B 446261
1
C
:
Mostephenson
1837
ARTES
SCIENTIA
VERITAS
LIBRARY
OF THE
UNIVERSITY OF MICHIGAN
LE PLURIBUS UMUNT
TUEBOR
•QUÆRIS-PENINSULAM AMŒNAM
CIRCUMSPICE
f
$
1814
QA
35
W683Z
THEORY
O F
INTEREST,
SIMPLE AND COMPOUND,
DERIVED FROM FIRST PRINCIPLES,
AND
APPLIED TO ANNUITIES OF ALL DESCRIPTIONS,
THEORY
O F
INTEREST,
SIMPLE AND COMPOUND,
DERIVED FROM FIRST PRINCIPLES,
1
AND
APPLIED TO ANNUITIES OF ALL DESCRIPTIONS
• J
.•
CONTAINING,
Arithmetical and Geometrical Progref- || Annuities on Lives, with Tables.
fions.
Simple and Compound Intereft, with
Tables.
M. De Moivre's Hypothefis of Equal De-
crements, with 22 Problems, fhewing
the Value of Life in all its Varieties.
Annuities at Simple and Compound In- || Infurance on Lives.
tereft, with Tables.
Of the Expectation of Life.
Of a Sinking Fund to extinguish the Of the Britiſh State Lottery.
National Debt.,
The Value of a Perpetuity.
Of Tontines, and Infurance from Fire,
&c.
WITH
AN ILLUSTRATION OF THE WIDOWS SCHEME
IN THE CHURCH OF SCOTLAND.
By the Reverend MR DAVID WILKIE,
MINISTER OF CULTS.
EDINBURGH:
PRINTED FOR THE AUTHOR.
AND SOLD BY PETER HILL, Edinburgh;
AND G. G. & J. ROBINSON, Londens
1794.
DEDICATION.
TO THE RIGHT HONOURABLE,
FRANCIS Lord NAPIER.
MY LORD,
IT
may be unuſual, that a perfon who hath
not had the honour of your Lordship's acquaint-
ance, fhould, in this public manner, crave your
protection, in his attempt to promote the know-
ledge of that ſcience, whereby human life, which
is liable to fo many accidents, is yet, in fome
meafure, reduced to rule. It flows, my Lord,
from the reſpect which I entertain for the memo-
ry of
your illuſtrious progenitor, John Napier,
Baron of Merchifton, in Scotland, whom I place
upon
vi
DEDICATION.
upon an equal footing with the moſt diſtinguiſh-
ed characters of the feventeenth century, for his
valuable invention of Logarithms, that I prefume
to aſk your patronage in the publication of the
following Treatife, and to profeſs myſelf, with all
reſpect,
My LORD,
Your Lordship's moſt obedient,
And humble fervant,
DAVID WILKIE.
PRE-
!
THE PREFACE.
SIMILAR to the fatisfaction which the mind
receives in the inveſtigation of geometrical pro-
pofitions, is the pleaſure which we derive from
the refolution of queſtions in algebra, both on
account of their ingenuity, certainty, and extenfive
uſefulneſs. In the following Treatife, wherein
ſo many algebraic theorems occur, it is propoſed
to inveſtigate, from first principles, the rules
adapted to one particular branch of ſcience, that
of Intereft, both fimple and compound, with
their application to Annuities of all defcriptions;
hoping thereby, not only to afford pleafure to the
ftudious, in the inveſtigation of the theorems, but
advantage alfo to men of bufinefs, in the applica-
tion of theſe to the affairs of property.
a
↑
The theorems, ſuch as being the value of a
perpetuity, are as beautiful as any which are to
be met with in the other branches of ſcience; and
to render them the more intelligible, a demon-
ſtration is fubjoined where it is thought necef-
fary. The tables are accurate, and feveral of them
new, fuch as Table IV. VIII. and XI. of annui-
ties on lives; the firft expreffing the amount of
L. I annuity upon a fingle life; in the conftruc-
tion
viii
PREFACE.
tion of which, the reafon is difcovered, why the
value of a fingle life is lefs, and its amount greater,
than that correfponding to its expectation; and
the two laſt ſhewing the probabilities of life ad-
apted to Scotland. The conftruction of the tables
is clearly pointed out; and, by three equations
annexed to each, all their ufes are diſcovered at
firſt fight. Several branches of education, con-
nected with this fcience, are treated of, fuch as
Arithmetical and Geometrical Progreffions, the
Extraction of the Square Root, Logarithms, the
Arithmetic of Infinities, Fluxions, and the rule of
Double Pofition.
PR— np.
72 r
In this Treatife there are many things new.
The application of geometry, for amufement, to
matters of intereft, moſt of the remarks annexed
to each fubject, the bearer's lofs in difcounting
bills, feveral of the theorems, fuch as P-
expreffing the value of a fingle life, a Tontine, and
Infurance from Fire, are all new. But what I
chiefly value in this refpect, are Prob. III. and
XVIII. of Annuities on Lives; the former difco-
vering the value, and the latter the expectation of
the joint continuance, or of the longeſt liver of
number of lives, by methods which are both
accurate, and fave many pages of algebraic inve-
ftigation. As the queſtions in Annuities on Lives
are various as the complections of men, I have,
any
in
PREFACE.
ix
n the courſe of twenty-two problems, delineated a
great number of thoſe which are marked with the
ſtrongeſt features. Particular attention hath been
paid to the Widows Scheme in the Church of
Scotland, wherein fo many are intereſted; and
the account given of its conftruction and opera-
tion may enable Contributors to judge for them-
felves, fhould the affairs of the Fund ever come
before them in a judicial capacity.
In the conftruction of the theorems of Intereſt
and Annuities certain, I acknowledge myſelf
much indebted to the labours of that able alge-
braiſt Mr Nicolas Vilant, St Andrew's. Where-
in I differ from others, as in the conftruction of
a widows fcheme, in Prob. VIII, let the learned
judge for themſelves; and wherein I am defective,
I crave their indulgence. But I hope it will not
be reckoned a defect, that I have not expreffed the
theorems in fo many words, being perfuaded that
an algebraic expreffion is much more intelligible
at firſt fight, than it can be made by any lan-
guage.
The ſcience of Annuities on Lives is yet in its
infancy. Dr Halley was the firſt who conftruc-
ted a table of the Probabilities of Life: Moivre,
Dodfon, Webfter, and Price, thefe fathers of the
fcience, dropped only a few years ago. Thofe theo-
b
rems
*
PREFACE.
rems of Mr De Moivre, which are built upon the
principle, "that the decrements of life are in a
"conſtant ratio," though beautiful in theory and
eafy in practice, are now exploded, becauſe they
give the value of life too fmall. In Scotland, we
have no writer who treats methodically of Annui-
ties; and Engliſh authors on theſe ſubjects are in
very few hands: therefore, till a better appears,
it is hoped that this Treatife, which has for its
object a more perfect Syftem of Intereft and An-
nuities than has hitherto appeared, will not be
deemed an unneceffary performance.
CULTS,
OAЯober 18. 1
}
CON-
CONTENT S.
CHAPTER I.
Pag.
I
Arithmetical Progreffion, with a Table and Ex-
amples,
CHAPTER II.
Simple Intereft, with Examples,
Rebate or Difcount,
Four Tables adapted to Simple Intereft,
Remarks on Simple Intereft,
Supplement to CHAP. II.
1. Of Intereft due upon Cafh-accounts,
2. Of the Equation of Payment,
3. Of the Premium of Commiffion,
}
CHAPTER III.
Annuities computed at Simple Intercft,
1. The amount of an Annuity in Arrears,
D
7
18
ය
21
27
སྔོན་མ་
30
33
34
35
35
2. The
*ii
CONTENT S.
Pags
2. The prefent worth of an Annuity,
38
3.
in Reverfion,
40
4.
of the Reverfion of an Annuity,
ib.
Two Tables of Annuities at Simple Intereft,
Remarks upon Annuities at Simple Intereft,
41
45
CHAPTER IV.
Geometrical Progreffion, with a Table and Ex-
amples,
Of a Geometrical Series continued indefinitely,
48
54
CHAPTER V.
Compound Intereft, with Examples,
The method of Calculation by Logarithms,
Three Tables, adapted to Compound Intereft,
Remarks upon Compound Intereſt,
!
55
57
59
65
CHAPTER VÍ.
Annuities computed at Compound Intereft, 67
t. The amount of an Annuity in Arrears,
The Rule of Double Pofition,
2. The Prefent worth of an Annuity,
3.
of an Annuity in Reverfion,
The Renewing of Leafes, with a Table,
ib.
69
73
77
81
82
84
94
96
Supplement
4. The prefent Worth of the Reverfion of an Annuity,
Three Tables of Annuities at Compound Intereſt,
Annuities computed at Compound Intereft when the Pay-
ments are made per advance,
Remarks upon Annuities at Compound Intereſt,
CONTENT S
xiii
Pag.
Supplement to CHAP. VI.
Of a Sinking Fund to extinguish the National Debt,
CHAPTER VII.
The Value of a Perpetuity or Freehold E-
ſtate,
Of a Reverſion in Perpetuity,
Remarks upon the Purchale of a Perpetuity,
Three Additional Tables of Compound Intereft,
CHAPTER VIII.
Annuities on Lives,
Dr Halley's Table of the Probabilities of Life,
Six Articles derived from it,
Mr De Moivre's Hypothefis of equal Decrements,
Twenty-two Problems derived moftly from it.
PROB. I. To find the Value of a fingle Life,
PROB. II.
PROB. III.
97
ΙΟΙ
103
105
108
IIO
II [
I 12
116
117
of two joint Lives,
119
of the joint continuance,
or of the Longest Liver of any number of Lives,
Ten Tables adapted to Annuities on Lives,
122
127
PROB. IV.
of the longeſt Liver of
two Lives,
144
PROB. V.
of the longeſt of three
Lives,
146
PROB. VI.
for a given term of
Years, on the Contingency of an affigned Life, with a
Table,
Supplement to PROB. I. II. III. IV. V. and VI.
To find, by a Geometrical Figure, the Value of Life, in
four Articles,
147
150
Art.
xiv
CONTENT S.
Art. 1. To find the Value or amount of a Single Life,
2.
3.
4.
3w
or
of the joint continuance of
Pag.
ib.
152
of the longeft Liver of ditto,
154
for a given term of Years, on
158
any number of Lives,
or
or
the Contingency of an Affigned Life,
Of REVERSIONS.
PROB. VII. To find the value of a Reverſion in Perpetui-
ty, after one or more affigned lives,
PROB. VIII. To find the Value of the Reverſion of an
Annuity during a given Life, after the Deceaſe of the
prefent Poffeffor,
162
165
Of a Widows Scheme,
167
Of a Marriage-Tax,
170
PROB. IX.
during two joint Lives, af-
ter the Deceafe of the prefent Poffeffor,
172
PROB. X.
during one Life, after two
joint ones,
173
PROB. XI.
during a Life of a given
Age, after the longest of two Lives,
174
PROB. XII,
during the longeſt of two
Lives after one,
175
PROB. XIII.
where a given Term of
Years is concerned,
176
PROB. XIV.
depending on Survivor-
fhip, and the Expectation of Life,
180
PROB. XV. Of Infurance on Lives,
184
PROB. XVI. To find the Value of fucceffive Lives,
PROB. XVII. Of a Copyhold, and renewing of Leafes,
PROB. XVIII. Of the Expectation of Life,
The Arithmetic of Infinities,
The Expectation of Life difcovered by Fluxions,
Supplement to PROB. XVII.
PROE. XIX. Of the Probability of Survivorship,
FROB. XX. Of Half-yearly and Quarterly Payments of
an Annuity,
187
189
192
194
199
200
206
212
PROB.
CONTENT S.
XV
PROB. XXI. Of the Britiſh State-Lottery,
PROB. XXII. Of a Tontine,
Supplement to CHAP. VIII.
Of Infurance from Fire,
CHAP. IX.
Of the Widows Scheme in the Church of
Scotland, contained in Twelve Articles,
Supplement to CHAP. IX.
A Differtation, refpecting the Difpofal of the
Surplus in the Fund of the Widows Scheme,
when the Capital is raiſed to L. 100,000;
addreſſed to the Truſtees for managing faid
Fund,
Pag.
214
220
223
225
254
The
( xvi)
The Signs which occur, without being ex-
plained in the body of the Work, are fuch as
theſe :
= The Sign of Equality.
+ 1 ×
a
,
16
or a÷b
Addition.
Subtraction.
Multiplication.
Diviſion; a divided by b.
+b, The Sum of the two Quantities a and b.
aub, The Difference of the two Quantities
a and b.
abcd, Ratio; a is to b as c is to d.
a, The Square Root of a.
"a, The n Root of a.
2
a+b, The Square of a+b.
R", The n Power of R.
I
R",
1 divided by the n power of R,
L. The Logarithm of a Quantity.
ARITH-
CHAPTER I.
Of ARITHMETICAL PROGRESSION.
1. WHEN a feries of quantities, exceeding two
in number, increaſes or decreafes by the addi-
tion or fubtraction of one common difference,
the terms of the feries are faid to be in continued
arithmetical progreffion. Thus,
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
10, 9, 8, 7, 6, 5, 4, 3, 2, I.
an increaſing feries.
a decreaſing ſeries.
a, atd, at2d, a†3d, a+4d...a+x=1Xd=v.
v, v-d, v-2d, v—zd, v—4d... v-n-X d=a.
v—2d,
atv, atv, atv, atv, atv
a+v=
Sum of
both.
2. If four quantities are thus proportional, the
fum of the extremes will be equal to the fum of
the means; Thus,
Sum of the extremes.
a, a+d, a+2d, a+3d; then 2 a+3d={Sum of the means.
3. If three quantities are thus proportional, the
fum
A
2
THEORY OF INTEREST,
fum of the extremes is double of the mean;
Thus,
=
Sum of the extremes.
a, a+d, a+2d; then 2 a+2d Double of the mean.
4. Whatever be the number of terms in an
arithmetical progreffion, the fum of the two ex-
tremes will be equal to the fum of any two terms
equidiftant from theſe extremes. Thus, 1+10
=5+6=3+8.
5. In a ferics of terms in arithmetical progref-
fion, let
a
the firſt term.
v=the laft term.
n
the number of terms.
d the common difference.
S= the fum of the feries; then as d is found
in all the terms except the firft, its co-efficient in
the laft term will be n-1, hence a+-1xd=v,
து . - 1x d
as
and
Hasv.
P = p x 1 − 1
6. In any number of terms in arithmetical
progreflion, the first and laft terms, multiplied
by the number of terms, are equal to double the
fum of the whole feries; Thus in the two feries
above (1) which are perfectly equivalent, it is
manifeft that the two taken together, that is the
double of any one of them, will be equal to the
fum of a ſeries of equals a+v, a+v &c. taken
to n terms a tv x n = 25.
7. Therefore, in a continued arithmetical pro-
greffion, if any three of the five quantitics, a, v,
d, a, and s, are fuppofed to be given, the remain-
ing two may be found, as in the following table:
Given.
AND ANNUITIES.
CA)
Given.
a, d, n,
I
บ
a, d, v,
+
a
S
2
a, n, v,
s=nXa+dx"= 1,
v-a
2
X +I, •
s=a+ux
2
Equation I.
2.
3.
d, n, v,
s =nX v — dXn—1,
4.
•
2
a, d, n,
a, d, s,
2 5
a, n, s,
พ
a,
ท
S
d, n, s,
V =
ཡ =
72
d, v, n,
S
d, n, s,
a
v=a+ni Xd,…….
v = √ axa―d + d x 2s +1 — 1d,..
+ ½ d× n−1,...
dx
a = v—ñ—1Xd,
-1dxn−1,..
Xn
कं
5.
6.
7.
8.
9.
10.
12
II.
n, v, s,
a
}}
d, v, s,
v Xv+d — d X 2s+d, •
12.
25
n
a = &d ± √ v Xv + d -
Soughts in Equation 1ft.
Sec. 6. 2s=nXatu
Sought v in Equation 6.
ato X n=25
a
Sec. 5.
12 =
+1
d
2
s = n xato
Sec. 5. va+dxn−1
td X-I
s=nX² a + d x =
2
Hence s=nXa+dX.
+v+a=25
v² — a²+dv+da =2ds
v²+dv=aXa-d+2ds
2
v²+dv+{d²=a Xa¬d+d'X 2044
v+3d = √ @ Xa-dded
A 2
25-
1
THEORY OF INTEREST,
Given.
v-a
a, d, v,
n =
+ 1,
a, d, s,
ď
S
a
n = √ ² s = a + q 2 + − +b….
Equation 13.
a, v, s,
d, v, s,
25
n =
n =
vja
14.
15.
ย
7
+ 2 + 1 - 2/2 + 1/1, . . .
25
16.
¿+ax v — a
a, v, s, d:
d=
17.
25
vta
25-zan
a, n, s,
d =
X
18.
n-
n
a, n, v, d
d =
19.
2 V
25
n
n, v, s, d=
20.
22 — [
Sought in Equation 14th.
Sec. 5. and 6.2 a+d n-d× n = 25
2 and n²-d n = 2.$
2 a
122 af
d
25
ď
!
I
+ *
d
25-
a
d
Sought a in 16th,
Sec. 5. and 6.2 v n—d n—d n² = 2 s
20
n+
+r-
d
2
I
2 V
d
25
}
V
d
+
2
+
+
a?
d2
+
4
Sec. 6.
Sought din 17th.
-a X +1=25
d
v+axv-a
d
+v+a=25
vta × v—a = 2ds—dx vte
Sought din 18th.
2a+n-1xdx n = 2s
2an+n-1xdn = 28
2s-2 an
22Ixd
n
Sought din 20th.
I
}
2 v-n-IXdx n = 2s
I
2
2 v n―n—Ix d n = 2 s
I
&cq
2
I
n
In
AND ANNUITIES.
5
In the above table, the expreffions belong to
an increafing progreffion, and may be applied
alfo to a decreaſing one, by making the quantities
a and v change places every where.
EXAMPLES.
iſt, Sought, the fum of the firſt hundred num-
bers in their natural order. Here, a=1, n=100,
v=100; by equation 3d, a+v× 3 = 101 × 50—
5050=fum fought.
n
2d, Suppofe a baſket and 500 apples were pla-
ced in a ſtraight line, a yard diſtant from each
other, it is required how far one muſt go before
he brings them, one by one, into the baſket?
Here, as the diſtance muſt be gone over twice in
bringing each apple, a=2, d=2, n=500; by e-
quation Ift, n xa+dx n=1=500 x 501250500
yards 142 miles.
I
3d, A man had 12 children; the youngeſt was
three years old, and the common difference of
their ages was four years; What was the age of
the eldeft? Here a=3, d=4, N=12. By equa-
tion 5th, a + ñ—1 x d = 3+44 = 47 years.
8. When a number of terms in arithmetical
progreffion begins with a cypher, (0), which is
the moſt regular feries, in this cafe, the fum of
the feries is cqual to the laft term multiplied by
half the number of terms. For ao, and, èqua-
tion
6
THEORY OF INTEREST,
tion 3d, s=
n
V X Thus 0, 1, 2, 3,...... 31
= 31 × 16496.
9. When a=2, and d=2 alfo, in this cafe, in
equation ift, s = n² + n = a pronic number,
which is produced by the addition of even num-
bers in an arithmetical progreffion, beginning at
2; and the pronic root n = √ 45+1−1.
I.
2
10. When a 1 and d=2, then sn². So that
all ſquare numbers may be confidered as arifing
from the addition of odd numbers in an arith-
metical progreffion. Thus.
Arithmetical Progreffion, 1, 3, 5, 7, 9, 11, 13, 15,
17, 19, &c.
Square Numbers, 1, 4, 9, 16, 25, 36, 49, 64,
81, 100, &c.
Hence may be determined how many odd
numbers muſt be added to produce a given power
of that number. Let the given number be
n, the
exponent of its power m, and the firft term of
the feries a; then fince d=2, and the number of
hence s=n" =n x a + 12-1
s=n™
and a=n"-
n
And thus it appears, how powers of all
dimenfions may be obtained by fumming up of
odd numbers. Thus, fuppofing m=3, and n=
3, 4, 5, 6, &c.
terms n,
·72-
33
3456
M
772
7+9+1 I
13 + 15 +17 + 19.
5 3 = 21+23+ 25 +27 + 29.
6³ = 31 +33 + 35 + 37 + 39 +41.
772 - I
1
CHAP.
AND ANNUITIES.
7
CHAP.
II.
Of SIMPLE INTEREST.
11. WHEN a fum of money is lent upon in-
tereſt, and that intereft is either kept in the bor-
rower's hands, or is paid regularly, without be-
coming a part of the principal, it is then faid to
bear fimple intereſt.
The intereſt of money was formerly very high;
but by an act of parliament, made in 12th Queen
Anne, it is now reduced to, or it muſt not ex-
ceed, 5 per cent. per annum. As the intereſt of
L. I for one year is fo neceffary in computations
both of fimple and compound intereft, it may be
found, for any rate of intereft per cent, by the
following proportions:
3
I
.03
3
33.8
I
3.5
.C35
در)
28.57
e
.04
25
As 100: 4.5, fo is 1:
.045
5
.05
11
6
.06
00
8
I
22.7
4
11
Intereft of L. 1 for one year at
I
20
-18
I
I().
.08
I
12.5
ΤΟ
.I
-19
10
T
4 z per cent,
LA
5
S
10
12.
THEORY OF INTEREST,
12. In fimple intereft, let
p, repreſent any principal lent out upon inte
reſt.
r,
*,
d,-
i,
5,
tereft.
||
the intereſt of L. 1 for one year.
the time.
any number of days.
the intereft, and
the amount of principal and in-
Then, r, 2r, 3 r, 4r,...tr, will be the in-
tereſt of L. 1 for 1, 2, 3, 4,...t times; but 1 :
tr::p: trp=i, the intereſt of p for t times at
the rate r; and drp
= intereft of p for d days at
365
the rate r Hence the following
TABLE.
Given.
t, r, p,
i = trp = tor
tp
+ P &c.
&c. ...
Equation 1.
20
25
i, t, r,
1=...
2.
tr
i, r, p,
t =
3.
i
p,
rp
ist, psr = //...
drp
d₂ r₁₁ i = 108....
d, r, pr
365
4.
5:
EX-
AND ANNUITIES.
9
EXAMPLES.
ift, Sought, the intereſt of L. 356 lent for 12
years, at the rate of 4 per cent. per annum.
Here trp = 356 ×.54 = 192.24 = L. 192, 4 S.
9 d. intereſt.
2d, Sought, the principal whofe intereft will a-
mount to L. 145 in the ſpace of 9 years, at the
rate of 5 per cent.
½
5 d.
principal.
3d, Sought, the
i
Here
tr
145
.45
L. 322, 4s.
time in which L.567, 10 s.
9s. of intereft, at the rate
306.45
will produce L. 306,
of 6 per cent. Here
9, the years
гр
34.05
required.
14s. for
4th, Sought, the intereft of I..378,
the ſpace of 127 days, at 5 per cent. Here
2404·7 — L. 6:11: 9
365
intereft required.
dr p
305
13. Again, the intereſt of any principal p, be-
ing manifeftly in the compound ratio of P the
principal, t the time, and r the rate; it will be,
as above, trp; to which adding p, we fhall have
s, the amount of principal and intereft =
p+
trpi+ir.xp. Hence the following table.
B
Given.
19
THEORY OF INTEREST,
Given.
p, t, r,
s=p+trp=1+tr xp.... Equation 1.
s, t, r,
R = i +tr..
-P
s, r, p,
rp
s, p, t,
T
↑ =
s-p.
#p
EXAMPLES.
2.
3.
4.
!
1, Sought, the amount of L. 156, lent for 12
years, at the rate of 4 per cent. Here+trx p
240.24 = L. 240 : 4 : 9½ d. — $•
1.54 × 156
2d, Sought, the principal which will amount
to L. 873, 19s. in 9 years, at 6 per cent.
S
873.95
1+tr 1.54
fought.
Here
L. 567, 10s. the principal
3d, Sought, the time in which L. 600, 143.
will amount to L. 871.015, at the rate of 4 per
Here Sp
cent.
rp
270.315
27.0315
= 10 years.
14. If the principal or intereft confifts of
pounds, fhillings, and pence, theſe parts of a
pound may be thrown into a decimal fraction,
a
and
AND ANNUITIES.
IF
and the intereft or amount will be found as a-
bove. There is an eafy method of converting the
decimals of a pound into fhillings and pence, viz.
double the firft decimal on the left hand, calling
it fo many fhillings, to which add 1, if the 20
decimal exceeds 5; prefix the remainder above 5,
in the 2d place of decimals, to the decimal in the
3d place, calling them fo many farthings, and
fubtract at or above 25, and 2 at or above 47,
and vice verfa. Thus, L. .7856 = = 15 s. 8 d.
81
and 11s, 6 d. L. .578.
15. If the time t confifts of years, months, and
days, or of days only, their number may be
collected from the following table, fhewing the
number of days from the 1ft, 10th, 20th, &c. of
one month, to the 1ft, 10th, 20th, &c. of any
other month; where the months on the left hand
column are the former dates, and thoſe at the
top are the latter. Thus, from ift of July to
ift of April following, are 274 days; and fron
20th of March to 10th of Auguft, are 153-10=
143 days.
B 2
TABLE.
12
THEORY OF INTEREST,
From Ift, &c. of.
1
TABLE.
To ift, &c. of.
