076 | 0.0506 076
0.0005 778
0.0505 778
0.0108 695 0.0005 494 0.0505 491
0.0103 426 0.0005 225 0.0505 225
0.0098 412 0.0004 970 0.0504 970
0.0093 641 | 0.0004 726 | 0.0504 726
0.0089 101 0.0004 495 0.0504 495
0.0504 275
0.0504 066
0.0503 868
0.0503 679
0.0503 499
0.0500 000
Page 37.
RATE OF INTEREST: 54 per Cent. per Ann.--CORRESPONDING VALUE of

Years.
P12:9-2
19
<+2.04
AISVAL
w
*
APPL
ແດ
10. Kog
~*~*~*~
Pla&-D
AKC-A-
Co
.
ست . سر
#PJ00898—1—
HAN.SE-d
11
མཛཏིཛཡཔཙྩཊསུམ -Y2]e G
13
14
- 1992 7*** 235
15
18
21
22
23
RATIO of INTEREST:
ANNUALLY, 0.0525.
24
Improved Cumulated Value of
Principal. Annuity. Annuity.
► ··
1.0525 000 1.0000 000
1.1077 563 2.0525 000
..
1.1659 135
1.2271 239 4.3261 697
1.2915 479 | 5.5532 936
1.3593 542 | 6.8118 415
....
1.4307 203
....
1.5058 331
3.1602 563
....
1.5848 893
1.7556 711
8.2041 957
9.6349 160
•
...
11.140 749
1.6680 960 12.725 638
14.393 734
1.8478 438 16.149 405
16 2.2575 333
17
2.3865 788
2.5118 742
2.6437 476
26.411 025
28.797 603
31.309 478
20 2.7825 413 | 33.953 225
1.9448 556 17.997 249
2.0469 605 19.942 105
2.1544 259 | 21.989 065
..
24.143 491
36.735 769
39.664 397
42.746 778
2.9286 279
3.0823 809
3.2442 059
3.4145 267 45.990 984
3.5937 893 49.405 511
0.9501 19
...
1.8528 44
2.7105 41
3.5254 55
4.2997 19
5.0353 63
··
5.7343 11
··
6.3983 95
7.0293 54
7.6288 40
8.1984 23
8.7395 91
9.2537 71
9.7423 01
10.206 46
...
10.647 47
RATIO of INTEREST:
HALF-YEARLY, 0.02625
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0262 500
1.0531 891
1.0808 353
1.1092 072
1.1988 703
1.1383 239 2.6347 408
1.1682 049 3.2039 027
I
3.7880 052
1.2303 406 | 4.3874 403
1.2626 371
1.2957 813
1.3297 955
1.3647 027
0.5000 000
1.0131 250
1.5397 195
2.0801 372
1.4005 261 7.6290 689
....
..
...
1.4372 899 | 8.3293 319
5.0026 106
5.6339 291
1.4750 1889.0479 769
6.2818 198
6.9467 175
...
1.5137 380 9.7854 863
1.5534 737 10.542 355
1.5942 523 11.319˚092
1.6361 015
12.116 218
1.6790 491 12.934 269
1.7231 242 13.773 794
1.7683 562 | 14.635 356
...
....
1.8147 755 15.519 534
1.8624 134 | 16.426 922
1.9113 017 | 17.358 128
1.9614 734 | 18.313 779
....
2.0129 621 19.294 516
2.0658 023 20.300 997
11.066 48 2.4132 804
11.464 59 2.5416 405
11.842 84 2.6768 280
12.202 22 2.8192 060
...
2.1200 296 21.333 898
2.1756 804 22.393 913
12.543 68 2.9691 569
12.868 10 3.1270 836
13.176 35
13.469 21
13.747 47
2.2327 920 | 23.481 753
2.2914 028 | 24.598 149
26.919 627
29.364 581
31.939 581
34.651 542
....
37.507 750
40.515 877
43.684 004
47.020 640
3.2934 102
3.4685 836
3.6530 74350.534 749
....
0.4872 11
0.9619 59
1.4245 64
1.8753 37
··
2.3145 79
...
2.7425 86
....
3.1596 46
....
3.5660 37
...
3.9620 34
....
4.3479 01
4.7238 99
1.
5.0902 79
5.4472 88
5.7951 65
6.1341 43
6.4644 52
6.7863 11
7.0999 38
··
7.4055 42
7.7033 30
7.9935 00
8.2762 49
8.5517 65
··
8.8202 34
9.0818 36
9.3367 46
..
9.5851 36
9.8271 73
....
10.063 02
10.292 83
...
10.516 77
10.734 97
11.154 79
11.553 40
11.931 88
12.291 24
12.632 46
12.956 44
13.264 06
13.556 15
13.833 48
RATIO of INTEREST:
QUARTERLY, 0.013125.
Improved Cumulated
Principal. Annuity.
1.0131 250 0.2500 000
1.0264 223 0 5032 813
1.0398 941 0.7598 868
1.0535 427 | 1.0198 603
1.0673 704
1.0813 797
1.0955 728
1.1099 522 | 2.0943 267
1.8204 335
1.1245 203
1.1392 796
1.1542 326
1.1693 820.
1.1847 301
1.2002 797
1.2160 333
1.2319 938
1.3149 937
1.3322 530
1.3497 388
1.3674 541
1.3854 020
1.4035 854
1.4220 074
1.4406 713
1.2481 637 4.7269 276
1.2645 451 | 5.0389 516
1.2811 430
1.2979 580
5.3551 050
5.6753 907
1.4595 801
1.4787 371
1.4981 455
1.5178 087
1.5377 299
1.5579 126
1.5783 602
1.5990 762
1.6200 641
1.6413 274
1.6628 698
1.6846 950
1.7068 066
1.7292 085
1.7519 043
1.7748 981
1.7981 936
1.8217 947
1.8457 060
1.8699 308
1.8944 737
1.9193 387
1.9445 300
1.9700 519
1.9959 089
2.0221 052
2.0486 453
2.0755 338
2.1027 751
2.1303 741
2.1583 352
2.1866 634
1.2832 460
1.5500 886
2.2153 633
2.2444 400
2.2738 983
2.3037 432
2.4270 917
2.5570 447
2.6939 557
2.8381 973
2.3718 147
2.6529 448
2.9377 647
3.2263 229
3.5186 684
3.8148 509
4.1149 208
4.4189 291
5.9998 802
6.3286 287
6.6616 919
6.9991 266
7.3409 901
7.6873 406
8.0382 370
8.3937 388
8.7539 067
9.1188 017
9.4884 860
9.8630 223
10.242 475
10.626 907
11.016 385
11.410 975
11.810 744
12.215 760
12.626 092
13.041 810
13.462 983
13.889 685
14.321 987
14.759 963
15.203 688
15.653 236
16.108 685
16.570 111
17.037 594
17.511 212
17.991 047
18.477 180
18.969 693
19.468 670
19.974 196
20.486 357
21.005 241
21.530 935
22.063 528
22.603 112
23.149 778
23.703 619
24.264 729
24.833 203
27.182 699
29.657 994
32.265 822
35.013 281
2.9901 619
3.1502 631
3.3189 366
3.4966 414
37.907 846
40.957 393
44.170 222
47.555 074
3.6838 609 51.121 159
Value of
Annuity.
0.2467 61
0.4903 26
0.7307 35
0.9680 29
1.2022 50
1.4334 36
1.6616 27
1.8868 62
2.1091 79
2.3286 16
2.5452 10
2.7589 98
2.9700 17
3.1783 02
3.3838 88
3.5868.11
3.7871 06
3.9848 05
4.1799 43
4.3725 53
4.5626 68
4.7503 21
4.9355 41
5.1183 63
5.2988 16
5.4769 31
5.6527 39
5.8262 69
5.9975 51
6.1666 15
6.3334 88
6.4981 99
6.6607 76
6.8212 47
6.9796 39
7.1359 80
7.2902 95
7.4426 10
7.5929 53
7.7413 48
7.8878 20
8.0323 95
8.1750 97
8.3159 50
8.4549 78
8.5922 05
8.7276 55
8.8613 50
8.9933 15
9.1235 68
9.2521 33
9.3790 34
9.5042 90
9.6279 23
9.7499 55
9.8704 06
9.9892 97
10.106 65
10.222 48
10.336 81
10.449 65
10.561 40
10.670 98
10.779 50
11.199 70
11.598 54
11.977 12
12.336 45
12.677 52
13.001 26
13.308 55
13.600 22
13.877 06
Page 38.
PERPETUAL STOCK: 3 p. C. 57,1429; 3½ p. C. 66,6667; 4 p. C. 76,1905; 5 p. C. 95,2381.

Years.
porok
سات سال است
→
རིན
ཙམ ཨིཏྟ པྤཡསྶཡཙ
5
*:*
ས
ساهمات:
∞
POKLAH
AK-AL
HAN*~
→
12:00
10
20:00 - and
GALELAL SAPN-AL
12
13
~]~*~~~~# →→
JFSHAL
HKO ZOO2 7847 3
17
18
19
20
22
23
£4
25
RATIO of INTEREST:
ANNUALLY, 0.0525.
Discounted Annuity for Ratio of
Principal. Cumulation
Cumulation Annuity.
0.9501 188
0.9027 257
0.8149 137
0.7742 644
0.7356 435
0.6989 487
0.8576 966 0.3164 300 0.3689 300
0.6309 589
………..
1.0000 000
0.5141 770
……….
0.4885 292
0.4641 607
..
0.4872 107 | 0.5397 107
0.6640 842 0.1037 892
0.4190 098
0.3981 091
0.3782 510
0.3593 833
……….
0.3414 568
0.3244 245
0.3082 418
0.2928 664
0.2782 578
0.2311 514 | 0.2836 514
0.5994 859 0.0785 815
0.1800 733 0.2325 733
0.1460 954 | 0.1985 954
..
1.0525 000
0.1218 889 0.1743 889
0.5695 828 0.0694 747
...
···
0.0897 606
0.5411 713 0.0619 218 0.1144 218
0.1562 892
...
0.1422 606
0.0694 747 | 0.1219 747
0.1310 815
0.0555 640 0.1080 640
0.0501 452 | 0.1026 452
0.4410 079 0.0414 190 0.0939 190
0.0378 630
0.0903 630
0.0347 251 0.0872 251
0.0319 392 0.0844 392
0.0291 524 0.0819 524
0.0454 771 | 0.0979 771
0.0272 214 0.0797 214
0.0252 115 0.0777 115
0.0233 936 0.0758 936
0.0217 434 0.0742 434
0.0202 407 0.0727 407
RATIO of INTEREST:
HALF-YEARLY, 0.02625.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9744 214
....
0.9494 971
0.8784 816
...
0.8560 142
....
0.9252 104 0.6494 690 0.7019 690
0.9015 448
0.4807 375
0.5332 375
..
....
..
0.8341 187 0.2639 912 0.3164 912
0.8127 830 0.2279 286
0.280+ 286
0.7919 932 | 0.1998 956
0.7717 352 0.1774 960
0.7140 174
0.6957 538
0.6437 187
0.6272 533
2.0000 000
2.0525 000
0.9870 450 1.0395 450
0.7519 953 0.1591 895 0.2116 895
0.7327 603 | 0.1439 529 | 0.1964 529
...
0.5510 324
0.5369 377
0.3795 440
...
0.3121˚193
...
··
0.4143 739
0.3934 467
0.3735 765
0.3547 098
0.6779 574 | 0.1105 219
0.1630 219
0.6606 163 0.1021 922 | 0.1546 922
0.3367 959
0.3197 868
0.3036 366
0.2883 021
………
0.6112 090 0.0825 340
0.5955 752 | 0.0773 140
|
0.1310 776 0.1835 776
0.1725 577
0.1200 577
···
0.4320 440
0.3646 193
0.0918 555
0.0883 463
....
0.5232 037 | 0.0576 099
0.0576 099
0.5098 208 0.0546 037
...
0.2523 956
0.2299 960
0.4716 915 | 0.0468 738
0.4596 263 | 0.0146 550
0.5803 412 0.0726 016 0.1251 016
0.5654 970 0.0683 277 | 0.1208 277
....
0.1473 555
...
0.1408 463
0.1350 340
……..
0.1298 140
0.0644 349 0.1169 349
0.0608 757
0.1133 757
0.4967 804 0.0518 282 0.1043 282
0.4840 734 0.0492 587 | 0.1017 587
0.1101 099
0.1071 0
037
0.4478 697 0.0425 863 0.0950 863
0.4364 139 | 0.0406 535
0.0406 535 0.0931 535
....
0.0993 738
0.0971 550
0.0266 612 | 0.0791 612
0.0246 817 0.0771 817
0.0228 917 0.0753 917
0.0212 673 0.0737 673
0.2737 420 0.0197 884 | 0.0722 884
0.0371 476 | 0.0896 476
0.0340 546 | 0.0865 546
0.0313 091
0.0838 091
0.0288 587 0.0813 587
RATIO of INTEREST:
QUARTERLY, 0.013125.
Ratio of
Discounted Annuity for
Principal. Cumulation Annuity.
0.9870 450
0.9742 579
0.9616 364
0.9191 781
0.9368 819
0.9247 446
0.9127 646
0.9009 397
0.8892 681
0.8777 477
0.8663 765
0.8551 526
0.8440 741
0.8331 392
0.8223 459
0.8116 924
0.8011 770
0.7907 977
0.7805 530
0.7704 409
0.7604 598
0.7506 082
0.7408 841
0.7312 860
0.7218 122
0.7124 611
0.7032 312
0.6941 209
0.6851 286
0.6762 527
0.6674 919
0.6588 445
0.6503 093
0.6418 846
0.6335 689
0.6253 611
0.6172 595
0.6092 630
0.6013 700
0.5935 792
0.5858 895
0.5782 993
0.5708 074
0.5634 126
0.5561 136
0.5489 092
0.5417 981
0.5347 791
0.5278 589
0.5210 128
0.5142 631
0.5076 008
0.5010 249
0.4945 341
0.4881 274
0.4818 038
0.4755 620
0.4694 011
0.4633 201
'0.4573 122
0.4513 932
0.4455 454
0.4397 734
0 4340 761
0.4120 138
0.3910 765
0.3712 013
0.3523 363
0.3344 301
0.3174 338
0.3013 013
0.2859 887
0.2714 544
4.0000 000
1.9869 606
1.3159 855
0.9805 264
0.7792 737
0.6451 245
0.8317 737
0.6976 245
0.5493 197
0.6018 197
0.4774 804 | 0.5299 804
:
0.4216 '181
0.3769 397
0.3403 919
0.3099 504
0.2115 539
0.1984 539
0.1867 377
0.1761 993
0.1666 700
0.1580 121
0.1501 120
0.1428 750
0.2841 984
0.2621 334
0.2430 181
0.2262 992 | 0.2787 992
0.1362 214
0.1300 831
0.1244 054
0.1191 364
0.1191 364
0.1142 347
0.1096 635
0.1053 909
0.1013 888
0.0976 327
0.0941 008
0.0907 739
0.0876 349
0.0846 687
0.0846 687
0.0818 615
0.0792 011
0.0766 765
0.0742 777
0.0719 959
0.0698 227
0.0677 508
4.0525 000
2.0394 606
1.3684 855
1.0330 264
|
0.0586 937
0.0571 063
0.0555 832
0.0541 208
0.4741 181
0.4294 397
0.3928 949
0.3624 504
0.3366 934
0.3146 334
0.2955 181
0.2640 539
0.2509 539
0.2392 377
0.2286 995
0.2191 700
0.2105 121
0.2026 120
0 1953 750
0.1887 214
0.1825 831
0.1769 054
0.1716 364
0.1667 347
0.1621 635
0.1578 909
0.1538 888
0.1501 327
0.1466 008
0.1432 739
0.1401 349
0.1371 687
0.1343 615
0.1317 011
0.1291 765
0.1267 777
0.1244 959
0.1223 227
0.1202 508
0.0657 735
0.0638 846
0.0620 783
0.1145 783
0.0603 496 | 0.1128 496
0.1182 735
0.1163 846
0.1111 937
0.1096 063
0.1080 832
0.1066 208
0.0527 157
0.1052 157
0.0513 646 0.1038 646
0.0500 646 | 0.1025 646
0.0488 130 0.1013 130
0.0476 072 0.1001 072
0.0464 448 0.0989 448
0.0453 237 0.0978 237
0.0412 417
0.0967 417
0.0431 970 0.0956 970
0.0421 877
0.0946 877
0.0412 121 0.0937 121
0.0402 687 | 0.0927 687
0.0367 881 0.0892 881
0.0862 177
0.0337 177
0.0309 925 0.0834 925
0.0285 606 0.0810 606
0.0263 798 | 0.0788 798
0.0244 156 0.0769 156
0.0226 397 0.0751 397
0.0210 282 0.0735 282
0.0195 614 0.0720 614
Page 39.
RATE OF INTEREST: 54 per
Ann.-CORRESPONDING VALUE of
Cent.
per

{
Years.
26
27
28
29
30
31
***3 85822 733$2 95338 58883 95888 58858 85882 K2**R OF228 ***** 85382 48343 85888
96
100
RATIO of INTEREST:
ANNUALLY, 0.0525.
Improved Cumulated
Principal. Annuity.
3.7824 633
3.9810 426
4.1900 473
4.4100 248
4.6415 511
4.8852 325 74.004 429
5.1417 072 78.889 662
5.4116 46984.031 369
5.6957 583 89.443 016
5.9947 856 95.138 774
6.3095 119
6.6107 613
6.9894 012
7.3563 448
7.7425 529
8.1490 369
8.5768 613
136.17 213
144.32 117
9.0271 466 152.89 803
9.5010 718 161.92 518
9-9998 780 | 171.42 625
10.524 872
11.077 427
11.658 992
12.271 089
12.915 322
17.556
18.478 212
19.448 319
20.469 355
21.543
997
52.999 300
56.781 763
60.762 806
64.952 853
69.362 878
13.593 376239.87 383
253.46 720
267.77 423
14.307 028
15.058 147
15.848 700 282.83 238
16.680 757 | 298.68 108
496 315.36 184
332.91 833
351.39 655
370.84 486
391.31 422
22.675 056
23.865 497
25.118 435
26.437 153
27.825
101.13 356
107.44 307
114.08 383
121.07 323
128.42 958
412.85 822
435.53 327
459.39 877
484.51 720
104 | 510.95 436
181.42 613
191.95 100
203.02 843
214.68 742
226.95 851
29.285 922 538.77 946
568.06 538
30.823 433
32.441 663| 598.88 882
631.33 048
665.47 533
34 144 850
35.937
455
48.851
51.416 445
54.115 809
56.956 888
59.947 125
37.824 171 701.41 278
39.809 940 | 739.23 696
41.899 962 | 779.04 690
44.099 710 820.94 686
46.414 915 865.04 657
63.094 349
66.406 803
69 893 160
73.562 551
77.424 584
729 | 911.46 151
960.31 324
1011.7 297
1065.8 455
1122.8 024
105.24 743
110.77 292
116.58 850
122.70 940
129.15 164
1182.7 495
1245.8 439
1312.2 507
1382.1 438
1155.7 061
81.489 375
85.767 567
90.270 365
95.009 559
1533.1 310
1614.6 203
1700.3 879
1790.6 583
99.997 561 | 1885.6 678
1985.6 654
2090.9 128
2201.6 857
2318.2 742
2440.9 836
2570.1 353
135.93 210
143.06 854 2706.0 674
150.57 964 2849.1 359
158.48 507 2999.7 156
166.80 553 3158.2 006
Value of
Annuity.
14.011 85
14.263 04
14.501 70
14.728 46
14.943 90
15.148 60
15.343 09
15.527 87
15.703 44
15.870 25
16.028 75
16.179 33
16.322 40
16.458 34
16.587 50
16.710 21
16.826 80
16.937 58
17.042 83
17.142 83
17.237 85
17.328 12
17.413 89
17.495 38
17.572 81
17.646 38
17.716 27
17.782 68
17.845 78
17.905 73
17.962 69
18.016 80
18.068 22
18.117 08
18.163 49
18.207 59
18.249 50
18.289 31
18.327 13
18.363 07
18.397 22
18.429 66
18.460 48
18.489 77
13.517 60
18.657 71
18.677 16
18.695 64
18.713 20
18.729 88
18.745 73
18.760 79
18.775 09
18.788 69
18.801 60
18.813 88
18.825 54
18.836 61
18.847 14
18.857 14
18.866 64
18.875 67
18.884 24
18.892 39
18.900 14
Value of Perpetual Ann. 19.047 62
RATIO of INTEREST:
HALF-YEARLY, 0.02625.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
3.8473 779
4.0520 163
4.2675 393
4.4945 257
4.7335 853
4.9853 603
5.2505 269
5.5297 975
5.8239 223
6.1336 912
6.4599 365
6.8035 345
7.1654 081
7.5465 294
7.9479 223
18.211 892
19.180 565
20.200 761
21.275 221
22.406 830
8.3706 618
140.39 361
8.8158 926 | 148.87 414
9.2848 017 157.80 575
9.7786 516| 167.21 241
10.298 769
177:11 941
23.598 628 |
24.853 817
26.175 768 |
27.568 033
29.034 351
10.846 551
187.55 335
11.423 469 | 198.54 226
12.031 072
12.670 991
13.344 952
210.11 566
222.30 464
235.14 191
30.578 660
32.205 111
33.918 070
35.722 141
37.622 168
14.054 757
248.66 205
14.802 317 | 262.90 127
15.589 638 | 277.89 787
16.418 836
293.69 212
310.32 645
17.292 139
54.235 770
58.133 644
62.238 843
66.562 394
71.115 910
75.911 624
80.962 417
86.281 857
91.884 233
97.784 595
18.544 04
735.68 106
18.569 15
775.82 437
18.593 02
818.10 287
18.615 70
46.288 081 | 862.63 012
18.637 24 48.750 101909.52 573
39.623 256
41.730 780
43.950 401
66.529 393
70.068 029
73.794 882
77.719 962 |
81.853 814
103.99 879
110.54 351
117.43 634
124.69 580
132.34 138
111.70 612
117.64 766
123.90 523
130.49 563
137.43 658
160.55 408
169.09 380
178.08 774
327.84 555
346.29 647
365.72 878
386.19 468
407.74 914
430.45 006
454.35 842
479.53 844
506.05 777
533.98 763
51.343 073 | 958.91 568
54.073 963 | 1010.9 326
56.950 106
59.979 229
63.169 468
1065.7 163
1123.4 139
1184.1 803
563.40 306
594.38 306
627.01 086
661.37 411
697.56 510
86.207 542
90.792 840
95.622 026
100.70 807
1623.0 008
1710.3 398
1802.3 243
1899.2 014
106.06 464 | 2001.2 312
1248.1 789
1315.5 815
1386.5 692
1461.3 326
1540.0 727
18.907 49 144.74 670 | 2738.0 323
18.914 48 152.44 564
18.921 12
18.927 43
18.933 43
2884.6 788
3039.1 253
3201.7 867
3373.0 998
2108.6 880
2221.8 602
2341.0 520
2466.5 835
2598.7 919
-
14.096 81
14.346 84
14.584 25
14.809 66
15.023 69
15.226 91
15.419 87
15.603 08
15.777 04
15.942 21
16.099 04
16.247 95
16.389 34
16.523 60
16.651 07
16.772 10
16.887 02
16.996 14
17.099 74
17.198 12
17.291 52
17.380 21
17.46+ 42
17.544 37
17.620 29
17.692 38
17.760 82
17.825 81
17.887 51
17.946 10
18.001 73
18.054 55
18.104 70
18.152 32
18.197 54
13.240 47
18.281 23
18.319 94
18.356 69
18.391 58
18.424 71
18.456 17
18.486 04
18.514 40
18.541 33
18.566 90
18.591 18
18.614 23
18.636 12
18.656 90
18.676 63
18.695 37
18.713 16
18.730 05
18.746 09
18.761 32
18.775 78
18.789 50
18.802 5ł
18.814 91
18.826 67
18.837 82
18.848 42
18.858 48
18.868 03
18.877 10
18.885 72
18.893 89
18.901 67
18.909 03
18.916 03
18.922 67
18.928 98
18.934 97
18.940 66
19.017 62
RATIO of INTEREST:
QUARTERLY, 0.013125.
Improved Cumulated
Principal. Annuity.
3.8811 046
4.0889 093
4.3078 404
4.5384 937
4.7814 967
5.0375 108
5.3072 326
.5.5913 960
5.8907 742
6.2061 820
6.5384 775
6.8885 651
7.2573 972
7.6459 776
8.0553 636
8.4866 693
8.9410 682
9.4197 969
9.9241 579
10.455 524
11.015 340
11.605 131
12.226 501
12.881 140
13.570 831
14.297 449
15.062 973
15.869 485
16.719 179
17.614 369
18.557 489
19.551 105
20.597 925
21.700 793
22.862 711
24.086 812
25.376 515
26.735 242
28.166 718
29.674 839
31.263 709
32.937 651
34.701 221
36.559 217
38.516 695
40.578 982
42.751 688
45.040 728
47.452 329
49.993 053
52.669 814
55.489 897
58.460 974
61.591 130
61.888 884
68.363 207
72.023 556
75.879 889
79.942 701
84.223 046
88.732 573
93.483 551
98.488 910
103.76 223
109.31 798
115.17 115
121.33 772
127.83 447
134.67 907
141.89 015
149.48 732
157.49 127
165.92 377
174.80 778
184.16 745
54.878 183
58.836 368
63.006 484
67.399 879
72.009 462
76.904 968
82.042 525
87.455 161
93.157 604
99.165 371
105.49 481
112.16 314
119.18 852
126 59 005
134.38 788
142.60 322
151.25 844
160.37 708
169.98 396
180.10 522
190.76 839
202.00 250
213.83 811
226.30 741
239.14 440
253.28 475
267.86 615
283.22 828
299.41 294
316.46 417
334.42 336
353.35 441
373.29 381
394.30 081
416.43 259
439.74 936
464.31 458
490.19 508
517.46 129
546.18 741
576.45 160
608.33 622
641.92 802
677.31 742
714.60 371
753.88 536
795.27 026
838.87 101
884.80 626
933.20 101
984.18 694
1037.9 028
1094.4 947
1154.1 168
1216.9 311
1283.1 087
1352.8 296
1426.2 836
1503.6 705
1585.2 009
1671.0 966
1761.5 915
1856.9 316
1957.3 766
2063.1 996
2174.6 886
2292.1 471
2415.8 947
2546.2 680
2683.6 218
2828.3 299
2980.7 861
3141.4 052
3310.6 243
3488.9 038
Value of
Annuity.
14.139 84
14.389 26
14.626 00
14.850 72
15,064 02
15.266 46
15.458 63
15.641 03
15.814 15
15.978 48
16.134 47
16.282 51
16.423 04
16.556 42
16.683 03
16.803 20
16.917 27
17.025 54
17.128 30
17.225 84
17.318 43
17.406 31
17.489 72
17.568 89
17.644 05
17.715 39
17.783 09
17.847 35
17.908 35
17.966 25
18.021 21
18.073 37
18.122 88
18.169 88
18.214 49
18.256 83
18.297 01
18.335 16
18.371 37
18.405 74
18.438 36
18.469 33
18.498 71
18.526 60
18.553 09
18.578 22
18.602 08
18.624 72
18.646 21
18.666 61
18.685 98
18.704 36
18.721 80
18.738 36
18.754 08
18.769 00
18.783 15
18.796 58
18.809 35
18.821 46
18.832 96
18.843 86
18.854 22
18.804 05
18.873 38
18.882 25
18.890 64
18.898 62
18.906 19
18.913 33
18.920 20
18.926 68
18.932 82
18.938 65
18.944 19
19.047 62
Page 40.
PERPETUAL STOCK: 3 p. C. 57,1429; 3½ p. C. 66,6667; 4p.C.76,1905; 5 p. C. 95,2381.

RATIO of INTEREST:
QUARTERLY, 0.013125.
Years.
O***A HA*** 85822 79**2 95938 58848 85888 68888 86882 F**** O*228 ***** 8*893 28313 85833
56
57
96
100
RATIO of INTEREST:
ANNUALLY, 0.0525.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.0188 682
0.0176 113
0.2643 780
0.2511 905
0.2386 608 0.0164 574
|
0.0153 958
0.0144 169
0.2267 561
0.2154 452
0.2046 986 | 0.0135 127
0.1944 879 0.0126 759
0.1847 866
0.0119 003
0.1755 693 0.0111 803
0.1668 116 0.0105 110
0.1584 909
0.1505 851
0.0098 879
0.0093 073
0.0087 655
0.1359 371 0.0082 593
0.1291 564 | 0.0077 864
0.1430 738
0.0073 436
0.0069 290
0.1227 139
0.1165 927
0.1107 770 | 0.0065 403
0.1052 513 0.0061 757
0.1000 012 | 0.0058 334
10%
0.0950 130 | 0.0055 119
0.0902 737 0.0052 097
0.0857 707 | 0.0049 254
0.0814 924 0.0046 579
0.0044 061
0.0774 274
0.0735 652 | 0.0041 689
0.0041 689
0.0698 957 | 0.0039 453
0.0664 092
0.0037 345
0.0630 957 | 0.0035 357
0.0599 493 0.0033 481
0.0569 590 0.0031 710
0.0541 178 0.0030 037
0.0514 183 0.0028 458
0.0488 535 0.0026 965
0.0464 166 | 0.0025 555
0.0441 013
0.0419 015
0.0398 114
0.0378 256
0.0359 388
0.0341 461 0.0018 560
0.0324 429 0.0017 604
0.0308 246 0.0016 698
0.0292 870
0.0015 840
0.0278 261 | 0.0015 027
0.0024 221
0.0022 960
0.0021 768
0.0020 639
0.0019 571
0.0264 381
0.0251 194
0.0238 664
0.0014 257
0.0013 527
0.0012 836
0.0226 759 | 0.0012 181
0.0215 448 0.0011 560
0.0204 701
0.0194 490
0.0184 789 | 0.0009 881
0.0175 571 | 0.0009 382
0.0166 814 0.0008 906
0.0010 971
0.0010 413
0.0008 455
0.0008 027
0.0158 493
0.0150 587
0.0143 076 | 0.0007 620
0.0135 939 | 0.0007 235
0.0129 158 0.0006 870
0.0095 014
0.0090 275
0.0085 772
0.0081 493
0.0077 428
0.0122 715 0.0006 523
0.0116 594 | 0.0006 193
0.0110 778 0.0005 881
0.0105 253
0.0100 002
0.0073 566
0.0069 897
0.0066 410
0.0713 682
0.0701 113
0.0689 574
0.0678 958
0.0669 169
0.0660 127
0.0651 759
0.0644 003
0.0636 803
0.0630 110
0.0623 879
0.0618 073
0.0612 655
0.0607 593
0.0602 864
0.0003 891
0.0003 695
0.0003 510
0.0598 436
0.0594 290
0.0590 403
0.0586 757
0.0583 334
0.0580 119
0.0577 097
0.0574 254
0.0571 579
0.0569 061
0.0566 689
0.0564 453
0.0562 345
0.0560 357
0.0558 481
0.0556 710
0.0555 037
0.0553 458
0.0551 965
0.0550 555
0.0549 221
0.0547 960
0.0546 768
0.0545 639
0.0544 571
0.0543 560
0.0542 604
0.0541 698
0.0540 840
0.0510 027
0.0539 257
0.0538 527
0.0537 836
0.0537 181
0.0536 560
0.0535 971
0.0535 413
0.0534 884
0.0534 382
0.0533 906
0.0533 455
0.0533 027
0.0532 620
0.0532 235
0.0531 870
0.0531 523
0.0531 193
0.0530 881
-0.0005 585 0.0530 585
0.0005 303 0.0530 303
0.0005 036 0.0530 036
0.0004 783 0.0529 783
0.0004 542 0.0529 542
0.0004 314 0.0529 314
0.0004 097 0.0529 097
0.0528 891
0.0528 695
0.0528 510
0.0528 334
0.0528 166
0.0063 097 0.0003 334
0.0059 950 | 0.0003 166
Ratio of Perpetual Ann. 0.0525 000
RATIO of INTEREST:
HALF-YEARLY, 0.02625.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2599 173 0.0184 380
0.2467 907 | 0.0172 017
0.2343 271 | 0.0160 671
0.2224 929 0.0150 235
0.2112 564 0.0140 616
0.0140 616
0.2005 873 0.0131 732 | 0.0656 732
0.1904 571 0.0123 514
0.1808 385 0.0115 899
0.1717 056 0.0108 833
0.1630 339 | 0.0102 266 |
0.0648 514
0.0640 899
0.0633 833
0.0627 266
0.1548 003 | 0.0096 155 0.0621 155
0.1469 824 | 0.0090 462 | 0.0615 462
0.1395 594 | 0.0085 153 0.0610 153
0.1325 112 0.0080 195 | 0.0605 195
0.1258 190 | 0.0075 562 | 0.0600 562
0.0071 228 | 0.0596 228
0.0067 171 0.0592 171
0.0063 369 | 0.0588 369
0.1022 636 | 0.0059 805 0.0584 805
0.0970 990 | 0.0056 459 | 0.0581 459
0.1194 648
0.1134 315
0.1077 029
0.0921 952 | 0.0053 318 0.0578 318
0.0875 391 0.0050 367 | 0.0575 367
0.0831 181 0.0047 593 0.0572 593
0.0789 204 0.0044 983 0.0569 983
0.0749 347 0.0042 528 | 0.0567 528
0.0040 215
0.0711 503
0.0675 570 | 0.0038 037
0.0565 215
0.0563 037
0.0641 452 | 0.0035 981 | 0.0560 984
0.0609 057
0.0578 297
0.0034 019
0.0032 224
0.0559 049
0.0557 224
0.0549 092
0.0521 361
0.0495 031
0.0470 030
0.0446 292
0.0423 753
0.0402 353
0.0382 033
0.0362 739 |
0.0344 420
0.0327 025
0.0310 510
0.0294 828
0.0279 938
0.0265 801
0.0252 377
0.0239 631
0.0227 529
0.0216 038
0.0205 128
0.0030 502
0.0028 877
0.0027 343
0.0025 894
0.0024 525
0.0709 380
0.0697 017
0.0685 671
0.0675 235
0.0665 616
0.0023 231
0.0022 009
0.0020 853
0.0548 231
0.0547 009
0.0545 853
0.0019 761 | 0.0544 761
0.0018 727 | 0.0543 727
0.0013 593
0.0012 890
0.0012 223
0.0011 592
0.0010 995
0.0017 749 | 0.0542 749
0.0016 824
0.0015 949
0.0015 120
0.0014 336
0.0541 824
0.0540 949
0.0540 120
0.0539 336
0.0194 768 0.0010 428
0.0184 932 | 0.0009 892
0.0175 592 | 0.0009 383
0.0166 724
0.0008 901
0.0158 304 0.0008 445
0.0150 309
0.0008 012
0.0142 718 0.0007 601
0.0135 511 | 0.0007 212
0.0128 667 | 0.0006 843
0.0122 169 | 0.0006 493
0.0115 999 0.0006 161 |
0.0110 141 0.0005 847
0.0104 578 0.0005 548
0.0099 297
0.0094 282
0.0005 265
0.0004 997
0.0089 521 0.0004 742
0.0085 000
0.0004 501
0.0080 707 0.0004 272
0.0076 631 0.0004 054
0.0072 761 | 0.0003 848
0.0555 502
0.0553 877
0.0552 343
0.0550 894
0.0549 525
M
0.0538 593
0.0537 890
0.0537 223
0.0536.592
0.0535 995
0.0535 428
0.0534 892
0.0534 383
0.0533 901
0.0533 445
0.0533 012
0.0532 601
0.0532 212
0.0531 843
0.0531 493
0.0531 161
0.0530 847
0.0530 548
0.0530 265
0.0529 997
0.0529 742
0.0529 501
0.0529 272
0.0529 054
0.0528 848
Ratio of
Discounted Annuity for
Principal. Cumulation Annuity.
0.2576 586
0.2445 640
0.2321 349
0.2203 374
0.2091 395
0.1985 107
0.1884 221
0.1788 462
0.1697 570
0.1611 297
0.1529 408
0.1451 681
0.1377 904
0.1307 877
0.1241 409
0.1178 319
0.1118 434
0.1061 594
0.1007 642
0.0956 432
0.0907 825
0.0861 688
0.0817 895
0.0776 329
0.0736 875
0.0699 425
0.0663 880
0.0630 140
0.0598 115
0.0567 718
0.0538 866 |
0.0511 480
0.0485 486
0.0460 813
0.0437 393
0.0319 860
0.0303 601
0.0288 174
0.0273 529
0.0259 628
0.0246 433 |
0.0233 909
0.0222 021
0.0210 738
0.0200 028
0.0182 222
0.0169 963
0.0158 714
0.0148 368
0.0138 834
0.0094 791
0.0089 156
0.0083 901
| 0.0078 995
0.0078 995
0.0074 411
0.0146 278
0.0138 843
0.0131 787
0.0125 090
0.0118 732
0.0130 031
0.0121 888
0.0114 344
0.0107 345
0.0100 842
0.0069 086 0.0003 652
0.0066 895
0.0065 597 0.0003 467
0.0063 496
0.0062 284 0.0003 290
0.0060 269
0.0059 139 0.0003 123 0.0528 123 0.0057 206
0.0056 152 0.0002 965 | 0.0527 965 0.0054 298
0.0528 652
0.0528 467
0.0528 290
0.0525 000
0.0070 125
0.0066 112
0.0062 353
0.0058 829
0.0055 523
0.0415 164 | 0.0022 740 | 0.0547 740
0.0021 537 | 0.0546 537
0.0394 065
0.0374 038
0.0355 029
0.0336 986
0.0545 400
0.0544 325
0.0543 309
0.0707 222
0.0694 963
0.0683 714
0.0673 368
0.0663 834
0.0052 420 | 0.0577 420
0.0049 504 | 0.0574 504
0.0045 764 | 0.0571 761
0.0044 188 0.0569 188
0.0041 763 | 0.0566 763
0.0020 400
0.0019 325
0.0018 309
0.0655 031
0.0646 888
0.0639 341
0.0632 345
0.0625 842
0.0039 481
0.0564 481
0.0037 332
0.0562 332
0.0035 307 0.0560 307
0.0033 399 | 0.0558 399
0.0031 599 | 0.0556 599
0.0189 862 | 0.0010 161
0.0180 213 0.0009 635
0.0171 054 | 0.0009 137
0.0162 361 0.0008 665
0.0154 110
0.0008 217
0.0619 791
0.0614 156
0.0608 901
0.0603 995
0.0599 411
0.0029 902 | 0.0554 902
0.0028 300 0.0553 300
0.0026 789 | 0.0551 789
0.0025 361 0.0550 361
0.0024 013 0.0549 013
0.0595 125
0.0591 112
0.0587 353
0.0583 829
0.0580 523
0.0542 348
0.0017 348
0.0016 438
0.0015 578
0.0541 438
0.0540 578
0.0014 764 | 0.0539 764
0.0013 994 0.0538 994
0.0086 827 0.0004 598
0.0082 415 0.0004 363
0.0078 226 | 0.0004 139
0.0074 251
0.0070 477
0.0003 927
0.0003 726
0.0013 265 | 0.0538 265
0.0012 574 0.0537 574
0.0011 921 0.0536 921
0.0011 302 0.0536 302
0.0010 716 0.0535 716
0.0003 536
0.0003 355
0.0003 183
0.0003 021
0.0002 866
0.0002 866
0.0007 794 | 0.0532 794
0.0007 392
0.0007 011
0.0006 650
0.0006 308
0.0532 392
0.0532 011
0.0531 650
0.0531 308
0.0112 698
0.0106 971
0.0530 677
0.0005 984 | 0.0530 984
0.0005 677
0.0101 534 0.0005 384
0.0096 374 | 0.0005 109
0.0530 384
0.0530 109
0.0091 476 | 0.0004 847 | 0.0529 847
0.0535 161
0.0534 635
0.0534 137
0.0533 665
0.0533 217
0.0529 598
0.0529 363
0.0529 139
0.0528 927
0.0528 726
0.0528 536
0.0528 355
0.0528 183
0.0528 021
0.0527 866
-0.0525 000
Page 41.