M. Jan. Feb. |Mar April May June July Aug. Sept. | Oct. Nov. Dec.
Fan. 365 31 59 90
120 151 181 212 243 273|304|334
Feb. 334 365 28 59
March. 306 337 365 31
89
120 150 181 212 242 273 303
61
92 122 152 184 214 245 275
April. 275 306 334 365 30 61 91
May. 245 276 304 335 365 31
276|304|335| |
122 153 183 214 244
61
92
123 153 184 214
June. 214 245 275 304 334 365
30
61
92
122 153 183
July. 184 215 243 274 304 335 365 31
Auguft. 153 184 212 243 273 304 334 365 31
Sept. 122 153 181 212 242 273 303 334 365
OЯ. 92 123 151 182 212 243 273 304 335 365
Νου. 61 92
Dec. 31
62
Days. 31 28
90 121 151 182 212 243 274 304 335 365
3I 30 31 30 31 31 30 3 I 30 31
62
92 123 153
30
སྦྱ「ཚུ
61 92122
61 | 91
31 61
120 151 181 212 242 273 304 334 365 30
16. Theſe days may be thrown into the deci-
mals of a year by the following Table.
Days. Decimals Days. Decimals. Days. Decimals.
I
.0027397
+423 +
.0054794
8 .0219178 60
9.0246575 70
•
1643835
1917808
.0082192
ΙΟ 0273972
80
.2191730
4
.0109589
20 .0547945
90
.2 4 6 57 53
.0136986
30.0821918
100
.2739726
.0164383
-5479452
40.1095890 200
7 .0191780 50.1369863 300 .8219178
Thus it is required to exprefs 143 days in the de-
cimal fraction of a year,
100
AND ANNUITIES.
13
}
100.2739726
40=.1095890
3.0082192
143.3917808 of a year.
Ex. Required the intereft of L. 356, 12 s. for
3 years and 143 days, at five per cent. per an-
Here
num.
trp=3.41178x.05×356.6=L. 60: 16: 7 Intereft.
60:16:
17. The intereſt of any fum of money, for any
number of years, at a given rate, may be collect-
ed from the following table, calculated for the
nine digits, at feveral rates of fimple intereft.
ερ
TABLE.
trp for years.
3 pr cent 3 pr cent 4 pr cent 44 pr cents pr cent (5 pr cent
I
.03
2 3
.05
.07
.035 .04
.08
.045
.05
.06
.09
.10
.12
.09
.105 . I 2
.135
.15
.18
4
.12
.14
.16
.18
.20
.24
5
.15
.175 .20
.225
.25
•30
6
.18
.2[
.24
.27
:30 .36
7
.2[
.245
.28
• 315
35
.42
• 2.4
.28
• 32
•36
•47
·48
9
.27
·31536
•405 •45
•54
In this table, the decimals, for inftance, cor-
refponding to 9 exprefs the intereft of L. 9 for 1.
year, or of L. 1 for 9 years, or of p
for t years,
whatever
14
THEORY OF INTEREST,
whatever theſe be, provided their product tp=9,
at a given rate. Of confequence the intereft-co-
lums are equal to trp, while the marginal one
correfponds to tp. And it is evident that the
tabular numbers will not only anfwer for units,
but alfo for tens, hundreds, or thoufands, by re-
moving the decimal point one, two, or three pla-
ces to the right hand. Hence, in ufing this ta-
ble, multiply the principal by the number of
years in queſtion, and take out the decimals cor-
refponding to their product; thefe being properly
pointed and added together will exprefs the inte-
reft required.
Ex. Required the intereft of L. 348, 12s. for
12 years, at four per cent. per annum. Here
3+8.6x12=4183.2.
By the Table.
4000
160.0
100
4.0
80
3.2.
3
.12
.008
Intereſt.
L. 167:6:64 = 167.328
If the given rate be not in the table, the inte-
reſt may be found by reduction.
18. The intereſt of any fum of money, for any
number of days or months, at a given rate, may
be collected from the following table, calculated
for the 9 digits, at feveral rates of fimple inte-
reft.
TABLE
AND ANNUITIES.
17
TABLE.
drp for Days.
365
dp
4 per cent. 4 per cent.
5 per cent.
I
mip for Months.
12
4 pr cent 4 p. cent 5 pr cent.
.00010959 .00012328.000136986.0033.00375.00416
2.00021918.00024657.co0273972.0060.00750.00833
3.00032876.00036986.000410958
.0100.01125.01250
4.00043835.00049315.00054794
0133.01500.0166€
.00054794.00061643.00068493
.0166.01875.c2083
6.00065753.00073972.000821918
.0200.022501.02500
7.00076712.00086301.00095890 023302625.02916
8.00087671.00098629 .0010959 .0256.03000.03333
.00098630.00110958.00123287
I
.03001.03375.03750
In this table alfo, the decimals, for inftance,
correfponding to 6, exprefs the intereſt of L. 6 for
1 day, at a given rate, or of L. 1 for 6 days, or of
p for d days, whatever thefe be, provided their
product dp-6. Of confequence the intereft-co-
lumns are equal to dip, while the marginal one
365'
correfponds to dp. The fame of months. In
computing intereft by days, which is more ufual
than by months, the year is always fuppofed to
confift of 365 days; the 29th of February, in leap
years, not being taken into the account. Alfo the
day computed from is not reckoned, but the day
computed to is: and in difcharging bills, three
days of
grace are allowed the acceptor to make
payment.
In ufing the above table, multiply the princi-
pal by the number of days or months in queftion,
and take out the decimals correfponding to their
product;
16
THEORY OF INTEREST,
product; theſe being properly pointed and added
together will exprefs the intereſt required.
Ex. Required the intereft of L. 564 for 238
days, at 41 per cent. per annum. Here 564×238=
4½
134232.
By the Table.
:
100000
12.3287
30000
3.6986
4000
.49315
200
.02465
30
.00369
2
.00024
Intereſt fought.
L. 16, 11 S.- 16.54903
If the given rate be not in the table, the inte-
reft be found by reduction; or by the fol-
lowing table.
may
19. The intereft of any fum of money, for any
number of days, and at any given rate, may be
collected from the following table.
TABLE.
drpd for Days.
365
I
23
.002739726
.005479452
.008219178
4 .010958904
5 .013698630
6 .016438356
7
.019178082
.021917808
9.024657534
In this table, the digits in the
firſt column are the products
drp; and the decimals are the
quotients of thefe products di-
vided by 365, and, of conſe-
quence, are the intereft of p for
d days, and at the rate r, what-
ever thefe factors be. There-
fore to find the intereft of any
fum of money by the table,
multiply
AND ANNUITIES.
17
;
multiply the principal, the number of days, and
the rate of L. I per annum, into one fum, and
from the table take out the intereft correſponding
to their product.
Ex. Required the intereft of L. 342, 15 s. for
260 days, at 5 per cent. Here 342.75 × 260 ×.05
=4455.75•
By the Table.
4000
10.9589
400
1.09589
50
.13698
5
.01369
.7
.00192
.05
.000137
L. 12:4:2 = 12.207517 Intereſt.
=
20. Or the intereft of any fum of money, for
any number of days, at five per cent. may be dif-
covered by the following proportion:
100x365:5:: pxd: px
equations,
pxd
7300°
Hence the following
ift, -Intereft of p for d days,
7300
2d, Intereſt of p for 100 days,
73
3d,
d
73
Intereft of L. 100 for d days,
d
P
at 5 per
cent.
4th, When 1, 2, 3, &c. then intereſt
73
of p for 73×1, 2, 3, &c. days.
100
Ex. ft. Suppofe p-L. 564, and d=238 days;
then p×
tereft.
7300
134232
7300
18.388
C
L. 18: 7:9, the in-
Ex. 2d,
18
THEORY OF INTEREST,
1
Ex. 2d, Suppofe p=L. 342.75, and d=260 days;
then,
2=342.75 -4.6952
73
1
73
2
9.3904 Int. for 200 days.
=
T = 2.81712
12.20752=Intereſt required.
Ex. 3d, Suppofe p-L. 448, and d236 days;
then,
d
=235=3.23287=Intereſt of L. 100.
73
A
5
4
12.93148
= 1.29314
= .25863
14.48325 = Intereſt of L. 448.
Of Rebate or Diſcount.
21. To find p, the prefent value of
any
fum
وک
due at any time t, and at any rate of intereſt r.
Here (13, Eq. 2.)
s-p-diſcount.
S
p the prefent value of s, and
Ex. What ready money will pay a bill of
L. 682, 18 s. due 3 years and 9 months hence,
diſcounting intereft at the rate of 5 per cent.
Here
S
682.0
1.1875
1+tr
682.9-
=575.07 = p, and
575.07= L. 107: 16:7
107:16:74
the diſcount.
If the
time
:
AND ANNUITIES.
I
time be days, they may be thrown into the deci-
mals of a year by a former table (16.) and then
you may proceed as above.
Ex. What is the prefent value of a bill of L. 285,
10s. due 95 days hence, diſcounting intereſt at 5
per cent.
95 = .26=
95.26=t
.05=r
285.5 281.841 p.
1.013+
1.013=1+tr, 285.5—281.841—3.66=diſcount.
d
73
Or thus, feeing intereft of L 100 for d days at
5 per cent. (20); hence, in the laſt example,
9
7
21.3 Intereft of L. 100 for 95 days;
then 101.3: 1.3:: 285.5: 3.66-diſcount.
3.66
281.84-prefent value.
But as the difcounting bills in this manner
is no better than lending money upon intereft,
bankers, and others who keep money for the pur-
pofe of diſcounting bills, confider the fum to be
paid in the bill as a principal, find the intereſt
thereof for the time to run, adding thereto
three days of grace, and deducting the intereft fo
found from the content of the bill, they pay the
balance to him for whom the diſcount is made :
fome, likewife, for their trouble, charge or
per cent. commiffion, which, before deduction, is
added to the intereft.
2
Ex. A bill is prefented the 5th of May for dif
count, of L. 350, 16 s. payable 27th of July;
how much money does the bearer receive, after
deduction
C 2
י
20
THEORY OF INTEREST,
1
deduction of intereft and per cent. commiffion?
Here p 350.8, d=83+3; hence
3 50.8×86=4.133=Intereſt.
7300
.877 Commiffion.
5.010 Sum to be deducted.
Hence L. 345.790 paid the bearer of the bill.
But the true diſcount being preciſely equal to
the intereft of the prefent value of the bill for the
time to run, as calculated in a former example, it
is evident that the bearer is a confiderable lofer by
this tranfaction; and it will be found, that, call-
ing the intereft of the bill its difcount, the bearer
thereof lofes exactly, befides commiffion, the in-
tereſt of its true difcount for the time to run,
For calling d the true diſcount found by 13. Equa-
tion 2. then ſeeing s=d+ p.
I-(intereft of s) intereft of d+p, for the time
of diſcount; but dintereft of p for the faid
time hence i-d, (the bearer's lofs when i is the
difcount) Intereft of d for the time of difcount.
Ex. A bill of L. 210 payable a twelvemonth
hence is prefented to be diſcounted; required the
bearer's lofs in this tranfaction :
L. 10 10 the intereſt of the bill.
10 oo the true diſcount.
of true diſcount
10 s. the bearers lofs-intereſt
per cent. per annum, upon the
prefent worth of the bill, for the time of difcount.
When the rate of intereft is not mentioned, 5 per
cent. the legal intereft, is always underſtood.
22. The
AND ANNUITIES.
21
22. The feveral particulars refpecting fimple in-
tereſt will more readily be found by the help of
the four following tables, wherein are calculated
the amount and prefent value of L. 1 for days or
years, at ſeveral rates of intereſt.
TABLE I. Shewing the Amount of L. 1 for
I
Days.
1
D
4 per cent. 4 per cent.[5 per cent. D. 14 pr cent. 41 pr cent.5 pr cent.
11 00010958 1.00012328 1.00013698|| 27| 1.002959 1.003328 1.003698
21.00021917 1.00024657 1.0002739 28 1.003068 1.003452 1.003835
31.00032876 1.00036986 1.0004109 29 1.003178 1 003575 I 003972
41.00043834 1.00049314 1.0005479 30 1.003287 1.003698 1.004109
51.00054794 1.00061643 1.0006849 31 1.003397 1.003822 1.004246)
61.000657521 c0073972 1.0008219
7 1.00076710 1.00086300 1.0009589
81.00087668 1.00098628 1.001958
91.00098628 1.0011095 1.0012328
101.co10958 1.00123281.0013698
11 1.0012059 1.0013561 1.0015068
12 1.0013501.0014794 1.0016438
131 0014246 1.0016027 1.0017808
14 1.0015342 100172601.0019178
15 1.0016438 1.0018493 1.0020548
32 1.003507.1.003945 1.004383
321 003616 1.004068 1.0045 20
34 1.003726 1.0041917 1.004657
35 1.003835 1.0043151.004794
36 1.003945 1.004438 1.004931
+
40 1.004383 1.004931 1.005479
50 1.005479 1.006164 1.006849
60 1.006575, 1007397 1.008219
701 007611 1.008630 1.009589
80 1.008767 1.0098621.010959
窄
​16 1.0017534 1.001972 1.0021917
90 1.009863 1.011095 1.012328
17 1.0018629 1.0020958 1.0023287 100 01058 1.012328 1013698
18 1.0019726 1.002219 1.0024657 OC 1.021917|1.02465 1.027397
19 1.0020821
1.002242
1.0026027 300 1.03287 1.03697
20 1.00219178 1.002465
1.0027397 310 1.033971 03822
21 1.0023013 1.002589
22 1.002410у 1.002712
23 1.0025205 1.002835
1.0028767 320 103506|1.03945
1.0030137 330 1.03616
1804068
24 1.0026301 1.002958
1
25 1.0027397 1.003082
26 1.0028493 1.003205
1.041095
1.042465
1.043835
1.045205
і
1.04315
104438
1.047945
1.049315
1.045
1.05
1.0031506 340 1.03726 1.041917 1.046575
1.0032876 350 1.03835
1.0034246 360 1 03945
1.0035616 1:65|1.04
TABLE
22
N
THEORY OF INTEREST,
A
4
I
TABLE II. Shewing the Amount of L. 1 for
Years.
Yrs 4pr cent. 41p.cen. 5 pr cent. 6pr cent. Yrs 4pr cent. 4p.cen.15 pr cent.6pr cert.
I
1.04
1.045 1.05 1.06 31 2.24
2
1.08
1.09
I.IO
I.12
32 2.28
2.395 2.55 2.86
2.44 2.60
2.92
31
1.12
1.135
1.15
1.18
33
2.32
2.485 2.65
2.98
4 1.16
1.18
1.20
1.24
34 2.36
2.53 2.70
3.04
5
1.20
1.225
1.25
1.30
35 2.40
2.575
2.75
3.10
6 1.24
1.27
1.30
1.36
36 2.44
2.62
2.80
3.16
7
1.28
1.315
1.35
1.42 37 2.48
2.665
2.85
3.22
8 1.32
1.36
1.40 1.48
38 2.52
2.71 2.90
3.28
9 1.36
1.405
I.45
1.54
39 2.56
2.755 2.95
3.34
IO 1.40
1.45
1.50
1.60
40 2.60
2.80
3.00
3.40
II
1.44
1.495
1.55
1.66
43 2.64
2.845
3.05
3.46
12 1.48
1.54
1.60
1.72 42 2.68
2.89
3.10
3.52
13 1.52
1.585
1.65
1.78 43 2.72
2935
3.15
3.58.
14 1.56
1.63
1.70
1.84 44 2.76
2.98
3.20
3.64
15 1.60
1.675 1.75
1.90 45 2.80
3.025
325
3.70
16 1.64'
1.72
1.80
1.96
46 2.84
3.07
3.30
3.76
17 1.68
1.765
1.85
2.02
47 2.88
3.115
3.35
3.82
18 1.72
1.31
1.90
2.08
48 2.92
3.16
3.40
3.88
19 1.76
1.855
1.95
2.14
49 2.96
3.205 3.45
3.94
20 1.80
1.90
2.00
2.20
50 3.00
3.25
3.50
4.00
21
1.84
1.945
2.05
2.26
51 3.04
3.295 3.55
4.06
22 1.88
1.99
2.10 2.32
52 3.08
3.34 3.60
4.12
23 1.92
24 1.96
2.035
2.15 2.38
53 3.12
3.385 3.65
4.18
2.08
2.20
25
2.00
2.125 2.25
2.44
2.50
54 3.16
3.43 3.70
4.24
55 3.20 3.475
3.75
4.30
26 2.04
27 2.08
2.17
28 2.12
2.30
2.215 2.35 2.62
2.26
2.56
56 3.24
352 3.80
4.36
57 3.28
3.565 3.85
4.42
3.6x 3.90
4.48
3.655
3.95
4.54
2.40 2.68
58 3.32
29 2.16 2.305 2.45 2.74 59 3.36
30 2.20 2.35 2.50 2.80 60 3.40 3.70 4.00 4.60
Tables I. and II. fhew the amount of L. 1 for days
or years, and are conftructed by Equation 1. 13.
whereby n the tabular number=1+ir. Or thus,
Table I. is conſtructed by taking 364 mean arithme-
tical proportionals between o and the rate of L. I
for one year, or by dividing this rate by 365,
the quotient gives the intereft of L. I for one day;
and this intereft added to itfelf continually gives
the intereft for any number of days, to which
prefixing 1, you have the amount of L. 1 for thefe
I
days.
AND ANNUITIES.
23
days. If a given number of days are not in the
table, divide them into fuch numbers as are to
be found there, and to the fum of the decimals
correfponding to theſe numbers prefix 1, the whole
will exprefs the amount of L 1. for the given
number of days at a given rate. Or remove the
decimal point, where it can be done, one figure
to the right hand, and you have the amount for
10 times the tabular number; thus 240=1.032876.
at five
In like manner Table II. is con-
per cent.
ſtructed by adding the intereſt of L. I continual-
ly to itſelf for any number of years, to which
adding 1, you have the amount of L. 1 for thefe
years. If the given time confifts of years and
days, to the tabular number correſponding to the
years in Table II. add the decimals or intereſt an-
fwering to the days in Table I. their fum will be
the amount of L. 1 for the given time. From the
nature of theſe two tables, I: n::p:np=s the a-
mount of p, for the time and at the rate corre-
fponding to n. Hence the following table:
TABLE.
Given.
p, t, r,
s, t, r,
s, r, p, l
|s, t, p,
s=np.
P
p
12.
The rate r being given, t
the time is found oppofite
ton; and t being given, r
is found above n in the ta-
bles.
Ex. ift, Sought, the amount, L. 356, 10 s. for
160 days, at 4 per cent. By Table I. np=1.017533
×356.5--L. 362.75; and intereft=L. 6.25
2d, Sought, the amount of L. 300 for 7 years
and 165 days, at five per cent. By Table II. and I.
22+n-1xp= 1.3726 × 300- L. 411.78. amount
fought.
3d,
24
THEORY OF INTEREST,
3d, Sought, the time in which L. 300 will a-
mount to L. 400 at 5 per cent. Here=
1.33333-6 years +200+40+3 days.
P
TABLE III. Shewing the prefent Value of L. i
for Days, at five per cent.
Dys
I
2
3
4
นา
5
6
7
8
9
0.99
9863
97209589
9452 9315 9179
9042 8905 8768
ΙΟ 8632
8495
8359
8277
8086
7950
7815
7678 7541 7403
20
7266
7128
6991
6857
6723 6587
6451
6365
63656179 | 6043
30
5907 5771 5635
5549 5463 5377
5291
5106 4921 4735
40
4550 4414
4279
4144 4009 3873
3738
3603 3468
|
3332
50
3197 3062
2927
2388
60
1848
1713
1578
70 0502
0367 0233
80.989163
9028 | 8893
8893
90 7823
7689 7556
7021
2792 2657 2522
1443 1309 1174
00999965
9965 | 9831
8758 8623 8489
8758 8623
7422 7288 7154
1010 0905 0771 0636
96979563 | 9430 | 9296
8356 | 8223 | 8090 7956
6887 6754
6621
2253
2118 1983
100
6488 6356 6225
6225
IIO 5156 5023 4891
120 3829 3696 3504
3299
130 2504 2372 2240 2107 1975
140 1184 1052
6093 5962
5827
5692
5692 | 5557 5422 5289
4758 4625
4410
4360 4227
4094
3961
3434
3166
3934 2901 2769
2686
1843
1711
1579 1447 1315
0920
0788
0657
0525
0394
0262
0131 9999
150.979868 9736
9605
9605 9473
9342
9210 9079
8947
8816 8684
1160
8553 8422
8291 8160
8029
7898 7767
7636
7605 7374
170
7243
7112
180
5936 5806
190 4633 4503
69826851 6720
5075 5545 5445
4373 4243 4113
6589 | 6449
6318
6187
6057
5284 5154
3983
5024 4893 4763
3853 3723 3593 3463
210
200 3333
2038
3203
2815 2685
1909
1521
2944
25562426
3074
1779 1650
2297
2167
1391
1262
1133 1003 0874
220 0745
0616 0487 0358
0229
ΟΙ Ι
9972
9843
9714 9585
230.969457
9328 9200
9071
8942
8814
8685
8557
84288309
240 8171 8043 7934
7786
7658
7529 7401 7273 7144 7016
2501
6388 67606632
6504
6376 | 6248 | 6121
5993 5865 5737
260
56095481 5354
5226 5098 4971 4843
4716 4588 4460
270
4333 4205 4078 3951 3824 3697 3570
280 3061 2934 2807 2680 2553 2426 2300
290 1792 1665 1539
|
1412 1286 1159 1033
3442
3315
3188
2173
2046 1919
0906
0770
07700653
1300
340
350
13601
5498 5373 5248
4249 4124 3999
3002: 2878. 2753
0527 0401 0274 0148 0022
310.959265 9139 9013 8387
1320 8006 7880 7755 7629
7378 7252 7127 7001 6875
330 6750 6625 6499 6374 6249 6124 5999
5123 4998 4873
3875 3750 3625
2629
2381
9896 9769 9643
9643
9517 9391
8761
8635 8509
86358509 | 8384
8384 8258 8132
7503
5873 5748 | 5623
4748
4624 4499 4374
3501
2505
3376 3251
TABLE
3127
AND ANNUITIES.
25
TABLE IV. Shewing the prefent Value of
L. I for Years.
I
I
I+tr
Ys4 pr cent.15 pr cent. 6 pr cent. Ys4 pr cent.5 pr cent.6 pr cent.
1.961538952381 943396 26.490196.434781.390625
2.925926.909091 .892857 27.480769.425532.381679
3.892857.869565 847457 28.471698.416666.373134
4.862069.833333.806451 29.462963.408163.364963
5833333.8
.769230 30.4545454
-357143
6.806451.769230.735294 31.446428.393157.349650
7.781250.740740 .704225 32.438596.384615.342466
8.757576.714286.675675 33.431034-377358 .335570
9.735294689655 .649350 34.422728.370370-328947
10.714286.666666 .625 35.416666.363636.322581
11.694444.645161.602409 36.409836.357143.316456
12.675675 625 .581395 37.403226350877.310559
13.657895.606060.561797 38 396825.344827 304878
14.641025.588235-543478 39-390625.338983.29940г
15.625
-57142852631540384615-333333.294117
16.609756.55555551020441-378788.327870.289017
17.595238540540.49504942.373134.322581.284091
18 581395.526315.480769 43.367647.317460.279330
19.568182 .512820.467289 44.362319.3125
20.555555 |·5
-454545
.274725
45.357143.307692.270270
21.543478.487804.44247746.352113.303030.265957
22.581915.476190.43103447-347222.298509.262383
23.520833.465116.420168 48.342466.294117.25 77 32
24.510204.454545409836 49.337838.289855.253807
1.285714.25
25.5
.444444 .4
150-33333
!
Tables III. and IV. fhew the prefent value of
L. 1 for days or years, and are conftructed by E-
quàtion 2d, 13. whereby n, the tabular number,
For they are the reciprocals of Table I. and
II. whereby of Table I. and II.=" of Table III.
and IV. From the nature of theſe two Tables,
1:n::s:ns=p,
Insns=p, the principal or prefent worth
S, for the time and at the rate correfponding
of
to 11.
2
D
When
26
THEORY OF INTEREST,
When the time of diſcount confifts of years and
days, in this caſe, nxns=p, the prefent worth.
Hence the following
TABLE.
Given.
s, t, r,
p=ns.
p, t, r,
s=²
s, r, p,
n=?
(s, t, p,
Ex. 1ft, What ready money will take up a bill
of L. 685, 12s. due 255 days hence, intereſt be-
ing reckoned at 5 per cent? By Table III ns=
•966248 × 685.6=L. 662.4596=p, and diſcount =
L. 23.14.
2d, Required the prefent worth of L. 250, dif-
counting for 10 years, at the rate of 5 per cent.
By Table IV, ns=.6666 × 250 = L. 166.666=p, and diſcount —
·L. 83.33.
3d, What ready money will pay a debt of
L. 150 due 5 years and 95 days hence, intereſt be-
ing reckoned at 5 per cent.? By Table IV. and
III. nns=.8×.987154×150= L. 118.45848=p, and
L. 31.541-difcount.
4th, Havings, p, and r, in the laft example,
fought, t the time. 18.45848.7897232=nn, and
7897232
85 yrs
£ 2 —
-.987154=95 days.
Theſe examples may with equal facility be de-
termined by the equations annexed to Table I.
and II.
Re-
AND ANNUITIES:
27
Remarks upon Simple Intereft.
23. 1ft, If the time be fought in which a fum
of money will double itſelf at a given rate of in-
tereft. In this cafe, the intereft being equal to
the principal, trp=p, which gives tr=1, t=‡ and
r=; ſo that t and are reciprocal of and diſcover
one another, when a ſum of money doubles it-
felf at ſimple intereſt. Or thus, feeing (12) inte-
reft=, or, &c. according to the feveral rates
of intereft; hence if t=20, 25, &c. when the
rates are 5, 4, &c. per cent. the intereft will e-
qual the principal
So that
25
33.33
28.57
25
3/12/2
334
પત
Intereſt➡p, when t― 22.24 years at 4 per cent. per ann.
20
16.68
12.5
10
6
8
10, &c.
2d, The fimple intereft of any fum of money
is, ceteris paribus, in the direct proportion of p, t,
or r; fo that doubling p, while and remain
the fame, the intereft will be doubled alfo, &c.
For as intereſt=trp, if you increaſe or diminiſh
any one of theſe three factors, the intereft will be
increaſed or diminiſhed in the fame proportion.