RATE OF INTEREST: 5½ per
5½ per Cent. per Ann.--CORRESPONDING VALUE OF

RATIO of INTEREST:
HALF-YEARLY, 0.0275.
Years.
1
20
Cas
10
-={****
4
مدرسه عشر
HENKE
CO
MKN-AIL
AKVAL
→
2:0}+
PICKKPH
R41234
***
===}<
10
HEAN-d
12
#464:0}< O
13
~<~~**
GAKAP
***** *** ****
16
17
RATIO of INTEREST:
ANNUALLY, 0.055.
Improved Cumulated Value of
Principal. Annuity. Annuity.
....
25
1.0550 000 1.0000 000
1.1130 250
1.1742 414
1.2388 247
11 1.8020 924
1.3069 600
1.3788 428
...
..
1.6190 943
1.7081 445
1.4546 792 8.2668 938
1.9012 075
2.0057 739
2.0550 000
1.5346 865 9.7215 730
3.1680 250
4.3422 664
2.3552 627
2.4848 021
2.6214 663
2.7656 469
19
20 2.9177 575
..
5.5810 910
6.8880 510
..
11.256 260
12.875 354
14.583 498
16.385 591
2.1160 915 20.292 572
18.286 798
2.2324 765 22.408 663
24.641 140
26.996 403
29.481 205
32.102 671
34.868 318
21
3.0782 342
22 3.2475 370
37.786 076
40.864 310
23 3.4261 516 | 44.111 847
24
3.6145 899
47.537 998
3.8133 923
51.152 588
0.9478 67
...
1.8463 20
2.6979 33
...
3.5051 50
...
4.2702 84
4.9955 30
5.6829 67
6.3345 66
6.9521 95
7.5376 26
8.0925 36
8.6185 18
9.1170 78
9.5896 48
10.037 58
10.462 16
10.864 61
11.246 07
11.607 65
11.950 38
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0275 000
0.5000 000
1.0557 563 | 1.0137 500
....
1.0847 895
1.1146 213
....
1.1452 733
2.6413 335
1.1767 684 3.2139 702
....
1.2091 295
1.2423 806
...
....
1.2765 460 5.0281 094
1.3116 510 | 5.6663 824
....
1.5416 281
2.0840 229
3.8023 544
4.4069 191
1.3477 214
6.3222 079
1.3847 838 6.9960 686
1.5859 559
1.6295 697
1.4228 653 7.6884 605
1.1619 941 8.3998 932
....
1.7677 402
1.8163 531
1.5021 990 9.1308 903
1.5435 094 | 9.8819 897
...
1.6743 829 12.261 507
1.7204 284
13.098 699
..
....
10.653 744
11.446 722
13.958 913
14.842 783
1.8663 028 15.750 960
1.9176 261 16.684 111
1.9703 608 | 17.642 924
2.0245 457 | 18.628 104
2.0802 208 19.640 377
2.1374 268 20.680 488
2.1962 061 | 21.749 201
2.2566 017 | 22.847 304
2.3186 583 23.975 605
2.3824 214 25.134 934
2.5152 563 27.550 114
2.6554 975 | 30.099 955
2.8035 581 32.791 965
2.9598 740 35.634 072
12.275 24 3.1249 055 38.634 645
12.583 17 3.2991 385 41.802 518
12.875 04 3.4830 861 45.147 019
13.151 70 3.6772 899 | 48.677 998
13.413 93 3.8823 218
52.405 850
0.4866 18
...
0.9602 12
1.4211 31
...
1.8697 14
2.3062 91
2.7311 83
3.1447 04
3.5471 57
...
3.9388 39
4.3200 38
..
4.6910 35
....
5.0521 02
5.4035 05
5.7455 04
6.0783 49
6.4022 87
6.7175 5+
7.0243 83
....
7.3230 01
..
7.6136 26
7.8964 73
8.1717 50
8.4396 59
8.7003 98
8.9541 59
9.2011 28
9.4414 87
9.6754 13
9.9030 78
10.124 65
10.340 29
...
10,550 16
10.953 20
11.334 96
11.696 55
12.039 05
12.363 46
12.670 74
12.961 79
13.237 46
13.498 58
RATIO of INTEREST:
QUARTERLY, 0.01375.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0137 500
1.0276 891
1.0418 198
1.0561 448
1.0706 668
1.0853 885
1.1003 126
1.1154 419
1.1620 89+
1.1780 681
1.1307 792 | 2.3778 033
1.1163 274 2.660+ 981
1.1942 666
1.2106 877
1.2273 347
1.2442 105
1.2613 184
1.2786 616
1.2962 432
1.3140 665
1.5693 457
1.5909 212
1.6127 994
1.6349 754
1.6574 563
1.6802 463
1.7033 497
1.7267 708
1.3321 349 6.0388 167
1.3504 518 6.3718 504
1.3690 205 6.7094 633
1.3878 445 | 7.0517 185
1.7505 139
1.7745 834
1.7989 840
1.8237 200
1.4069 74 7.3986 796
1.4262 726 | 7.7504 114
1.4458 839 | 8.1069 796
1.4657 648 | 8.4684 505
0.2500 000
0.5034 375
0.7603 598
1.0208 147
1.4859 190 8.8348 917
1.5063 504 | 9.2063 715
1 5270 628 | 9.5829 591
1.5480 599 9.9647 248
1.9525 964
1.9794 446
2.0066 620
2.0342 536
1.2848 509
1.5525 176
1.8238 647
2.0989 429
2.0622 246
2.0905 802
2.1193 257
2.1484 664
2.1780 078
2.2079 554
2.2383 148
2.2690 916
2.9170 800
3.2376 023
2.3002 916 |
2.3319 206
2.3639 846
2.3961 893
3.5321 194
3.8306 860
4.1333 579
4.4401 916
2.5310 398
2.6731 445
2 8232 277
2.9817 373
4.7512 442
5.0665 739
5.3862 392
5.7103 000
1.8487 961 | 15.432 657
1.8742 171 15.894 856
1.8999 876 | 16.363 410
1.9261 124 16.838 407
10.351 740
10.744 076
11.141 807
11.545 007
11.953 751
12.368 115
12.788 177
13.214 014
13.645 707
14.083 335
14.526 981
14.976 727
17.319 935
17.808 084
18.302 946
18.804 611
19.313 174
19.828 731
20.351 376
20 881 207
21.418 324
21.962 826
22.514 815
23.074 393
23.641 666
24.216 739
24.799 719
25.390 715
27.837 087
30.420 810
33.149 595
36.031 587
39.075 388
42.290 083
3.5126 897 | 45.685 257
3.1491 464
3.3259 546
3.7099 090
49.271 072
3.9182 011
53.058 202
0.2466 09
0 4898 73
0.7298 38
0.9665 48
1.2000 47
1.4303 80
1.6575 88
1.8817 14
2.1028 01
2.3208 89
2.5360 18
2.7482 30
2.9575 64
3.1640 58
3.3677 51
3.5686 82
3.7668 87
3.9624 04
4.1552 69
4.3455 18
4.5331 87
4.7183 10
4.9009 23
5.0810 58
5.2587 50
5.4340 32
5.6069 37
5.7774 96
5.9457 42
6.1117 06
6.2754 19
6.4369 12
6.5962 14
6.7533 55
6.9083 65
7.0612 73
7.2121 06
7.3608 94
7.5076 64
7.6524 42
7.7952 58
7.9361 36
8.0751 03
8.2121 86
8.3474 09
8.4807 98
8.6123 78
8.7421 73
8.8702 07
8.9965 05
9.1210 90
9.2439 86
9.3652 14
9.4847 98
9.6027 60
9.7191 22
9.8339 06
9.9471 33
10.058 82
10.169 00
10.277 68
10.384 89
10.490 64
10.594 96
10.998 28
11.380 16
11.741 74
12.084 09
12.408 25
12.715 17
13.005 78
13.280 94
13.541 47
Page 42.
PERPETUAL STOCK : 3 p. C. 54,5455; 3½ p. C. 63,6364; 4p. C. 72,7273; 5p. C.90,9091.

Years.
минера
HK${2:0
∞
{ve$¥p{!
Co
PICINDAH
*
10
FONPA…
2.
CO
-}
{0.1]-R
HE-29+
❤
A-A
FAA Cold
11
12:42
12
COADS-PL
13
HABE
GIAKETAX
* 700 20** *
16
17
18
19
20
22
23
24
25
RATIO of INTEREST:
ANNUALLY, 0.055.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9478 673
0.8984 524 0.4866 180
0.8072 167
1.0000 000
0.8516 136 0.3156 541
··
0.7651 344 0.1791 764 | 0.2341 764
0.2302 945 0.2852 945
0.7253 458 0.1451 790 0.2001 790
0.6515 989 0.1028 640
0.6874 368 0.1209 614 0.1759 644
...
0.6176 291 | 0.0888 395
0.4985 607
0.5549 105 | 0.0685 707
0.5259 815 0.0610 292
1.0550 000
0.5416 180
0.2918 727
0.2766 565
0.2622 337
0.3706 541
0.5854 306 0.0776 678 0.1326 678
0.4725 694 0.0492 791
0.4479 330 | 0.0446 256
0.1578 640
0.1438 395
{
0.1235 707
0.0546 843 0.1096 843
****
0.1160 292
0.0446 256 0.0996 256
0.1042 791
0.4245 811 0.0405 825 0.0955 825
0.4024 465 | 0.0370 420
0.3814 659
0.3615 791
0.3427 290
0.0339 200
0.0311 501
0.0286 793
0.0920 420
0.0889 200
0.0861 501
0.0836 793
0.3248 616
0.0264 648
0.3079 257 | 0.0244 712
0.0244 712
0.0226 696
0.0226 696
0.0210 358
0.0195 494 0.0745 494
RATIO of INTEREST:
HALF-YEARLY, 0.0275.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
....
0.9732 360
0.9471 883
0.8731 540
0.8497 849
0.9218 378 0.6486 649 | 0.7036 649
0.8971 658 0.4798 412 0.5348 412
0.7419 931
0.7221 344
...
0.7028 072
0.6839 972
...
2.0000 000
0.9864 365
··
0.8270 413 0.2629 950 | 0.3179 950
0.8049 064 0.2269 159 0.2819 159
···
0.3785 966 | 0.4335 966
0.3111 417
.311.
0.7833 639 -0.1988 819
0.7623 979 | 0.1764 794 | 0.2314 794
0.6656 908 0.1095 183
0.6478 742 0.1011 942
...
2.0550 000
1.0414 365
0.1300 651
0.1190 491
....
....
0.5656 940 0.0716 388
0.5505 537 | 0.0673 728
0.0814 648 0.3200 097
0.0794 712 0.3031 094
0.0776 696 0.2871 017
0.0760 358 0.2719 394
0.2575 778
DIS
0.3661 417
0.1581 726 0.2131 726
0.1429 374 0.1979 374
0.5075 212 | 0.0566 800
0.4939 380 0.0536 823
...
....
....
0.2538 819
0.6305 345 0.0938 628 | 0.1488 628
0.6136 589 | 0.0873 613 | 0.1423 613
0.0815 560 0.1365 560
0.5972 350
0.5812 505 0.0763 435 | 0.1313 435
0.1850 651
0.1740 491
...
0.1645 183
...
0.1561 942
0.5358 188 | 0.0634 882
0.1184 882
0.5214 781 0.0599 373 | 0.1149 373
··
0.1266 388
0.1223 728
0.4807 182 0.0509 155 | 0.1059 155
0.0483 548 0.1033 548
....
0.4678 523
0.1116 800
0.1086 823
0.4553 307 0.0459 787 | 0.1009 787
0.4431 442
0.0437 688 0.0987 688
0.0417 091 | 0.0967 091
0.4312 839
0.4197 410
0.0397 853 0.0917 853
0.3975 738 0.0362 975 | 0.0912 975
0.3765 773 0.0332 226 0.0882 226
0.3566 896 0.0304 953 | 0.0854 953
0.3378 522 | 0.0280 630 0.0830 630
0.0258 835 0.0808 835
0.0239 220 | 0.0789 220
0.0221 499 | 0.0771 499
| 0.0205 432 | 0.0755 432
0.0190 818 | 0.0740 818
RATIO of INTEREST :
QUARTERLY, 0.01375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9864 365
0.9730 570
0.9598 589
0.9468 398
0.9339 974
0.9213 291
0 9088 326
0.8965 058
0.8843 459
0.8723 511
0.8605 190
0.8488 473
0.8373 340
0.8259 768
0.8147 737
0.8037 225
0.7928 212
0.7820 678
0.7714 602
0.7609 965
0.7506 747
0.7404 929
0.7304 493
0.7205 418
0.7107 687
0.7011 282
0.6916 185
0.6822 377
0.6729 842
0.6638 561
0.6548 519
0.6459 699
0.6372 082
0.6285 655
0.6200 399
0.6116 300
0.6033 342
0.5951 509
0.5870 785
0.5791 156
0.5408 925
0.5335 561
0.5263 192
0.5191 805
0.5121 386
0.5051 922
0.4983 400
0.4915 808
0.4849 133
0.4783 361
0.4718 482
0.4654 483
0.4591 352
0.4529 077
0.4467 617
0.4107 050
0.4347 275
0.4288 312
0.4230 146
0.4172 770
0.3950 945
0.3740 913
0.3542 045
0.3353 749
4.0000 000
1.9863 440
1.3151 668
0.9796 097
0.3175 464
0.3006 656
0.2846 821
0.7783 005
0.6441 151
0.5482 863
0.4764 304
0.4205 562
0.3758 695
0.3393 189
0.3088 705
0.2831 161
0.2610 499
0.2419 340
0.2252 155
0.5712 608
0.5635 126
0.0732 831
0.0710 059
0.5558 693 | 0.0688 374
0.5483 298
0.0667 703
0.2104 712
0 1973 721
0.1856 583
0.1751 222
0.1655 954
0.1569 403
0.1490 432
0.1418 094
0 1351 593
0.1290 254
0.1233 505
0.1180 854
0.1131 876
0.1086 204
0.1043 519
0.1003 540
0.0647 977
0.0647 977
0.0629 134
0.0611 120
0.0593 880
4.0550 000
2.0413 440
1.3701 668
1.0346 097
0.8333 005
0.6991 151
0.6032 863
0.5314 304
0.0517 781
0.0504 319
0.0491 367
0.0478 900
0.4755 562
0.4308 695
0.3943 189
0.3638 705
0.0466 890
0.0455 315
0.0444 152
0.0433 381
0.3381 161
0.3160 499
0.2969 340
0.2802 155
0.2654 712
0.2523 721
0.2406 583
0.2301 222
0.0966 021 | 0.1516 021
0.0930 745
0.0807 520
0.0866 175
0.2205 954
0.2119 403
0.2040 432
0.1968 094
0.1901 593
0.1840 254
0.1783 505
0.1730 854
0.0836 557
0.0808 531
0.0781 972
0.1386 557
0.1358 531
0.1331 972
0.0756 772 | 0.1306 772
0.1681 876
0.1636 204
0.1593 319
0.1553 540
0.1480 745
0.1447 520
0.1416 175
0.1282 831
0.1260 059
0.1238 374
0.1217 703
0.0577 368
0.0561 543
0.1127 368
0.1111 543
0.0546 360 | 0.1096 360
0.1081 784
0.0531 784
0.1197 977
0.1179 134
0.1161 120
0.1143 880
0.1067 781
0.1054 319
0.1041 367
0.1028 900
0.1016 890
0.1005 315
0.0994 152
0.0983 381
0.0422 982
0.0412 937
0.0403 230
0.0393 845
0.0393 845 | 0.0943 845
0.0972 982
0.0962 937
0.0953 230
0.0359 233 0.0909 233
0.0328 722 0.0878 722
0.0301 663 | 0.0851 663
0.0277 534 | 0.0827 534
0.0255 916 | 0.0805 916
0.0236 462
0.0236 462 | 0.0786 462
0.0218 889 0.0768 889
0.2695 484 | 0.0202 959 | 0.0752 959
0.2552 192 0.0188 472 | 0.0738 472
Page 43.
RATE OF INTEREST: 5½ per Cent. per
-CORRESPONDING VALUE OF
Ann.

Years.
€8898 9788** ***** ***** ****2 22332 28898 ***82 20*** **OOO CAELA ***** *** co co co co co co co CERRA
66
100
RATIO of INTEREST:
ANNUALLY, 0.055.
Improved Cumulated Value of
Principal. Annuity. Annuity.
4.0231 289
4.2444 010
4.4778 431
4.7241 244
4.9839 513
5.2580 686
5.5472 624
5.8523 618
6.1742 417
6.5138 250
6.8720 854
7.2500 501
7.6488 028
8.0694 870
8.5133 088
8.9815 408
9.4755 255
9.9966 794
10.546 497
11.126 554
11.738
13.065
515
12.384 133
260
849
961
13.783
14.541
15.341 769
16.185 566
17.075 773
18.014 940
19.005 762
20.051 079
21.153 888
22.317 352
23.544 806
24.839 770
26.205 958
27.647 285
29.167 886
30.772 120
32.464 587
44.763 562
47.225 558
49.822 963
52.563 226
55.454 204
58.504 185
61.721 915
65.116 620
68.698 034
72.476 426
76.462 630
80.668 074
85.104 818
89.785 583
94.723 791
99.933 599
105.42 995
111.22 859
117.34 617
123.80 021
54.965 981
58.989 109
63.233 510
67.711 354
72.435 478
77.419 429
82.677 498
88.224 760
94.077 122
100.25 136
145.37 132
153.36 675
161.80 192
106.76 519
113.63 727
120.88 732
128.53 613
136.60 561
145.11 892
154.10 046
163.57 599
173.57 267
184.11 917
34.250 139
604.54 798
36.133 896638.79 812
38.121 261
674.93 201
713.05 327
40.217 930
42.429 916753.27 120
195.24 572
206.98 423
219.36 837
232.43 363
246.21 748
260.75 944
276.10 121
292.28 677
309.36 255
327.37 749
346.38 325
366.43 433
387.58 821
409.90 557
433.45 037
458.29 014
484.49 610
512.14 339
541.31 127
572.08 339
795.70 112
8+0.46 468
887.69 024
937.51 320
990.07 613
1045.5 306
1104.0 348
1165.7 567
1230.8 734
1299.5 714
1372.0 478
1448.5 104
1529.1 785
1614.2 833
1704.0 689
1798.7 927
1898.7 263
2004.1 563
2115.3 848
2232.7 310
130.60 922 | 2356.5 312
137.79 272 | 2487.1 404
2624.9 332
2770.3 045
2923.6 712
13.662 50
13.898 10
14.121 42
14.333 10
14.533 74
14.723 93
14.904 20
15.075 07
15.237 03
15.390 55
15.536 07
15.674 00
15.804 74
15.928 66
16.046 12
16.157 46
16.263 00
16.363 03
16.457 85
16.547 75
16.632 92
16.713 66
16.790 20
16.862 75
16.931 52
16.996 70
17.058 48
17.117 05
17.172 55
17.225 17
17.275 04
17.322 32
17.367 12
17.409 60
17.449 85
17.488 01
17.524 18
17.558 47
17.590 96
17.621 77
17.650 96
17.678 64
17.704 87
17.729 74
17.753 30
17.775 64
17.796 82
17.816 89
17.835 91
17.853 95
17.871 04
17.887 24
17.902 60
17.917 16
17.930 95
17.941 03
17.956 43
17.968 18
17.979 32
17.989 87
18.042 61
18.049 87
18.056 75
18.063 27
18.069 45
170.70 102
180.08 958
189.99 451
3085.4 732
3256.1 742
3436.2 638
200.44 420 3626.2 583
211.46 864 3826.7 025
Value of Perpetual Ann. 18.181 82
RATIO of INTEREST:
HALF-YEARLY, 0.0275.