D 2
3d, The
,
28
A
1
:
THEORY OF INTEREST,
3d, The intereſt of any fum of money, for a
given time, and
at a given rate,
may be diſcover-
ed by means of a
right-angled tri-
D
trp
E
I
angle.
Thus.
In conftructing
the right-angled
triangle
ABC,
make AB, and
B
I
P
BC=p; in AB
C
take the point D,
above or below
BC, making AD=t, and join AC; then DE drawn
parallel to BC will reprefent the intereft fought,
For AB, or being equal to the time in which p
will double itſelf at the rate r, BC-intereſt of p at
the end of that time; but AB: BC:: AD : DE,
therefore, by the laſt remark, DE=intereſt of p for
the time t, and at the rate r.
Ex. Suppofe p=L. 460, t=12 years, and the
rate 5 per cent.; then AB-20, BC=460, AD=12,
and DÊ-L. 276=intereſt.
4th, The different amounts are, ceteris paribus,
in the direct proportion of their principals or pre-
fent worths; ſo that doubling p, while t and r re-
main the fame, the amounts will be doubled al-
fo, &c. For as the amount s=p+trp, ſo 2s=
2p+trx2p.
But the different amounts of the fame princi-
pal, or the different prefent worths of the fame
bill, are not in the direct proportion of t or r; for
2s is greater than p+2trp; and though,
yet
AND ANNUITIES.
29
yet is greater than ip.
Hence the different
I+2tr
diſcounts of the fame bill are not in the direct
proportion of the times or rates.
5th, The amount of any fum of money, for a
given time, and at a given rate, may be diſcover-
ed by a geometrical figure, two of whofe oppoſite
fides are parallel, and two of its angles right.
Thus, in conſtructing the annexed figure, make
E
A
G
F
H
I
Þ
Þ
C
B
D
AB=1, CD, bifected in B, = 20; in AB take AF
=t; and having completed the parallelogram AC
and triangle ABD, GH drawn through F paral-
lel to CD will repreſent the amount fought. For
as GF=p, and FH, by remark 3d, is equal to
trp, hence GH=s the amount.
Ex. Suppoſe p=L. 300, t=9 years, and the rate
5 per cent.; then AB=20, AF-9, CD=600, GF
=300, and GH-L. 435, the amount.
6th, As in fimple intereft, ftrictly speaking,
the intereft is fuppofed not to be uplifted while
the
30
THEORY OF INTEREST,
the principal remains in the borrower's hands; it
muft alter the cafe greatly when that intereft is
paid yearly, and make it partake of the nature of
compound intereft, feeing one hath the uſe, and
can diſpoſe of that interelt.
Supplement to Chapter II.
ift, Of intereſt due upon caſh-accounts and par-
tial payments.
24. In calculating intereft on cash-accounts,
or where ſeveral partial payments are made at dif-
ferent periods on the fame fum, multiply. the
principal and the ſeveral balances into the num-
ber of days they are at intereft; the fum of theſe
products divided by 7300 (20) will give the total
intereſt at 5 per cent.
Ex. 1. Lent A B, the 11th of November 1788,
L. 800, which I received in the following partial
payments; what intereſt is due at 5 per cent?
1738
Nov. 1. Principal lent,
1789.
L. d
800 65 52000
Jan. 15. Received in part,
250
Balance,
550 92 50600
April 17. Received in part,
150
Balance,
400 62 2480o
June 18. Received in part,
100
Balance,
300 62 18600
Aug. 19. Received in part,
125
Balance,
175 37 6475
Sept. 25. Received in full of principal, 175
}
1524.75=20.887
Int.
O
73
Bankers
AND ANNUITIES.
31
Bankers fometimes receive at 4 and lend at 5
per cent.
In this cafe it
may be proper to confi-
der the money lent by the banker as debtor, and
the money received by him as creditor, and to
make two columns for products, where the intereſt
arifing from the debtor products is to be compu-
ted at five, aad the other at 4 per cent.; the dif-
ference of theſe two is the balance of intereft due
to or by the banker.
Ex. 2. A B runs a cafh-account with a bank-
er to the extent of L. 500, at 5 per cent. for the
balances due by him, and has allowed him
4 per cent. for fuch balances as may be due to
him. What intereft will be due, and by whom,
in confequence of the following tranſactions?
1789.
L. d. Dr
Jan. 8. Dr 200
March 10. Cr: 350
May 15. Dr 310
Cr
6I
12200
Cr 150
66
9900
Dr 160
66 10560
July 20. Cr
240
Cr 80 52
4160
Sept. 10. Dr 410
Dr 330 52 17160
Nov. 1.Cr 420
Cr 90 | 49
4410
Dec. 20.Dr 250
1790.
Jan. 8.Dr 160
19
3040
42960 | 18470
મ
32
THEORY OF INTEREST,
1
429.60—4) 184.7-L. 3.86 Int.
73
5 73
}
due to the banker.
160.00 Prin. Š
When partial payments are made on bills or
bonds at any interval greater than a year, the le-
gal and ufual method is to add the intereſt at the
times of payment to the principal, and from that
amount to deduct the payment.
Ex. 3d. Borrowed on bond, June 1. 1781, the
fum of L. 1000, at 5 per cent. and made partial
payments as follows; required, the ſtate of the af-
fair on the 14th of Auguſt 1785.
1781.
June 1. Principal borrowed,
Interest for 1 year and 129
L.
1000
67.671
days,
1782.
Amount,
Oct. 8. Paid in part,
Balance,
1067.671
250
817.671
Intereft for 1 year and 85
days,
50.4
1784.
Amount,
868.071
Jan. 1. Paid in part,
408.071
Balance,
460.
Intereſt for 1 year
and 225
days,
37.178
1785.
Amount,
497.178
Aug. 14. Paid in full,
497.178
2. Of
1
AND ANNUITIES:
33
2d. Of the equation of payment.
25. When two or more debts are payable at
different times; the finding a mean time at which
all the debts may be paid at once, without loſs to
debtor or creditor, is called equating the terms
of payment, and may be performed by the fol-
lowing
RULE.
Multiply the feveral debts into their reſpective
times, and divide the fum of the products by the to-
tal debts; the quotient is accounted the mean time.
Ex. A owes B L. 800, whereof L. 200 are to be
paid at the end of 2 months, L. 300 at the end of
6 months, and L. 300 at the end of 10 months;
required the equated time for paying the whole.
200 x 2
2 = 400
6 1800
300 × 6 =
300 × 10 = 3000
800
)5200(6.5 months, anſwer.
The above method is eaſy, but not accurate :
for a perfon, by keeping money unpaid after it be-
comes due, gains the intereſt thereof for that time ;
but by paying money before it is due, he does
not loſe the intereft, as the Rule fuppofes, but
only the diſcount thereof for that time, which is
always leſs than the intereft, by the intereſt of
the diſcount for the time; therefore take the fol-
lowing
RULE.
Find, by Equation 2d, 13. the
of each debt, and by Equation 3d,
time the fum of the prefent worths
to the fum of the debts.
E
prefent worth
find in what
will amount
EN. A
15
34
THEORY OF INTEREST,
Ex. A owes B L. 250, whereof L. 50 is pay-
able 1 year hence, L. 100 3 years hence, and
L. 100 5 years hence; what is the equated time
for paying all theſe debts at once, intereſt reckon-
ed at 5 per cent.? Here,
50
1.05
100
1.15
100
1.25
гр
47.619 then 35.428-3.302-3 years
and 110 days.
86.956
80.000 =p•
214.575
3d. Of the premium of commiſſion.
It is ufual with bankers, in granting a draught
upon their correfpondents either at home or a-
broad, payable to the bearer or order, for value
received, to charge fo much per cent. upon the
tranfaction, in name of commiffion; which com-
miffion varies from to 2 per cent. according to
the nature of the cafe; hence the following
ठ्ठ
2
TABLE.
1 15
Per cent.
Com. on L. I.
Per cent.
Com. on L. I.
L. S. d.
02 6
L.
S.
.005
0 15
I I O
.0075
.01
0 3
3 4
4
5
.00125
.0015
.00168
.002
.0025
6 8 .0038
Ο 1Ο
141237
1 2 2 2 2 3 3)
5
.0125
I 10
.015
.0175
2 O
22
5
2 IO
23
2 15
.02
.0225
.025
.0275
3 Q .03
3/12/13 10 .035
Ex. Sought, the commiffion upon L. 542 at 1-
Here 542×.015=L. 8.13, the commif-
per cent.
fion.
CHAP.
CHA P. III.
Of ANNUITIES computed at SIMPLE INTEREST.
26.
AN annuity, rent, or penfion, is a fum of
money payable yearly, every half year, or quar-
terly, to continue for a certain number of years,
for life, or for ever. When an annuity conti-
nues unpaid after it falls due, it is then faid to be
in arrears. When the purchaſer does not imme-
diately enter upon poffeffion, the annuity not
commencing till fome time after, it is faid to be
in reverfion; and the reverfion of an annuity is
the preſent value of a perpetuity which is to com-
mence after the expiration of the annuity, being
the fame with a reverſion in fee-fimple.
Ift. Of the amount of an annuity in arrears.
27. In computing annuities at fimple intereft,
let,
a repreſent the annuity, rent, or penfion.
P
t
S
its prefent value.
the time of its continuance.
the number of years before it com-
mences.
the intereſt of L. I for one year.
the amount of an annuity and its in-
tereſt.
E 2
Then,
36
THEORY OF INTEREST,
*
Then, fince L. 1:r.:a: ar, and as the firſt
year's annuity bears no intereft, being due only
at the end of the year, we fhall have, from an
yearly annuity, a,
axito
a × 1+r
a × 1+ 2r
axitzr
amount of the 1ſt year.
= amount of the 2d year.
= amount of the 3d year.
= amount of the 4th year,
@ × 1+1=1×r = amount of the t year.
a
Here the number of terms of this feries being
†, the fum s=tax1+r (8).
ing
Hence the follow-
TABLE,
Given.
a, t, r,s=taxı
= taxi+=r=ta+xtra-ra. Equation 1.
s, t, r,
a
txi+=r.
25
=7
20
s, t, a,
-IX a
s, r, a, t = √ 121/4 + 1.
ar
Sought r, in Equation 3d.
Equ. 1. sta+12-14.
2
25 — 2ta=12a-taxr.
25
2a
24t- I xar.
2
}
Sought t, in Equation 4th.
25 = 2att²a — to xr.
Equ. I.
25
2a1
+12a-te.
21
23
ar
23
ar
2.
3,
ள்
4.
Ex. ift.
AND ANNUITIES.
37
Ex. 1ft, Required the amount of an annuity of
L. 20 per annum, being in arrears for 12 years,
at 5 per cent. fimple intereft. Here a=20, t=12,
r=.05, and tax ir=240x1.275=306=s.
2
2d, Sought, the annuity, which being in
arrears for 9 years will amount to L. 250, at
5 per cent. Here s 250, t = 9, r = .05•
250 —L. 23.148
S
t-I
tx1+10.8
X I
2
annuity.
3d, Sought, the rate of intereft at which an
annuity of L. 20 per annum, being in arrears for
12 years, will amount to L. 306.
7.5
20
t =.055 per cent.
-Ixa
220
4th, In what time will an yearly annuity of
L. 85.5 amount to L. 2522.25, at the rate of 5
per cent. per annum?
years.
√1180+380.25-19.5=20
28. When an annuity is paid every half year,
quarterly, or monthly; then for half years pay-
ments, take half the yearly annuity, half of the
ratio, and twice the number of years; and for
quarterly payments, take the fourth part of the
yearly annuity and of the ratio, and four times
the number of years, &c. and work as directed
above.
To Extract the Square Root of any Quantity.
Let the given quantity =g, and its
fquare root r+e, then
rr+are+ee=g
are+ce=8—is d
d
Example.
Let g
1560.25(39.5=rth
9
۲۲
621
2r+c=69)660=d
ar+c=785)3925 = d
3925
Ex. What
38
THEORY OF INTEREST,
Ex. What will an yearly annuity of L. 70, pay-
able every quarter of a year, amount to in 5
years, at 5 per cent. Here a=17.5, t=20, r=
.0125, and 350x1.11875=391.56=s.
2d, Of the preſent worth of an annuity.
29. When an annuity is to continue for a cer-
tain number of years, its prefent value p, at a gi-
ven rate of fimple intereft, may be found thus:
Since 1+tr, the amount of L. i for any time, hath
the fame ratio to L. 1, as tax 1+r, the amount
of
any annuity a, hath to p its prefent value;
therefore 1+trxp=tax1+r, from which all the
varieties in this cafe will be refolved, as in the fol-
lowing
TABLE.
t-I
I+
2
I+tr
Equ. 1.
Given.
a, r, t,p = tax
p, r, t, a = = 2x-
I+tr
t-I
T.
a
t
p, t, a,
t-I
a—p.
2
гр
I
+
P
I
ar
2
2.
3.
p, r, a,
Sought r in Equation 3d.
Equ. 1.p+trp=ta+
a12. -at
r.
2
1+rp=a+¹==¹ar.
t
24
t-I
I
a
2
4.
Sought in Equation 4.
Equ. 1.2p+2rpt=2at+t²—Ixar.
12+221-21-10-1
་
12
+
नाक है।
a
I
21
t=
ar
Ex. ift.
AND ANNUITIES.
39
Ex. 1. Required the preſent value of a yearly
annuity of L. 20, to continue 12 years, allowing
diſcount at 5 per cent. fimple intereſt. Here,
L.
ta x
t-I
1+r
2
I+tr
= 240x275=L. 191.2488.
2. A perfon is willing to lay out the fum of
300 in the purchaſe of an yearly annuity, to
continue 15 years, being allowed diſcount at 6
cent; fought the annuity. Here x itir
1.338=L. 26.76
=
annuity.
per
C20X
3. There is an annuity of L. 50 a-year, to con-
tinue 20 years, to be fold for L. 737.5, ready mo-
ney; fought, the rate of fimple intereft at which
the diſcount is given.
t
t
t-I
2
a
1312505=5 per cent.
-262.5
4. There is an yearly annuity of L. 30, to con-
tinue a certain number of years, which can be
fold for L. 405, diſcounting at 5 per cent; fought,
the time of its continuance. ✔540+36—6=18
years.
30. When an annuity is paid every half year,
quarterly, or monthly, and its prefent value be
fought. See 28.
Ex. What will be the prefent value of an year-
ly annuity of L. 70, payable every quarter of a
year, and to continue 5 years, difcounting inte-
reft at 5 per cent. Here a=17.5, t=20, r=.0125,
and 350x.895=313.25=p.
In like manner a, t, or r may be found.
3d, Of
40
THEORY OF INTEREST,
3d. Of the prefent worth of an annuity in re-
verfion.
31. When an annuity is to commence after a
certain number of years (n), and then to continue
for a certain time (t), its prefent value may be
found thus: Find its amount at the end of the
time t, (Equation 1. 27.) and divide this amount
by 1+t+r, the quotient will give p, the preſent
value. Thus ta x
t-I
1+
I
2
↑
ititur = p.
Ex. Required the prefent value of an yearly
annuity of L. 30, which is to commence 6 years
hence and to continue 15 years, difcounting in-
tereſt at 5 per cent. Here a=30, t=15, n=6,
r=.05, and 450x1:35=L. 296.3=p.
4th. Of the preſent worth of the reverſion of
an annuity.
32. The prefent value of the reverfion of an annui-
ty, being the fame with that of a perpetuity which
is to commence after a certain number of years (n),
may be found thus: From the value of the perpe-
tuity (fee Remark 3d), fubtract that of the annui-
ty, the remainder gives the prefent value of the
reverfion: Or thus, divide the value of the per-
petuity, viz. by 1+nr, the quotient gives the
preſent value of the reverfion, thus rx = p.
Ex. Sought, the reverfion of an yearly annuity
of L. 20, which is to continue 12 years, allowing
diſcount at 5 per cent. Here
a
2x14-115
20
.08
a
L. 250 the reverſion.
33. The
AND ANNUITIES.
41
*
33. The feveral particulars refpecting annui-
ties at fimple intereft may be difcovered by means
of the two following Tables, wherein are calcula-
ted the amount and prefent value of L. 1 an-
nuity for years, at ſeveral rates of intereſt.
TABLE V. Shews the Amount of L. I Annuity
for Years.
Yrs
-$4 pr cent.5
pr
cent. 6
pr cent. Yrs 4 pr cent. 5 pr
nt.s
cent. 6
pr
cent.
1 2
I 1.00
1.00
1.00 26 39.00
42.25
45.50
2.04
2.05
2.06
27 41.04
44.55
48.06
3
3.12
3.15
3.18
28 43.12
46.90
50.68
4
4.24
4.30
4.36
29| 45.24
49.30
53.36
5
5.40
5.50
5.60
30 47.40
51.75
56.10
6
6.60
6.75
6.90
31 49.60
54.25
58.90
7
7.84
8.05
8.26
32 51.84
56.80
61.76
8
9.12
9.40
9.68
33 54.12
59.40
64.68
9| 10.44
10.80
11.16 34 56.44
62.05
67.66
IO 11.80
12.25
12.70 35 58.80
64.75
70.70
II 13.20
12 14.64
13.75
14.30 36 61.20
67.50
73.80
15.30
13 16.12
16.90 17.68 38 66.12
15.96 37 63.64
70.30
76.96
73.15
80.18
14 17.64
18.55 19.46 39 68.64
76.05
83.46
15 19.20
20.25 21.30
40 71.20
79.00
86.80
16 20.80 22.00 23.20
41 73.80
17 22.44 23.80 25.16 42 76.44
18 24.12 25.65 27.18
82.00
90.20
85.05
93.66
20 27.60
19 25.84 27.55 29.26 44 81.84
43 79.12
88.15
97.18
91.30
100.76
29.50
31.40 45 84.60
94.50
104.40
2 2
46 87.40
97.75
108.10
47 90.24
101.05 III.86
48 93.12
J04.40
115.68
107.80
119.56
III.25 123.50
21 29.40 31.50 33.60
22 31.24 33.55 35.86
23 33.12 35.65 38.18
24 35.04 37.80 40.56 49 96.04
25 37.00 40.00 43.00
50 99.00
Table V. fhews the amount of L, 1 annuity,
and is conſtructed by equation 1. 27. whereby n
the tabular number=tr; Or thus, feeing
a the amount for 1 year,
I
a+i+r = b
b + 1 + 2r — c
c+ I + 3r
d
F
for 2 years,
for 3 years,
for 4 years, &c.
P
42
THEORY OF INTEREST,
Hence, to L 1, the first year of this Table, add
the firſt year of Table II. their fum is the ſecond
year
of this Table; to which add the fecond year
of Table II. their fum is the third year of this
Table, &c. From the nature of the Table,
1:n::a: nas the amount of any annuity a, for
the time and at the rate correfponding to n: hence
the following
TABLE.
Given.
a, t, r,
s = na.
s, t, r,
a =
Л
s, t, a,
n =
a
s, r, a,
Ex. 1. what will an yearly annuity of L. 35 a-
mount to in 16 years, at the rate of 5 per cent.
fimple intereft?
na = 22 × 35 =L. 770, the a-
mount.
2. What yearly annuity will amount to L. 471
in the ſpace of 20 years, at 6 per cent. fimple in-
tereſt?
471
31.4
=L. 15, the annuity.
TABLE
AND ANNUITIES.
43
TABLE VI. Shews the preſent Value of L. 1
Annuity for Years.
Ys4 per cent.5 per cent.16 per cent. Ys 4 per cent. | 5 per cent. | 6
per cent.
10.96154
0.95238
0.9433926 19.11764 18.36956
17.77343
2
1.88888
3 2.78571 2.73913
1.86363
1.83928 27 19.71154
18.95744
18.34351
2.69491 28 20.33962
19.54166
18.91044
4 3.65517
3.58333
3.5161329 20.92592
20.12245 19.47262
5 4.5
4.4
4.3076930 21.54545
20.7
20.03571
6 5.32258
6.125
8 6.90909
9 7.67647
5.19231
5.0735331 22.14285
21.27451
20.59440
5.96296
5.81690 32 22.73688
21.84615
21.15067
6.71428
6.54054 33 23.32758
22.41509
21.70469
7.44827
10 8.42857
8.16666
7.24675 34 23.91525
7.9375 35 24.5
22.98148 22-35526
23.54545
22.80645
II 9.16666
8.87096
8.6144536 25.08197
24.10714 23.35443
12 9.8189 9.5625
13 10.60526 | 10.24242
9.2790737 25.66129
9.93258 38 26.23809
24.66666
23.90062
25.22414
24.44512
25-77965 24.98802
14 11.30769 10.91176 10.85326 39 26.8125
11.5714311.21052 40 27-38461
I5 12.
16|12,68392|12.22222
16 12.68392 12.22222 11.83673 ||41| 27.95454
17 13.35714 12.86486 12.45544 42 28.52238
18 14.02325 13.5 13.0670643 29.08823
19 14.6818214.1282013.67289 44 29.65217
2015-33333 14.75 14.2727245 30.21428
21 15.97826 15.36585 14.8650446 30.77465
22 16.61702 15.97619 | 15.45689 ||47| 31.33333
23 17.24948 16.58139 16.04201 48 31.89041
24 17.87755 17.18182 16.6024649 32.44594
2518.5
17.77777 17.2
50 33.
26.33333 25.52941
26.88524
26.00936
27.43548
26.60795
27.98412 27-14525
28.53125 27.68132
29.07692 28.21621
29.62121
28.75
30.16418
29.28272
30.70588
31.24638
29.81444
30.34518
31.78571 30.87.5
Table VI. fhews the prefent value of L. 1 an-
nuity, and is conftructed by Equation 1ſt, 29.
whereby n the tabular number = tx
thus, ʼn Table VI. = T
n
", T. V.
n, T. II.•
t. I
2
+ r
Itiri Or
This Table is conftructed upon the fuppofition,
that the preſent value of an annuity is equal to
that of the amount of the feveral annual pay-
ments, with their intereft at the expiration of
the annuity. From the nature of the Table,
F 2
I
2
44
THEORY OF INTEREST,
1:n::a: na=p, the preſent worth of any annui-
ty a, for the time and at the rate correfponding
to n. Hence the following
TABLE.
Given.
a, r, t,
Р
na.
p, r, t,
a
n
p, t, a,
p, r, a, S.
12
Ex. ft. What is the prefent worth of an year-
ly annuity of L. 50, to continue 15 years, dif-
counting intereft at 5 per cent?
na=11.57143 × 50
=L. 578.5715=p.
2d. A perfon is willing to lay out the fum of
L. 300 in the purchaſe of an annuity to continue
18
years, being allowed diſcount at 5 per cent.;
fought the annuity.
300
n
13.5
L. 22.222 = annuity.
3d. A gentleman hath L. 160, which he would
lay out in the purchaſe of an annuity of L. 20 per
annum; fought, how many years the faid annui-
ty muſt continue, difcounting intereft at 6 per
cent.
1 = 160
a
20
1
8n. the next lefs to which below 6
per cent. is 7.9375, correfponding to 10 years, and
for the odd days fay
8.61445
7.93750
8.0000
7.9375
as .67695:1:: .0625:.09233, correfponding to
34 days, (16).
4th. A
AND ANNUITIES.
45
P
4th. A hath an annuity of L. 20 per annum, to
continue 7 years; B hath one of L. 5, 10 s. to
continue 21 years; fought, which of thefe is moſt
valuable, diſcounting intereft at 6 per cent.
A's annuity 5.8169 × 20 =
5.8169 × 20=116.3380
B's annuity =14.865 × 5.5= 81.7575
A—B=L. 34.5805
5th. A lends to B L. 360, upon a mortgage of
land, whofe rent is L. 75 per annum; B keeps
the money 5 years, during which time A receives
the faid rent; fought, the ſtate of their affairs, dif-
counting intereft at 6 per cent.
By Table II.
Table V.
360 × 1.3 = L. 468
75 × 5.6 =
420
Balance due by B to A = L. 48
Remarks upon Annuities at Simple Intereſt.
34. Ift, The amount of an annuity for a given
time, and at a given rate of fimple intereft, may
be expreſſed by the area of a rectangle and triangle.
Thus,
45
THEORY OF INTEREST,
A
a
C
Barxt-1
D
Thus, in conftructing the annexed figure, make
AB=t, BC, raiſed at right angles to AB, a, and
BD = intereſt of a for the time t-1; laftly, join
AD, and complete the rectangle AC; then will
AC + ABD exprefs the amount of the annuity
a, for the time t, at a given rate of intereft; for
AC + ABD = = at + × tru—ra amount of the
annuity.
=
t
2
Ex. Suppofe a L. 20, t 12 years, and the
rate 5 per cent. ; fought the amount.
Here AB 12, BC
20, BD =!!.
II. Hence,
AC+ABD = 240+66= L. 306 amount.
2d. The amount of an annuity is, cæteris pari-
bus, always lefs, and its prefent value greater, up-
on the principles of fimple than of compound in-
tereft, as is manifeft from infpecting the Tables ;
and
AND ANNUITIES.
47
and as the tabular numbers expreffing the preſent
worth of an annuity may be confidered as the
years purchaſe of ſaid annuity for the correſpond-
ing years, theſe at fimple intereft are unlimited.
Thus an annuity to continue 99 years, at five per
cent. is valued, Equation 1ft, 29. at 57.4 years
purchaſe; whereas the years purchaſe at com-
pound intereſt are always leſs than . For theſe
reafons the purchaſe of annuities upon the prin-
ciples of fimple intereft is very unequitable, and
compound intereft, that both parties may employ
their money to the greateſt advantage, is uſually
admitted in thefe tranfactions.
I
3d. The value of a perpetuity cannot be ob-
tained upon the principles of fimple intereft, be-
cauſe the number of years purchaſe is indefinite.
For, as a finite quantity can make no alteration
upon an indefinite one, by any operation of arith-
metic, the fractional part of the annuity, (Equa-
-I
+r
2
tion 1ſt, 29.) viz.