Improved Cumulated
Principal. Annuity.
4.0987 855
4.3273 184
4.5685 934
4.8233 211
5.0922 514
7.0516 671
7.4418 416
7.8599 380
8.2981 787
8.7608 540
5.9924 003
6.3265 141
6.6792 568
5.3761 762 | 79.566 840
5.6759 316 | 85.016 939
90.770 914
96.845 710
103.25 921
9.2493 264
9.7650 341
10.309 496
10.881 315
11.491 183
12.131 889
12.808 317
13.522 461
14.276 423
15.072 422
15.912 804
16.800 042
17.736 750
18.725 684
19.769 758
20.872 046
22.035 793
23.264 426 |
24.561 563
25.931 024
61.778 626
65.223 170
68.859 770
72.699 132
76.752 563
17.999 88
18.009 36
18.018 35
106.28 570
112.21 180
118.46 831
18.026 88 125.07 365
18.034 95
132.04 729
81.031 998
85.550 039
90.319 988
95.355 892
100.67 258
56.341 554
60.496 698
64.883 517
69.514 928
74.404 570
139.40 975
147.18 272
155.38 908
164.05 299
173.19 997
18.075 31
182.85 695
18.080 86 193.05 236
18.086 12
18.091 11
18.095 84
110.03 031
117.17 894
124.72 615
132.69 416
141.10 644
27.376 841
28.903 271
30.514 809
32.216 200
479.57 892
507.33 219
536.63 288
567.56 727
34.012 454 | 600.22 644
149.98 775
159.36 426
169.26 356
179.71 481
190.74 879
35.908 861
37.911 005
634.70 657
671.10 918
40.024 780 | 709.54 146
42.256 412 750.11 658
44.612 471 | 792.95 401
202.39 797
214.69 667
227.68 111
241.38 950
255.86 222
47.099 895 | 838.17 991
49.726 008
52.498 544
55.425 666
58.515 993
885.92 743
936.33 717
989.55 757
1045.7 453
271.14 189
287.27 350
304.30 454
322.28 517
341.26 833
361.30 993
382.46 896
404.80 775
428.39 206
453.29 134
1105.0 659
1167.6 940
1233.8 140
1303.6 206
1377.3 193
1455.1 272
1537.2 734
1623.9 998
1715.5 617
1812.2 287
1914.2 855
2022.0 327
2135.7 874
2255.8 846
2382.6 780
2516.5 410
2657.8 676
2807.0 741
2964.5 998
3130.9 085
3306.4 900
3491.8 612
203.81 624 3687.5 680
215.18 027 3894.1 867
227.17 792 | 4112.3 257
Value of
Annuity.
13.745 92
13.980 18
14.202 08
14.412 25
14.611 33
14.799 89
14.978 50
15.147 67
15.307 91
15.459 69
15.603 45
15.739 62
15.868 59
15.990 76
16.106 47
16.216 07
16.319 89
16.418 22
16.511 36
16.599 58
16.683 14
16.762 29
16.837 25
16.908 26
16.975 52
17.039 23
17.099 57
17.156 72
17.210 86
17.262 14
17.310 71
17.356 71
17.400 29
17.441 56
17.480 66
17.517 69
17.552 76
17.585 98
17.617 45
17.647 25
17.675 49
17.702 23
17.727 56
17.751 54
17.774 27
17.795 79
17.816 18
17.835 49
17.853 78
17.871 10
17.887 51
17.903 05
17.917 78
17.931 72
17.944 93
17.957 44
17.969 29
17.980 52
17.991 14
18.001 21
18.010 75
18.019 79
18.028 34
18.036 45
18.044 13
18.051 40
18.058 29
18.064 81
18.070 99
18.076 84
18.082 39
18.087 64
18.092 61
18.097 32
18.101 78
18.181 82
2 RATIO of INTEREST:
QUARTERLY, 0.01375.
Improved Cumulated
Principal. Annuity.
4.1381 878
4.3705 255
4.6159 078
4.8750 671
5.1487 768
5.4878 539
5.7431 612
6.0656 099
6.4061 624
6.7658 351
7.1457 017
7.5468 957
7.9706 147
8.4181 234
8.8907 573
9.3899 272
9.9171 228
14.536 169
15.352 300
152.54 413
162.12 951
10.473 918 | 172.25 305
182.94 498
11.061 974
11.683 046
194.23 721
57.057 959
61.282 282
65.743 779
70.455 765
75.432 305
12.338 989
206.16 343
13.031 759 | 218.75 925
13.763 425
232.06 226
80.688 253
86.239 294
92.101 997
98.293 861
104.83 337
36.791 623
38.857 282
41.038 916
43.343 038
45.776 525
111.74 003
119.03 447
126.73 845
134.87 497
143.46 831
16.214 252
276.62 276
17.124 598 | 293.17 451
18.086 055 | 310.65 555
19.101 493 | 329.11 806
20.173 943 | 348.61 714
21.306 605 369.21 100
22.502 860 | 390.96 110
23.766 279 | 413.93 235
25.100 632 | 438.19 332
26.509 903 | 463.81 641
246.11 217
260.95 091
27.998 296 | 490.87 811
29.570 255
31.230 471
32.983 900
34.835 775
519.45 918
549.64 493
581.52 546
615.19 591
109.70 295
115.86 221
189.43 186
200.06 747
211.30 022
650.75 678
688.31 421
48.346 639 860.84 799
51.061 052 | 910.20 095
53.927 865962.32 482
56.955 635 1017.3 752
60.153 398 1075.5 163
727.98 030
769.87 343
814.11 864
63.530 699
67.097 618 |
70.864 801
74.843 492
1136.9 218
1201.7 749
1270.2 691
1342.6 089
79.045 565 | 1419.0 103
83.483 564 | 1499.7 012
88.170 732 | 1584.9 224
93.121 061
98.349 325
103.87 113
1674.9 281
1769.9 877
1870.3 842
1976.4 174
2088.4 037
122.36 727 | 2206.6 776
129.23 755
136.49 357
2331.5 919
2463.5 195
144.15 698
2602.8 541
152.25 064 | 2750.0 117
160.79 873 | 2905.4 314
169.82 674 3069.5 771
179.36 163
3242.9 387
3426.0 337
3619.4 085
3823.6 404
223.16 363 | 4039.3 387
235.69 311 | 4267.1 475
Value of
Annuity.
13.788 15
14.021 72
14.242 87
14.452 26
14.650 53
14.838 25
15.016 00
15.184 29
15.343 64
15.494 52
15.637 38
15.772 64
15.900 71
16.021 98
16.136 80
16.245 51
16.348 44
16.445 90
16.538 19
16.625 56
16.708 29
16.786 63
16.860 79
16.931 02
16.997 51
17.060 47
17.120 08
17.176 52
17.229 97
17.280 56
17.328 48
17.373 84
17.416 79
17.457 45
17.495 97
17.532 43
17.566 95
17.599 64
17.630 59
17.659 89
17.687 64
17.713 91
17.738 78
17.762 33
17.784 63
17.805 75
17.825 74
17.814 67
17.862 59
17.879 56
17.895 63
17.910 84
17.925 25
17.938 89
17.951 80
17.964 03
17.975 61
17.986 57
17.996 95
18.006 78
18.016 08
18.024 89
18.033 23
18.041 13
18.048 61
18.055 69
18.062 40
18.068 75
18.074 76
18.080 45
18.085 84
18.090 94
18.095 77
18.100 35
18.1041 68
18.181 82
Page 44.
PERPETUAL STOCK: 3 p. C. 54,5455; 3ž p. C. 63,6364; 4 p.C.72,7273; 5 p.C. 90,9091.

Years.
26
27
28
29
30
31
32
33
85889 793‡9 95938 58833 85888 58868 86882 FRPER ORF228 ***** ***38 **836 85883
i
0.19 1 839
0.1802 691
0.1708 712
34 0.1619 632
0.1535 196
35
93
RATIO of INTEREST:
ANNUALLY, 0.055.
100
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2485 628
0.2356 045
0.2233 218
0.2116 794
0.2006 440
0.1455 162
0.1379 301
0.1307 394
0.1239 236
0.1174 631
0.1113 395
0.1055 350
0.1000 332
0.0948 182
0.0898 751
0.0651 815
0.0617 834
0.0585 625
0.0555 095
0.0526 156
0.0498 726
0.0472 726
0.0448 082
0.0424 722
0.0402 580
0.0381 593
0.0361 699
0.0342 843
0.0324 969
0.0308 028
0.0851 897 0.0051 218
0.0807 485 0.0048 313
0.0045 585
0.0043 023
0.0040 615
0.0765 388
0.0725 487
0.0687 665
0.0291 970
0.0276 748
0.0262 321
0.0248 645
0.0235 683
0.0223 396
0.0211 750
0.0200 711
0.0190 247
0.0180 329
0.0181 931
0.0169 523
0.0158 144
0.0147 686
0.0138 054
0.0129 167
0.0120 952
0.0113 347
0.0106 296
0.0099 749
0.0170 928
0.0162 017
0.0093 663
0.0093 663
0.0087 999
0.0082 722
0.0077 799
0.0073 203
0.0068 909
0.0064 893
0.0061 134
0.0057 613
0.0054 313
0.0038 350
0.0036 219
0.0034 213
0.0032 325
0.0030 546
0.0028 870
0.0027 290
0.0025 801
0 0024 396
0.0023 071
0.0012 568
0.0011 898
0.0011 265
0.0010 667
0.0010 100
0.0731 931
0.0719 523
0.0708 144
0.0697 686
0.0688 054
0.0679 167
0.0670 952
0.0663 347
0.0656 296
0.0619 749
0.0130 783 0.0007 288
0.0123 965 | 0.0006 904
0.0117 502 0.0006 539
0.0111 376 0.0006 195 |
0.0105 570 | 0.0005 868
0.0643 663
0.0637 999
0.0632 722
0.0627 799
0.0623 203
0.0618 909
0.0614 893
0.0611 134
0.0607 613
0.0604 313
0.0076 564 | 0.0004 244
0.0072 573 | 0.0004 021
0.0068 789 0.0003 810
0.0065 203 0.0003 610
0.0061 804 0.0003 420
0.0601 218
0.0598 313
0.0595 585
0 0593 023
0.0590 615
0.0021 820
0.0020 640
0.0019 526
0.0018 474
0.0571 820
0.0570 640
0.0569 526
0.0568 474
0.0017 480 | 0.0567 480
0.0588 350
0.0586 219
0.0584 213
0.0582 325
0.0580 546
0.0016 541 0.0566 541
0.0015 654 | 0.0565 654
0.0014 816
0.0014 024
0.0013 275
0.0564 816
0.0564 024
0.0563 275
0.0578 870
0.0577 290
0.0575 801
0.0574 396
0.0573 071
0.0009 565
0.0009 058
0.0559 565
0.0559 058
0.0153 571 0.0008 578
0.0558 578
0.0145 565 0.0008 124 0.0558 124
0.0137 976 | 0.0007 695 | 0.0557 695
0.0557 288
0.0556 904
0.0556 539
0.0556 195
0.0555 868
0.0100 066 | 0.0005 559 | 0.0555 559
0.0094 850
0.0005 267
0.0089 905 0.0004 990
0.0085 218 0.0004 727
0.0080 775 | 0.0004 479
0.0555 267
0.0554 990
0.0554 727
0.0554 479
0.0554 244
0.0554 021
0.0553 810
0.0553 610
0.0553 420
RATIO of INTEREST:
HALF-YEARLY, 0.0275.
0.0058 582
0.0055 528
0.0003 241
0.0003 071
0.0052 633 | 0.0002 910
0.0049 889 0.0002 753
0.0047 288 | 0.0002 613
0.0553 241
0.0553 071
0.0552 910
0.0552 758
0.0552 613
Ratio of Perpetual Ann. 0.0550 000
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2439 747
0.2310 900
0.2188 858
0.0177 489
0.0177 489 0.0727 489
0.0165 298 0.0715 298
0.0154 122 0.0704 122
0.2073 260 | 0.0143 854 | 0.0693 854
0.1963 768
0.0684 400
0.0134 400
0.1860 058 0.0125 680
0.1761 825 0.0117 623
0.1668 780 0.0110 167
0.1580 649 | 0.0103 257
0.1497 173 | 0.0096 814
0.1418 104
0.1343 212
0.1272 275
0.1205 084
0.1141 441
0.1081 160
0.1024 062 |
0.0969 980
0.0918 753
0.0870 232
0.0780 743
0.0739 510
0.0700 456
0.0663 463 |
0.0562 568
0.0561 898
0.0212 315
0.0201 102
0.0561 265 0.0190 481
0.0180 422
0.0170 893
0.0560 667
0.0560 100
0.0479 110
0.0453 807
0.0429 841
0.0407 140
0.0385 638
0.0824 274 | 0.0049 408 | 0.0599 408
0.0046 577 | 0.0596 577
0.0043 921 0.0593 921
0.0041 427 0.0591 427
0.0039 084 | 0.0589 084
0.0628 425 | 0.0036 881
0.0595 237 0.0034 810
0.0563 801 0.0032 862
0.0534 026 0.0031 028
0.0505 823 0.0029 302
0.0365 272
0.0345 982
0.0327 710
0.0310 403
0.0294 010
0.0278 483
0.0263 776
0.0249 845
0.0236 650
0.0224 153
0.0161 868
0.0153 320
0.0145 223
0.0137 553
0.0130 289
0.0123 408
0.0116 891
0.0110 717
0.0104 870
0.0099 332
0.0090 884
0.0085 340
0.0080 176
0.0075 361
0.0070 868
0.0094 086
0.0089 117
0.0084 411
0.0079 953
0.0075 730
0.0071 731
0.0067 943
0.0064 355
0.0060 956
0.0057 737
0.0066 672 0.0616 672
0.0062 749 | 0.0612 749
0.0059 079 | 0.0609 079
0.0055 644 0.0605 644
0.0052 425 0.0602 425
0.0020 852
0.0019 711
0.0018 635
0.0017 619
0.0016 660
0.0675 680
0.0667 623
0.0660 167
0.0653 257
0.0646 844
0.0015 755
0.0014 901
0.0014 094
0.0013 331
0.0012 611
0.0540 884
0.0635 340
0.0630 176
0.0625 361
0.0620 868
0.0027 677 | 0.0577 677
0.0026 146 | 0.0576 146
0.0024 703 | 0.0574 703
0.0023 343
0.0022 061
0.0573 343
0.0572 061
0.0006 872
0.0006 505
0.0006 158
0.0005 829
0.0005 518
0.0586 881
0.0584 810
0.0582 862
0.0581 028
0.0579 302
0.0570 852
0.0569 711
0.0568 635
0.0567 619
0.0566 660
0.0011 931 0.0561 931
0.0011 288
0.0561 288
0.0010 680 0.0560 680
0.0010 106 | 0.0560 106
0.0009 563 0.0559 563
0.0565 755
0.0564 901
0.0564 094
0.0563 331
0.0562 611
0.0009 049 | 0.0559 049
0.0008 564 0.0558 564
0.0008 105 0.0558 105
0.0007 671 0.0557 671
0.0007 260 | 0.0557 260
N
0.0556 872
0.0556 505
0.0556 153
0.0555 829
0.0555 518
0.0004 682
0.0004 433
0.0004 197
0.0005 224 | 0.0555 224
0.0004 946 | 0.0554 946
0.0554 682
0.0554 433
0.0554 197
0.0003 974
0.0553 974
0.0003 762 | 0.0553 762
0.0003 562 | 0.0553 562
0.0003 373
0.0553 373
0.0003 194 0.0553 194
0.0054 688 | 0.0003 024 0.0553 024
0.0051 799 0.0002 864 | 0.0552 864
0.0049 064 | 0.0002 712 | 0.0552 712
0.0046 473
0.0044 018
0.0002 568 | 0.0552 568
0.0002 432 | 0.0552 432
· 0.0550 000
RATIO of INTEREST:
QUARTERLY, 0.01375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2416 517
0.2288 054
0 2166 421
0.2051 254
0.1942 209
0.1838 961
0.1741 201
0.1648 639 |
0.1560 997
0.1478 014
0.1399 443
0.1325 048
0.1254 608
0.1187 913
0.1124 764 |
0.1064 971
0.1008 357
0.0810 439
0.0767 356
0.0726 563
0.0687 939
0.0651 368
0.0469 338
0.0444 388
0.0420 764
0.0398 396
0.0377 218
0,0954 753 0.0058 054
0.0903 998 | 0.0054 661
0.0855 941 0.0051 483
0.0357 165
0.0338 178
0.0320 200
0.0303 178
0.0287 061
0.0271 801
0.0257 352
0.0243 671
0.0230 718
0.0218 453
0.0175 260 0.0725 260
0.0163 179 0.0713 179
0.0152 106 | 0.0702 106
0.0141 933
0.0132 569
0.0616 741
0.0036 150 0.0586 150
0.0583 955 | 0.0034 109 | 0.0584 109
0.0552 912
0.0523 519
0.0495 689
0.0032 190
0.0030 384
0.0028 685
0.0582 190
0.0580 384
0.0578 685
0.0119 784
0.0113 416
0.0107 387
0.0101 678
0.0096 273
0.0123 934
0.0673 934
0.0115 956 0.0665 956
0.0108 575 | 0.0658 575
0.0101 736 0.0651 736
0.0095 389 | 0.0645 389
0.0091 155
0.0086 309
0.0081 721
0.0077 377
0.0073 264
0.0089 493 0.0639 493
0.0084 009 | 0.0634 009
0.0078 903 | 0.0628 903
0.0074 143 0.0624 143
0.0069 702 | 0.0619 702
0.0065 555 | 0.0615 555
0.0061 679 0.0611 679
0.0608 054
0.0604 661
0.0601 483
0.0052 789
0.0049 983
0.0047 326
0.0044 810
0.0042 428
0.0048 505 | 0.0598 505
0.0045 712 | 0.0595 712
0.0043 092 | 0.0593 092
0.0040 632 | 0.0590 632
0.0038 321 0.0588 321
0.0691 933
0.0682 569
0.0027 085
0.0025 578
0.0024 159
0.0022 821
0.0577 085
0.0575 578
0.0574 159
0.0572 821
0.0021 560 | 0.0571 560
0.0020 372
0.0019 251
0.0019 251
0.0018 194
0.0017 196
0.0016 255
0.0015 367
0.0014 528
0.0013 737
0.0012 989
0.0012 283
0.0157 404 0.0008 796
0.0149 037 0.0008 321
0.0141 114 0.0007 872
0.0133 612 | 0.0007 448
0.0126 509
0.0206 840
0.0195 844
0.0185 433
0.0011 616 | 0.0561 616
0.0010 987
0.0560 987
0.0010 392
0.0560 392
0.0175 575 | 0.0009 829 | 0.0559 829
0.0166 242
0.0009 298
0.0559 298
0.0558 796
0.0558 321
0.0557 872
0.0557 448
0.0007 047 | 0.0557 047
0.0005 060
0.0004 788
0.0004 532
0.000+ 289
0.0004 059
0.0004 059
0.0069 369 0.0003 842
0.0065 681 0.0003 636
0.0052 190 0.0003 442
0.0058 884 0.0003 258
0.0055 753 0.0003 084
0.0570 372
0.0569 251
0.0568 194
0.0567 196
0.0566 255
0.0565 367
0.0564 528
0.0563 737
0.0562 989
0.0562 283
0.0006 668 | 0.0556 668
0.0006 309
0.0556 309
0.0005 970
0.0555 970
0.0005 650
0.0005 346
0.0555 650
0.0555 346
0.0002 919
0.0002 763
0.0002 615
0.0002 476
0.0002 343
0.0555 060
0.0554 788
0.0554 532
0.0554 289
0.0554 059
0.0553 842
0.0553 636
0.0553 442
0.0553 258
0.0553 084
0.0552 919
0.0552 763
0.0552 615
0.0552 476
0.0552 343
0.0550 000
Page 45.