I+tr =1, when t is indefi-
nite; and of confequence, the number of years
purchaſe of a perpetuity, viz. tx1, is alfo indefi-
nite. Or thus, feeing the value of an annuity at
fimple intereſt is fuppofed to be equal to the pre-
fent value of the amount of the feveral annual
payments, with their intereft, collected into one
fum; but as this, when t is indefinite, cannot be
done, therefore the value of a perpetuity up-
on the principles of fimple intereft muſt remain
impracticable, and its only value that can be ob-
tained, muſt be derived from the principles of
compound intereft, viz. . (56.)
a
СНАР-
i
CHA P. IV.
Of GEOMETRICAL PROGRESSION.
35. WHEN a feries of quantities, exceeding
two in number, increaſes by the fame common
multiplier, or decreaſes by the fame common di-
vifor, the terms of the feries form a continued
Geometrical Progreffion; and the common mul-
tiplier or divifor is called the common ratio of
the feries. Thus,
a, ar, ar², ar³, art...
arn-I—z. an increaſing ſeries.
Z Z
Z
Z
Z
Z,
a. a decreaſing feries.
r3, 74,
rn-I
1, 2, 4, 8, 16, 32, 64: 2 the common { Dultiplier.
I
36. In a geometrical feries, the product of the
extremes is equal to the product of any two
· terms
AND ANNUITIES.
49
terms equally diſtant from the extremes; thus
Z
azar X ; and the ſquare of any one term is
r
equal to the rectangle under any two other terms
equally diſtant from it; thus, ar2 x ar² = dr ×
ar³.
37. In a geometrical feries, let
a repreſent the 1ſt term.
the laft term.
the number of terms.
n
r
the common ratio
$
2
x
quot. of the
greater by the leffer contig. term.
the fum of the feries, and L, the
Logarithm of any quantity.
Then, as r is always increaſing from the 2d.
term, its exponent in the laſt term will be n
and ar"-1= %.
38. In a geometrical feries, the ſum of all the
terms wanting the leaft is equal to the fum of all
the terms wanting the greateſt multiplied by the
common ratio.
that is, s―as-xxr, in an increaſing feries.
and s—a=s—≈x, in a decreafing feries.
for ar, ar², ar³... Aprox
any
rxa, ar, ar²
Hence,
-2
39. In a continued geometrical progreffion, if
three of the five quantities, a, x, n, r, and s
be given, the other two may be determined, as
in the following table:
G
TABLE.
50
THEORY OF INTEREST,
Given.
TABLE.
g" I
a, r, n,
s = ax
Equation 1.
a, r, z,
s=
rz-
† - 1
a
z-a
r-I
+ z,...
ભેં
2.
z-a
a, n, z,
+ 2,...
3.
S ==
I
z
r, z, n,
X
4.
a, r, n,
z = ar”—1,..
5.
a, r, s, 2=
rs—sta=s—sta...
6.
r
r, n, s, z=r - sx
rn-I
Soughts in Equation 1.
385-9=5-2 X 1.
38.
37
z = ar
rs
1.
-or
Soughts in Equation 3.
$7
ar
— Z,
Sought z in Equation 7.
-I
zr"
n-I
-I
=
Subftit. this in Equ. 4-
Given.
AND ANNUITIES.
51
Given.
Z
r, z, n,
a
T, Z, S,
a = rz-sx r—I, •
r, n, s,
• • • • 1 = 1/4 x 5 = 0
a, r, z, 12 =
L.Z
L.
Equation 8.
9.
IO.
a
L.r
L.z-L. a
L.r
+1,.
II.
L.FIXsta
a, r, s,
12=
a
L. r
L.r—1xs+a—L. a
L.r
12.
L.z- L. a
a, z, s,
12
+1,.
13
L.s-
L.5-%
L. Z
L.rz
T, Z, S,
n =
+1,
14.
L.r
Sought in Equation 11.
AI
37.
38.
L.
a
L.r.
Sought n in Equation 12.
-ar".
ar"=rsasta.
Sought, in Equation 13.
Equ. 15.
Subftit. this in Equ. 11.
Soug ht n in Equation 14.
Equ. 9.
art-$xr-I.
Subſtit. this in
Equ. II.
G 2
Given.
52
1
THEORY OF INTEREST,
Given.
a
a, z, s,
↑ =
Equation 15.
S
a, n, s,
I,.
16.
a
a
n-I
z
a, n, z,
T=
S
Z
12, Z, S,
r
S-Z
S Z
17.
18.
In the above table, the expreffions belong to an
increafing progreffion, and may be applied alfo
to a decreaſing one, by making the quantities a
and ≈ change places every where.
EXAMPLES.
ift, In the geometrical feries, 1, 2, 4, 8, 16,
&c. containing fixteen terms, fought, the laſt
term and the fum of the feries.
n=16. Sought, ≈ and s.
Here a=1, r=2,
n
T I
Equation I. ax
==IX ×
65526-I
L.r=0.3010300
=65525=s.
16
r-I
I
4.8164800=
65526=r
7-I
Equation 5. ar
I×32768=z.
r
Sought in Equation 16.
r.
Sought in Equation 18.
38.
·ar".
સ
38.
n-
=
-I
$s-ar
n
a.
***
n
I.
}
7-.I
FLI
+
ง
r
2d,
1
AND ANNUITIES.
53
2d, A gentleman, who had a daughter mar-
ried, gave the huſband towards her portion 4 s.
promifing to triple that fum the firſt day of every
month, for nine months after the marriage. The
fum paid on the firſt day of the ninth month
was 26244 fhillings. What was the lady's portion?
Here are given a, r, z; fought s.
Equ. 2.
rz—a__78728
2
39364 fh. L. 1968, 4 s. S.
3d, A gentleman buys a fine houſe, in which
were 20 thresholds, and, in name of price, was
to lay a farthing on the first threſhold, a half-
peny on the ſecond, a penny on the third; doub-
ling the fum on every following threfhold; what
would the houſe coft him? Here are given, a, r,
n; fought s.
I
Equ. 1. ax =1048575 Far. L. 1092, 5 s. 4 d. Price.
r-I
4th, A Gentleman fold his eftate for L. 21844,
which was diſcharged by ſeveral payments in geo-
metrical progreffion; the firft was L. 4, the laſt
L. 16384; what was the ratio, and how many
payments were there? Here are given a, z, 5;
fought r, n.
Equation 15. —4=2
a
2184
546
=4= Ratio.
L.z-L.a
+1=3.61226 +1=7=n.
Equation 13..+1=
Of
54
THEORY OF INTEREST,
Of a Geometrical Series continued indefinitely.
I
40. As in a geometrical feries s-as-zxr,
(38), hence I r::s'—z:s—a, and rir—1::
s-%%-a. but as the quantities a, z, change
places every where, when applied from an increa
fing to a decreafing feries, we fhall have 1: r—1
::s—a:a−z; therefore fuppofing z to vaniſh,
or the decreafing feries to have no laft term, then
I
1 : r— 1 — s—a: a; which gives s =
the following
r
ra
; hence
Given.
TABLE.
I
a, r,
S = r² =ra x 1, ...
Equation 1.
ƒ— I
I
a, s,
SX.
2.
sa
a
r, s,
a =
sxr-I
↑
SXI
ள்
3.
EXAMPLES.
, Required the fum of the feries, 1, 4, 4, 4,
이
​T'o, 3'2, &c.
Here a = 1, r=2, and s=
2XI
2, the fum.
I
2d, Required the ſum of the ſeries .333 = +
3
120+ T200, &c.
3ΧΙΟ
r=10, ands=
IOX 10-I
3
Here a = 3,
=, the fum.
I
I
3d, A+
+
R2
&c.
RS
RX/
R-I
I
R—I
CHAP.
CHAP. V.
Of COMPOUND INTEREST.
41.
WHEN the intereſt of money at the end
of each year is added to the principal fum, and
both bear intereft the following year, money is
ſaid to be at compound intereſt: In which let
p repreſent any principal fum lent out.
t
R
S
:
the time.
the intereſt
the amount
}
of L. 1 for one year.
the amount of principal and inte-
reft; and L the logarithm of a
quantity.
Then L. 1, at the end of the year, will become
1+r=R, hence the geometrical feries 1: R: R²
: R3 R4..... R. So Rt will be the amount of
L. I, for the time t, and at the rate r.
But I p
:: R': pR's, the amount of p, at the end of t
years, at the rate r; hence the following table.
TABLE,
THEORY OF INTEREST,
TABLE.
در
Given.
p,
P, R, t,
spR. or L.p+L.Rxt=L.s... Equa. 1.
S
s, R, t,
p=.
R.
s, R, p, t =
or L.-L.Rxt = L.p, ....
L.s~ L.p
L.R
s, p, t,
R='√!.
• or L. R=L.s—L.p,
t
2.
3.
4.
Ex. 1. Required, the amount of L. 25 foreborn
12 years, at 5 per cent. compound intereft. Here,
p=25, t = 12, R= 1.05.
L. p = 1.3979400
+L. Rxt= .2542716
Sum = 1.6522116L. 44.896
L. R 0.0211893
t =
- 12
.2542716
2d. Sought, the principal which, in the ſpace
of 12 years, will amount to L. 300, reckoning
intereſt at 4 per cent.
L. s = 2.4771213
L. Rxt
Diff.
.2044002
Here,
L. 1.04 0.0170333
t=
12
.2044002
2.2727211 = L. 187.379
3d. Sought, the time in which L. 15.875 will
amount to L. 31.9436 at the rate of 6 per cent.
L.s- L. p
L. R
.3036708 = 12 years.
.0253059
I
4th. At what rate of intereft per cent. will
L. 480 amount to L. 643.246, in the fpace of 6
years.
L. s — L. p = .1271358 = .0211893
cent.
= L. 1.05. = 5 per
0211893 = L.
=
5th.
AND ANNUITIES.
$7
5th. Required the amount of one farthing lent
out at 5 per cent. compound intereſt for the ſpace
of 1000 years? Here,
L. 1.05 x 1000 = 21.1893
+L..001041683.0177287
L. PRt-
places of figures.
18.2070287 L. 1610752, &c. to 19
If p the principal confifts of pounds, fhillings,
and pence, or t the time, of years, months, and
days, thefe parts of an integer may be thrown in-
to
OF LOGARITHM S.
Logarithms are a moſt valuable invention for folving queſtions in
trigonometry, or where the higher powers of numbers, as here, are
concerned; and for which the learned world are indebted to John
Napier, Baron of Merchiefton, in Scotland, about the year 1614.
ift. The index of the logarithm of any number is always one lefs
than the number of integers it contains; of confequence the index of
the logarithm of any one of the 9 digits is o; and the index of the
logarithm of a decimal fraction is negative, though ftill equal to the
diftance of its firft fignificant figure from the units place, being
—I, −2, —3, &c. or the complement thereof to 10, viz. 9, 8, 7,
&c.
zd. The logarithm of any number to 7 places of figures may be
found by Sherwin's Tables, namely of 5 figures by the Tables, and
of two by the column pts, nearly exact, and vice verfa. Thus,
L. 6484.300 3.8118631
L.
L.
.080 =
.006 =
53
4
L. 6484.386
3.8118688
3d. The multiplication or divifion of natural numbers is performed by
the addition or fubtraction of their logarithms, or, in place of fubtrac
tion, the addition of the complement of a logarithm may be uſed,
providing you caft 10 out of the index of the fum. Thus,
L. 20 1.3010300
L. 21 1.3222193
Co. L. 220 = 7.6575773
20 X 21
L.
220
= 0.2808266 = 1.pog
H
4th. A
58
THEORY OF INTEREST,
1
to a decimal fraction, and the anſwer found as a-
bove.
Ex. Sought the amount of L. 310: 12:6, for
5 years and 91 days, at 4 per cent. compound in-
tereſt. Here,
pr² = L. 1.04 × 5.25 + L. 310.625 L. 381, 12 s.
11 d. Amount.
42. The feveral particulars refpecting com-
pound intereſt may more readily be found by the
help of the three following Tables, wherein are
calculated the amount and prefent worth of L. I
for any given time, and at feveral rates of com-
pound intereft.
4th. A given power of any number may be found by multiplying
the logarithm of the number by the index of the power, the produc
will be the logarithm of the power fought. Thus,
3
L. 32′ = 1.50515 × 3 = 4.51 54500
32768, the cube of 32.
In a decimal fraction, the firft fignificant figure of the power muft
be put fo many places below the place of units, as the index of its
logarithm wants of 10 x index of the power. Thus,
L..25*9.39794×4
=40.
37.5917600.00390625, feeing index x 10
5th. The root of any power may be extracted by dividing the lo
garithm of the number by the index of the power, the quotient gives
the logarithm of the root fought. Thus,
L. √√2304 3.36248251.681241248, the root fought.
In a decimal fraction, prefix to the index of its logarithm a figure
lefs by one than the index of the power, and divide as above. Thus,
29.8750613
L. 3√.75 =
=9·9583533.90856, the root fought.
3
6th. Any number of mean proportionals,
numbers, may be found thus: Let a
number of proportions; then a
L.-L.
times, gives the proportionals.
1 I
between any two given
ift, z= laſt number, n =
L.r, which added to L. a,
TABLE
AND ANNUITIES.
59
TABLE I. Shewing the Amount of L. 1 for
Days.
Dys 3 per cent. 13 per cent. 4 per cent. 4 per cent. 5 per cent.
|
per cent.
11.0000809 1.0000942 1.0001074 1.0001206 1.0001336 1.0001596
2 1.0001619 1.0001880 1.0002149 1.0002412 1.0002673 1.00031931
31.0002429 1.0002827 1.0003224 1.0003618 1.0004011 1.0004790-
4 1.0003240 1.0003770 1.0004299 1.000482 1.0005348 1.0006387
5 1.0004050 1.0004713 1.0005374 1.000603 1.0006685 1.0007985
6 1.000486 1.000565 1.000645
71 1.000357 1.000660 1.000752
8 1.000648 1.000754
1.000860
1.000729 1.000848 1.000967
10 1.000810 1.000943 1.001075
1.000724
1.000802 1.000958
1.000844 1.000936 1.001118
1.000965 1.001070 1.001278
1.001086 1.001203 1.001437
1.001206 1.001337
1.001597
20 1.001621 1.001886 1.002151
30 1.002432 1.002831
1.002415 1.002677
1.003198
1.003228
1.003624 1.004018
1.004801
1.004307
1.004835 1.005361
1.006806
1.005387
1.006048 1.006705
1.008014
1.006468
1.007262 1.008052
1.009624
1.007550
1.003777
40 1.003244
50 1.004057 1.004723
60 1.004871 1.005671
70 1.005685
1,006619
80 1.006499 1.007568 1.008633
90 1.007315 1.008518 1.009717
100 1.008131 1.009469 1.010803
120 1.009765 1.011374 1.012978
1.008477 1.009401 1.011237
1.009694 1.010751 1.012853
1.014471
1.010912
1.012103
1.012132 1.013457
1.016092
1.014576 1.016169 1.019341
1.018890 1.022601
1.021617 1.025871
1.024352 1.029152
1.024412 1.027095 1.032443
1.026885 1.029844 1.035745
1.039057
1.042380
1.045713
1.049057
1.052412
1.055778
1.059154
1.06
140 1.011402 1.013282 1.015157 1.017026
160 1.013041 1.015194 1.017341 1.019482
180 1.014683 1.017109 1.019530 1.021944
200 1.016328 1.019028 1.021723
220 1.017976 1.020951 1.023921
240 1.019626 1.022877 1.026124
260 1.021278 1,024807
280 1.022934 1,026741
300 1.024592 1,028638 1.032761
320 1.026253 1.030619 1.034983
340 1.027916 1.032564 1.037210
360 1.029583 1.034512
1365 1.03 1.035
1.029365 1.032601
1.028332 1.031851 1.035365
1.030544 1.034343 1.038137
1.036840 1.040916
1.039344 1.043703
1.041854 1.046497
1.049298
1.039441
1.04
1.044370
1.045
1.05
H 2
TABLE
бо
THEORY OF INTEREST,
TABLE II. Shewing the Amount of L. 1 for Years.
Yrs. 3 per cent. 13 per cent.14 per cent. 14½ per cent. 5 per cent.
6 per cent.
1
1234
I
1.03
2 1.0609
1.035
1.071225
1.04
1.0816
1.045
1.092025
1.05
1.06
1.1025
1.1236
1.092727 1.108718
1.124864 1.141166
1.157625
1.191016
1.125509 1.147523 1.169858
1.192518
1.215506
1.262476
5
1.159274 1.187686 1.216653 1.246182
1.276281
1.338225
6 7∞ a
I.194052 1.22925 1.26532
1.30226
1.34009
1.41852
1.229874 1.27228
8 1.266770 1.31681
9 £.304773 1.36289
1.31592
1.36086
1.40710
1.50363
1.36857
1.42210
1.47745
1.59385
1.42321
1.48609
1.55133
1.68948
IO
1.343916 1.41059
1.48024
1.55296
1.62889
1.79084
ннннн
ΙΙ 1.38423 I.45997
1,53945 1.62285
1.71034
1.89830
12
1.42576 1.51107
1.60103
1.69588
1:79585
2.01220
13
J.46853 1.56395
1.66507
1.77219
1.88565
2.13293
14 1.51259 1.61869
1.73167
1.85194
I.97993
2.26090
15 1.55796 1.67535
1.80094
1.93528
2.07893
2.39656
16
17
18
19
6 7∞ a
1.60471
1.73398 1.87298 2.02237
2.18287
2.54025
1.65284 1.79467
1.70243 1.85749
1.94790 2.11337
2.29202
2.69277
2.02581
2.20848
2.40662
2.85434
1.75350 1.92250
2.10685 2.30786 2.52695
3.02560
20 1.80611 1.98978
2.19112 2.41171 2.65329
3.20713
21
1.86029
2.05943 2.27877 2.52024
2.78596
3.39956
22
1.91610 2.13151
2.36992 2.63365
2.92526
3.60353
23
1.97358 2.20611 2.46471 2.75216
3,07152
3.81975
24 2.03279 2.28333
2.56330 2.87601
3.22510
4.04893
25 2.09378 2.36324
2.66583 3.00543
3.38635
4.29187
222
67∞
26 2.15659 2.44595
2.77247 3.14068
3:55567
4.54938
27
28
2.22129
2.531567
2.88337 3.28201
3:73345
4.82234
2.28792
2.62017 2.99870 3.42969
3.92013
5.11168
29 2.35656
2.711878 3.11865
3.58403
4.11613
5.41838
30 2.42726
2.80679 3.24339
3.74531
4.32194
5.74349
31 2.50008
2.905031
3.37313 3.91385
4.53804
6.08810
32 2.57508
33
3.00671
2.65233 3.11194 3.64838
34 2.73190 3.22086 3.7943I
35 2.81386 3-33359
3.50806 4.08998
4.76494
6.45338
4.27403
5,00319
6.84059
4.46636 5.25335
7.25102
3.94609 4.66734
5.51601
7.68608
362.89828
3.45026 4.10393
4.87738
5.79181
8.14725
37 2.98522
3.57102 4.26809
5.09686
6.08141
8.63608
38 3.07478
3.69601 4.43881 5.32622 6.38547
9.15425
39 3.16703 3.82537 4.61636
5.56589
6.70475
9.70351
40 3.26204 3.95926 4.80102
5.81636
7.03998
10.28571
41
3.35989 4.09783 4.99306
6.07810
7.39198
10.90286
42
43 3.56451 4.38970 5.40050
3.46069 4.24126 5.19278 6.35161
7.76158
11.55703
6.63744
8.14966 12.25045
44
3.67145 4.54334
5.61651
6.93612
8.55715
12.98548
45
3.78559 4.70236
5.84117
7.24825
8.98500
13.76461
46 3.89504 4.86694
6.07482
7.57442
9.43426
14.59048
47 4.01189 5.03728
6.31781
7.91527
9.90597 15.46591
48
4.13225 5.21356 6.57053
8.27145
10.40127
16.39387
49
4.25622 5.39606 6.33335
8.64367
10.92133
17.37750
50
4.38390 5.58494 7.10668
9.03263
II.46740
18.42015
!
AND ANNUITIES.
61
Table I. and II. fhew the amount of L. 1 for
days or years, and are conſtructed by Equation
1ſt, 41. whereby the tabular number n = R. Or
thus, Table I. is calculated by taking 364 mean
geometrical proportionals, betwixt one, and the
amount of L. 1 for one year, according to the
feveral rates of intereft, or, which is the fame
thing, by dividing the logarithm of R by 365,
the natural number anfwering to the quotient
will be the amount of L. 1 for one day; and this
amount multiplied into itſelf continually will give
the amount for any number of days. If a given
number of days be not in the Table, divide it into
two fuch numbers as are to be found there, multiply
their tabular numbers into one another, and their
product will be the amount of L. 1 for the given
time. In like manner, Table II. is conſtructed
by multiplying the amount of L. I for one year,
at a given rate, by itſelf continually, the products
of which fhew the amount of L. 1 for their cor-
refponding times. And if the time confifts of
years and days, their tabular numbers muſt be
multiplied into one another to form the amount
of their fum. From the nature of the Tables,
1: np: np = s, the amount of any principal p₂
for the time and at the rate correfponding to n.
Hence the following
I
Given.
p, t, r,
s, t, r,
TABLE.
t, r,p,
1 s, t, p₁ S
p,
np.
Þ
n =
r being given, t is found
oppofite ton; and t being
given, r is found above n
in the tables.
Ex. Ift,
62
THEORY OF INTEREST,
{
Ex. ift. What will L. 500 amount to in 15
years, at five per cent. compound intereft? Here,
—
np=2.07893 × 500 L. 1039.465.
Ex. 2d. What will be the amount of L. 419.2965
in the ſpace of 20 years and 9 months, at 41 per
cent.
Here,
n×n = 2.49265 = 0.3966608
p=419.2965 = 2.6224885
L. nnp
3.0191493 = L. 1045.08.
Ex. 3d. What principal will amount to
L. 1039.464 in 15 years, at 5 per cent? Here,
44
1039.464 = L. 500.
2.07893
Ex. 4th. At what rate of intereft will L. 25 a-
mount to L. 44.896 in the ſpace 12 years? Here,
44.896 = 1.79585, found under 5 per cent.
25
Ex. 5th. In what time will L. 300 amount to
L. 500, at the rate of 5 per cent? Here,
P
500
300
1.66666
1.66666, correfponding to 10 years.
and =1.0232, correfponding to 172 days.
1.62889
TABLE
AND ANNUITIES.
63
ند
TABLE HI. Shewing the prefent Worth of
L. 1 for Years.
Yrs. 13 per cent.13 pr cent.4 per cent. 4 pr cent.[5 per cent.6 per cent.
I
2
3
4
5
.970874.966183 | .961538 | .956938 | .952381 | .943396
.942596.933510
.942596 .933510 | 924556 .955729 .907029.889996
.915141 .901942.888996 .876296 .863837.839619
.888487 .871442 .854804.838561.822702792093
.862608 .841973.821927.802451.783526.747258
6.837484.813500
7
|
790314 .767895 .746215 704960
.813091.785991.759918.734828.710681.665057
8.789409 .759411 -730690 .703185 | .676839 | .627412
9.766416.733731 .702586 .672904 .644609.591898
.702586.672904 |
IO .744093 708918.675564 | .643927.613913.558394
IL
12
14
.722421.684945 | .649581.616198|| .584679 | .526787
.701379 .661783 624597 589663556837 .496969
13.680950 .639404 .600574 564271
|
•530321 .4688 39
.661117 617781 -577475 -539973 •505068 .442301
15 .641860.596190 555264 516720.481017.417265
16 .623167 .576706533908 494469 .458111.393646
17.605016 .557203 513373 473176.436296.371364
18.587394.538361 | .493628.452800 .415520 .350344
|
19 570286 -520155474642 .433302 .395734 -330513
20 .553675 .502566 | .456387.413643.376889
21
|
22.521892 .469150 .421955
23.506691.453285.405726.363350
.311804
-537549.485571.438833.396787.358942.294155
.37970I .341849 277505
.325571 .261797
24 .491933 .437957 .390121 .347703 .310068 .246978
25.477605 .423147 .375117 .332730 .295303 .232998
26.463694.408837.360689.318402.281240
.219810
27 .450189 .395012 .346816.304691.267848.207368
28 .437076.381654.333477 .291571 255093.195630
29.424346.368748.320651 •279015.242946 | ·184556
30 .411986 .356278 .308318.267000 .231377 174110
| |
31.399987 .344230
.344230 | .296460 | .255502 .220359.164255
32.388337 332589 .285058
.332589.285058 | .244500 | .209866
.209866.154957
33 377026 .321342 .274094 2339?I .199872.146186
34.366045.310476 .263552 .223896.190355 .137911
35 -355383 .299977 .25.3415 .214254.181290
36
37
.130105
.II5793
.345032 | .289833
.289833 .243668.205028.172657.122741
.334983 .280031.234297
.234297.196199 | .164435
38 .325226 | .270562 | .225285 .187750.156605 | 109238
39 .315753 .261412.216620.179665.149148.103055
40.306556.252572.208289.171928 .142045 .097222
41.297628 .244031 .200278 .164525 .13528x .091720
42.288959
.288959.235779 .192575 .157440 .128840 .086527
43 .280543.227806 .185168.150660 .122704.081629
44 .272372 .220102 .178046 .144173 .116861 .077009
45 264438.212660.171198 .137964 .111296 .072650
|
46.256736.205468 | .164614 .132023.105996 | .068538
47 .249258.198520
.198520 .158282 .126338 .100949.064658
| |
48.241998.191806.152195 .120897.096142.060998
.234950.185320.146341.115691 091564 || 057545
|
50.228107 .179053 .140713 110709 1.087204.054288
49
64
THEORY OF INTEREST,
I
Rt
Table III fhews the prefent value of L. 1 for
years, and is conſtructed by Equation 2d, 41.
whereby the tabular number n = Or it is the
reciprocal of Table II. whereby n of Table III. =
I of Table II. From the nature of the Table
In::s: ns = p, the principal or prefent worth
of
S, for the time and at the rate correſponding
Hence the following
to n.
TABLE.
Given.
s, t, r,
p, t, r,
s, r, p, l
s, t, p, S
р
= ns.
上
​S= 1.
n
n =
오
​Ex. 1. Required the prefent worth of L. 250,
due 10 years hence, diſcounting at the rate of 4½
per cent. compound intereſt.
ns=.
.643927 × 250 = L. 160.98175 preſent worth.
Ex. 2. In what time will L. 350 amount to
L. 571.458, at 4 per cent compound intereft?
Here
350
=
571.458
years and an half.
.61247 correfponding to
12
Theſe examples may, with equal facility, be.
determined by the equations annexed to Table II.
Re-
1
:
65
AND ANNUITIES.
:
Remarks upon Compound Intereft.
43. Ift. A fum of money put out at compound
intereft ought, by the doctrine of fluctions, to
accumulate every inftant, at the rate of the in-
ſtant; ſo as to reſemble an organized body in its
gradual and conſtant increaſe. Upon this fuppofi-
tion let r be the fluction of the rate, or the rate of
the inſtant,
Then 1+r amount of L. 1 in one inſtant
multiplied by 1+r
produces 1+2r+rr
the fluction of that
amount at the end of the firſt year; whoſe fluent,
viz. 1 + 2r = amount of L. I at the end of the
year. Hence the intereſt of money, when the
times and rates are the fame, will, when it accu-
mulates every inftant, be double to what it would
be when it accumulates only once in the year.
But as this mode might elude the grafp of calcu-
lation, and for the benefit of all concerned, the
accumulation of money hath been extended to
the twelvemonth, and only takes place once in the
year.
2d. Although there be no particular theorem
for diſcovering the compound intereft of a fum
of money diftinct from the amount, yet it may
eafily be found, being the difference betwixt the
principal and the amount. In like manner, was
it lawful to diſcount money on the principles of
compound intereft, the difcount might be found
by fubtracting the prefent worth from the fum of
the bill.
I
3. If
J
1
1
66
THEORY OF INTEREST,
3d. If the time be fought in which a fum of
money will double itſelf at the feveral rates of
compound intereft; in this cafe, as the amount is
equal to twice the principal, pR 2p, which
gives R' = 2, and t. Hence,
✔ = 2p, when t=
L. 2
L. R
=
2
23.45-
20.149-
17.67+
=
3/1/2
334
15.8-years at 44 per cent..
14.206+
11.98
34.886+
2
4th. The amount of a fum of money, ceteris
paribus, is always greater, and its prefent value
lefs, upon the principles of compound than of
fimple intereft, in proportion of +r, to 1+tr;
and of, to. And as this difference is
I
I
I
L.
L.
very confiderable, feeing 12 years intereſt of the
one is equal to years intereft of the other, the
charging compound intereft upon any bill hath
been exprefsly prohibited by acts of parliament, at
the rifque of forfeiting both principal and intereſt.
Notwithſtanding of this, compound intereft en-
ters deeply into the tranfactions of men in their
affairs of property. Thus, not only the fale of an-
nuities, and the purchafe of perpetuities are cal-
culated upon the principles of compound intereft;
but alſo where cafh-accounts are balanced, or the
intereft of money and the rent of an eftate recei-
ved yearly, there you have the advantage of com-
pound intereft, feeing you have the ufe, and can
difpofe of that intereft.
CHAP.
CHA P. VI.
Of ANNUITIES computed at COMPOUND INTEREST.
ift, Of the Amount of an Annuity in Arrears.
IN computing annuities at compound inte-
44.
reft, let a reprefent an annuity, penfion, or rent,
&c. in arrears; then, fince the first year's annui-
ty bears no intereft, the last year's annuity will
be reprefented by aR-1; and deriving a feries
from the quantity a, in the ratio of 1: R, we fhall
have a + aR + aR² + aR³ + aR.. + a R¹ — ¹ = s
+aR$−¹=s
which gives s = ax
R— I
R— I•
3
Hence the following
TABLE.
Given.
Rt 1
a, t, R, s = ax
R*
R-I
s, t, R, a=rs×··
s, t, a, = R— R
a
R-I
I
a
a
R'— Equat. 1.
2.
3.
L.rs
+1
a
s, R, a, t=
L. R
Soughts in Equation ft.
38.|s—a=s—art—IX R.
s—a=Rs-art.
4.
Sought R in Equation 3.
38. | Rt-IXa = R s — s
RI
R
a
a.
Rs-saRta.
R-IX s = aXRt~1.
Sought t in Equation 4th.
but R1+r, &c.
Eq. 2.
R-1 Xars.
R'− 1 = " £ & R² = 1 + 1.
rs
IT
I 2
Ex. Ift.
68
THEORY OF INTEREST,
Ex. ft. It is required to find the amount of an
annuity L. 5.25 per annum, at the end of 30
years, reckoning intereſt at 4 per cent? Here
a = 5.25, t = 30, R= 1.04 and r=.04.
2.243
Rt
ах
=5.25× .04
L. 2
L. 297.42 the amount.
L. R=.0170333
t =
30
L. Rt = .5109990 = 3.2433
I
Rt -
=
2.2433
Ex. 2d. Sought, the yearly annuity which in
the ſpace of 15 years will amount to L. 1000, at
the rate of 4 per cent. Here, by Equation 2d,
L.R=.0191163
I
I
rs X
Rt-1
=45×93528
t=
15
45 × 1.0692 =L. 48.114
ΙΟ
.2867445=1.93528
1.
L..935289.9709416
·93528 = R— I
I
.02905841.0692 = RtI
Ex. 3d. In what time will an yearly annuity of
L. 48.114 amount to L. 1000 at the rate of 4 per
cent. compound intereft?
L.TS
+1
Equation 4th,
a
L. 1.045
L. R
L. 1.93528 .2867445
0191163
=
15
years.
45. Given, the annuity, time, and amount;
fought, the rate of compound intereft. In this
cafe, as R, in Equation 3d, is involved to a high
power, and the Equation is of a mixed nature,
the
AND ANNUITIES.
69
a
aR¹¹
I
2
t
aRt-I
Ι
the value of R may
be found in this man-
ner. The amount of
an annuity may be
conceived to be near-
ly equal to the area
of a geometrical fi-
gure, having two of
its fides parallel and
two of its angles right;
whofe leaft breadth =
a, its greateſt = aR-1, and its height=t. Hence
s = at R, nearly; which gives = R', and
= √ ÷ = R = i +r, nearly.
t-I
2
at
1
at
2)
Ex. 4th. An annuity of L. 34.4 in arrears 12.5
years, amounts to L. 614.4328; it is required to
find the rate of intereſt allowed. Here
S
L. = √ =.0269575 = L.s=2.7884743
2
at
1.064 =R=1+r= L. at = 2.6334685
6
t-I
2
per cent. nearly. == 5.75)0.1550058(.0269575
If greater exactneſs be defired, let
√ub+2ib-b=r corrected. Or thus *
= b, then
Affume
* The Rule of DOUBLE POSITION.
Let P and p, the two pofitions, and the number fought, be e-
qually increafed or diminished, by any operation of arithmetic, pro-
ducing thereby M, m and 2, their refpective refults; then it is ob-
vious, that
Mom: Psp ::
{
M in Q: P is i
m is 2: pix.
X.
Let
70
THEORY OF INTEREST,
Affume two pofitions, find their reſpective re-
fults, and work by the following proportion.
As the difference of the refults of the two pofi-
tions is to the difference of the poſitions :: fo is
the difference of the reſults of the firſt or of the
fecond pofition, and of the number fought to
the difference of the firſt or of the ſecond pofition
and number fought.
Ex. 5th. An annuity of L. 20 per annum, 7
years in arrears, is offered to be fold for L. 180.5;
what rate of intereft is allowed to the purchaſer
in this bargain? Here R is fought.
Ift pofition, R1.08
2d poſition, R = 1.085.
refult 178.456... error 2.044
refult 181.210... error + 0.710
By the queſtion
refult 180.500, diff. of ref. = 2.754
hence
Let M s 2= R, and m ✪ 2=r be the errors of the two reſults
M and m, which are either + or
Msm=
{
; then
:}
when the errors are
Rar
R+rs
Ex. Suppoſe 2x + = + 2 + 1 = 20; fought x.
2
=
{
ift pofition, P9... M 25... R= +5.0
2d poſition, p =6….. m = 17.5 ...
fought x
-2.5
2=20... Mcm=7.5
alike.
unlike.
hence, 7.5: 3:5:2 = P c x, which gives x = 7, when the errors
+
—
and are equal,
P+P = x.
2
When the higher powers of the unknown quantity, x, are concern-
ed in the queftion, this method can only give an anfwer near the
truth; feeing P2; p is in a duplicate, and P: p3, in a triplicate ratio
of P : p.
2
The above proportion is alſo the foundation of a fecond well-known
rule in double pofition.
་
RULE
AND ANNUITIES.
71
hence 2.754:.005:: 2.044: .0037, to be added
2.044.0037,
to the firit pofition, making R = 1.0837; which
ſhews that the rate of intereſt is L. 8.37 per cent.
46. When an annuity is paid, which is ufual-
ly the cafe, every half-year, quarterly, or month-
ly, its amount may be found by the Equation,
s = a × R; Here a is the half-year's, or quar-
terly payment, R' is the fame as if the annuity
had
x
Rt I
RULE.
Divide the difference of the products of the two pofitions, multi-
plied into their alternate errors, by the difference of their reſults,
that is, by the difference of the errors, when they are alike; or di-
vide the fum of the products by the difference of their refults, that is,
by the fum of the errors, when they are unlike; the quotient in ei-
ther cafe will give the anſwer. Thus,
rP
Mm
rP
Rp
Ror
Rp
Mcm = R + r
x
{
x{.
=x
when the errrors are alike, that is both +,
or both
when the errors are unlike, that is one + and
the other
Ex. When the errors are both +, or both
ſt poſition, = P, its error + R.
2d pofition, p its error
x is fought.
N
A-
D-
b
F-
a
P
+ R
+
十八
​C
B
Let the right line AB be conceived to terminate the true refult by
the queſtion; then the errors will fall above, and below that
line; the rectangles CD + CFP+ Rp, are the products of the
two
72
THEORY OF INTEREST,
had been paid yearly, feeing R' for years is equal
to R4 for quarters; but the divifor muft always
correfpond to the payments, and may be difcover-
ed by the following
TABLE. Shewing the Amount of L. 1 for the
even Parts of a Year.
M. W. D. 3 per cent. 13 pr cent.4 per cent. 14 pr cent. 5 per cent. 6 per cent.
I 7 1.000567
2 3 4
3
4
11.00008098 1.0000942 1.0001074 1.0001206 1.0001336 1.0001596)
1.000660 1.0007524 1.0008445 1.0009361 1.001118
1.001320 1.001505 1.001690 1.001873 1.002238
1.0015051.001690 |
1.0019811.002258 1.002535 1.0028111.003360
1.002642 1.003012 1.003382 1.003750 1.004480
14 1.001134
211.001472
281.002270
1.002871 1.003274 1.003675 1.004074 1.004867
1.008637 1.0098531.011065 1.012272 1.014674
1.017350 1.0198401.022253|1.024694 | 1.029563
1.026137 1.029852 1.033564 1.037270 1.044671
1.05
1.06
I
313
6126
939
1.002466
1.007417
1.014889
1.022417
1252 365 1.03
1.035
1.04
1.045
In this Table, which is the fame in conftruction
with Table I. whilft the tabular numbers fhew R
the amount, the decimals point out r the intereſt
of L. 1 for their correfponding days, months, or
quarters, at the feveral rates of intereft.
I
Ex. 1ft, A fervant, who was engaged to ferve
at 5 s. per week, after 7 years received his wages,
with intereft upon the weekly payments, at the
в
a
as b
Ror
two poſitions multipled into their alternate errors, and a cob is the
difference of thefe products. But a=px Rr, which gives R
p and b = P∞ pxr. Again, by the above proportion, Mum=
Ror: Pop::ripx; which gives x. Hence
Rear
= (x =); that is, the difference of the products, when
the errors are alike, divided by the difference of the errors, will quot
the anſwer. In like manner it might be fhewn, when the errors are
P + p
S. x = x.
unlike, that
*
P-P
2
:)
rate
AND ANNUITIES.
73
rate of
5 per cent. per annum ; fought, the amount
of his wages, and intereſt thereon. Here a = .25,
R' = 1.4071 and r = .000936. Hence,
ax
Rt-I
= .25
.4071
.000936
× = L. 108.734.
Ex. 2d. A failor, being engaged for a voyage of
6 years, at the rate of L. 1, 10 s. per month, at
his return to port, received his wages, with inte-
reſt upon his monthly payments, at the rate of 4
per cent: per annum : Sought, the amount of the
whole.
Here a
6
1.5, R=1.04 1.26532, and r=.003274.
=
R— I
Hence a x
= 1.5 ×
26532
.003274
=L. 121.557
Ex. 3d. What will an annuity of L. 17.5, pay-
able every quarter of a year, amount to in 5 years,
at five per cent per annum? Here,
5
a = 17.5, Rt = 1.05 = 1.27628, and r = .012272.
X
Hence 17.5 ×
.27628
.012272
L. 393.977.
In like mannner, mutatis mutandis, may a, t, and
R be found from the other equations of 44th,
when the payments are made every half
quarterly, &c.
year,
2d. Of the prefent Worth of an Annuity computed at
Compound Intereft.
I
R
47. Let p reprefent the prefent worth of an an-
nuity a; then, fince 1: the prefent worth of L. 1,
as a, this at the end of one year will be the
prefent worth of a; and deriving a feries from
K
the
·
74
THEORY OF INTEREST,
the quantity
have +
a
R
a
R
a
R
+
fent worth of the
I
R-
I
Rt
= ax
I
I
in the ratio of 1:, we fhall
a
R3
R².
I Rt
a
α
R
+ Ri... +7=p the pre-
R4
Rt р
annuity a, which gives pax
1- Or thus, the feller, when the
annuity ceaſes, has pR'; the purchaſer, at the
end of the time, has a × , but by fuppofition
Rt- I
pRt
Rt- I
=ax
which gives pax
Rt-I
=ax
ах
r Rt
I
Rt
Hence the following
TABLE.
Given.
a, t, R, pax
f, t, R, a=rp x
f, t, a, i+
↑, r, a,t=
I..
a
I
I
Rt
I
R
*
× R² —— R²+ I
i.. R
1
TRt
Equation 1.
- R² + 1 = 1/2 •
Sought in Equation iſt.
a x R.
38
a
P - R
Rt
p = Rp-a.
R$
Rp—p=a — •
pt
I
x=ixp=axi-
R**
rh
a
Rp+ a+p.
Rt
2.
3.
4.
}px R¹+ ¹ + a=aR$ +pRª•
R² + 1 + 7 = // R$ + Rˆ•
Sought in Equation 4th.
I
• Eg. 2. rp=a-at
Sought R in Equation 3d.
381 - X
= Rp — a.
a
I
Rt
=a
a
a ----
a-rp.
rpx Rt, and t=&c.
Ex. ift.
4.
AND ANNUITIES.
75
Ex. ft. What is the prefent worth of an year-
ly annuity of L. 20 to continue 20.75 years, rec-
koning intereft at 4 per cent?
I
I
Rt
ax
=20 x X
.598822
.045
— L. Rt = .3966632 I.CC
L._1=9.6033368
I
Rt
L. 266.14 the prefent value. 1-
I
Rt
0.401178
.598822
If the years purchaſe which an annuity is
worth, be fought; they are manifeftly equal to
the number of times the annuity is contained in
its preſent value. So that 2, the number of years
purchaſe, = £
the laſt example,
purchaſe.
-
I
I
Ꭱ
266.14
20
=n of Table V. Thus in
-59882
.045
= 13.307 = years
Ex. 2d. A gentleman is willing to fpend L. 1000
upon the purchaſe of an annuity, to continue 15
years, being allowed difcount at the rate of 4
per cent.; fought, the annuity.
I
rpx
45×2.096 L. Rt = .2867445 1.00
1-
R
L.
=9.7132555 0.51672
Rt
L.93.114 annuity.
I
Rt
.483289.6841988
2.0962.3158012
1
Ex. 3d. Sought the duration of an yearly annui-
of L. 80, whofe prefent value is L. 800; dif-
counting intereſt at 5 per cent. :
ty
L. a
L.
Here
L. R
=2010300 = 14 years 75 days.
.0211593
48. Given, the annuity, its preſent worth and
time; fought, the rate of intereft per cent.
Cafe ift, If the number of years be great, as
40 or upwards, and if the rate of intereft be high;
in this cafe, will be of fmall value, and r,
I
Ri
K 2
a
P
with
76
THEORY OF INTEREST,
with this rate find the prefent value of the rever-
a
rR
fion of the annuity, viz. = x; then=r, the
rate fought.
Ex. 4. Suppofe a =
L. 70, p = L. 1321.3028,
and t = 59 years; fought r, the rate of intereſt.
a
a
70
=/=.052978= =r; R = 11139 = L.62.844=x, the
reverfion; andx=.05057=r corrected = 5 per
cent.
Cafe 2. If the number of years is ſmall, as 20
or under, the rate may be found thus:
The preſent worth of an annuity may alfo be
conceived to be nearly equal to the area of a geo-
metrical figure, having two of its fides parallel,
and two of its angles right; its greateſt breadth
=, its leaſt, and its height=t; hence p=
at
a
MT, which gives "V=R=1+r, nearly.
2
Ex. 5. Suppofe a=L. 20, p=L. 220, and t=
21 years. Sought, the rate of intereft allowed.
L. at,
2.62 3 2 4 9 3
+co. L. p. =7.6575773
1- I
2
11) 0.2808265 (.0255297=1.0605=
R=1+r=6 per cent. Hi greater accuracy be de-
fired, let, then b-br corrected.
Theſe two cafes may alfo be refolved by the Rule
of double pofition.
Ex. 6. Suppofe a=L. 20, p=L. 100, and t=
7 years; fought, R.
Ift
AND ANNUITIES.
77
}
Ift pofition, R=1.09... refult 100.659... error +.659
2d poſition, R=1.0925... refult 99.82... error .180
refult 100. diff. of refult.839
Per queftion.
Hence, .839: .0025: : .180: .000536-difference
between the 2d pofition and R; but feeing, in this
queſtion, the greater the intereſt is, cæteris paribus,
the leſs will be the refult, this difference muſt be
deducted from 1.0925, making R = 1.091964;
which gives p=L. 99.9989.
49. When an annuity is paid every half year,
quarterly, or monthly, &c. its prefent value may
I
I
p=ax R¹.
be found by the equation pa×¹-
Rt
Ex. What is the prefent worth of an annuity
of L. 5 paid every quarter, to continue 20 years,
reckoning intereft at 4 per cent.? Here a=5, R$
= 1.045 20.75 and r=.011065; hence
hence 5X
L. 270.59, the prefent worth.
598822
.011005
By comparing this with Ex. 1ſt, 47. it appears,
that an annuity paid every half year or quarter-
ly is more valuable than when paid yearly. See
Chap. VIII. Prob. XX.
In like manner may a, t, and R, be found by
the other equations of 47th, when the payments
are made every half year, quarterly, or monthly.
3d. Of the prefent Worth of an Annuity in reverfion.
50. When an annuity is to commence after a
certain number of years, (») and then to continue
for a certain time, () its prefent value is found
by
78
THEORY OF INTEREST,
by fubtracting the prefent worth for 2 years,
1
n
I
from that for ten years: thus, a×
Rt + n
a
T
I
Rn
X
r
p. Or, divide the preſent value of the
annuity at its commencement by R", the quotient
gives p: Thus, ax
the following
I
I
Rt
Rt-I
=ax
ах
=p.
rpt+n
Hence
TABLE.
I
I
I
R*
Ri
=ax
R" + _R"
Equation I.
Given.
a, n, t, R,
p, n, t, R,
p, n, t, a,
p= ax
r R"
a=r RpX
a
I
Rt
RM
n
1
R
J
р
I
R
L. a
La
L R
r "p
L.a-
Rt
L.rp
p, R, n, a, t
:
2.
4.
3.
p, R, t, a, n =
L, R
in
Ex. 1. There is an yearly annuity of L. 30,
which, commencing after the expiration of a leafe
of 10, is to continue 19 years; it is required to
find the prefent value thercof, reckoning intereſt
at 5 per cent. Here a=30, n=10, t=19, and R
.604266
=1.05: Hence p = 30 X 0814447 L. 222.58, the
prefent value.
Ex. 2. An annuity, which, commencing 12
years hence, is to continue 20 years, being ex-
pofed
AND ANNUITIES.
79
pofed to fale, fold for L. 250. Reckoning intereſt
at 4 per cent. fought the annuity.
Herep=250, n=12, t=20, R=1.04, fought a
L. rRp 1.2044002
Equ. 2. L.=.2647102
I
R$
1.4691104=L. 29.4517, the ann.
Ex. 3. When 12 years of a leafe of 21 years
were expired, a renewal for the fame term was
granted for L. 500; 8 years are now expired, and
for what fum muſt a correſponding renewal be
made, reckoning 5 per cent. compound intereft?
Here,
1ſt, Givenp=500, n=9, t=12, R=1.05, fought a
Equ. 2. px=500X 139298-L. 87.5, the ann.
r Rt + n
-79585
2. Given a=87.5, n=13, t=8, fought p.
I
.47746
Equ. 1. ax=87.5X139298 =L. 299.9,
r Rt n
the fine.
When a, p, n, and t are given, R may be found
by approximation: Thus, let T=2n+t+1, then
T at
2 р
=R=1+r nearly. If greater exactneſs be
defired, let 66b, then b-bb-2rb is exceed-
ing near the value of r.
Ex. 4th, An annuity of L. 40 per annum, being
in poffeflion for the term of 21 years, for L. So
paid down can be prolonged for 10 years more, or
to 31 years; what is the rate of intereſt required?
Here a=40, p=80, n=21, and t=10, fought R.
L.a
80
THEORY OF INTEREST,
L. at
2.6020600
co. L. p
8.0959100
T
2
T=42+11=53
26.5) 0.6989700 ( 0.0263762 =L.1+r=1.062616.
b=67+6
=3.24
tt
2r= .125232.
b-2rXb=3.114768 X 3.24 10.091848bb-2rb.
63.24
√ bb—2rb = 3.176767
.063233r, the rate fought.
Ex. 5. Given a L. 87.5, p=L. 500, n=9 years,
R=1.05, fought t, the time of the annuity's con-
tinuance.
.2542796
Equ. 4. 0211893
=12 years.
Ex. 6. Given a = L. 20, p=L. 145.836, t=12
years, R=1.05, fought n, the number of years
prior to the commencement of the annuity.
.0847572
Equ. 5. 4 years.
0211893
Of
AND ANNUITIES.
81
Of the Renewing of Leafes for a certain Number of
Years.
A TABLE, fhewing the Fine for renewing of
any Number of Years elapfed in a Leaſe of 21
Years.
Years L.II : 11':84
elapfed.
per cent.
+
•
5 per cent. 6 per cent. 18
per cent. 10 per cent.
I
попро
23
0.1200
0.2118
0.3590
0.2942
c.1987
0.1352
0.7359
0.6060 C.4122
0.2848
0.3364
1.1317
0.9365 | 0.6489
0.4473
4
0.4754 1.5471 1.2869 | 0.8942
c.6272
5 0.6306 | 1.9834 | 1.6583
1.1544
0.8250
6 0.8035
2.4415 2.0419
14563
1.0426
78
9
ΙΟ
1.0000
1.2121
1.4525 3.9576 3.3803
1.7210 4.5148 3.8772
2.9226 2.4691
1.7726 1.2820
3-4276 2.9114
2.1120 I.5454
2.4807
1.8350
2.8779 2.1437
II
2.0205 5.0995 4.4040
3.3067
2.5042
12
2.3446
5.7134 4.9624 3.76939
2.8897
13
2.7275
6.3580 5.5543 4.2702
3.3038
14
3.1435
7.0349 6.1817 4-8104
3.7803
15
3.6077
7.7456 6.7468 5.3939
4.2935
16
4.1258
8.4917 7-5417 6.0241
4.8579
17
4.7038
92752
8.2990
6.7047
5.4759
18
5.3488 10.0980
9 0911
7.4397 6.1619
19
6.0686 10.9618
,9.9207
8.2356 | 6.9132
20
6.8716 11.8678 10.8107
9.0309 7-7396
Total
2 I 7.7682 12.8212 11.7641 10.0168 8.6487
value.
In the conſtruction of this table, let n be the
number of years in eſſe, or yet to run by the
leaſe, t the time elapfed fince its commencement ;
then from the prefent value of L. 1 annuity for
L
n+t
82
THEORY OF INTEREST,
5
nt years, at a given rate, fubtract that of an
annuity for n years; the remainder = N, the ta-
bular number expreffing the fine for renewing
the leafe; the yearly rent being L. 1.
Ex. There is a leaſe of church-lands, for the
ſpace of 21 years, at L. 70 per annum, whereof
14 years are expired: It is required to know what
fine ought to be paid, to have the fame renewed,
reckoning intereft at L. 11: 11: 8 per cent.
Let a exprefs the rent, and N the tabular num-
ber for 14 years; then
II:
4
a N=70×3.1435=L. 220.045, the fine required.
When the length of the leafe is different from
21, ſuppoſe 31 years, whereof t years are elapſed:
In this cafe, from the prefent value of L. 1 an-
nuity for nt (=31) years, at a given rate, Tab.
V. fubtract that of an annuity for n years yet
to run; the remainder will fhew the fine for re-
newing the leafe, fuppofing the rent to be L. 1.
When the fine for the whole leafe is a given
ſum, for inſtance L. 500; fee Ex. 3d; find a, the
annuity which that fum will purchaſe, at a given
rate, during the currency of the leafe; then a
multiplied by N, correfponding to the years elap-
fed, will produce the fine.