RATE OF INTEREST: 52 per Cent. per Ann.--CORRESPONDING VALUE OF

Years.
0
20
={{*<<<<][6#39}<
4
PK:ZHAN
-)
HAHADE
ગ
PIS:NLA
======{^2>0}<#
0
AKCHAL
DAKLAH
→
>><<*~*~*
POKKAL
~~~40
10
SAMo8==MauབྱུbS}L
11
12
CARILAH
13
H{*~{NA
ܢܚ ܐܥܪܢܝܟܬܪܘܐܗ
16
*** **** *
17
18
19
20
21
22
RATIO of INTEREST:
ANNUALLY, 0.0575.
23
Improved Cumulated Value of
Principal, Annuity. Annuity.
25
1.0575 000 1.0000 000
1.1183 063
1.1826 089 3.1758 063
1.2506 089
2.0575 000
....
1.3225 189 5.6090 240
4.3584 151
1.3985 637 6.9315 429
1.4789 811 8.3301 066
1.5640 225 | 9.8090 877
14 2.1873 852 | 20.650 177
1.6539 538 11.373 110
1.9559 805
1.7490 562 13.027 064
1.8496 269 14.776 120
··
2.4461 665
2.5868 211
2.7355 633
2.8928 582
3.0591 975
16.625 747
2.0684 493 18.581 728
2.3131 598 22.837 562
3.2351 014
38.871 329
3.4211 197 42.106 430
3.6178 341 45.527 550
24 3.8258 596 49.145 384
4.0458 465 52.971 243
25.150 722
27.596 888
30.183 710
32.919 273
35.812 131
0.9456 27
1.8398 36
....
2.6854 24
3.4850 35
4.2411 68
4.9561 87
5.6323 28
……..
6.2717 05
6.8763 17
7.4480 54
7.9887 03
8.4999 56
8.9834 10
9.4405 77
9.8728 86
....
10.281 69
10.668 26
11.033 82
11.379 50
11.706 38
12.015 49
12.307 79
12.584 20
12 845 58
13.092 75
RATIO of INTEREST:
HALF-YEARLY, 0.02875.
Improved Cumulated Value of
Principal. Annuity. Annuity.
1.0287 500 0.5000 000
1.0583 265
...
1.0887 535
1.1200 551
....
....
1.1522 567 2.6479 426
1.1853 841
1.2545 235
1.2905 910
1.3276 955
..
..
1.0143 750
1.2194 639 3.8167 630
4.4264 949
1.5435 283
2.0879 150
...
...
3.2240 709
***
1.3658 667 6.3628 999
1.4051 354
7.0458 333
1.9192 277
1.9744 055
1.4455 331
7.7484 010
1.4870 921 8.4711 675
5.0537 566
5.6990 522
1.5298 460 | 9.2147 136
1.5738 2919.9796 366
....
1.6190 767 10.766 551
1.6656 251 11.576 089
..
12.408 902
1.7135 119
1.7627 753 13.265 658
....
1.8134 551 14.147 046
1.8655 920 15.053 773
15.986 569
16.946 183
2.0311 697 17.933 386
··
2.0895 658 18.948 971
2.1496 408 | 19.993 754
2.2114 430 | 21.068 574
2.2750 220 22.174 295
2.3404 289 23.311 806
..
2.4077 162 24.482 021
2.4769 380 25.685 879
2.6214 093
2.7743 071
28.198 423
30 857 515
2.9361 229 | 33.671 703
3.1073 769 36.650 033
3.2886 195
3.4804 334
3.6834 351
3.8982 772
4.1256 503 51.359 135
39.802 078
43.137 971
46.668 436
50.404 820
0.4860 27
0.9584 71
1.4177 11
...
1.8641 18
2.2920 49
2.7198 53
3.1298 70
3.5284 27
3.9158 47
4.2924 39
4.6585 07
5.0143 45
5.3602 38
5.6964 64
6.0232 95
6.3409 91
6.6498 09
6.9499 97
7.2417 95
7.5254 39
7.8011 56
8.0691 67
8.3296 88
8.5829 29
8.8290 93
9.0683 77
9.3009 74
9.5270 71
9.7468 49
9.9604 85
10.168 15
..
10.370 01
10.756 97
11.122 60
11.468 08
11.794 52
RATIO of INTEREST:
QUARTERLY, 0.014375.
Improved Cumulated Value of
Principal. Annuity. Annuity,
1.0143 750
1.0289 566
1.0437 479
1.0587 518
1.0739 713
1 0894 097
1.1050 699
1.1209 553
1.1370 690
1.1534 144
1.1699 947
1.1868 134
1.2038 739
1.2211 795
1.2387 340
1.2565 408
1.2746 036
1.2929 260
1.3115 118
1.3303 648
1.3494 888
1.3688 877
1.3885 654
1.4085 261
1.4287 736
1.4493 123
1.4701 461
1.4912 795
1.5127 166
1.5344 619
1.5565 198
1.5788 918
1.6015 914
1.6246 143
1.6479 681
1.6716 576
1.6956 877
1.7200 632
1.7447 891
1.7698 705
1.7953 124
1.8211 200
1.8472 986
1.8738 535
1.9007 901
1.9281 140
1.9558 306
1.9839 457
2.0124 649
2.0413 941
2.0707 392
2.1005 060
2.1307 008
2.1613 296
2.1923 987
2.2239 145
2.2558 832
2.2883 116
2.3212 060
2.3545 734
2.3884 204
2.4227 539
2.4575 810
2.4929 087
2.6393 715
2.7944 393
2.9586 175
3.1324 415
12.102 97
12.394 43
12.669 81
12.930 03
3.3164 780
3.5113 270
3.7176 236
3.9360 406
13.175 90 4.1672 899
0.2500 000
0.5035 938
0.7608 329
1.0217 699
1.2864 578
1.5549 507
1.8273 031
2.1035 706
2.3838 094
2.6680 766
2.9564 302
3.2489 289
3.5456 323
3.8466 007
4.1518 956
4.4615 791
4.7757 143
5.0943 652
5.4175 967
5.7454 747
6.0780 659
6.4154 381
6.7576 600
7.1048 014
7.4569 329
7.8141 263
8.1764 544
8.5439 909
8.9168 108
9.2949 899
9.6786 054
10.067 735
10.462 459
10.862 857
11.269 010
11.681 002
12.098 917
12.522 839
12.952 855
13.389 052
13.831 520
14.280 348
14.735 628
15.197 452
15.665 916
16.141 113
16.623 142
17.112 099
17.608 086
18.111 202
18.621 551
19.139 235
24.146 441
24.743 546
25.349 235
25.963 630
0.2464 57
0.4894 22
0.7289 43
0.9650 70
28.510 809
31.207 640
34.062 913
37.085 940
1.1978 51
1.4273 33
1 6535 63
1.8765 87
40.286 574
43.675 251
47.263 020
51.061 575
55.083 303
2.0964 51
2.3131 99
2.5268 75
2.7375 23
2.9451 86
3.1499 05
3.3517 25
3.5506 84
3.7468 23
3.9401 83
4.1308 03
4.3187 21
4.5039 76
4.5866 06
4.8666 48
5.0441 39
5.2191 14
5.3916 10
5.5616 61
5.7293 02
5.8945 68
6.0574 91
6.2181 06
6.3761 45
6.5325 39
6.6864 22
6.8381 24
6.9876 76
7.1351 09
7.2804 53
7.4237 36
7.5649 90
7.7042 41
7.8415 19
7.9768 52
8.1102 67
19.664 362
9.2290 58
20.197 037 9.3447 28
20.737 369 9.4587 58
21.285 469
9.5711 72
8.2417 91
8.3714 52
8.4992 74
8.6252 86
21.841 448 9.6819 94
22.405 419 9.7912 45
22.977 496 9.8989 47
23.557 798
10.005 12
8.7495 12
8.8719 77
8.9927 07
9.1117 26
10.109 80
10.212 98
10.314 71
10.414 99
10.802 12
11.167 76
11.513 12
11.839 31
12.147 40
12.438 39
12.713 23
12.972 83
13.218 02
Page 46.
PERPETUAL STOCK: 3 p. C. 52,1739; 31 p. C. 60,8696; 4p. C. 69,5652; 5p. C.86,9565.

Years.
Q!
#{<<<={**
2
HO
~*~*
HAND
um
HAURIA
t
PENAL
{{N}<
HENR
♡
5 0.7561 329 0.1782 841
PENTAX
SANKAY
=423le# {7 4537*cY
11
ܢܚܙܐܠܫܢܚܝh
13
RATIO of INTEREST:
ANNUALLY, 0.0575.
2685 CARLA GALA BALAL
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9456 265 1.0000 000 1.0575 000
17
18
0.8942 094 0.4860 268 | 0.5435 268
8 0.6393 770 0.1019 463 0.159± 463
19
20
0.8455 881
21
22
23
24
25
0.7996 106 0.2294 412 | 0.2869 412
0.3148 807 | 0.3723 807
0.7150 192 0.1442 680
0.6761 412
0.6046 118
10 0.5717 369 | 0.0767 633 0.1342 633
....
0.2357 841
0.1200 465 | 0.1775 465
0.2017 680
...
0.0879 267 0.1454 267
***
12 0.5112 526 0.0601 477 0.1176 477
··
0.5406 496 | 0.0676 768 | 0.1251 768
14 0.4571 669 0.0484 257 0.1059 257
0.4834 539 | 0.0538 163 0.1113 163
0.4323 091 | 0.0437 875 | 0.1012 875
0.4088 029 0.0397 603 | 0.0972 603
0.3865 749 0.0362 360 0.0937 360
0.3655 554 0.0331 305 | 0.0906 305
0.3456 789 0.0303 773
0.3268 831 0.0279 235
0.0878 773
0.0854 235
0.3091 093 | 0.0257 259
0.0257 259
0.0237 493
0.0832 259
0.0812 493
0.2764 085 0.0219 647 | 0.0794 647
0.2923 019
0.2613 792
0.0203 478 | 0.0778 478
0.2471 671 0.0188 782 | 0.0763 782
RATIO of INTEREST:
HALF-YEARLY, 0.02875.
Discounted Annuity for Ratio of
Principal. Cumulation Aunuity.
..
0.9720 535
0.9448 879
0.9184 816
0.8928 132
0.8200 325
0.7971 154
0.4351 517
0.8678 621 0.3776 517
0.8136 084 0.3101 669 0.3676 669
0.7748 388
0.7531 848
0.6917 863
0.6724 533
0.6536 605
0.6353 930
2.0000 000 2.0575 000
0.9858 287
1.0433 287
0.7321 359 0.1571 610
0.7116 751 0.1419 279
0.5478 621 0.7053 621
0.4789 467 | 0.5364 467
••
0.2620 021
0.2259 123
0.4923 272
0.4785 683
0.4651 940
0.4521 934
0.1978 726 0.2553 726
0.1751 678 | 0.2329 678
0.4395 562
0.4272 721
0.1571 610 0.2146 610
0.6176 360 0.0928 803
0.6003 752
0.0863 850
0.1419 279 0.1994 279
0.1290 589
0.1180 475
0.1085 221
0.1002 040
0.5514 335 0.0706 861
0.5360 229 0.0664 285
0.5210 429 0.0625 525
0.5064 816 0.0590 103
0.3195 021
0.2834 123
0.5835 967 0.0805 873 0.1380 873
0.5672 873 0.0753 826
0.0753 826 | 0.1328 826
...
0.1865 589
| 0.1755 475
...
0.1660 221
0.1577 040
...
0.4153 313
0.4037 243
0.3814 742
0.3604 504
0.3405 852
0.3218 149 | 0.0272 851
...
0.1503 803
0.1438 850
0.2423 860 0.0183.962
0.1281 861
0.1239 285
0.0557 619 | 0.1132 619
0.0527 733 0.1102 733
....
0.1200 525
0.1165 103
...
0.0500 156 | 0.1075 156
0.0174 611
0.1049 641
0.0450 973 | 0.1025 973
0.0128 967 | 0.1003 967
0.0408 463 0.0983 463
0.0964 319
0.0389 319
0.0354 630 | 0.0929 630
0.0324 070 0.0899 070
0.0296 985
0.0296 985 | 0.0871 985
0.0817 851
0.3040 790 0.0251 243 0.0826 243
0.2873 205 | 0.0231 814 0.0806 814
0.2714 858 0.0214 278 0.0789 278
0.0198 394 | 0.0773 394
0.2565 236
0.0183.962 | 0.0758 962
RATIO of INTEREST:
QUARTERLY, 0.014375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9858 287
0.9718 583
0.9580 858
0.9445 083
0.9311 235
0.9179 282
0.9049 201
0.8920 963
0.8794 541
0.8669 911
0.8547 047
0.8425 924
0.8306 518
0.8188 804
0.8072 758
0.7958 357
0.7410 213
0.7305 201
0.7201 677
0.7099 620
0.6999 009
0.6899 824
0.6802 045
0.6705 651
0.6610 623
0.6516 942
0.6424 589
0.6333 544
0.7845 577 | 0.2093 928
0.7734 395 | 0.1962 953
0.7624 788 0.1845 847
0.7516 735 | 0.1740 500
0.6243 790 |
0.6155 307
0.6068 079
0.5982 086
0.5897 312
0.5813 740
0.5731 352
0.5650 131
0.5260 970
0.5186 415
0.5112 917
0.5040 461
0.4969 031
0.4898 613
0.4829 193
0.4760 757
4.0000 000
1.9857 276
1.3143 409
0.9786 939
0.4432 854
0.4370 034
0.4308 105
0.4247 054
0.4186 868
0.4127 534
0.4069 042
0.4011 378
0.8348 283
0.7773 283
0.6431 072 | 0.7006 072
0.5172 516 | 0.6047 546
0.4753 821 0.5328 821
0.4194 966
0.3748 018
0.3382 458
0.3077 938
0.2820 372
0.2599 698
0.2408 538
0.2241 359
0.3015 247
0.2847 926
0.2689 890
0.2540 624
0.2399 641
0.5570 061 0.0722 987
0.5491 126 0.0700 263
0.5413 310 | 0.0678 627
0.5336 596 | 0.0658 005
0.1341 034
0.1279 734
0.1223 024
0.1170 413
0.1121 477
0.1075 848
0.1033 207
0.0993 272
0.1615 260 | 0.2220 260
0.1558 740
0.2133 740
0.1479 820 0.2054 820
0.1407 499 0.1982 +99
0.0955 798 |
0.0920 568
0.0887 389
0.0856 091
0.0826 520
0.0798 541
0.0772 031
0.0747 437
0.4693 291 0.0508 534
0.4626 782 | 0.0495 122
0.4561 214 0.0482 223
0.4496 576 0.0469 804
0.0567 921
0.0552 144
0.0537 012
0.0522 487
0.0457 845
0.0446 320
0.0435 209
0.0424 488
4.0575 000
2.0432 276
1.3718 409
1.0361 939
0.0394 489
0.0385 154
0.3788 781 0.0350 744
0.3578 535 0.0320 434
0.3379 957 0.0293 574
0.3192 398 | 0.0269 611
0.4769 966
0.4323 018
0.3957 458
0.3652 938
0.3395 372
0.3174 698
0.2983 538
| 0.2816 359
0.0248 222
0.0228 963
0.0211 582
0.0195 842
0.0181 548
0.2668 928
0.2537 953
0.2420 847
0.2315 500
0.1213 328
0.0638 328
0.0619 536
0.0601 571
0.1191 536
0.1176 571
0.0584 382 | 0.1159 382
0.1915 034
0.1854 734
0.1798 024
0.1745 413
0.1696 477
0.1650 848
0.1608 207
0.1568 272
0.1530 798
0.1495 568
0.1462 389
0.1431 091
0.1401 520
0.1373 541
0.1347 031
0.1322 437
0.1297 987
0.1275 263
0.1253 627
0.1233 005
0.1142 921
0.1127 144
0.1112 012
0.1097 487
0.0414 140
0.0989 140
0.0404 146 0.0979 146
0.0969 489
0.0960 154
0.1083 534
0.1070 122
0.1057 223
0.1044 804
0.1032 845
0.1021 320
0.1010 209
0.0999 488
0.0925 744
0.0895 434
0.0868 574
0.0844 644
0.0823 222
0.0803 963
0.0786 582
0.0770 842
0.0756 543
Page 47.
RATE OF INTEREST: 5 per Cent.
CORRESPONDING VALUE of
per
Ann.-

Years.
ON 26 M****
26
4.2784 827
27 4.5244 951
28
36
37
38
39
40
RATIO of INTEREST:
ANNUALLY, 0.0575.
#9933 95393 5883K 85888 786J5 85882 N***R OF22% ***** 86888 *88*5 85838
Improved Cumulated Value of
Principal. Annuity.
Annuity.
29 5.0597 715
30 5.3507 084
95
96
57.017 090
61.295 573
4.7846 539 | 65.820 068
70.604 722
75.664 493
31 5.6583 741
32
33
5.9837 306
6.3277 951
6.6916 433
35 7.0764 128
34
7.4833 066 112.75 316
7.9135 967 120.23 646
8.3686 285 128.15 006
8.8498 247 136.51 869
9.3586 896145.36 851
9.8968 142
10.465 881
11.067 669
11.704 060
12.377 044
13.088 724
13.841 325
14.637 201
15.478 841
16.368 874
17.310 084
18.305 414
19.357 975
20.471 059
21.648 145
30.276 310
32.017 197
33.858 186
35.805 032
37.863 821
81.015 202
86.673 576
92.657 307
98.985 102
105.67 674
52.954 967
55.999 877
59.219 870
62.625 013
66.225 951
154.72 720
164.62 402
175.08 990
186.15 757
197.86 163
380.74 631
22.892 913
24.209 256 | 403.63 923
25.601 288
27.073 362
28.630 080
427.84 848
453.44 977
480.52 313
210.23 867
223.32 740
237.16 872
251.80 592
267.28 476
283.65 364
300.96 372
319.26 913
338.62 711
359.09 817
40.040 991 | 678.97 376
42.343 348 719.01 475
44.778 091761.35 810
47.352 831 806.13 619
50.075 619
853.48 902
161.99 970
171.31 469
181.16 528
191.58 228
202.59 827
509.15 321
539.42 952
571.44 672
605.30 491
641.10 994
903.56 464
956.51 961
1012.5 195
1071.7 394
1134.3 644
1200.5 903
70.033 943
74.060 895 1270.6 243
78.319 396 1344.6 852
82.822 762 1423.0 046
87.585 070
1505.8 273
92.621 212 | 1593.4 124
97.946 932 | 1686.0 336
103.57 888 1783.9 805
109.53 467 | 1887.5 594
115.83 291 | 1997.0 941
122.49 330 2112.9 270
129.53 667 2235.4 203
136.98 502 2364.9 570
144.86 166 2501.9 420
153.19 121 2616.8 036
13.326 47
13.547 49
13.756 49
13.954 13
14.141 02
214.24 767
3708.6 551
226.56 691 | 3922.9 027
4149.4 696
4389.0 641
239.59 450
253.37 119
100 267.94 003 | 4642.4 353
14.317 75
14.484 87
14.642 91
14.792 35
14.933 66
15.067 29
15.193 66
15.313 15
15.426 15
15.533 00
15.634 04
15.729 59
15.819 94
15.905 38
15.986 18
16.062 58
16.134 83
16.203 15
16.267 75
16.328 84
16.386 61
10.441 24
16.492 90
19.541 75
16.587 94
16.631 62
16.672 93
16.711 99
16.748 93
16.783 86
16.816 88
16.848 12
16.877 65
16.905 58
16.931 99
16.956 97
16.980 58
17.002 92
17.024 03
17.044 00
17.062 89
17.080 74
17.097 63
17.113 60
17.128 70
17.142 98
17.156 48
17.169 25
17.181 32
17.192 74
17.203 54
17.213 75
17.223 40
17.232 53
17.241 16
17.249 33
17.257 05
17.264 35
17.271 25
17.277 78
2799.9 948 17.283 95
2961.9 946 17.289 79
3133.3 092 17.295 31
3314.4 745 17.300 53
3506.0 568 17.305 46
17.310 13
17.314 54
17.318 72
17.322 66
17.326 40
Value of Perpetual Ann. 17.391 30
4.3662 853
4.6209 557
4.8904 801
5.1757 250
5.4776 073
RATIO of INTEREST:
HALF-YEARLY, 0.02875.