4th, Of the prefent Worth of the Reverſion of an An-
nuity.
51. The reverfion of an annuity being the fame.
with a reverfion in fee-fimple, which is to com-
mence after a certain time, t, the duration of the
annuity, its prefent value may be found by fub-
tracting
!
AND ANNUITIES.
88
ļ
tracting the value of the annuity from that of the
perpetuity, the remainder gives the reverfion: Or
rather thus, Divide the prefent value of the per-
petuity, viz. (56.) by R', the quotient gives the
a
*
reverfion; thus, the reverfion.
Ex. 1ft, Sought the reverfion of an yearly an-
nuity of L. 20, which is to continue 12 years,
reckoning intereft at 5 per cent.
a
20 = L 222.735, the reverfion.
TRt .0897925
Ex. 2d, Sought, the reverfion of an yearly an-
nuity of L. 40, which is to commence 21 years
hence, and to continue 10 years, reckoning inte-
reft at 6 per cent. Here a=40, n+t=31, and
R=1.06.
a
40
L. 109.5, the reverfion.
rRn+t—.365286
See reverfions more fully treated of in 57.
52. The feveral particulars refpecting annuities.
at compound intereft may more readily be found
by the help of the three following tables, where-
in are calculated the amount and prefent value of
L. I annuity for years, at feveral rates of com-
pound intereft.
L2
TABLE
84
THEORY OF INTEREST,
TABLE IV. Shewing the Amount of L. 1 Annuity
for Years.
Yrs. 3 per cent. 13 per cent.4 per cent. 14 per cent.'5 per cent. (6 per cent.
I
1 2 3 4 5
I.
2.03
3.0909
4.18363
I.
2.035
I.
I.
I.
I.
2.04
2.045
2.05
2.06
3.10623
3.1216
3.13703
3.1525
3.1836
4.21494
4.24646
4.27819
4.31013
4.37460
5.30913
5.36246
5.41633
5.47071
5.52563
5.63709
6.46341
6.55015
6.63297
6.71689
6.80191
6.97532
78
7.66246 7.77941 7.89829
8.01915
8.14201 8.39384
8.39233 9.05168
9.21423
9.38001
9.54911 9.89747
9
10
10.15911 10.36849
10.58279
JI
11.46388
12.80779
11.73139 12.05611
10.80211 11.02656 11.49131
12.28821 12.57789 13.18079
I 2 14.19203
13
15.61779
13.14199 13.48635 13.84118
14.60196 15.02580
16.11303 10.62684
14.20678 14.97164
15.46403
15.91712 16.86994
17.15991
17.71298 18.88214
14
17.08632
17.67698 | 18.29191
18.93211
19.59863 21.01506
15
18.59891
19.29568 | 20.02358
20.78405
21.57856 23.27597
16
20.15688
20.97103
21.82453
22.71933
23.65749 25.67253
17
21.76158
22.70501
23.69751
24.74171
25.84036 28.21288
18
23.41443 24.49969
25.64541
26.85508 28.13238 30.90565
19
20
25.11687
26.35718 27.67123
26.87037 28.27968 | 29.77810
31.37142
29.06356 30.53900
33.06595
33-75999
36.78559
2 2 2 2 N
22
23
24
25
26 38.55304
27
41.31310 44.31174 47.57064
40.70963 43.75906 | 47.08421
28
20
45.21885 48.910Ɛn | 52.96628
30
47.57541 50.62267 56.08494
33
31
34
33
21 28.67648 | 30.26947 31.96920
30.53678 32.32890 34.24797 36.30338 | 38.50521 43.39229
32.45288 34.46041 | 36.61789 38.93703
|
34-42047 | 36.66653 39.08260 41.68919 44.50199 50.81557
|
36.45926 38.94985 | 41.64591 | 44.56521
50.71132
42.93092 | 46.29063 | 49.96758 | 53.99333 58.40258 68.52811
50.00263 54.42947 59.32833
52.50276 57-33450 62.70147
55.97784 60.34121 66.20953
34 57.73017 63.45315 69.85791
35
57.42303 62.32271 73.63979
61.00707 66.43884
79.05818
64.75238
64.7523870.76079
70.76079 84.86167
68.66624 75.2988390.88977
72.75623 80.06377
97.34316
77.03025 85.06696 104.18375
60.46208 66.67401 | 75.65222 | 81.4966 2
36 63.27594 70.00760 77.59831
37
66.17422 73-45787 81.70224
38 69.15945 77.02889
39 72.23423 80.72490
86.16396
90.32031111.43478
86.16396 | 95.83632 |119.12086
9104134 101.62814 127.26812
85.97033 96.13820 107.70954 135.90420
90.40915 101.46442 114.09502 145.05845
40 75.40126 | 84.55028 95.02551 107.03032 120.79977 154.76196
Af
78.66330 | 88.50954 | 99.82654 112.84668 127.83976 165.04767
42 82.02320 92.60737 104.81959 118.92478 135.23175 175.95053
85.48390 | 96.84863 110.01238 125.27640 142.99333 187.50756
44 89.04841 101.23816 115.41287 131.91384 151.14300 199.75801
45 92.71986 105.78167 121-02939 138.84996 159.70015 212.74349
43
|
46 96.50146 110.48403 126.87056 146.09821 168.68516 226.50810
47 100.39650 115.35097 132.94539 153.67263 178.11942 241.39858
48 104.40839 120.38825 139.26320 161.58790 188.02539 256.56449
49 108.54065 125.60184 145.33373 169.85935 198.42666 272.95836
30 |112.79686-[130.99791 (152.66708 178.50303 259.34799 (290.33586
33.78313
33.78313 35.71925 39.99273
41.43047 | 46.99583
47-72709 54.8645 1
51.JI345 59.15638
54.66913 63.70576
ན་
AND ANNUITIES.
833
Table IV. Shews the amount of L. 1 annuity;
and is conſtructed by Equation 1ft, 44; whereby
Rt—— I
N=1X
r
Or thus, feeing
=a=amount for 1 year.
I
a+R
= b =
b + R²
= c =
c + R³
= d =
for 2 years.
for 3 years.
for 4 years, &c.
Hence, to L. 1, the first year of this table, adđ
the first year of Tab. II. their fum is the fecond
year of this table, to which add the fecond year
of Tab. II. their fum will be the third year of this
table, &c. From the nature of the table, 1: N::
a: Nas, the amount of any annuity a, for the
time and at the rate correfponding to N.
the following
Hence
TABLE.
Given.
a, t, r,
s = Na.
s, t, r,
a=
st, a,
= ·
N=
s, r, a,
40
r being given, t is found op.
pofite to N, and t being given,
ris found above N in the Ta-
ble.
Ex. ft. Given, a=L. 80, t=20 years, and the
rate 4 per cent.; fought the amount. Here,
Na=29.7781 × 80 L. 2382.248, the amount.
Ex
86
THEORY OF INTEREST,
Ex. 2d. Given, s=L. 1000, t=15 years, and the
rate of intereft 5 per cent.; fought, 'the annuity.
S
N
1000 L. 46.3423, the annuity.
21.57856
Ex. 3d. Given, a=L. 48.114, s=L. 1000, and
t=15 years; fought, the rate of intereſt.
a
48000=20.7840=N for 15 years, correfpond-
ing to 4 per cent.
Ex. 4th, Given a L. 5.25, s = L. 419.297,
= s=
and the rate 4 per cent.; fought, the time of con-
tinuance.
== 419,297 = 79.8661=N for 4 per cent. corref
a
5.25
ponding to 34 years; and for the odd days ſay,
as
81.49662
77.03025
79.86610
77.03025
4.46637 1: 2.83585635232 days.
:
TABLE
AND ANNUITIES.
87
TABLE V.
Yrs. | 3 per cent.
Shewing the prefent Worth of
L. I Annuity for Years.
3 pr ceut. | 4 per cent. | 4 pr cent. 5 per cent. ¡6
per cent
I
0.97087
0.26618 0.96154 0.95694 0.95238
C.94339
2 1.91347
1.89969 1.88609
1.87267 1.85941
1.83339
2.82861
2.80164 2.77509
2.74896
2.72325
2.67301
4
3.71710
3.67308
3.62989
3.58752
3.54595 3.46510
5
4.57971
4.51505
4.45182
4.38997
4.32947 4.21236
IO
679 a o
5.41719
8
7.01969 6.37395
7.78611
5.32855 5.24214
6.23028 6.11454 6.00205
5.15787
5.07569
4.91732
5.89270 5.78637
5.58238
6.73274
6.59588 6.46321
6.20979
7.60768 7.43533 7.26879
7.10782
6.80169
8.53020
8.31660 8.11089 7.91272
7.72173
7.36008
II
9.25262 9.00155 8.76047 8.52892
8.30641
7.38687
I 2 9.95-100 9.66333 9.38507
13
14
15
10.63495 10.30274
11.29607 10.92052
11.93793 II.51741
9.11858
9.98565 9.68285
10.56312 10.22282
11.11838 10.73954
8.86325
8.38384
9.39357 8.85268
9.89364 9.29498
10.37966
9.71225
16
a
12.56110
17
18
19
12.09412 11.65229
13.16612 12.65132 | 12.16567
13-75351 13.18968 12.65929
14.32380 13.70983
11.23401
10.83776 10.10389
11.70719
11.27406 10.47726
12.15999
11.68958 10.82760
13.13394 12.59329
12.08532 11.15812
20
14.87747 14.21240 13.59032 13.00793
12.46221 11.46992
2 2
21
22
15.93691
15.41502 14.69797 | 14.02916 | 13.40472
15.16712 | 14.451II 13.78442
12.82115 11.76407
13.16300 12.04158
23
16.44360
24
16.93554
25
17.41315
15.62041 | 14.85684 14.14777 13.48857 12.30338
16.05836 | 15.24696 | 14.49548 | 13.79864 | 12.55036
16.48151 15.6220814-82821 14-09394 12.78335
17.87684 16.89035 15.98276 | 15.14661|| 14.37518 | 13.00316
|
18.32703 17.28536 | 16.32958 | 15.45130 | 14.64303 13.21053
18.76411 17.66702 16.66306 15.74287 14.89813 13.40616
19.18845| 18.03576 | 16.98371 | 16.02189 IS.14107 13.59072
18.39204 17.29203 16.28888
29
30
CV N N
81 a
26
er ca
27
28
19.60044
15.37245
13.76483
زرع
31
20.00043 18.73627
17.58849
16.54439
15.59281
13.92910
32
|
20.38876 19.06886| 17.8735 5
16.78889 | 15.80267
14.08404
33
20.76579 19.39020
18.14764
17.02286 16.00255 14.23023
34
21.13183
19.70068
18.41119
17.24676
16.19290 | 14.368 14
35
21.48722 20.00066
18.66461 | 17.46102
16.37419 14.49824
36
37
38
39
40
21.83225 20.29049 18.90828 | 17.6660+
22.16723 20.57052 19.14258 17.86224
22.49246 20.84108 | 19.36786 | 18.04999
22.80821 21.10250 19.58448 | 18.229 65
23.11477 21.35507
16.54685 14.62098
16.71128 14-73678
16.86789
14.84602
17.01704 14.94907
19.79277 18.40158
17.15908 15.04629
4 I
23.41240
21.59910
19.99305 18.56611
17.29436 | 15.13802
42
20.18562 18.72355
43
44
45
23.70136 21.83488
23.98190 | 22.06269 20.37079 18.87421 17.54591 15.30617
24.25427 22.28279 | 20.54884 | 19.01838
24.51871 22.49545 20.72004 19-15634
17.66277 15.38318
17.42320 15.22454
17-77406 | 15-455 8 3
46
47
24.77545 22.70092 20.88465 19.28837 17.88006
25.02471 22.89943 21.04293
48 25.26671 23.09124 21.19513 19.53560 18.07715
15.52437
19.41471 17.98101
15.58902
15.65002
49
25.5016623.27656 | 21.34147 19.65130 | 18.16872
501 25.72976 | 23.45562 | 21.48218 | 19.76201 18.25592 | 15.76186
15.70757
83
THEORY OF INTEREST,
Table V. Shews the prefent value of L. 1
nuity, and is conftructed by Equation ft, 47.
I
whereby NIX-the number of years pur-
+
chafe an annuity is worth: Or thus, Seeing
I
Ꭱ
I
=a=Prefent worth for 1 year.
a + /12/2₁ = b =
6 + /1/₁ = c ===
I
گا
c+ R₁ = d =
for 2 years.
for
3 years.
for 4 years, &e.
Hence the first year in this table and in table
III. are the fame, the fum of the firſt year in this
and of the fecond in Table III. makes the fecond
in this, and the ſum of the fecond in this, and of
the third in Table III. makes the third year in
this table, &c. From the nature of the table,
IN::a: Na=p, the prefent worth of any an-
nuity a, for the time, and at the rate correfpond¬
ing to N. Hence the following
TABLE.
Given.
a, t, r,
Na
p, t, r,
p
a
N
p, t, a,
\p, r, a, S
N=t
a.
Ex. 1ft, Given a L. 50,. t=15 years, and the
rate 5 per cent.; fought the prefent worth. Here,
Na=10.37966×50=L. 518.983=p.
Ex.
AND ANNUITIES.
89
Ex. 2d, Given, p=270, t=59, and the rate 3
per cent.; fought, the annuity. Here,
N
=25-729764=L. 10.493685, the annuity.
Ex. 3d, Given, a=L. 100, p=L. 1037.966, and
t=15 years; fought, the rate of intereft.
£ 1037.966 = 10.37966=N for 15 years, corre-
a
100
ſponding to 5 per cent.
Thus, as far as the tables extend, may theſe
difficult equations be refolved, where Ris fought
in Equation 3d, 44. and 47. and where it is in-
volved to a high power.
M
TABLE
90
THEORY OF INTEREST,
TABLE VI. Shewing the Annuity which L. I will
purchaſe.
Yrs. 13 per cent
per cent. 4 per cent. 4per cent. 5 per cent 5
per cent.
I
1.03
1.035
1.04
1.045
1.05
1.06
2
.522510
·353530
.356934
4
.269027
5
·526400 .530196
.360348
.272251 .275490
.218354 .221481 22.1627
.533997 537805
.545437
363773 ·367208
.374109
.278743 .2820II .288591
.227791
.230974 .237396
879
.184597
.187668 .190762
.193878
.197016 .203362
.160506
.142456
9
.128434
·
163544 .166609
-145476 .148527
.131446 •T34493
·
•
169701
.172820
.179135
.151609
.154722
.161036
137574 .140690 .147022
ΙΟ
.117230
.120241 .123291
.126378 .129504 .135868
IJ
.108077 .II1092
.114149
117248
.120389 .126793
I 2 100462
.103484
106552
.109666
.112825
.I19277
13
.094029
.097061
.ICO143
.103275
.106455
.112960
14
.083526
.091570 .094669
.097820
.101024 .107584
15
.083766
.086825 .089941
.093114
096342.102962
16
.079611
II .082684 .085820
.089015
.092269 098952
17
075952
18 072708
*079043
075816
.082198 .085417
.088699
.095444
.078993 .082237
.085546
.092356
19
.069813
.072940 .076138
.079407
.082745
.089620
20
.067215
.070361 073582 .076876
.080242
.087184
2 I .c64871
.063036 .071280
.074600
.077996
.085004
23
22 .062747
.060813
065932 .c69198
.064018 .067399
.072545
.075970 .083045
.070682 074136 081278
24
.059047 062273
.065586
.068987 •072471
.079679
25
.057427 .060674
.064012
.067439
.070952
.078226
26
.055938 .059205 .062567
.066021
.069564
.076904
27
.054564 .057852 .061238
.064719
.068292 075697
•
28
.053293
.056602
.060013
.063521
.067122
.074592
29
.052114 .055445
.058879
.062414 .066045
.073579
30
.051019
.054371
.057830
.061391
.065051
.072649
31
.049999
953372
.056855
060443
.064132
.071752
32
.049046
.052441
.955948
.059503
.063280 .071002
33
048156
.051572
.055104
.058744
.062490 .070273
34
0 4 7 3 2 2
.050759
0543 15
.057982
.061755
.069598
35
.046539
.049998
.053577
.057270
.061072 .068974
36
.045204 .049284 .052887
.056606
.060434 .068395
ONSUI .048613 052239
.055984
059840 057857
334
38
044459 .047982
.051632 .055402
.059284
.067358.
39
.043344 047387 .051061 054857
058764
.066894
40 .043262 .046827 .050523 054343
.058278
.066462
.042712
.046298 .050017 .053861
057822
.066059
42
.042191
.045798 .049540 .053403
.057395
.065683
43
.041698 .045325 .049090 .052982
.056993
065333
44 .041230 .044877 .048664 .052581
.056616
.065006
45
.040785 .044453 .048262
.052202
.056262
.064701
46
.040362
.044051 .047882
.051845
.055928
.064415
47 .039960
48.
.043669 .047522
.051507
.055614
.064148
.039577
49 .039213
.042962
50 -038865 042634
043306 .047181 .051188
.055318
.063927
.046857 .050887
.055039
.063664
.046550 .050602
.054777
.063444
AND ANNUITIES.
91
4
Table VI. fhews the annuity which L. I will
purchaſe, and is conſtructed by Equation 2d, 47.
whereby N=
Or thus, it being the reci-
I
Rt
I
N
procal of Table V. N of Table VI. of Table V.
From the nature of the table, 1: N::p: Np=a,
the annuity which p will purchaſe, for the time
and at the rate correfponding to N. Hence the
following
TABLE.
p, t,R,
a, t, R,
p, t, a, 2
p, R, a, S
a =
Np.
р
N=
Ex. Ift, Given, p=L. 1000, t= 20 years, and
the rate 5 per cent.; fought, the annuity. Here,
Np=.080242X 1000 L. S0.242, annuity.
Ex. 2d, Given, a=L. 30, t=19 years, and the
rate 4 per cent.; fought, the prefent worth. Here,
a
N=.070138=L. 394.021, the prefent worth.
Ex. 3d, Given, a = L. 80.242, p=L. 1000, and
t=20 years; fought, the rate of intereft. Here,
a
P 1000
=== 80.0242 = .080242=Nfor 20 years, correſpond-
ing to 5 per cent. the rate fought.
More
M 2
92
THEORY OF INTEREST,
More Examples.
ift, A gentleman purchaſes an eftate for L. 1000,
and is allowed to retain the purchaſe-money in his
own hands, upon condition of paying to the for-
mer proprietor 4 per cent. per annum: He fets
the eſtate at fuch a yearly rent as to have 6
cent for his money; fought, the time in which
the eſtate will clear itfelf.
per
Here the purchafer's yearly profit-rent is 2 per
cent. upon the purchaſe-money, and the time is
fought in which this intereft will equal the prin-
cipal. By Remark 3d, 43,
s=2p when t
L. 2
L. 1.02
3010334.886 years.
2d, A puts L. 1500 out at intereft, B hath an
annuity of L. 75 to continue 50 years; which of
them will amount to the greateſt fum at the end
of the 50 years; and what is the prefent worth
of the difference, at the rate of 4 per cent. per
annum? Here, by Table II. and IV.
The amount of L. 1500 for 50 years at 4 per cent. = L. 13548.945
of L.75 annuity for 50 years
A's amount exceeds that of B by
***
The prefent worth of which by Table III. L. 17.848.
13387.727
161.218
3d, A hath a term of 7 years in an eftate of
L. 50 per annum; B hath a reverfion of the fame
eftate of 14 years after A; and C hath a further
reverfion of 21 years after A and B; fought, the
prefent worths of the feveral terms, reckoning
Here aX
1
intereſt at 4 per cent.
prefent worths refpectively. Or thus,
Rt
their
rR”
By
}
AND ANNUITIES.
93
By Table V. 42 years
18.72355×50=936.1775
21 years
7 years
-
13.40472 X 50=670.236
5.89270×50=294.635
fubtracting thefe from each preceding term, we
have
A's term L. 294.635, B's L. 375.601, C's L. 265.9415.
4th, A perſon having 7 years to run in a leaſe
of an eſtate of L. 80 per annum, it is required
what fum he ought to pay at prefent to have the
leafe renewed by having 20 years added thereto,
reckoning intereft at 6 per cent.? Here,
I
R
ax
TR
=80×7.62815=L. 610.25=p. Or thus,
=
†, of L. 1 annuity for 27 years 13.21053
for 7 years
5.58238
7.62815× 80=
L. 610.25, the fum he ought to pay.
5th, A leaſe of an eſtate, to continue 14 years,
is offered for L. 250 fine, and L. 44 yearly rent;
but the tenant wants to reduce the rent to L. 20
per annum; what ought the fine to be at 6
per cent?
Here,
44—20=24, and Tab.V. 9.29498 × 24=223.0795
+ 250.
The whole fine L. 473.0795
6th, The proprietor of an eftate, having, 20
years ago, granted the leafe of a farm for 38 years,
is defirous of recovering the fame: It is required
to find what he ought to pay for the recovery of
faid
!
94
THEORY OF INTEREST.
!
faid leafe, upon the fuppofition that the farm is
worth L. 40 of yearly additional rent, and reckon-
ing intereft at 5 per cent. Here are given,
a=L. 40, t=18, and the rate 5 per cent. fought p.
By Tab. V. 11.6896 × 40=L. 467.584=p.
53. Of Annuities computed at Compound Intereft,
when the first Payment is either due, or
paid per
Advance.
ift, To find the amount of an annuity in ar-
rears, the firft payment being due per advance,
at a given rate of compound intereft. Here, as
the firſt year's annuity (a) bears intereft, the laſt
year's annuity will be reprefented by aR'; and
deriving a feries from the quantity aR, in the ratio
of 1: R, we ſhall have aR+aR²+a R³...+aR' =
R++ ' ____ R
s; hences=ax =axRxRaN× R_of
Table IV.
Σ
r
r
Ex. It is required, to find the amount of an
annuity of L. 5, at the end of 30 years, the firſt
payment being due per advance, and reckoning
intereſt at 4 per cent. Here,
ах
R++¹ _R
r
=5x 2.33313—L. 291.64, the amount.
.04
If the feries expreffing the amount of an annuity,
when the firft payment is due at the end of the
year, (44.) be fubtracted from the above feries,
their difference will be a R—a, in favour of the
latter, which, of confequence, will exceed the
former, by the accumulated intereft of one year's
annuity for the time of its continuance. Or
thus: The amount of an annuity for t years, when
the first payment is due from the beginning of the
year,
1
AND ANNUITIES.
95
year, will be equal to its amount for t+1 years—a,
when the firſt payment is due at the end of the
year.
2d, To find the prefent value of an annuity,
the firſt payment being made per advance, at a
given rate of compound intereſt. Here we ſhall
have,
a
a +++, the prefent value of
R R
the annuity, which gives pax
R-
I
= ax
ах
I
I
Rt
-XR=aNx R of Table V.
r
Ex. It is required, to find the prefent value of
an annuity of L. 20, to continue 19 years, the
firſt payment being made per advance, and rec-
koning intereſt at 4 per cent. Here,
Rt-I
R-
I
ax
r
=20X-5464 =L. 273.4, the value.
.04
If the feries expreffing the prefent value of an
annuity, when the firft payment is made at the
end of the year, (47.) be fubtracted from the a-
bove feries, their difference will be a-
a
R
,
in fa-
vour of the latter, which, of confequence, will
exceed the former by the difference betwixt the
annuity and the prefent value of one payment
thereof due at the end of years. Or thus: The
prefent value of an annuity for t years, when the
firſt payment is made per advance, will be equal
to its prefent value for t-1 years-a, when the
firſt payment is due at the end of the
year.
Remarks
96
THEORY OF INTEREST,
Remarks upon Annuities at Compound Intereſt.
I
Rt
r
54. 1ft, The expreffion in Equation Ift,
47. is always leſs than, except when t, the time,
is indefinite; of confequence, the years purchaſe
of an annuity for a certain number of years, at
compound intereft, are always lefs than the years
purchaſe of a perpetuity: Hence compound inte-
reft, by which the value of a perpetuity is deter-
mined, is the proper ſtandard for afcertaining the
value of an annuity.
2d, The time in which the preſent value of an
annuity is equal to its reverfion may be found by
a
the equation TR
r
a
a
,
rRt
rRt
which gives -
23
r
rRt
2
I =
Rt
R' = 2, and
R2,
t== the time in which a fum of money doubles.
L. 2
L. R
itſelf at a given rate of compound intereft.
3d, Befides the two methods mentioned in the
conſtruction of Tables IV. and V. the amount
and preſent value of an annuity may alſo be cal-
culated thus:
ift, To find the amount of L. I annuity for years.
Let ia the amount for 1 year.
Then aR+1=b=
2.
bR+1=c=
cR+1=d=
3.
4 years, &c.
2d,
AND ANNUITIES.
97
2d, To find the prefent value of L. 1 annuity for
years.
Leta=the preſent value for 1 year.
Then "+¹=b=
R
I
111 = c =
R
c+1 = d =
R
2.
3.
4 years, &c.
SUPPLEMENT to CHAPTER VI.
Of a Sinking Fund to extinguish the National Debt.
WHEN a debtor's funds are infufficient
55.
to pay the intereft of his debt, this muſt increaſe
yearly by the accumulation of the deficient inte-
reft; but if he can fpare a little, after paying this
intereft, that little may, with proper œconomy,
become a finking fund to extinguifh his debt in
time.
Any part of the national debt is faid to be
bought at par, when the legal intereft of the
pur-
chafe money, at 5 per cent. is equal to the inte-
reft of the ſtock purchaſed, or when the purcha-
fer hath 5 per cent for his money.
Thus, pr cents. L. 3, 3, 4, 4, 5, 5/1/ 6,
Pars, L. 60, 70, 80, 90, 100, 110, 120.
As a debtor hath it always in his power to dif-
charge his debt, by paying the fum borrowed,
when he is able, and is fo difpofed, Government, it
is prefumed, is not obliged to pay above the
par
of 5 per cent for each L. 100 of national debt.