Improved Cumulated
Principal. Annuity.
5.7970 973
6.1352 221
6.4930 685
6.8717 868
7.2725 945
7.6967 800
8.1457 067
8.6208 178
9.1236 404
9.6557 910
10.218 980
10.815 018
11.445 821
12.113 416
12.819 950
18.013 766
19.064 447
20.176 411
21.353 231
22.598 692
23.916 796
25.311 781
26.788 130
28.350 589
30.001 182
55.975 560
59.240 422
62.695 712
66.352 538
70.222 653
13.567 694218.56 859
14.359 051
15.196 565
16.082 928
17.020 990
232.33 132
246.89 678
262.31 180
278.62 592
74.318 499
78.653 242
83.240 815
88.095 965
93.234 300
58.544 092
62.973 142
67.660 524
72.621 305
77.871 431
83.427 779
89.308 210
95.531 626
102.11 803
109.08 860
116.46 574
124.27 316
132.53 596
141.28 070
150.53 550
160.33 009
170.69 597
181.66 645
193.27 681
205.56 435
31.754 222
33.606 337
534.85 604
567.06 673
35.566 479 601.15 616
37.640 950
39.836 417
637.23 391
675.41 595
273.74 737
289.71 411
295.89 158
314.16 430
333.50 280
353.96 924
375.62 943
42.159 938
715.82 501
44.618 982 | 758.59 100
47.221 454
49.975 719
52.890 631
803.85 138
851.75 164
902.44 576
398.55 298
422.81 358
448.48 921
475.00 242
504.42 055
956.09 670
1012.8 769
1072.9 689
1136.5 659
1203.8 722
1275.1 043
1350.4 912
1430.2 750
1514.7 124
1604.0 748
98.672 336 1698.6 493
104.42 755 | 1798.7 401
110:51 845 1904.6 688
116.96 462 | 2016.7 759
123.78 676 2135.4 219
131.00 682 2260.9 881
138.64 799 | 2393.8 782
146.73 485 2534.5 192
155.29 339 2683.3 634
164.35 112 2840.8 891
173.93 716 3007.6 028
184.08 232 3184.0 403
194.81 921 3370.7 688
206.18 234 3568.3 885
218.20 825 3777.5 347
230.93 558 3998.8 797
244.40 526 | 4233.1 350.
258.66 058 4481.0 536
4743.4 324
5021.1 149
Value of
Annuity.
13.408 22
13.627 73
13.835 15
14.031 14
14.216 32
14.391 30
14.556 64
14.712 86
14.860 48
14.999 96
15.131 75
15.256 28
15.373 94
15.485 12
15.590 18
15.689 44
15.783 23
15.871 86
15.955 60
16.034 72
16.109 48
16.180 13
16.246 89
16.309 95
16.369 55
16.425 86
16.479 07
16.529 34
15.576 85
16.621 73
16.664 14
16.704 22
16.742 09
16.777 87
16.811 68
16.843 62
16.873 80
16.902 32
16.929 27
16.954 74
16.978 80
17.001 53
17.023 01
17.043 31
17.062 49
17.080 61
17.097 73
17.113 91
17.129 20
17.143 64
17.157 29
17.170 19
17.182 38
17.193 89
17.204 77
17.215 05
17.224 77
17.233 94
17.242 62
17.250 81
17.258 55
17.265 87
17.272 78
17.279 31
17.285 49
17.291 32
17.296 83
17.302 04
17 306 96
17.311 61
17.316 00
17.320 15
17.324 07
17.327 77
17.331 27
17.391 30
RATIO of INTEREST:
QUARTERLY, 0.014375.
Improved Cumulated Value of
Principal. Annuity.
Annuity.
4.4121 256
4.6713 458
4.9457 956
5.2363 698
5.5440 158
5.8697 365
6.2145 939
6.5797 123
6.9662 821
7.3755 635
7.8088 909
8.2676 770
8.7534 176
9.2676 964
9.8121 900
10.388 673
10.999 026
11.645 239
12.329 417
13.053 792
13.820 725
14.632 717
15.492 416
16.402 622
17.366 305
18.386 607
19.456 852
20.610 564
21.821 471
23.103 521
24.460 894
25.898 015
27.419 569
29.030 517
30.736 111
32.541 912
34.453 807
36.478 029
38.621 178
40.890 240
43.292 614
45.836 132
48.529 086
51.380 255
54.398 936
57.594 970
60.978 776
64.561 387
68.354 483
72.370 430
76.622 320
81 124 017
85.890 197
90.936 308
96.279 072
101.93 564
107.92 454
114.26 529
120.97 858
128.08 629
135.61 158
143.57 900
152.01 452
160.94 565
170.40 149
180.41 288
191.01 245
202.23 477
214.11 642
226.69 614
240.01 494
254.11 624
269.04 602
284.85 295
301.58 856
59.341 315
63.849 492
68.622 532
73.675 997
79.026 362
84.691 070
90 688 590
97.038 475
103.76 143
110.87 936
118.41 549
126.39 438
134.84 205
143.78 602
153.25 548
163.28 128
173.89 611
185.13 459
197.03 334
209.63 117
222.96 914
237.09 074
252.04 201
267.87 169
281.63 140
302.37 577
321.16 265
341.05 329
362.11 254
384.40 907
408.01 555
433.00 395
459.47 076
487.48 725
517.14 976
548.55 500
581.80 534
617.00 920
654.28 136
693.74 331
735.52 373
779.75 882
826.59 280
876.17 836
928.67 715
984.26 035
1043.1 092
1105.4 154
1171.3 823
1241.2 249
1315.1 708
1393.4 612
1476.3 512
1564.1 113
1657.0 273
1755.4 024
1859.5 572
1969.8 312
2086.5 840
2210.1 963
2341.0 710
2479.6 348
2626.3 395
2781.6 634
2946.1 128
3120.2 239
3304.5 644
3499.7 351
3706.3 725
3925.1 503
4156.7 815
4402.0 216
4661.6 699
4936.5 730
5227.6 272
13.449 60
13.668 33
13 874 92
14.070 05
14.254 35
14.428 43
14.592 84
14.748 13
14.894 81
15.033 34
15.164 19
15.287 78
15.404 50
15.514 75
15.618 89
15.717 24
15.810 14
15.897 88
15.980 75
16.059 02
16.132 95
16.202 78
16.268 73
16.331 02
16.389 86
16.445 44
16.497 92
16.547 50
16.594 32
16.638 55
16.680 32
16.719 77
16.757 04
16.792 23
16.825 48
16.856 88
16.886 53
16.914 54
16.941 00
16.965 99
16.989 59
17.011 88
17.032 94
17.052 82
17.071 60
17.089 34
17.106 10
17.121 93
17.136 88
17.151 00
17.164 32
17.176 92
17.188 82
17.200 06
17.210 67
17.220 69
17.230 16
17.239 10
17.247 55
17.255 53
17.263 06
17.270 18
17.276 90
17.283 25
17.289 25
17.294 91
17.300 26
17.305 31
17.310 08
17.314 59
17.318 84
17.322 87
17.326 66
17.330 25
17.333 64
17.391 30
Page 48.
PERPETUAL STOCK: 3 p. C. 52,1739; 3½ p. C. 60,8696; 4 p. C. 69,5652; 5 p. C. 86,9565.

Years.
OMODO MOMIE OMA *** 9**98 78886 95898 688JU 85882 NOMER CE
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
46
47
48
49
50
52
56
57
62
63
64
66
69
71
73
74
75
76
77
78
81
82
***8 85888 ****a 8588g
RATIO of INTEREST:
ANNUALLY, 0.0575.
96
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2337 277
0.2210 191
0.2090 015
0.1976 374
0.1868 911
0.1767 292
0.1671 198
0.1580 329
0.1404 401
0.1413 145
0.1336 308
0.1263 648
0.1194 939
0.1129 966
0.1068 526
0.0577 698
0.0546 286
0.0516 583
0 0488 495
0.0461 933
0.0436 816
0.0413 065
0.0390 605
0.0369 367
0.0349 283
0.0903 533
0.1010 426 0.0064 630
0.0955 486 | 0.0060 744
0.0057 114
0.0053 718
0.0050 540
0.0854 404
0.0807 947
0.0764 016
0.0722 474
0.0683 191
0.0646 043
0.0610 916
0.0330 291
0.0312 332
0.0295 350
0.0279 290
0.0264 104
0.0249 744
0.0236 105
0.0223 323
0.0211 181
0.0199 698
0.0142 788
0.0135 024
0.0127 682
0.0120 740
0.0114 175
0.0107 967
0.0102 096
0.0096 545
0.0091 295
0.0086 331
0.0081 637
0.0077 198
0.0073 001
0.0069 031
0.0065 278
0.0175 386
0.0163 144
0.0151 929
0.0141 634
0.0132 162
0.0061 729
0.0058 372
0.0055 198
0.0052 197
0.0049 359
0.0123 434
0.0115 375
0.0107 925
0 0101 025
0.0094 628 |
0.0046 675
0.0044 137
0.0041 737
99 0.0039 468
100 0.0037 322
0.0035 254
0.0033 227
0.0031 322
0.0029 531
0.0027 848
0.0026 264
0.0024 775
0.0023 373
0.0022 053
0.0020 811
0.0047 565 | 0.0622 565
0.0044 777 0.0619 777
0.0042 164 0.0617 164
0.0039 713 0.0614 713
0.0612 413
0.0037 413
0.0088 689 0.0663 689 0.1299 245
0.0083 169 | 0.0658 169 0.1227 641
0.0078 034
0.0653 034
0.1159 983
0.0073 250 0.0648 2500.1096 054
0.0068 791 0.0643 791
0.1035 648
0.0017 499
0.0016 521
0.0015 598
0.0750 386
0.0738 144
0.0726 929
0.0716 634
0.0707 162
0.0188 840
0.0010 455
0.0178 572
0.0168 862 0.0009 876
0.0009 331
0.0159 681
0.0008 816
0.0150 998
0.0698 431
0.0690 375
0.0682 925
0.0676 025
0.0669 628
0.0019 640
0.0594 640
0.0018 538 0.0593 538
0.0592 499
0.0591 521
0.0590 598
0.0008 329
0.0007 870
0.0007 437
0.0007 027
0.0006 641
0.0639 630
0.0635 744
0.0632 114
0.0628 718
0.0625 540
0.0014 728 | 0.0589 728
0.0013 908
0.0013 134
0.0012 405
0.0011 717
0.0588 908
0.0588 134
0.0587 405
0.0586 717
0.0005 298
0.0005 007
0.0610 254
0.0608 227
0.0606 322
0.0604 531
0.0602 848
0.0004 733
0.0004 473
0.0004 228
0.0003 997
0.0003 778
0 0601 264
0.0599 775
0.0598 373
0.0597 053
0.0595 811
0.0011 067 | 0.0586 067
0.0585 455
0.0584 876
0.0584 331
0.0583 816
0.0002 696
0.0002 549
0.0006 276 | 0.0581 276
0.0005 931
0.0005 605
0.0580 931
0.0580 605
0.0580 298
0.0580 007
0.0583 329
0.0582 870
0.0582 437
0.0582 027
0.0581 641
0.0579 733
0.0579 473
0.0579 228
0.0578 997
0.0578 778
0.0003 571 0.0578 571
0.0003 376 | 0.0578 376
0.0003 192 0.0578 192
0.0003 017 0.0578 017
0.0002 852 0.0577 852
0.0577 696
0.0577 549
0.0002 410 0.0577 410
0.0002 278 | 0.0577 278
0.0002 154 | 0.0577 154
Ratio of Perpetual Ann. 0.0575 000
RATIO of INTEREST:
HALF-YEARLY, 0.02875.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2290 £76
0.2164 055
0.2044 789
0.1932 096
0.1825 615
0.1725 001
0.1629 933
0.1540 104
0.1455 226
0.1375 025
0.0978 571
0.0924 640
0.0873 681
0.0825 531
0.0780 034
0.0737 045
0.0696 425
0.0658 013
0.0621 777
0.0587 510
0.0314 919
0.0297 563
0.0281 164
0.0237 192
0.0224 120
0.0211 768
0.0200 097
0.0189 069
0.0170 811
0.0158 798
0.0147 797
0.0137 701
0.0128 417
0.0555 131
0.0524 537
0.0495 628 | 0.0029 985
0.0033 796
0.0031 830
0.0468 313
0.0442 503
0.0028 251
0.0026 622
0.0178 649
0.0168 804
0.0159 501
0.0150 710
0.0142 404
0.0119 864
0.0111 972
0.0104 677
0.0097 926
0.0091 669
0.0418 116
0.0025 091
0 0395 073 | 0.0023 651
0.0373 300
0.0022 297
0.0352 726 0.0021 023
0.0333 287 0.0019 825
0.0085 862
0.0080 468
0.0075 451
0.0070 781
0.0066 430
0.0018 697
0.0017 635
0.0016 635
0.0265 6680.0015 693
0.0015 693
0.0251 027 0.0014 806 |
0.0101 346
0.0095 760
0.0090 483
0.0085 496
0.0080 784
0.0062 371
0.0058 584
0.0055 046
0.0051 739
0.0048 647
0.0045 752
0.0043 043
0.0040 503
0.0038 123
0.0035 890
0.0057 492
0.0054 324
00051 330 |
0.0048 501
0.0045 828
0.0013 970
0.0013 182
0.0012 440
0.0011 741
0.0011 081
0.0134 556 | 0.0007 842
0.0127 140
0.0120 133
0.0113 513
0.0107 257
0.0007 405
0.0006 992
0.0006 602
0.0006 234
0.0010 459
0.0009 873
0.0009 320
0.0008 798
0.0008 307
0.0745 811
0.0733 798
0.0722 797
0.0712 701
0.0703 417
0 0694 86+
0.0686 972
0.0679 677
0.0672 926
0.0666 669
0.0650 862 ||
0.0655 468
0.0650 451
0.0645 781
0.0641 430
0.0637 371
0.0633 584
0.0630 046
0.0626 739
0.0623 647
0
0.0620 752
0.0618 043
0.0615 503
0.0613 123
0.0610 890
0.0608 796
0.0606 830
0.0604 985
0.0603 251
0.0601 622
0.0600 091
0.0598 651
0.0597 297
0.0596 023
0.0594 825
0.0593 697
0.0592 635
0.0591 635
0.0590 693
0.0589 806
0.0588 970
0.0588 182
0.0587 440
0.0586 741
0.0586 081
0.0585 459
0.0584 873
0.0584 320
0.0583 798
0.0583 307
0.058% 842
0.0582 405
0.0581 992
0.0581 602
0.0581 234
0.0005 887 0.0580 887
0.0005 559 0.0580 559
0.0005 250 0.0580 250
0.0004 958 0.0579 958
0.0004 683 | 0.0579 683
0.0076 332 0.0004 423 0.0579 423
0 0072 125 0.0004 177 0.0579 177
0.0068 150 0.0003 946 | 0.0578 946
0.0064 394 0.0003 727
0.0c60 845 0.0003 520
0 0578 727
0.0578 520
0.0003 325 0.0578 325
0.0003 141 0.0578 141
0.0002 967 0.0577 967
0.0002 802 0.0577 802
0.0002 647 0.0577 647
0.0002 501 0.0577 501
0.0002 362 | 0.0577 362
0.0043 302
0.0040 916
0.0038 661 | 0.0002 232 0.0577 232
0.0036 530 0.0002 108 0.0577 108
0.0034 517 0.0001 992 | 0.0576 992
-0.0575 000
RATIO of INTEREST:
QUARTERLY, 0.014375.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2266 481
0.2140 711
0.2021 919
0.1909 720
0.1803 746
0.1703 654 | 0.0118 076
0.0118 076
0.1609 116 0.0110 267
0.1519 823
0.1435 486
0.1355 828 | 0.0090 188
0.0103 052
0.0096 375
0.1280 592
0.1209 530
0.1142 411
0.1079 017
0.1019 140
0.0962 587
0.0909 171
0.0858 720
0.0811 068
0.0766 061
0.0543 874
0.0513 694
0.0485 188
0.0458 264
0.0432 834
0.0408 816
0.0386 130
0.0364 703
0.0344 465
0.0325 350
0.0307 296
0.0290 244
0.0274 138
0.0258 925
0.0244 557
0.0723 551
0.0683 400
0.0645 477 | 0.0039 676
0.0609 659
0.0575 828
0.0037 331
0.0035 133
0.0168 517
0.0156 618
0.0145 725
0.0135 729
0.0126 540
0.0130 510
0.0123 268
0.0116 428
0.0109 967
0.0103 865
0.0084 448
0.0079 117
0.0074 161
0.0069 548
0.0065 251
0.0073 740
0.0069 648
0.0065 783
0.0062 133
0.0058 685
0.0061 244
0.0057 506
0.0054 015
0.0050 753
0.0047 703
0.0055 428
0.0052 353
0.0049 447
0.0046 704
0.0044 112
0.0044 849
0.0042 178
0.0041 664
0.0039 352
0.0037 168
0.0035 106
0.0033 158
0.0230 986 0.0013 596
|
0.0218 169 0.0012 824
0.0206 062 | 0.0012 098
0.0194 627
0.0183 827
0.0011 413
0.0010 768
0.0033 071
0.0031 137
0.0029 321
0.0027 616
0.0026 014
0.0173 626
0.0163 991
0.0154 891
0.0010 160
0.0009 587
0.0009 016
0.0146 296 | 0.0008 537
0.0138 178
0.0008 057
0.0024 509
0.0023 094
0.0021 764
0.0020 513
0.0019 337
0.0018 230
0.0017 188
0 0016 207
0.0015 284
0.0014 415
0.0098 101 0.0005 697
0.0005 378
0.0092 657
0.0087 516 0.0005 077
0.0082 659 0.0004 793
0.0078 072 | 0.0004 524
0.0007 604
0.0007 176
0.0006 773
0.0006 393
0.0006 035
0.0003 205
0.0003 026
0.0002 857
0.0002 698
0.0002 548
0.0002 406
0.0002 272
0.0002 145
0.0002 026
0.0001 913
0.0743 517
0.0731 618
0.0720 725
0.0710 729
0.0701 540
-
0.0693 076
0.0685 267
0.0678 052
0.0671 375
0.0665 188
0.0659 448
0.0654 117
0.0649 161
0.0644 548
0.0640 251
0.0636 244
0.0632 506
0.0629 015
0.0625 753
0.0622 703
0.0619 849
0.0617 178
0.0614 676
0.0612 331
0.0610 133
0.0608 071
0.0606 137
0.0604 321
0.0602 616
0.0601 014
0.0599 509
0.0598 094
0.0596 76+
0.0595 513
0.0594 337
0.0593 230
0.0592 188
0.0591 207
0.0590 284
0.0589 415
0.0588 596
0.0587 824
0.0587 098
0.0586 413
0.0585 768
0.0585 160
0.0584 587
0.0584 046
0.0583 537
0.0583 057
0.0001 272 0.0579 272
0.0004 033 0.0579 033
0.0003 808 0.0578 808
0.0003 595 0.0578 595
0.0003 394
0.0578 394
0.0582 604.
0.0582 176
0.0581 773
0.0581 393
0.0581 035
0.0580 697
0.0580 378
0.0580 077
0.0579 793
0.0579 524
0.0578 205
0.0578 026
0.0577 857
0.0577 698
0.0577 548
0.0577 406
0.0577 272
0.0577 145
0.0577 026
0.0576 913
0.0575 000
Page 49.
RATE OF INTEREST: 6 per Cent.