N
la
1
98
THEORY OF INTEREST,
In the following particulars, units are affumed
in place of millions, when the national debt is
concerned; which debt may confift either of
money borrowed at 3, 4, or 5 per cent. or in
the value of annuities paid by Government, for a
certain number of years, at 5 or 6 per cent.; and
may be in all about L. 262, at 3 per cent. anno
1792; at which period there was, in the ſpace of
fix years, the fum of L. 8 recovered, and the na-
tional finking fund was raifed to L. 1.4, which is
fuppofed to accumulate till it rifes to L. 4.
ift, It is required, to find the time in which a
finking fund of L. 1.4 will extinguiſh L. 25-8=
L. 17 of 3 per cents, the purchaſes being made at
the market-price of 90.
S
a
Here, 100: 90 :: 17: 15.3; and, by Table IV.
1.4
15.310.928=N, at 3 per cent. correſpond-
ing to 9.59 years, the time required; being that
period at which Government can, if it fhall be
thought proper, point all the artillery of the fink-
ing fund against the 4 and 5 per cents, for the
reduction of the fame.
2d, It is required, to find the time in which
the national finking fund will rife to L. 4; beyond
which it is fuppofed it needs not to accumulate,
being then fufficient to anſwer all the purpoſes of
its deftination.
On or before the year 1808, to which period
the long annuities of 1778 extend, it is prefumed
that L. 12 of annuities, at 5 per cent. will ceafe;
whofe intereft L. .60.
Therefore, to the prefent finking fund 1.40
Add the intereft of L. 17 recovered in 9.59 years = 0.51
yo
Their fum L. 1.91
And
AND ANNUITIES.
99
And find the capital of 4 per cents, which a ſinking
fund of L. 1.91 will recover in 16 —9.59=6.41
years. Na 7.152 X 1.91 L. 13.66, whofe in-
tereſt, at 4 per cent.
to which add 1.91+.60
= .546
= 2.510
Their fum is the finking fund, an. 1808 = 3.056
04
Again, 4—3.056=.944; and 244=L. 23.6, a
capital of 4 per cents, which L. 3.056 will reco-
ver in the ſpace of 7 years. Hence the national
finking fund will rife to L. 4 anno 1815.
3d, It is required, to find the amount of the
national debt diſcharged when the finking fund
rifes to L. 4.
Prior to an. 1792, there was redeemed, in 6 yrs, a fum=L. 8.co
From hence, in the space of 9.59 years, a fum
During 6 41 years, prior to anno 1808,
In anno 1808, of annuities,
During ſeven years, prior to anno 1815,
= 17.00
13.66
12.00
23.60
In all, L. 74.26
That a finking fund may have its proper ef-
fect, it ought to accumulate yearly, with the in-
tereſt of the capital which it hath recovered. But
when it is limited, for inftance, to L. 4, the na-
tion may be relieved from taxes to the amount
of faid intereſt.
4th, It is required, to find the time in which a
finking fund of L. 4, without accumulation, will
reduce the national debt to the fame ftate in
which it was anno 1755, the medium rate of
intereft being 3 per cent. and the purchafes made
at 100.
Let L. 9.45 be the intereft of the national debt,
anno 1786, correfponding to a capital of L. 270,
N 2
at
¿
100
THEORY OF INTEREST,
at 3 per cent. Upon this fuppofition :
2
To the amount of debt anno 1755, = L. 72.00
Add of capital recovered in 29 years
prior to anno 1815, as above, a fum=
And of annuities expiring,
74.26
10.00
Their fum L. 156.26
Hence 270-156.26 L. 113.74, the debt to be
extinguiſhed; fought, the time. Here,
4
II3.74 =28.43 years, the time required, extending
to the year 1844 at which period the national
taxes, including one million of a finking fund,
will be 10.45-2.52=7.93 millions lefs, yearly,
than they were anno 1786.
5th, It is required, to find the effect which a
finking fund of L. 1.4 will have upon a debt of
L. 270, in the ſpace 25 years, reckoning intereſt
at 3 per cent.; and what will be the amount of
faid fund at the expiration of that period.
By Table IV. Ña=38.95 × 1.4=L. 54.53, the
debt diſcharged in 25 years; and 1.4+1.9=L. 3.3,
the finking fund.
6th, It is required, to find the time in which
a finking fund of L. 1.4 will extinguiſh a debt of
L. 270, reckoning intereft at 3 per cent. Here,
By Equation 4. 44.
the time required,
a
L. = +1
.8893017
=59.52 years,
L. R
.0149403
:
A...
. 1
CHAP.
1
:
CHAP. VII.
Of the Value of a Perpetuity, or Freehold Eftate.
56.
IN treating of a perpetuity_or_freehold-
a
a
R3 R49
eftate; let s reprefent the value, and a the yearly
rent thereof; then the progreffion indefinitely
continued, viz. tått, &c. = s. (47.)
which progreffion, having no laft term, gives
S==== (40.) from whence all the varieties in
the buying and felling of eftates in fee-fimple may
be refolved, as in the following
R-I
Given.
a, R,
s, r,
a
S=
R— 1
a =rs.
TABLE.
a
==/ &R=1+ /=//=
r =
a
S
38.
Soughts.
S=5- 1 x R.
R
Rs - a.
S=RS
R—1×s=ɑ.
Or thus:
Rx
a
R =
40.
S=
R-I
a
R-I.
Let
}
102
THEORY OF INTEREST,
I
I
Let N repreſent the number of years purchaſe
a is worth, then Nas; and for a fubftituting
its value, rs, from the fee-fimple equation, we
fhall have Nrss; which gives Nr 1, N= &
r=. Or thus: Seeing s=Na, and a, the rent,
may be confidered as the intereft drawn yearly
for the purchaſe-money, hences: a= Na: a=N
117, which give sNa, rsa, and Nr=1.
from whence the following
EQUATION S.
= Na
a
N
a
==/
r
rs.
N=
S
a
a
I
S
N
2
a
150
.035
Ex. 1ft, Sought, the value of a freehold eftate,
whofe yearly rent is L. 150, diſcounting intereſt
at 3 per cent. Here L. 4285.7=s, the
value. Had the purchaſe been made at 28.57
years purchaſe, correfponding to 3 per cent. then
150×28.57=L. 4285.5=s, the value.
Ex. 2d, A gentleman purchaſes an eftate for
L. 14,000; at what yearly rent muſt he let it, to
have 4 per cent. upon his purchaſe-money? Here,
rs=.04X 14000=L.560=a.
Ex. 3d, An eftate, which coft L. 8000, is let in
tack
AND ANNUITIES.
103
tack at the rent of L. 360 a-year; fought, at what
number of years purchaſe it is bought, and what
rate of intereſt the purchaſer hath for his money?
Here
8000
360
41 per cent.
22.24=N; and
360
8000 = .045=r =
Of a Reverfion in Perpetuity.
57. A reverſion in perpetuity, or in fee-fimple,
being the fame with the reverfion of an annuity
when the time prior to its commencement is e-
qual to the duration of the annuity, its value
may be found thus: Divide the value of the per-
petuity, viz. by R, t being the time prior to
its commencement, the quotient gives the rever-
fion: Thus Rs, the reverfion in perpetuity.
Hence the following
r
a
a
TABLE.
Given,
a, t, R,s=
a
rRt
s, t, R, a=rs× R'.
s, t, a, R+¹ —R' =÷•
a—L.
s, R, a, t-La-L. rs
t=
L. R
Equ. 1.
2.
3.
4.
Ex. 1ft, Sought, the prefent value of a reverfion
in perpetuity, which is to commence 40 years
hence, and whofe yearly rent is L. 70, difcounting
intereft at 4 per cent. Here,
a
ƒ Rt
70 -L. 364.5, the value fought.
.19204
Ex.
104
THEORY OF INTEREST,
Ex. 2d, There is a reverfion in fee-fimple fold
for L. 500, which is to commence 25 years hence;
fought, the yearly rent thereof, difcounting intereſt
at 5 per cent.
rs × Rt=25×3.3863=84.6575, the rent fought.
Ex. 3d, The reverfion of an annuity of L. 20
a-year is fold for L. 222.735; fought, the dura-
tion of the annuity, reckoning intereft at 5 per
cent. Here,
L.a-L.rs .2542716
L. R .0211893
= 12 years.
Ex. 4th, There is an annuity of L. 35, to con-
tinue 19 years, whofe reverfion is fold for L. 256.9
ready money; fought, the rate of intereſt allowed
the purchaſer. Here,
α
35
256.9
=.13624=R²°—R19, found by Table II.
betwixt 5 and 6 per cent.; therefore take two po-
fitions, &c.
1ft Pofit. R=1.05
2d Pofit. R1 055
By the queſtion,
Refult 277.652
Error+20.744
Refult 230.095 Error -26.813
Refult 2569 Diff of Refult of 1ft & 2d=47.557
Then 47.557.005 :: 20.744: .00218 to be added to the 1ft po-
fition, making the rate 5.218 per cent. nearly.
Ex. 5th, Which is moft advantageous, a term
of 16 years in an eftate of L. 50 per annum; or
the reverfion of the fame eftate for ever, after the
expiration of the faid term, reckoning intereſt at
4 per cent.? Here,
By Equ. 1ft, the Reverfion
By Table V. the Annuity
13.3477 X 50667.385
11.11838 X 50555.919
The Reverfion is better than the Term of 16 years by L. 111.466
}
Ex.
AND ANNUITIES.
105
Ex. 6th, A and B purchaſe an eftate for
L. 6700, which, at the term of Martinmas 1790,
they agree to divide into two equal parts.
A fets his part for the yearly rent
of L. 180, and has, befides, tim-
ber, and an old manfion-houfe,
valued at L. 260.
B's part is under tack for 11 years
to run, at the yearly rent of
L. 143, and the timber upon it
is valued at L. 22.
pur-
Sought, firſt, At what rate of intereſt the
chafe was made; fecondly, How much A's part
is better than B's; thirdly, At what yearly rent
muft B fet his part, 11 years hence, to make it
equal to A's in its prefent value, reckoning inte-
reft at 5 per cent.
=.05825=L. 5: 16: 6 per cent.
a
180
S
3090
a 180
A's Part=
=3600
.95
a
B's Part=
143
=2860
r .05
His timber and houſe 260
His timber=
22
A's value=
L. 3860
So that A
57. Equ. 2d,
978.00
=L. 83.636
11.6935
B's value=
L. 2882
exceeds B in value L.
978. By
B's rife of
part e-
rent at the end of 11 years, to make his
qual in prefent value to that of A.
Remarks upon the Purchafe of a Perpetuity.
58. 1ft, A freehold-eftate or perpetuity, purcha-
fed at the rate of intereft r, is bought at a num-
ber of years purchaſe=, and the rate of intereſt
r, at which the purchaſe-money of a freehold e-
ſtate is valued for N years purchaſe: So that
N and r are reciprocal of, and difcover one ano-
ther.
Thus :
If
Ich
THEORY OF INTEREST,
If N=35, then r=.02857=2.857 per cent.
Or, If r=.04, then N=25 years purchaſe.
2d, N, the number of years purchaſe at which
a perpetuity is bought or fold at a given rate of
compound intereft, is always equal to the num-
ber of years in which money doubles itſelf at
the fame rate of fimple intereft, feeing either
of them, (23. & 56.) Thus :
3
140
33.3
3 //
When a purchaſe
28.57
4
is made at
45
per cent. then N=
25
22.7
years purchaſe.
20
16
16.6
12.5, and vice versa.
3d, The value of a perpetuity may be expreffed
by the area of a right-angled parallelogram AC,
A
D
B
a
a
C
whoſe breadth AB==, and its length BC=a the
rent of the eſtate; for ABX BC===s, the value
of the perpetuity; and half of the parallelogram,
r
viz,
AND ANNUITIES.
107
viż. BD==prefent value, or the reverſion of
an annuity a, whofe duration is equal to the time
in which a fum of money will double itſelf at
the rate r of compound intereſt: For in Equation
4th, 47. in place of p, put its value in this cafe,
a
a
viz. 27, then t— L. a—L. _ L. 2
L. R
2
L. R
—the time in which
a fum of money will double itſelf at the rate r
of compound intereſt.
4th, The method of buying and felling eſtates,
at a certain number of years purchaſe, is very
proper, being both fimple, and well adapted to
calculations upon the principles of compound in-
tereſt, by which alone the value of a perpetuity
can be aſcertained.
5th, The purchaſer of a perpetuity, if he can
afford it, may venture to give, at leaſt, 10 years
purchaſe more for an eſtate than what would a-
rife from the legal intereft of 5 per cent. on ac-
count of the fuperior fecurity, the chance of the
rife of ground in its value; and becauſe he is
allowed compound intereft upon the purchaſe-
money, and other advantages.
Q 2
A
108
THEORY OF INTEREST,
*
A TABLE, Shewing the prefent Worth of L. 1 for Years.
I
Yrs. 13 per cent. 4 per cent. 5 per cent. Yrs. 3 per cent. 4 per cent. 15 per cent.
51.221463135301
083051
52.215013.130097.079096
53.208750.125093.075330
54.202670.120282 | .071743
56.191036.111207.065073
71|.122619|.061749 |.031301
72.119047.059374.029811
73.115580.057091.028391
74.112214.054895.027039
55.196767115656 068326 75.108945052784025752
76.105772050754 024525
57 185472106930.061974 77.102691048801.023357
58.180070.102817.059023 78.099700c46924022245
59.174825.c98963 | .056212 79|.096796 | .045120.021186
60.169733C95060053536 80.093977.043384 | .020177
61.164789.091404.050986 || 81 |.091240.041716.0192.16
62.159990.087889.048858 82.088582 040111.018301
63155330 084508.046246 83.086002038569.017430
64.150806.081258 | .044044- 84083497037085 016600
65146413078133.041946 85 081065035659 .015809
60142149.075128039949 86078704.034287015056
67.138009.072238 | .038047 87.076412 032968014339
68.133989.069460.036235 88.074186.031700.013657
69.130086.066738.034509 89 072027.030481.013006
70.126297.c64219.032866 90.c69928029309 012387
A TABLE Shewing the prefent
Yrs. 13 per cent. 4 per cent. 15 per cent.
5125.9512 21.6174 18.3389
52 26.1562 21.7475 18.4180
5326.3749 | 21.8726 | 1 8.4934
54 26.5776 21.9927 18.5651
55 26.7744 22.1086 18.6334
56 26.9654 22.2198 18.6985
57 27.1509 22.3267 18.7605
58 27-3310 22.4295 18.8195
|
59 27.5058 22.5284 18.8757
60 27.6755 22.5234 18.9292
1.067891.028182 011797
Worth of L. 1 Annuity for Years.
Yrs. 3 per cent. 14 per cent. 15 per cent.
7129.2460| 23.4562 | 19.3739
7224.3550 23.5156 | 19.4037
73 29.4806 23.5727 19.432 1
74 29.5938 23 6276
23 6276 | 19.4592
75 29 7018 23.6804 19.4849
76 29.8076 23.7311 19.5094
77 29.910223-7799 19-5328
78 30.c099 23.8268 19.5550
|
79 50.1067 23.8720 19.5762
8030,2007 23.9:53 | 19.5964
81
6127.8403 | 22.7:48 | 18.9802
62 28.co03 22.8027 19.0288 82
63 28.1556 22 8872 19.075083
64 28-3064 22.968; 19.191
55 28.4528 23.0456 19 1610
66 28.5950 23.1218 19.2010
|
|
|
|
30.2920 23.9571 | 19.6156
30.3305 23.9972 19.6339
|
30.4665 24.0357 19.6514
| |
84 30.5500 24.0728 19.6680
85 30.6311 24.1085 19.6838
86 30.7098 24.1428 19.6988
|
87 30.7862 24.1757 19.7132
88 30.8604 24.2074 | 19.7268
29 30.9324 | 24.2379 19.7398
90 | 31.0024 24-2672 | 19.75 22
perpetuity 33.333325.0000
5.0000 20.0000
28.7330 23.1940 | 19.2390
28.8670 23.2635| 19.2753
69|28.9971 23.3302 19.30 9 8
70 29.1234 23.3945 19 3425
AND ANNUITIES.
10
A TABLE Shewing the Amount of L. 1 Annuity
for Years.
Yrs. 13 per cent. 4 per cent.15 per cent. Yrs. 13 per cent.4 per cent.5 per cent.
|
71238.512379.862
51 117.181 159.774 220.815 71 238.512 379.862
52 121.696 167.165 232.856 72 246.667 396.057
53 126.347 174.851 245-499 73 255.067 412.899
54 131.138 182.845 258.774 74 263.719 430.415
55 136.072 191.159 272.713 75 272.631 448.631
56 141.154 199.806 287.348 76 281.810 467-577
57 146.388 208.798 302.716|| 77 291.264 487.280
58 151.780 218.150 318.851 78 301.002 507.771
59 157-333 227.876 335-794 79 311.032 529.082
60 163.053 237-991 353-584|| 80 321.363551.245
|
618.955
650.903
684-448
719.670
756.654
795-486
836.261
879.074
924.027
971.229
61 168.945 248.510 372.263 81 332.004 574-295 1020.790
62 175.013 359.451 391.876 82 342.964 598.267 1072.830
63 181.264 270.829 412.47 83 354-253 623-197 1127.471
64 187.702 282.662 434.093 84 365.880 649.125 1184.845
65 194.333 294.968 456.789 85 377-857 676.090 1245.087
66 201.163 307.767 480.638 || 86 390.193 704.134 1308.341
67 208.198 321.078 505.670 87402.898 | 733.299 | 1374.758
68 215.444 334-921 531.953|| 88415.985 | 763.631 | 1444-496
69 222.907 349-318 559-551|| 89 429.465 795.176 1517.731
70 230.594 364.290 588.529 90 443.349 827.983| 1594.607
|
Theſe three Tables are continuations of Tab. III.
IV. and V. of compound intereft, and may be necef-
fary, befides other purpoſes, in the folution of
queſtions in annuities on lives, when n, the com-
plement of life, exceeds 50.
CHAP.
!
1
CHA P. VIII.
Of ANNUITIES on LIVES.
!
59.
WAS human life certain of continuing to
a given period, the fum of the feries++
I
I
Rn
R3 (22) = '—_R″ (47.) would expreſs, at a given rate,
↑
the value of a perfon's life who had yet n years
to live; but feeing there is much contingency in
the cafe, as we find by daily experience, Tables
of the Probabilities of Life, called alfo Tables of
Obfervations, have been conftructed from the bills
of mortality; from whence the value of life, at a
given age and rate of intereft, might be compu-
ted. Of thefe take the following Table, conftruc-
ted by Dr Halley from the bills of mortality at
Breflaw, as an example.
TABLE
1
OF ANNUITIES, &c.
III
TABLE I. Shewing the Probabilities of Life
at Breſlaw, Conftructed by Dr Halley from the
Bills of Mortality in that City.
31 523
32515
33507
37 472
7
63c 10
39 454
670
ΙΙ
653
98 7
40
41436
Age. Living. Dead. Age. Living. Dead. Age. Living.Dead.
Ok
2344O DO a
6
1238 239
I 1000 145
855 57
798
760
38
28
M2 2 -
7∞ ∞ 2 CO N
5 732 22
710
692
9
ΙΟ 661
18
12
34499
35
36 481
∞ ∞ ∞ a
229
8
62
222
ΙΟ
8
63
212
ΙΟ
8
64
202
ΙΟ
65
192
10
490
9
66
182
ΙΟ
67172
IO
68
162 ΙΟ
38 463 9
69
152 IO
9
70
142
I I
445
9
71
131
I L
9
72
120
[ I
42427
10 73
109
II
12
646
6
43417
ΙΟ
7+
98
10
13
640
6
44407
1Ο
75
88 10
14
634
6
45 | 397 IO 76 78 10
15
628
6
46387
1Ο
77
16
622
6
47 377 10 78
17
616
6
48357
IO
18
610
6
49 357
I I
80
19
604 6
50 346 11
I
81
20 598
6
51
335
I I
21
592
6
52
324 ΙΙ
8༣
22 586 7
53
313 11
23
579 6
54 302 10
24 573
6
55 292 10
25
567 7 56 282 1Ο
26
27
28
16 700
560
7 57 272 ΙΟ
3+56
~~~ 7∞ ∞ ∞ ∞ ∞ ∞ ∞
∞ ∞
78
68 IO
58
96
79
49
4I
34
76
82
28
5
23
84
85
86
87
88
19
15
4
II 3
8
3
5 2
co
553
546
29
7788
58 262
-
89 3
1
I
MMN N
O
O
539
30 531
59 252 IC 90
бо 242 10 91
61 232 10
Such a table as this of the probabilities of life,
is formed on the principle, "that the number
dying annually, after every particular age, is
equal to the number living at that age," and,
(C
of
112
OF ANNUITIES
of confequence, is poffeffed of the following pro-
perties:
ift, The ſeveral numbers in column 2. or 3.
from infancy to the utmoſt extent of life, will
repreſent the perfons living, or who die yearly,
at their correfponding ages.
2d, The fum of all the perfons alive or dead
in column 2. or 3. will exprefs the number of
inhabitants, or the number of perfons who die
yearly, and theſe laft, it is manifeft, will be e-
qual to 1238 who are born in any one year; al-
fo the fum of the perfons dying yearly, from a-
ny one age and upwards, will be equal to the
number living at that age.
3d, As the 1000 children, 1 year old, were not all
born at the beginning of the year, but fucceffively
from day to day, they, as well as the perfons living
at any fucceeding age, are reprefented in the table
half a year older, at a medium, than they are in
fact; of confequence, in queſtions refpecting the
expectation of a fingle life, (See Art. 4.) half a
year ought to be deducted from the anfwers gi-
ven by the Table. Hence may be derived the
following Articles:
Art. 1. To find the chance or probability which
a perſon of a given age hath of living a propofed
number of years. This is difcovered by the pro-
portion which the living, at the propofed age,
have to thofe who have died fince the given one.
Thus, Suppofing a perfon of 30 years of age,
then his chance of living is to that of dying,
As 523: 8=65: 1, in 1 year.
and 472:59 8: 1, in 7 years.
Hence,
ON LIVES.
113
Hence, to find how long a perſon of a given age
hath the chance of living, take half the number
of perfons living at the given age, and oppofite
to it in col. Ift, you have the years required.
Thus, 531-265, which is found in the table be-
tween 57 and 58 years; fo that a perfon aged 30
hath an equal chance of living between 27 and
28 years longer.
Art. 2. To find the probability of life's conti-
nuing, or of its failing in 1, 2, 3, &c. years, in
a perſon of a given age, when the certainty of
the prefent moment, or of death, is expreffed by
unity. Here, fuppofing a perſon of 30 years of
age, out of 531 chances, he hath 523 of living
one year, and eight chances of dying in that time.
But by the doctrine of chances, the probability
for the happening of an event is expreffed by a
fraction, whofe numerator is the number of chan-
ces for its happening, and denominator, the fum
of the chances both of its happening and fail-
ing. Hence,
523
531
8
531
Probability that fuch a perſon
will live 1 year.
will die in that time.
But the fum of thefe fractions is equal to uni-
ty; therefore, having one, we can find the other
by fubtraction. Hence, 523 515 507
507 will exprefs
531′ 531 53 1
the probabilities that fuch a perfon will live 1, 2, 3
years longer; and
ing in that time.
8
531' 531' 531'
16
24
thofe of his dy-
P
Art.
114
OF ANNUITIES
Art. 3d. Out of a certain number of perfons
of a given age, to find how many of them have a
chance of dying yearly.
As the tabular number, at the given age, is to
thoſe who die out of them, fo is the number of
perfons alive, to thofe who die yearly.
Ex. Out of 135 perfons, whoſe age, at a me-
dium, is 55 years, it is required, to find how
many have a chance of dying yearly, till the laſt
is extinct at the age of 91 years.
292:10:135: 4.623 who die the rft year;
and 282: 10: 130.377: 4.623
2d year, &c.
Or rather thus: As thefe perfons are in life from
30 years of age and upwards, nearly in the fame
proportion as the tabular numbers from the fame
age; hence,
15031:531::135 4.77 who die the 1ft year;
and 14500: 523:: 130.23: 4.70
2d year,
&c.
Art. 4th. To find the expectation of life, that
is, the number of years which mankind enjoy,
taken one with another, either from birth or any
age propofed.
Divide the number of perfons alive at the gi-
ven age and upwards, by the number alive at the
given age, the quotient, minus .5, gives the an-
fwer. Thus,
35173 (minus .5)=28, the expectation of an infant.
1238
Or thus: The expectation of life being equal to
the fum of the probabilities that a perfon fhall live
from the prefent inftant, 1, 2, 3, &c. years; the
expectation
ON LIVE S.
115
expectation of life in a perſon, for inſtance, of 30
years of age, is expreffed by the fum of the frac-
tions, viz.
531 523 515
+ + &c. carried on to the utmoſt ex-
531 531 531'
tent of life, (minus .5)=27.87.
Art. 5th. To calculate the value of L. 1 annui-
ty upon a life of a given age, and at a given rate
of intereft.
By Art. 2d. the probabilities of the continuance
of life in a perfon, for inftance, of 30 years of
age, for 1, 2, 3, &c. years, are expreffed by a fe-
ries of fractions, viz.
523 515 507, &c. But independent of chance,
531' 531' 531'
the terms of the feries, viz.
I
I
I
R R2 R3*
I
R"
, fhew the preſent value of L. 1
due at the end of 1, 2, 3,
・・・ 7, years. There-
fore, combining theſe two feries together,
523
531R
+
515
53112+
507
531 Res
&c. carried on to the ut-
I
moſt extent of life, will exprefs the value of L. 1
annuity, upon the life of a perfon aged 30 years,
at a given rate of intereft. But as the calculation
of 60 terms of this feries must prove a work of la-
bour, let N=1.44 denote at 4 per cent. the value
of a life aged 86, which is eafily found by the above
method, and =, the probability that a life,
aged 85, will furvive 1 year; then,
a
BR
a
b
II
X1+N=1.72, is the value of a life aged 85
years. Proceeding thus, ftep by ſtep, the values
P 2
of
',
116
OF ANNUITIES
of lives, by any table of obfervations, may be
found.