OF
per
-CORRESPONDING VALUE of
Ann.-

Years.
gand
={{{6}}}]]
D16:13HAH
Q
[C¢85%-**$==
∞
POSLAL
→
*V*{e*
=**=***}**
∞
*f**=420]#
PRAK
CANAL SPANLAL HA SA
13
~46831-****
& ANNA BEZ BASAL GAPA
RATIO of INTEREST:
ANNUALLY, 0.06.
19
20
Improved Cumulated Value of
Principal. Annuity. Annuity.
21
22
1.0600 000 1.0000 000
23
24
1.1236 000 | 2.0600 000
1.1910 160 3.1836 000
1.2624 770 4.3746 160
1.3382 256 | 5.6370 930
1.4185 191 6.9753 185
1.5036 303 8.3938 376
1.5938 481 9.8974 679
12 2.0121 965 16.869 941
1.6894 790 11.491 316
1.7908 477
1.8982 986 14.971 643
2.1329 283
•
2.2609 040
13.180 795
15 2.3965 582 23.275 970
2.5403 517 25.672 528
17
2.6927 728
28.212 880
18 2.8543 392
30.905 653
3.0255 995 33.759 992
3.2071 355 36.785 591
18.882 138
25 4.2918 707
21.015 066
3.3995 636 | 39.992 727
3.6035 374 43.392 290
3.8197 497 46.995 828
4.0489 346 50.815 577
54.864 512
0.9433 96
1.8333 93
2.6730 12
3.4651 06
4.2123 64
4.9173 24
5.5823 82
6.2097 94
6.8016 91
...
7.3600 86
7.8868 74
8.3838 43
8.8526 82
9.2949 83
9.7122 48
11.764 08
12.041 58
12.303 38
12.550 36
12.783 36
RATIO of INTEREST:
HALF-YEARLY, 0.03.
1.0300 000
1.0609 000
Improved Cumulated Value of
Principal. Annuity. Annuity.
0.5000 000
1.0150 000
...
1.0927 270 | 1.5454 500
1.1255 088 2.0918 135
1.1592 741 | 2.6545 679
1.1940 523 3.2342 049
1.2298 739 3.8312 311
1.2667 701 4.4161 680
1.3047 732 | 5.0795 530
1.3439 164 5.7319 397
1.3842 339 6.4038 983
1.4257 6097.0960 148
1.1685 337 7.8088 950
1.5125 897 | 8.5431 620
1.5579 674 | 9.2994 570
1.6047 064 | 10.078 441
1.7535 061
1.8061 112
....
1.6528 476 | 10.880 794
1.7024 331 11.707 218
12.558 434
13.435 187
1.8602 946 | 14.338 243
1.9161 034
15.268 390
··
1.9735 865 16.226 442
2.0327 941 17.213 235
2.0937 779 | 18.229 632
2.1565 913 | 19.276 521
2.2212 890 20.354 817
2.2879 277 | 21.465 461
2.3565 655 | 22.609 425
2.4272 625 | 23.787 708
....
2.5000 803 25.001 339
2.5750 827 26.251 379
10.105 89
10.477 26 2.7319 053 | 28.865 088
10.827 60 2.8982 783
11.158 12 3.0747 835
11.469 92
31.637 972
34.579 725
37.700 630
3.2620 378
3.4606 959 41.011 598
3.6714 523 44.524 204
3.8950 437 48.250 728
4.1322 519 | 52.204 198
4.3839 060 56.398 434
0.4854 37
0.9567 35
1.4143 06
1.8585 49
2.2898 54
2.7085 96
3.1151 42
3.5098 46
3.8930 55
4.2651 01
4.6263 12
4.9770 02
··
5.3174 78
5.6480 37
5.9689 68
6.2805 51
6.5830 59
6.8767 57
7.1619 00
7.4387 37
7.7075 12
7.9684 58
8.2218 04
8.4677 71
8.7065 74
8.9384 21
9.1635 16
9.3820 54
9.5942 27
9.8002 21
10.000 21
10.194 38
10.565 92
10.916 13
11.246 23
11.557 39
11.850 68
12.127 14
12.387 72
12.633 35
12.864 88
RATIO of INTEREST:
QUARTERLY, 0.015.
Improved Cumulated Value of
Principal. Annuity. Annuity,
0.2500 000
1.0150 000
1.0302 250
1.0456 784
0.5037 500
0.7613 063
1.0613 636 | 1.0227 258
1.0772 840
1.0934 433
1.1098 449
1.2880 667
1.5573 877
1.8307 484
1.1264 926 | 2.1082 098
1.1433 900
1.1605 408
1.1779 489
1.1956 182
1.2135 524 | 3.5592 074
1.2317 557
3.8625 955
1.2502 321
1.2689 855
4.1705 345
4.4830 924
1.2880 203
1.3073 406
1.3269 507
1.3468 550
1.3670 578
1.3875 637
1.4083 772
1.4295 028
1.6344 792
1.6589 964
1.6838 813
1.7091 395
2.3898 329
2.6756 804
2.9658 156
3.2603 028
1.4509 454
7.5157 560
1.4727 095 | 7.8784 920
1.4948 002
1.5172 222
8.2466 695
8.6203 695
1.7347 767
1.7607 983
1.7872 101
1.8140 184
8.9996 750
1.5399 805
1.5630 802
9.3846 705
1.5865 264 | 9.7754 405
1.6103 243
10.172 072
1.8412 287
1.8688 471
1.8968 798
1.9253 330
4.8003 388
5.1223 437
5.4491 790
5.7809 168
6.1176 305
6.4593 950
6.8062 860
7.1583 800
2.0741 305
2.1052 424
2.1368 211
2.1688 734
10.574 653
10.983 273
11.398 022
11.818 992
12.246 278
12.679 971
13.120 168
13.566 973
14 020 478
14.480 785
14.947 997
15.422 217
1.9542 130
15.903 550
1.9835 262 | 16.392 103
2.0132 791
2.0434 783
16.887 985
17.391 305
17.902 174
18.420 707
18.947 018
19.481 223
2.2014 065 20.023 441
2.2344 276 20.573 793
2.2579 440 21.132 400
2.3019 631 | £1.699 385
2.3364 926 | 22.274 877
2.3715 400
2.4071 131
2.4432 198
22.859 000
23.451 885
24.053 663
2.4798 681 24.664 468
2 5170 661 25.284 435
2.5548 221 25.913 701
5.5931 444 || 26.552 407
2.7522 690 29.204 483
2 9211 580 32.019 300
3.1004 106 35.006 843
3.2906 628 | 38.177 713
3.4925 895 41 543 159
3.7069 072 | 45.115 120
3.9343 762
4.1758 034
48.906 270
52.930 057
4.4320 45657.200 761
0.2463 05
0.4889 71
0.7280 50
0.9635 96
1.1956 61
1.4212 97
1.6495 53
1.8714 81
2.0901 29
2.3055 46
2.5177 79
2.7268 76
2.9328 83
3.1358 45
3.3358 08
3.5328 16
3.7269 12
3.9181 40
4.1065 42
4.2921 60
4.4750 34
4.6552 06
4.8327 15
5.0076 01
5.1799 03
5.3495 58
5.5169 04
5.6816 79
5.8410 19
6.0039 59
6.1615 36
6.3157 85
6.4697 38
6.6201 32
6.7688 98
6.9151 71
7.0592 82
7.2012 63
7.3411 46
7.4789 61
7.6147 40
7.7485 12
7.8803 08
8.0101 55
8.1380 84
8.2641 22
8.3882 08
8.5106 38
8.6311 71
8.7499 22
8.8669 18
8.9821 85
9.0957 49
9.2076 35
9.3178 67
9.4264 69
9.5334 67
9.6388 84
9.7427 43
9.8450 67
9.9458 79
10.045 20
10.143 06
10.239 46
10.611 06
10.961 17
11.291 03
11.601 83
11.894 66
12.170 56
12.430 50
12.675 42
12.906 18
Page 50.
KAN KEHIDU
PERPETUAL STOCK: 3 p. C. 50; 3ž
31 p. C. 58,3333; 4p. C. 66,6667; 5p. C. 83,3333.

Years.
pnja 20{d
Q
མ་ནས་
03
====&N:C{~*
flag
>{<==|2402}+
-2
Con
6
*
fagleg
=*={2}{20}<
HENNE
*****
$$\{} d
11
DABULAL
12
ZAKLAL CAPI
TEINDELO
15
16
17
18
19
20
21
22
23
£4
25
RATIO of INTEREST:
ANNUALLY, 0.06.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9433 962
0.8899 965 0.4854 369 0.5154 369
|
0.8396 193
0.7920 937
0.7472 582
0.6650 571
0.6274 124
0.7049 606 0.1433 626
0.5267 875
1.0000 000 | 1.0600 000
0.4688 390
0.3141 098 0.3741 098
0.4423 010
0.2285 915 0.2885 915
0.4172 650
0.1773 964 0.2373 964
0.5918 975 0.0870 222
0.3936 463
0.3713 644
0.3503 438
0.3305 130
0.3118 047
0.5583 948 0.0758 680
0.1191 350 0.1791 350
0.2033 626
0.1010 359 0.1610 359
0.0870 222 0.1470 222
0.4969 694 0.0592 770 0.1192 770
|
0.1358 680
0.0667 929 | 0.1267 929
0.0529 601 0.1129 601
0.0475 849 | 0.1075 849
0.0429 628 0.1029 628
0.0389 521
0.0989 521
0.0354 448 0.0954 448
0.0323 565 0.0923 565
0.0296 209
0.0896 209
0.0871 846
0.0271 846
0.2941 554
0.0250 045
0.0230 456
0.2775 051
0.0850 045
0.0830 456
0.0812 785
0.2617 973 0.0212 785
0.2469 786 0.0196 790 | 0.0796 790
0.2329 986 0.0182 267
0.0782 267
RATIO of INTEREST:
HALF-YEARLY, 0.03.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9708 737
0.9425 960
0.9151 417
0.8884 871
0.8626 088
0.8374 843
0.8130 915
0.7891 093
0.7664 167
0.7440 939
0.7224 213
0.7013 798
0.6809 513
0.6611 178
••
0.6418 620
0.6231 669
0.6050 164
0.5873 946
0.5702 860
0.5536 758
··
0.5375 493
0.5218 925
0.5066 918
0.4919 338
...
0.4776 056
0.4636 917
0.4501 891
0.4370 768
0.4243 464
0.4119 868
2.0000 000
2.0600 000
0.9852 217 1.0452 217
0.2889 592
0.2723 718
0.2567 365
0.2419 988
0.2281 071
0.6470 607 0.7070 607
0.4780 541
0.5380 541
0.3767 092 | 0.4367 092
0.3091 950
...
0.3691 950
0.2610 127 0.3210 127
0.2249 128 | 0.2849 128
0.1968 677 0.2568 677
0.1744 610 | 0.2344 610
0.1561 549 | 0.2161 549
0.1409 242 | 0.2009 242
0.1280 591
0.1880 591
0.1170527 | 0.1770 527
0.1075 332 | 0.1675 332
0.0992 217 | 0.1592 217
0.0919 051 0.1519 051
0.0854 174 | 0.1454 174
0.0796 278 0.1396 278
0.0744 314
0.1344 314
0.0697 435 | 0.1297 435
0.0654 948 0.1254 948
0.0616 278
0.0580 948
0.1216 278
0.1180 948
0.0548 557
0.1148 557
0.0518 766 | 0.1118 766
0.0491 284
0.1091 281
0.0165 865 | 0.1065 865
0.0412 293
0.0420 385
0.1042 293
0.1020 385
0.3999 871 0.0399 979 | 0.0999 979
0.3883 370 0.0380 932 | 0.0980 932
0.3660 449
0.0346 439 0.0946 439
0.3450 324 0.0316 076 | 0.0916 076
0.3252 262 0.0289 187 | 0.0889 187
0.3065 568 0.0265 248 0.0865 248
0.0243 833 0.0843 833
0.0224 597 | 0.0824 597
0.0207 251 | 0.0807 251
0.0191 555 | 0.0791 555
0.0177 310 | 0.0777 310
RATIO of INTEREST:
QUARTERLY, 0.015.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.9852 217
0.9706 618
0.9563 170
0.9421 841
0.9282 603
0.9145 421
0.9010 267
0.8877 111
0.8745 922 0.4184 393
0.8616 672 | 0.3737 367
0.8489 332 0.3371 754
0.8363 874 | 0.3067 200
4.0000 000
1.9851 117
1.3135 318
0.9777 792
0.8240 270 | 0.2809 614
0.8118 493 0.2588 933
0.7998 515 0.2397 774
0.7880 310 0.2230 603
0.7314 979
0.7206 876
0.7100 371
0.6995 439
0.7763 573
0.6421 008
0.5462-247
0.4743 361
0.7763 853 0.2083 186
0.7649 116 | 0.1952 231
0.7536 074 0.1835 139
0.7424 704 | 0.1729 829
0.6892 058 0.1330 568
0.6790 205 0.1269 278
0.6689 857 0.1212 611
0.6590 993 | 0.1160 043
0.5764 431
0.5679 242
0.5595 313
0.5512 623
0.1634 620
0.1548 133
0.1469 230
0.1396 964
0.6493 589 0.1111 151
0.6397 624 | 0.1065 568
0.6303 078 0.1022 972
0.6209 929 | 0.0983 084
0.6118 156 0.0945 658
0.6027 741 0.0910 475
0.5938 661 0.0877 345
0.5850 897 0.0846 096
0.0846 096
0.4821 297
0.4750 047
0.4679 849
0.4610 689
0.0816 575
0.0788 645
0.0762 185
0.0737 081
0.5431 156 | 0.0713 242
0.5350 893 0.0690
0.0690 570
0.5271 815 0.0668 986
0.5193 907 | 0.0648 415
0.5117 149
0.5041 527
0.4967 021
0.0628 790
0.0610 050
0.0592 137
0.4893 617 | 0.0575 000
0.0558 591
0.0542 867
0.0527 788
0.0513 315
0.4542 550 | 0.0499 415
0.4475 419 | 0.0486 055
0.4409 280 0.0473 207
0.4344 118 0.0460 843
0.4279 919 | 0.0448 936
0.4216 669 0.0437 465
0.4154 354 0.0426 405
0.4092 960 | 0.0415 737
0.4032 473 0.0405 442
0.3972 879 | 0.0395 500
0.3914 167 0.0385 896
0.3856 322 | 0.0376 614
0.0376 614
0.3633 366 | 0.0342 413
0.3423 300 0.0312 312
0.3225 380 | 0.0285 658
0.3038 902 | 0.0261 933
0.2863 205 | 0.0240 714
0.2697 666 | 1).0221 655
0.2541 699 | 0.0204 473
0.2394 748 0.0188 929
0.2256 295 | 0.0174 823
4.0600 000
2.0451 117
1.3735 318
1.0377 792
0.8363 573
0.7021 008
0.6062 247
0.5343 361
0.4784 393
0.4337 367
0.3971 754
0.3667 200
0.3409 614
0.3188 933
0.2997 774
0.2830 603
0.2683 186
0.2552 231
0.2435 139
0.2329 829
0.2234 620
0.2148 133
0.2069 230
0.1996 964
0.1930 538
0.1869 278
0.1812 611
0.1760 043
0.1711 151
0.1665 568
0.1622 972
0.1583 081
0.1545 658
0.1510 475
0.1477 345
0.1446 096
0.1416 575
0.1388 645
0.1362 185
0.1337 081
0.1313 242
0.1290 570
0.1268 986
0.1248 415
0.1228 790
0.1210 050
0.1192 137
0.1175 000
0.1158 591
0.1142 867
0.1127 788
0.1113 315
0.1099 415
0.1086 055
0.1073 207
0.1060 843
0.1048 936
0.1037 465
0.1026 405
0.1015 737
0.1005 442
0.0995 500
0.0985 896
0.0976 614
0.0942 413
0.0912 312
0.0885 658
0.0861 933
0.0840 714
0.0821 655
0.0804 473
0.0788 929
0.0774 823
Page 51.
RATE OF INTEREST: 6 per Cent.
CORRESPONDİNG VALUE OF
per
Ann.

Years.
26
27
28
29
30
36
37
38
39
40
31
6.0881 006
32 6.4533 867
33
6.8405 899
7.2510 253
7.6860 868
34
35
41
19323 95393 HOOD ON 08 78876 86882 ****R OKO
46
48
10.902 861
42 11.557 033
12.250 455
12.985 482
45 13.764 611
43
4.4
49
51
14.590 487
47 15.465 917
16.393 872
17.377 504
50 18.420 154
52
53
57
19.525 364
20.696 885
21.938 698
54 23.255 020
55 24.650 322
58
59
60
56 26.129 341
27.697 101
29.358 927
31.120 463
32.987 691
66
67
69
70
71
72
61 34.966 952
62 37.064 969 |
39.288 868
41.646 199
63
64
65
73
74
75
76
77
RATIO of INTEREST:
ANNUALLY, 0.06.
78
Improved Cumulated Value of
Principal. Annuity.
Annuity.
79
4.5493 830
4.8223 459
5.1116 867
5.4183 879
5.7434 912
***** ****8 58333 85888
97.
8.1472 520
8.6360 871
9.1542 523
100
119.12 087
127.26 812
135.90 421
145.05 846
10.285 718 | 154.76 197
9.7035 075
80 105.79 599
81
66.377 715
70.360 378
74.582 001
79.056 921
59.156 383
63.705 766
68.528 112
73.639 798
79.058 186
83.800 336
88.828 356
94.158 058
99.807 5+1
81.801 677
90.889 778
97.343 165
104.18 375
111.43 478
566.11 587
601.08 282
638.14 779
677.43 666
44.144 972 719.08 286
165.04 768
175.95 054
187.50 758
199.75 203
212.74 351
46.793 670 | 763.22 783
49.601 290 810.02 150
52.577 368 859.62 279
55.732 010 912.20 016
59.075 930 967.93 217
150.07 364
159.07 806
168.62 274
178.74 010
189.46 451
226.50 812
241.09 861
256.56 453
272.95 840
290.33 590
62.620 486 | 1027.0 081
1089.6 286
1156.0 063
1226.3 667
1300.9 487
200.83 238
212.88 232
225.65 526
239.19 458
253.54 625
308.75 606
328.28 142
348.97 831
370.91 701
394.17 203
418.82 235
444.95 169
472.6+ 879
502.00 772
533.12 818
112 14 375 1852.3 959
118.87 238 1964.5 396
126.00 472 2083.4 120
133.56 500 2209.4 167
141.57 890
2342.9 817
2484.5 606
2634.6 343
2793.7 123
2962.3 351
3141.0 752
13.003 17
13.210 53
13.406 16
13.590 72
13.764 83
3330.5 397
3531.3 721
3744.2 544
3969.9 097
4209.1 042
13.929 09
14.084 05
14.230 23
14.368 14
14.498 25
14.620 99
14.736 78
14.846 02
14.949 08
15.046 30
15.138 02
15.224 55
15.306 18
15.383 19
15.455 84
15.524 37
15.589 03
15.650 03
15.707 58
15.761 86
15.813 08
15.861 40
15.906 98
15.949 98
15.990 55
16.028 82
16.064 92
16.098 98
16.131 12
16.161 43
15.190 03
16.217 01
16.242 46
16.266 47
16.289 13
1380.0 056 16.467 78
1463.8 059
16.479 04
1552.6 343 16.489 66
16.499 68
16.509 13
1646.7 924
1746.5 999
16.310 50
16.330 66
16.349 68
16.367 62
16.384 55
RATIO of INTEREST:
HALF-YEARLY, 0.03.
Improved Cumulated
Principal. Annuity.
4.6508 859
4.9341 248
5.2346 130
5 5534 010
5.8916 031
4462.6 505
4731.4 095
268.75 903
284.88 457
301 97 765
5016.2 941
320.09 631
5318.2 718
339.30 208 | 5638.3 681
Value of Perpetual Anu. 16.666 67
8.4000 172
8.9115 783
9.4542 934
10.030 060
10.640 891
6.2504 017
6.6310 512
7.0348 822
7.4633 065
7.9178 219 | 115.29 703
87.506 695
93.850 853
100.58 137
107.72 177
15.171 365
16.095 302
17.075 506
18.115 404
19.218 632
20.389 047
21.630 740
22.948 052
24.345 588
25.828 234
11.488 921
11.976 416
12.705 780
171.48 201
182.94 027
195.09 633
13.479 562 | 207.99 270
14.300 467 221.67 445
32.718 434
34.710 987
36.824 886
39.067 522
41.446 734
43.970 840
46.648 664 |
66.509 841
70.560 290
74.857 412
16.443 20 79.416 228
16.455 85 84.252 676
16.400 52
16.415 58
16.429 79
49.489 568
52.503 482
55.700 945
59.093 132
62.691 904
27.401 174
29.069 905
440.01 956
467.83 175
30.840 262 | 497.33 771
528.64 057
561.81 979
120.12 417
127.43 973
135.20 081
16.541 89 143.43 454
16.548 95
16.518 05
16.526 46
16.534 40
152.16 971
60.848 098
65.568 746
70.576 883
75 890 016
81.526 718
16.555 61 161.43 684
16.561 90 171.26 834
16.567 83 181.69 859
16.573 42 192.76 403
16.578 70
204.50 336
16.583 68
16.588 38
16.592 81
216.95 761
230.17 033
244.18 771
16.596 99 259.05 874
16.600 94
274.83 541
123.33 362
131.85 964
140.90 489
150.50 100
160.68 151
236.18 942
251.58 836
267.92 509
285.25 673
303.61 387
323.15 078
343.84 566
365.80 086
389.09 313
413.80 391
597.08 144
634.45 870
674.11 223
716.18 067
760.81 107
89.383 664 1473.0 611
94.827 130 1563.7 855
100.60 210 1660.0 350
106.72 877 1762.1 462
1870.4 759
113.22 855
808.15 946
858.59 137
911.68 241
968.21 887
1028.1 984
1091.8 307
1159.3 382
1230.9 569
1306.9 371
1387.5 446
1985.4 028
2107.3 289
2236.6 802
2373.9 090
2519.4 951
2673.9 473
2837.8 057
3011.6 431
3196.0 672
3391.7 227
3599.2 936
3819.5 055
4053.1 284
4300.9 790
4563.9 236
16.604 66 291.57 289 4842.8 815
16.608 17 309.32 968 5138.8 280
16.611 48 328.16 786 5452.7 976
16.614 60 348.15 328 5785.8 880
16.617 55 369.35 582 | 6139.2 636
-
Value of
Annuity.
13.083 12
13.288 83
13.482 73
13.665 50
13.837 78
14.000 17
14.153 24
14.297 52
14.433 52
14.561 71
14.682 54
14.796 44
14.903 80
15.004 09
15.100 38
15.190 29
15.275 04
15.354 93
15.430 23
15.501 20
15.568 11
15.631 17
15.690 61
15.746 64
15.799 45
15.849 23
15.896 16
15.940 39
15.982 08
16.021 38
16.058 42
16 093 34
16.126 25
16.157 27
16.186 51
16.214 07
16.240 05
16.264 54
16.287 63
16.309 38
16.329 90
16.349 23
16.367 45
16.384 63
16.400 82
16.416 08
16.430 46
16.444 02
16.456 80
16.468 85
RATIO of INTEREST:
QUARTERLY, 0.015.
16.563 43
16.569 35
16.57+ 93
16.580 21
16.585 17
Improved Cumulated
Principal. Annuity.
16.589 84
16.594 26
16.598 41
16.602 33
16.606 02
4.7040 116
4.9926 565
5.2990 342
5.6242 018
5.9693 228
6.3356 216
6.7243 978
7.1370 308
7.5749 843
8.0398 102
8.5331 637
9.0567 889
9.6125 456
10.202 406
10.828 461
11.492 934
12.198 181
12.946 705
13.741 161
14.584 367
15.479 316
16.429 182
17.437 335
18.507 351
19.643 029
20.848 395
22.127 726
23.485 562
24.926 719
26.456 311
28.079 765
29.802 839
31.631 647
33.572 677
35.632 816
37.819 372
40.140 103
42.603 242
45.217 528
47.992 236
50.937 210
54.062 898
57.380 389
60.901 454
61.638 583
16.480 20
92.401 037
16.490 91
98.071 092
16.501 00 104.08 908
16.510 51 110.47 636
16.519 47
117.25 558
68.605 036
72.814 885
77.283 065
82.025 428
87.058 800
16.527 92
16.535 89
16.543 39 140.19 290
16.550 47 148.79 564
16.557 14
167.61 718
177.90 277
188.81 951
200 40 615
212.70 378
225.75 604
239.60 923
254.31 251
269.91 802
286.48 115
61.733 527
66.544 411
71.650 570
77.070 030
82.822 017
16.609 51 304.06 065
16.612 79
322.71 889
16.615 88 342.52 207
16.618 79 363.54 044
16.621 54
385.84 857
16.666 67
88.927 026
95.406 630
102.28 385
109.58 307
117.33 020
125.55 273
134.27 981
143.54 243
153.37 343
163.80 769
174.88 223
186.63 636
199.11 175
212.35 268
226.40 612
241.32 193
257.15 303
273.95 558
291.78 919
310.71 714
330.80 658
352 12 877
374.75 937
398.77 866
424.27 186
451.32 941
480.04 731
510.52 745
542.87 795
577.21 359
613.65 619
652.33 504
693.38 736
736.95 880
783.20 393
832.28 683
884.38 163
939.67 316
998.35 757
1060.6 431
124.45 080
2057.5 133
132.08 754 | 2184.7 924
2319.8 817
2463.2 606
157.92 627.2615.4 378
1126.7 506
1196.9 147
1271.3 844
1350.4 238
1434.3 133
1523.3 506
1617.8 515
1718.1 514
1824.6 060
1937.5 930
2776.9 530
2948.3 794
3130.3 252
3323.4 358
3528.3 963
3745.9 340
3976.8 205
4221.8 751
4481.9 671
4758.0 192
5051.0 109
3361.9 816
5692.0 345
6042.3 407
6414.1 429
Value of
Annuity.
13.123 59
13.328 44
13.521 44
13.703 28
13.874 61
14.036 04
14.188 13
14.331 43
14.466 44
14.593 65
14.713 50
14.826 43
14.932 82
15.033 06
15.127 51
15.216 50
15.300 34
15.379 34
15.453 77
15.523 89
15.589 96
15.652 21
15.710 86
15.766 12
15.818 19
15.867 24
15.913 46
15.957 01
15.998 04
16.036 70
16 073 12
16.107 44
16.139 77
16.170 23
16.198 93
16.225 98
16.251 45
16.275 46
10.298 08
16.319 39
16.339 47
16.358 38
16.376 21
16.393 00
16.408 82
16.423 78
16.437 78
16.451 01
16.463 48
16.475 22
16.486 29
16.496 72
16.506 55
16.515 81
16.524 53
16.532 74
16.540 49
16.547 78
16.554 65
16.561 13
16.567 23
16.572 98
16.578 40
16.583 50
16.588 31
16.592 84
16.597 11
16.601 13
16.604 92
16.608 49
16.611 85
16.615 02
16.618 01
16.620 82
16.623 47
16.666 67
Page 52.
PERPETUAL STOCK: 3 p.C. 50; 34 p. C. 58,3333; 4p. C. 66,6667; 5 p. C. 83,3333.

Years.
Ona ma 29 29*** G**38 58833 UAAA8 588*6 85082
28
26
0.2198 100
27 0.2073 680
0.1956 301
29 0.1845 527
0.1741 181
32
31 0.1642 548
36
37
38
39
40
0.1549 574
33 0.1461 862
34
35
41
42
43
44
0.0770 091
45 0.0726 501
54
46
0.0685 378
47 0.0646 583
48
49
0.0609 984
0.0575 457
0.0542 884
50
59
60
51 0.0512 154
52 0.0483 164
0.0455 816
0.0430 015
55 0.0405 674
62
63
56 0.0382 712
57 0.0361 049
58 0.0340 612
0.0321 332
0.0303 143
64
66
67
68
69
RATIO of INTEREST:
ANNUALLY, 0.06.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
-7777
10 KO
71
0.0213 704
0.0201 608
0.0190 196
0.0179 430
70 0.0169 274
72
73
0.0117 922 0.0717 922
0.0110 023 0.0710 023
0.0102 729 0.0702 729
0.1379 115 0.0095 984 | 0.0695 984
|
0.1301 052
0.0689 739
0.0089 739
75
76
77
78
79
80
0.1227 408
0.0083 948
0.1157 932 | 0.0078 574
|
0.1092 389
0.1030 555
0.0972 222
0.0917 190 | 0.0060 589
0.0865 274 | 0.0056 834
0.0316 296 | 0.0053 331
0.0050 061
0.0017 005
81
82
83
*** 8888 58838 85888
0.0285 984
0.0269 797
0.0254 524
0.0240 /18
0.0226 526
0.0159 692 | 0.0009 737
0 0150 653 0.0009 177
0.0008 650
0.0008 154
0.0007 687
0.0142 125
74 0.0134 081
0.0126 491
0.0119 331
0.0112 577
0.0106 204
0.0100 193
0.0094 522
0.0089 171
0.0084 124
0.0079 362
0.007+ 870
0.0070 632
0.0066 634
0.0062 862
0.0059 304
0.0055 947
0.0052 780
0.0049 793
0.0046 974
0.0014 315
0.0041 807
0.0039 411
0.0169 043
0.0156 972
0.0145 926
0.0135 796
0.0126 489
I
p
I
0.0073 581
0.0068 938
0.0064 617
0.0044 149
0.0041 477
0.0038 977
0.0036 636
0.0034 443
0.0032 388
0.0030 462
0.0028 655
0.0026 960
0.0025 370
0.0023 876
0.0022 474
0.0021 157
0.0019 920
0.0018 757
0.0017 664
0.0016 637
0.0015 670
0.0014 762
0.0013 907
0.0013 102
0.0012 345
0.0011 653
0.0010 963
0.0010 331
0.0007 246
0.0006 832
0.0006 441
0.0006 072
0.0005 75
0.0005 398
0.0005 090
0.0004 800
0.0004 526
0.0004 208
0.0004 025
0.0003 796
0.0003 579
0.0003 376
0.0003 184
0.0769 043
0.0756 972
0.0745 926
0.0735 796
0.0726 489
0.0037 208 0.0002 241
0.0035 102 0.0002 114
98 0.0033 115 0.0001 994
0.0031 241 0.0001 880
100 0.0029 472, 0.0001 774
0.0683 948
0.0678 574
0.0673 581
0.0668 938
0.0664 617
0.0660 589
0.0656 834
0.0653 331
0.0650 061
0.0647 005
0.0644 149
0.0641 477
0.0638 977
0.0636 636
0.0634 443
0.0632 388
0.0630 462
0.0628 655
0.0626 960
0.0625 370
0.0623 876
0.0622 474
0.0621 157
0.0619 920
0.0618 757
0.0613 102
0.0612 345
0.0611 633
0.0610 963
0.0610 331
0.0609 737
0.0609 177
0.0608 650
0.008 154
0.0607 687
0.0607 246
0.0006 832
0.0606 441
0.0606 072
0.0605 725
0.0605 398
0.0605 090
0.0604 800
0.0004 526
0.0604 268
0.0604 025
0.0603 796
0.0003 579
0.0603 376
0.0603 184
0.0003 003
0.0002 832
0.0002 671 0.0602 671
0.0002 519 0.0602 519
0.0002 376 | 0.0602 370
RATIO of INTEREST:
HALF-YEARLY, 0.03.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
Ratio of Perpetual Ann. 0.0600 000
0.2150 128
0.2026 702
0.1910 361
0.1800 698
0.1697 331
0.0617 664 0.0271 555
0.0616 637 0.0255 967
0.0615 670 0.0241 274
0.0614 762 0.0227 423
0.0613 907
0.0214 368
0.1599 897 0.0114 277 0.0714 277
0.1508 057 0.0106 552 | 0.0706 552
0.1421 488 0.0099 422
0.1339 889 | 0.0092 832
0.1262 973 0.0086 733
0.0699 422
0.0692 832
0.0686 733
0.1190 474
0.1122 135
0.1057 720
0.0997 003
0.0939 771
0.0885 824
0.0834 974
0.0787 013
0.0741 864
0.0099 278
0.0659 136
0.0621 299
0.0585 634
0.0552 016
0.0520 328
0.0490 459
0.0462 305
0.0135 767
0.0410 752
0.0387 173
0.0364 948
0.0343 998
0 0324 251
0.0305 638
0.0288 093
0.0202 063
0.0190 464
0.0179 530
0.0169 224
0.0159 510
0.0125 919
0.0118 691
0.0164 344
0.0152 512
0.0764 344
0.0752 512
0.0141 689 | 0.0741 689
0.0131 770
0.0122 659
0.0731 770
0.0722 659
0.0099 401
0.0093 695
0.0088 317
0.0081 081 0.0681 081
0.0075 838 0.0675 838
0.0070 970 | 0.0670 970
0.0066 445 0.0666 445
0.0062 235 0.0662 235
0.0603 003
0.0046 092
0.0602 832 0.0043 446
0.0040 952
0.0038 601
0.0036 385
0.0058 315
0.0054 663
0.0051 257
0.0048 079
0.0045 111
0.0042 339
0.0642 339
0.0039 747 0.0639 747
0.0037 324 0.0637 324
0.0035 055.0.0635 056
0.0032 933
0.0632 933
0.0030 945 0.0630 945
0.0029 083 0.0629 083
0.0027 337
0.0627 337
0.0025 701
0.0625 701
0.0024 166 | 0.0624 166
0.0022 726 |
0.0021 375
0.0020 107
0.0018 916
0.0017 798
0.0009 159
0.0008 626
0.0133 587 0.0008 124
0.0150 354
0.0141 723
0.0007 651
0.0007 207
0.0016 748
0.0015 761
0.0014 834
0.0013 963 |
0.0013 144
0 0083 247
0.0078 468
0.0073 964
0.0005 037
0.0004 745
0.0004 471 |
0.0069 718 0.0004 212
0.0005 716 0.0003 969 |
0.0358 315
0.0654 663
0.0651 257
0.0648 079
0.0645 111
0.0012 374
0.0612 374
0.0011 650 0.0611 650
0.0010 969 | 0.0610 969
0.0010 328 0.0610 328
0.0009 726 | 0.0609 726
0.0061 944 0.0003 740
0.0058 388
0.0055 036
0.0051 877
0.0048 899
00003 524
0.0003 320
0.0003 129
0.0002 948
0.0111 877
0.0606 789
0.0006 789
0.0105 455 | 0.0006 395 | 0.0606 395
0.0006 024 0.0606 024
0.0005 675 | 0.0605 675
0.0005 346
0.0605 346
0.0002 778
0.0002 618
0.0002 467
0.0002 325
0.0002 191
0.0622 726
0.0621 375
0.0620 107
0.0618 916
0.0617 798
0.0616 748
0.0615 761
0.0614 834
0.0613 963
0.0613 114
THE END.
0.0609 159
0.0608 626
0.0608 124
0.0607 651
0.0607 207
0.0605 037
0.0604 745
0.0604 471
0.0604 212
0.0603 969
0.0603 740
0.0603 524
0.0603 320
0.0603 129
0.0602 918
0.0602 241 0.0031 297
0.0602 114 0.0032 328
0.0601 994 0.0030 472
0.0002 055 0.0602 065
0.0001 946
0.0601 946
0.0001 834 0.0601 834
0.0001 880 0.0028 723 0.0001 728 0.0601 728
0.0601 774 0.0027 074 0.0001 629 | 0.0601 629
- 0.0600 000
0.0602 778
0.0602 618
0.0602 467
0.0602 325
0.0602 191
RATIO of INTEREST:
QUARTERLY, 0.015.
Discounted Annuity for Ratio of
Principal. Cumulation Annuity.
0.2125 845
0.2002 938
0.1887 136
0.1778 030
0.1675 232
0.1578 377
0.1487 122
0.1401 143
0.1320 135
0.1243 810
0.0870 100
0.0819 794
0.0772 397
0.0727 741
0.0685 666
0.1171 898 | 0.0079 648 | 0.0679 648
0.1104 144 0.0074 471 0.0674 471
0.1040 307 | 0.0069 666 | 0.0669 666
0.0980 151 | 0.0065 200
0.0665 200
0.0661 047
0.0923 492 0.0061 047
0.0573 482
0.0540 326
0.0509 086
0.0646 023
0.0041 438
0.0608 673 | 0.0038 887
0.0036 502
0.0264 415
0.0249 127
0.0234 724
0.0221 153
0.0208 367
0.0196 320
0.0184 970
0.0174 276
0.0164 200
0.0154 706
0.0161 987 | 0.0761 987
0.0150 279
0.0139 566
0.0129 752
0.0120 741
0.0750 279
0.0739 566
0.0729 752
0.0720 741
0.0112 452 0.0712 452
0.0104 814 0.0704 814
0.0097 767 0.0697 767
0.0091 255 | 0.0691 255
0.0085 230 0.0685 230
0.0108 224
0.0101 967
0.0096 072
0.0090 517
0.0085 284
0.0479 653
0.0451 922
0.0030 229
0.0028 399
0.0630 229
0.0628 399
0.0425 793 | 0.0026 684 | 0.0626 684
0.0401 176
0.0625 077
0.0377 982
0.0623 570
0.0025 077
0.0023 570
0.0057 181
0.0053 580
0.0050 223
0.0017 091
0.0044 168
0.0022 157
0.0020 831
0.0019 588
0.0622 157
0.0620 331
0.0356 128
0.0335 538
0.0316 139
0.0619 588
0.0297 861
0.0019 420 0.0618 420
0.0280 640 | 0.0017 325
0.0017 325 0.0617 325
0.0059 660
0.0056 210
0.0052 961
0.0049 899
0.0047 014
0.0044 296
0.0041 735
0.0039 322
0.0037 048
0.0031 906
0.0034 271
0.0032 184
0.0032 888
0.0030 987
0.0029 195
0.0027 507
0.0025 917
0.0657 181
0.0653 580
0.0650 223
0.0647 091
0.0644 168
0.0641 438
0.0638 887
0.0636 502
0.0634 271
0.0632 184
0.0145 762 0.0008 875 0.0608 875
0.0137 335 0.0008 355 | 0.0608 355
|
0.0129 394
0.0121 913
0.0114 865
0.0007 865 | 0.0607 865
0.0007 405 0.0607 405
0.0006 972 | 0.0606 972
0.0016 296 | 0.0616 296
0.0015 330
0.0014 422
0.0013 569
0.0615 330
0.0614 422
0.0613 569
0.0012 768 | 0.0612 768
0.0012 015 0.0612 015
0.0011 307
0 0611 307
0.0010 642 0 0610 642
0.0010 016 | 0.0610 016
0.0009 428 0.0609 428
0.0004 860
0.0004 860
0.0604 860
0.0080 353
0.0075 7670 0004 577 i 0 0604 577
0.0071 330 00004 311 0.0604 311
|
0.0067 206 0.0004 060 0.0604 060
0.0063 321 | 0.0003 823
0.0503 823
0.0006 564 0.0506 564
0.0006 181 0.0606 181
0.0005 820 0.0605 820
0 0005 481
0.0605 481
0.0005 161 0.0605 164
}
0.0003 601
0.0003 392
0.0603 601
0.0603 392
0.0003 195 | 0.0603 195
0.0003 009 0.0603 009
0.0002 834
0.0602 834
0.0002 670 0.0602 670
0.0002 515 | 0.0602 515
0.0002 369 0.0602 369
0.0002 231 0.0602 231
0.0002 102 | 0.0602 102
|
0 0001 980 0.0601 980
0.0001 865. 0.0601 865
0.0001 757 0.0601 757
0.0001 655 | 0.0001 655
0.0001 559
0.0601 559
0.0600 000
Page 53.
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