Art. 6th. To find the amount of the population
in any city or county in the kingdom, from their
annual births or burials.
If the annual average of births and burials be
equal, either of them multiplied by the expecta-
tion of an infant's life taken from a table adapted
to that diſtrict, will produce the amount of the
population thereof. If this annual average be
unequal, which is moſt likely; in this cafe, as the
population is as much increaſed by the fuperior
number of births, as it is diminiſhed by the com-
parative defect in the number of burials; there-
fore, the half of their fum, multiplied by an in-
fant's expectation of life, call it 40, (fee Table
VIII.) will produce the number of inhabitants.
Ex. By the collector's accounts of the duty up-
on baptifms, &c. within the bounds of the fy-
nod of Fife, anno 1790,
The annual average of baptifms=2484
of burials=1536
Half their fum=2010
Multiplied by
40
Produces 80400, the num-
ber of inhabitants within the bounds of the fynod
of Fife.
60. Of the Hypothefis of Equal Decrements.
Mr de Moivre hath invented an hypothefis,
whereby the values of annuities on lives may
determined
be
I
ON LIVES.
117
determined nearly as accurately as if they had
been deduced year by year from a table of ob-
fervations; namely, "That the probabilities of
66
life, after a certain age, decreaſe in arithmeti-
"cal progreffion." Thus, fuppofing, for inftance,
36 perfons, each of the age of 50, if, after one
year expired, there remained but 35, after two
34, after three 33, &c. it is evident that fuch
lives would be extinct in 36 years; and that the
probabilities of living 1, 2, 3, &c. years from the
age of 50, would be expreffed by the fractions,
35 34 33 &c, which decreaſe in arith-
36° 36′ 36′
metical progreffion: And fuppofing further, that
the age of 86 is the utmoſt probable extent of hu-
man life; the complement of life of a given age
will be the number of years which that age wants
of 86. Thus, 36 will be the complement of a life
aged 50 years.
Admitting, then, that the decrements of life,
after 10 years of age, are nearly equal; the fol-
lowing Problems will more eaſily be refolved, than
by a table of obſervations.
PROBLEM I.
61. Suppofing the decrements of life to be equal,
it is required, to find the value of L. I annuity upon
a life of a given age, and at a given rate of com-
pound intereſt.
Let n reprefent the complement of life.
R"
—the value of L. 1, due at the end of n years.
P===
}
of L. 1 annuity for ʼn years.
N=
!
i
1
118
OF ANNUITIES
N≈the value of L. 1 annuity upon a life of a
given age. Now, fince the probabilities of life.
decreaſe in arithmetical progreffion, they may be
repreſented by the terms of the following feries,
viz.
12- I N—2 22-3
>
12
72
N
,
NN2 which expreffions
12
may alſo be confidered as the values of each
ment of the annuity in the parts of L. 1.
I
I
the terms of the feries,
I
,
K² K² K³ 3
the prefent values of L. 1,
I
pay-
But
Rū
fhew
due at the end of 1, 2,
3,.... n years; therefore combining theſe two
feries together,
72→→ I
n R
+
77-2
273
(n)=N, will expreſs the va-
+
n R² F i K ³
2
3
lue of an annuity upon a life, whofe complement
is n, at a given rate of intereft.
Which feries may be divided into two other
feries, viz.
ท
= x + + +
N R
× 1/2 + 1/2 + 1/32
X
R
R³
1
R3
I
+
+ ( ) = = = = 2; (47) and-
(22) X
RP 321
(») = -1/2 ×
12
I
12
Hence,
* Sought, the value of n terms of the feries,
I
2
If the terms of the feries + +
2.
R
R
by R-1 = R ² + 2 R +ı
3
I
R
+ 1/2 + 1/2/3, &c.
༢
&c. be multiplied
The productR + o to. Hence,
is the value of the feries indefinitely continued.
¡
R
2
RI
R
2
ľ
Now
}
1
ON LIVES.
119
I
I
+1=N; but =+==
RP
12 r
r
RP.
Hence,
I
RP
n
Hence,
mir
r
Or, P-
RP-np
nr
i
= N, the value
fought.
Ex. It is required, to find the prefent value of
L. I annuity upon the life of a perfon aged 36
years, reckoning intereft at 4 per cent. Here,
n=50, P=21.482, R=1.04, and r=.04; hence,
1- Rp
I
r
•5532 13.83, the value fought,
.04
PROBLEM II.
Suppofing the decrements of life to be equal,
it is required to find the prefent value of L. 1 an-
Now the terms, which follow the n term, in the given feries,
Or,
are
I
n + 1 1 + 2 22+3
R+TR+²
+ +
Ru+² TR#+3'
X
R"
77
R
+
2
&c. al infinitum.
" + ² + " + 3, &c. ad infinitum.
R²
R39
Which feries may be divided into two other ſeries, viz.
I
n
71
n
X + + &c. =
R" R R² R3,
12
X
I
R”
I R
Rp
X
R"
× 1 + 2/2 + 3³;, &c. =
R
X
R"
(40) = "; and,
r
by the above. Hence,
if the fum of theſe two feries be fubtracted from the former, there
will remain
R
Rp
np
, or,
I pXR πρ
RP
пр
the value of
I
2
the feries
R
R
++
+ 233, (”).
nuity,
1 20
OF ANNUITIES
nuity, to continue during the joint lives of two
perfons, whofe refpective complements of life are
n and m, (m being the greater number) at a given
rate of compound intereft.
62. Since the probability of the continuance of
the life, whofe complement is n, for one year, is
and that of the life whofe complement is m,
22-I
n
m-
m
I
; and fince thefe two events are independent
of each other, it follows, that the terms of the
ſeries, "-1×"=1, "=2×m2, n=3x=3, &c. will re-
n
m
ทะ
I
n
172
12
-3
m-3,
m
ſpectively repreſent the probabilities of their con-
tinuing together, 1, 2, 3, &c. years which pro-
babilities may be confidered as the values of the
feveral payments of the annuity, in the
parts of
L. 1, which will become due at the end of 1, 2, 3,
&c. years; and the value of L. I annuity upon
the two joint lives, will be expreffed by the fol-
lowing feries.
11-
nm R
· 2 >< m
nm R2
2
+
11-3 Xiri-
·3 m·
nm R 3
3 &c.
of which feries n terms only will be ufeful, n
being the complement of the oldeft life. Now if
the numerators of the fractions in the above feries
be expanded by multiplication, they will appear
in this form.
12— I × 122- I = nm―m+nXII.
n—2 × m—2 = nm―m+nx2 +4.
nm―m+n x3+9, &c.
1—3×m—3 =
·3
Hence, the above feries may be divided into
>
three other feries,
viz,
ON LIVES.
121
nm
I
nm
(n)=??
viz. ×++, (")=x (47)
n m
m+n
nm
I
3
R2 R³,
nm
2
x + + (n) =
m+n
+ mm × 1/2 + 1/2 + 1/23 (12) = 1×
n
I
R2
R3
X
r
I-PRX np,
(By note, Prob. 1.)
1—p ×R+1×R—2nRp n²p
PXR+IX R—2nRp 12p
g3
↑
The fum of which three feries, being properly
ranged, may ſtand thus,
n m
n m
+m+n
1-1.
X
пр
m+ n
r
nm
I
nnp
1
2ND R
X
X
r
nm
nm
nm
X
Their fum=
X' + pXR⋅
r2
r²
+
m+nXR—m—nXRp
nmXr 2
the value of the annuity.
I
X
—PXR+1XR.
nm
r3
+
1~pXR+1XR
nmXr³ 3
Ex.
* If the feries+2+ &c. be multiplied
3
R³
by R-1'= R³+3R²+3R−1
the product
R²+R+0 = RIXR; therefore,
R+1 X R the value of the above feries indefinitely continued.
до 3
Now the terms which follow the n term in the feries are,
k" + 2
• &c.
Rr + 3
n2+12
74-2²
Rn+s
+
+ #+3²
nn +2n+1
Or,
R" + I
nn +4n+4, nn+6n +9, &c.
+
R" +
R" + 3
Which ſeries may be divided into three others, viz.
nn
I
I
I
>
R = × 1 + 2 + 3 &c. =
R"
277
R"
Ι
RR
X
X
X
R R2
[
++
3
x ÷ +
R
十​歳
​R3
,
-? &c. =
9
,
R³
&c. =
е
R
nn.
I
X
(40.)
R
(laſt note)
24
X
1x+iX®, as above.
R^
g. 3
In
122
OF ANNUITIES
Ex. It is required to find the prefent value of
L. I annuity, to continue during the joint lives of
two perfons, whofe refpective ages are 54 and 43
years, reckoning intereft at 4 per cent. Here n = 32,
m=43, p=.285, R=1.04 and r=.04. Hence,
I
+1 — pXR+iXR _ 1.5168
nmXr 3
nm Xr 2
=25.000
17.224
.088
42.224
33.947
m+nXR—m—n× Rp 74.730
2.4
The value of the annuity = 8.277
When the fingle lives are equal, m+n=2n, m—n
×Rp=0, nm=n2, and the negative quantity
2 R
nr2
Ex. Sought, the value of L. I annuity to con-
tinue during the joint continuance of two lives,
each aged 48 years, reckoning intereſt at 4 per
Here,
-2 + 1 =
cent.
—
r
- 1 X R TX
x² + 1 x R = 42.785.
11² X r³
R
2
nr² = 34.215.
The value fought=8.570,
1
In which three feries, making på and fubtracting their fum
Ru
from the above found value of the whole feries, there will remain
J−pXR+iXR 2np R пир
the value of " terms thereof.
PROB-
ON LIVE S.
123
PROBLEM
III.
To find the prefent value of L. I annuity, either
upon the joint continuance, or during the longeſt
liver, of any number of lives, at a given rate of
intereſt.
63. 1ft, As the value of joint lives enters an in-
gredient into almoſt every queſtion reſpecting life-
annuities, this value, at 4 per cent. may be dif-
covered by the following concife and accurate
method. Let n, m, p, and t, repreſent the com-
plements of life; c, the arithmetical mean com-
plement of all the lives, excepting the oldeft, whoſe
value, at 4 per cent. is N; and N-4N, the va-
lue of the joint lives. Then,
In the caſe of 2, 3, 4, equal joint lives, with
a fmall correction, 3
,, the ratio of the
value of the joint lives to that of the fingle one;
hence,
2
2, with a ſmall correction,,,
=1—the
ratio of the value of the joint lives to that of the
fingle one.
Again, in the caſe of 2
of
3 or 4
Sm.
equal joint lives, "=
71
C.
But when the lives are unequal, or 2 will
172
C
correct the irregularity arifing from the inequality
of the lives. Hence the following
&
TABLE
124
:
OF ANNUITIES
1
TABLE. ·
Corrections at 4 per cent.
−1 + 1+ 3+ 4+ 5+3
Lives.
n = 76
66
2
N.
n—12 N
4772
·7
56 51 46 35 26 16 6
-3
3
N-
n + 8
3c
4N-
26
6
N+9+10+10 +10 +10 + 9 + 6 +3
N -3
0+ 1+ 2+ 3+ 2+ı
Ex. Sought, the value of two joint lives, whofe
reſpective ages are 46 and 30 years, at 4 per cent.
Here n = 40, m= 56, and N—"+22 N = 12.135
-2.286 9.849, the value.
4772
Or, the prefent value of joint lives, at a given
rate of intereft, may be found univerfally. Thus:
Let a reprefent the expectation of their joint
continuance, (fee fupplement to problem 18.);
b, that of the oldeft; and A, the number of years,
by Table V. of annuities certain, correfponding
to N, the value of the oldeſt life *. Then, ſeeing
A A, the value by the faid table, an-
b
:а:
a
a
b
A
b
fwering to 4 years certain †, (or in the caſe of
two joint lives, the value correfponding to A-
A years certain), will be that of their joint con-
tinuance.
n
3m
Ex. Let there be five equal lives, each aged 32
* Let the age be 32, whofe value, at 4 per cent is 14.411; the
next lefs, to which, by Table V. is 14.029, anfwering to 21 years;
their difference is .382, and the increaſe of value for one year is .422 ;
.382
hence, =.905, to be annexed to 21, making, in all, 21.905 years.
422
+ Let the number of years certain be 32.96, the value of 32 years,
at 4 per cent. is 17.873, and the increaſe of value for one year is .274;
hence .274.86.235 to be added to 17.873, making their fum
18.108, the value fought.
years;
ON LIVES.
125
!
a A
years; it is required to find the value of their
joint continuance, at 4 per cent. Here, a=9,
b=27, and A=21.9; hence, 4 = 7.3 years,
whofe value, viz. 6.46, is that of the joint conti-
nuance of theſe five lives.
b
Or, the prefent value of L. 1-annuity upon
the
joint continuance of any number of lives, at a
given rate, may be found univerfally, thus:
2d, Let N expectation of the oldeſt life, whofe
complement is n; x=expectation of the joint con-
tinuance of all the lives; and A=number of years
certain, correfponding to the value of the oldeſt
life. Then,
x
Seeing N:x:: A: A, the value correſpond-
X
A
N
ing to years certain, (or in the caſe of two
N
A—
3772
lives, the value anſwering to A-A years), will
be that of their joint continuance.
Ex. Let there be three lives, whoſe reſpective
ages are 46, 36, and 26 years; it is required to
find the value of their joint continuance, reckon-
ing intereft at 5 per cent. Here,
N=20, x=12, and A=16.35; hence,
20
12× 16.35=9.81 years, whofe value, viz. 7.6, is
that of the joint continuance of thefe three lives.
In the caſe of m equal joint lives,
2
=1
N m
23d, To find the value of L. 1 annuity upon the
longeft liver of any number of lives, at a given rate.
Let "complement of the mean life, whofe ex-
pectation is N; X or expectation of the longeſt
liver, or of joint continuance of all the lives; and
d=N—the number of years certain, correfpond-
ing
126
OF ANNUITIES
- 1
ing to the value of the mean life, at a given rate.
Then,
X
Seeing n-N-N:n—X=x::d : N Xd, the
N
value correfponding to X- -xd years certain,
will be that of the longeſt liver.
Ex. Let there be four equal lives, whofe com-
mon age is 36 years; fought, the value of the
longeft liver, reckoning intereft at 4 per cent.
Here,
n=50, N=25, X=40, x=10, and d 4.46.
Hence,
و تم
40 — 4.45 x 10 = 38.2 years, whofe value, viz.
25
10=38.2
19.4, is that of the longeſt liver of thefe 4
2d
lives.
= the
In the cafe of m equal lives, X-
number of years certain, correfponding to the
value of the longeſt liver.
Theſe two methods of finding the value of the
joint continuance, &c. of any number of lives,
are more accurate than Mr de Moivre's Theorms
for the fame purpofe; and more concife than thoſe
of Mr Dodfon in his Mathematical Repofitory,
vol. 2. and may be applied in finding the value
of lives, by a table of obfervations, as well as by
the hypothefis.
Ex. It is required to find, by Table VIII. and
IX. the value of the joint continuance of two lives,
whoſe reſpective ages are 30 and 25 years, at 4
per cent. Here,
n
n=65.08, m=69.98, and A=27.244; hence,
A— A = 18.8 years, whofe value, viz. 13,
is that of the joint continuance of thefe two lives.
3771
TABLE
ON LIVES.
127
TABLE II. Shewing the prefent Values of L. I
Annuity on a fingle Life, according to Mr de
Moivre's Hypothefis of equal Decrements.
14.6c7 48
Age. 3 per cent. 4 per cent. 5 per cent. Age.3 per cent. 14 per cent. 15 per cent.
9-10 19.868 16.882
8-11 19.736| 16.791
7-12 19 604 16.698
14.544 49
13.012 11.748 10.679
12.764
|
11.548 10.515
14.480 50
12.511
11.344
10.348
13 19.469|| 16.604 | 14.412
6-14
19.331 16.508
14.342
15 19.192
16.410 14.271
ཀ ཀ ༦ད
51
12.255 11.135
10.176
52
11.994
10.921 9.999
53
11729
10.702
9.817
16
19.050 16.311
14.197 54
11.457
10.478
9.630
5-17 18.905
18 18.759 16.105
16.209
14.123 55
11.183
10.248
9.437
14.047
56
10.902
10.014
9.239
19 18.610
15.999 13.970 57 10.616
9.773
9.036
4-20 18.458 15.891
21 18.305 15.781 13.810 59
22 18.148 15.669
13.891
|
58
10.325
9.527
8.826
10.029
9.275
8.611
13.727 60
9.727 9.017 8.389
23 17.990 15.554
13.642 61
9.419
8.753
8.161
3-24 17.827 15.437 13.555
62
9.107
8.482
7.926
25 17.664
15.313 13.466
|
63
8.787
8.205
7.684
26 17.497
15.197 13.375
64
8.462
7.921
7.435
27 17.327 15.073 13.282
65
8.132
7.631
7.179
28 17.154 14.946 13.186 66
7.794
7.333
6.915
29 16.979 14.816 13.088 67
7.450
7.027
6.643
30 16.800 14.684 12.988
68
7.099
6.714
6.362
2-31 16.620 14.549 12.885
32 16.436 14.411 12.780
33 16.248 14.270 12.673 7! 6.coS
69
6.743 6.394
6.073
70
6.378 6.065
5.775
5.728 5.468
34 16.057 14.126
12.562 72
5.631
5-383 5.152
35 15.867 13.979
36 15.666 13.829
12.449 73
5.246
37 15.465 13.676
38 15.260 13.519 12.091 76
39 15.053
6 73
12.333 74
12.214 75
40 14.842
'3.359 11.956 77
13.196 | 11.837 78
4.453 4.293 4.143
4.046
3.912 3.784
3.632 3.520 3.415
3.207
41
14.626 13.028
|
11.705
79
42 14.407 12.858
11.570
80
43 14.185 12.683
44 13.958 12.504
11.431
11.288
81
8:
45 13.728 12.322
46 13.493
11.142
83
47 13.254
12.135
11.944
10 992
10.837
23+ in
85
-00 co co ∞0 CC co
3.111 3.034
2.776 2.707 2.641
2.334 2.284 2.235
1.886
1.850
1.916
1.429 1.406 1.384
0.961
0.950 0.937
0.484
0.481
0.476
0.000
O.OCO
0.000
4.854 4.666 4.489
5.029 4.826
128
OF ANNUITIES
}
Table II. is conftructed by the theorem
n
which expreffes the value of a fingle life, whofe
complement is n.
Or thus, let the fum of the following feries, viz.
I
R•
I
R
I
R
I
R
+
+
+
R2
I
R2
I
R²
+
I
R³
I
+ +
R$
&c.
I
R+
Their fumn
be divided by
1
J
R
+ n
R2
72-2 + n-3 R³
I
Τ
+
R-1 =the
n
preſent worth of L. 1 annuity upon a ſingle life,
whofe complement is n, at a given rate of intereſt,
and is the fame with the ſeries of equal decrements,
Problem I. wanting the laſt term, viz. ""R"-0.
12
Hence the fum of n-1 terms, in Table V. of an-
nuities certain, divided by n, will give the value
of a life whofe complement is n.
Ex. Let the age be 70, whofe value is fought,
at 4 per cent. Here, n=16, and
15 terms of Table V.
97.04
= 6.065
S the value
fought.
divided by
16
This affords an accurate and eafy method of
finding the value of ſingle or conjoint lives, &c. at
a given rate, not only by the hypothefis, but alſo
by
i
ON LIVES.
129
by a table of obfervations, when the expectation
of life is known.
Ex. By Table VIII. 22.42=expectation of the
joint continuance of two lives, whofe refpective
ages are 30 and 25; fought, their value at 4 per
cent. Here, n = 44.84 and 43.84 terms of
Table V. = 582.96
the value of the joint
continuance of theſe
divided by 44.84
=13,
Sth
two lives.
Again, let N denote the value of a fingle life of
a
7
n
n + 1 I
a given age, the probability that a life
will continue for one year preceding that correſ‐
ponding to N; then,
a
bR
x+N= the value of a fingle life one year
younger than that correfponding to N.
ท
As the difference of the terms in Table V. of an-
nuities certain doth conftantly decreaſe, and feeing
n-1 terms are divided by n, in finding the value
of a life whofe complement is n; hence, the pre-
ſent value of a ſingle life will always be leſs than
that of its expectation, confidered as fo many years
certain.
Whereas the amount of an annuity on a given
life (fee Table IV.) is greater than that during its
expectation; yet fuch is the accuracy of calcula-
tion, that the fum of Nx n, improven during
life, will exactly balance the n number of life-an-
nuities, each of whofe values is N, whether they
be paid regularly as they become due, or are dif
charged by fingle payments at the cloſe of each
TABLE
life.
R
}
130
OF ANNUITIES
TABLE III. Shewing the Value of L. 1 Annuity
on the Joint Continuance of two Lives, fup-
pofing the Decrements of Life to be equal.
Age Age.3 pr cent. pr cent. 5 pr cent. Age. Age. 3 pr cent.4 procent 15 procent.
30
12.434 11.182 10.133
35 12.010 10.838 9.854
40 11.502 10.428
10 15.206
13.342 11.855
15
14.878
13.093 11.661
20
14.503
12.808
II.430
9.514
25 14.074
12.480 11.182
45 10.898
9.936 9.112
30
13.585
12.102
10.884
30
50
10.183
9.345 8.620
35
13.025
11.665 10.537
55 9.338
8.634
8.018
ΙΟ 40
12.381
11.156 | 10.128
60
8.338
7.779
7.280
45
11.644
10.564 9.646
65
7.161
6.748
6.373
50
10.796
9.871 9.074
70
5.777
5.505
5.254
55
9.822 9.059
8.391
60 8.704
8.105
7.572
35
11.632
10.530
9.600
65 7.417
6.980
6.585
40 11.175
10.157
9.291
༡༠
5.936
5.652
5.391
45
10.622
9.702
8.913
50
9.955 9.149
8.450
15 14.574 12.860
11.478
35 55 9.156 8.476
7.879
20
14.225
25
13.822
12.593 11.266
12.281 11.022
30 13.359 11.921
60
8.202 7.658
7.172
65
7.066
6.662
6.294
10.736
70
5.718
5.450
5.203
35
12.824 11.501
10.402
15
40
12.207
11.013
10.008
40
10.777
9.826
9.014
45
II.496
I0.440
9.54I
45
10.283
9.418
8.671
50
10.675
9.76.7 8.985
50
9.677 8.911
8.244
55
9.727 8.975
40
8.318
55
8.936
8.288
7.710
60
8.632 8.041
60
7.515
8.038
7.510
7.039
65 7.377 6.934
6.544
65
6.951 6.556
6.198
70 5.932
5.623
5.364
70
5.646
5.383
5.141
20
13.904 12.341
11.067
45
9.863
9.063
8.370
25
13.531
12.051
10.840
50
9.331
8.619
7.987
30 13.098
11.711
10.565
55
8.662
8.044
7.500
35
12.594
11.314
10.278
45
60
7.831
7.332 6.875
40
12,008 10.847
9.870
65
6.807 6.425
6.080
20
45
11.325 10.297
9.420
70
5.556
5.300
5.063
50
10.536 9.648
8.880
55 9.617
8.879
8.233
60 8.549
7.967 7.448
6.495
25
65
7.308 6.882
70 5.868 5.590 5.333
25
13.192 11.786 10.621
30 12.792 11.468 10.367
35 12.333 11.095 10.067
40 11.776 10.655 9.708
45 11.130 10.131 9.278
10.374 9.509 8.761
9.488 8.766
50
55
8.134
60 8.452
7.880 7.37I
65 7.241 6.826 6.440
55 7.849 7.332 6.873
60 65 6.043 5.730 5.444
70 5.081 4.858 4.653
50
you you con
50
8.892 8.235
7.660
5'5
8.312
7.738 7.230
60
7.568
7.091 6.664
65
6.623
6.258
5.926
༡༠
5.442
5.193
4.964
60
7.220 6.781
6.386
65
6.379
6.036
5.724
55
༡༠ 5.291
5.053
4.833
60
6.737 6.351
6.001
1
70 5.826
5.551
5.294
65
65
70
5.547
4.773 4.571
5:277
5.031
4.385
70
70
4.270
4.104 3.952
In
ON LIVE S.
131
In the conftruction of Table III. Dr Price made
v + 1 x m—n — 20 — I X
ufe of the theorem V— "+
n
m
2บ
2+20=NM, expreffing the value of the joint
continuance of two lives, whofe complements are
n and m. Here,
}
repreſents
Sthe perpetuity=
Τ
at a given rate.
the value of L 1. annuity for n years certain.
If only one, or neither of the ages, are to be
found in the table, the value of their joint conti-
nuance, at a given rate, may be found by the
theorem, N-"+N.
n+x
4772
Ex. Let there be two lives, whofe refpective
ages are 38 and 22 years; fought, the value of
their joint continuance, at 4 per cent. Here N-
" + :3 N = 13.5 19 — 2.55=10.969, the value fought.
4111
Or thus,
ift, Let there be two lives, aged 35 and 22
years; fought, the value of their joint continuance,
at 4 per cent.
From the value of 20 by 35=H.314
Subtract that of 25 by 35=11.095
The remainder
×
= .4X.219.088; which
product, fubtracted from 11.314, leaves 11.226=
the value fought.
2d, Let there be two lives, aged 38 and 22
years; fought, the value of their joint continuance,
at 4 per cent.
R 2
From
}
132
OF ANNUITIES
From the value found above 11.226
Subtract that of 22 by 40=10.770
The remainder X &
*
=.6.456.274, which
product, fubtracted from 11.226, leaves 10.952,
the value fought.
TABLE IV. Shewing the amount of L. 1 An-
nuity foreborn, during a Life of a given Age,
at 4 per cent.
Ages.
Amount.
Ages. Amount. Ages Amount. ||Ages. Amount.
9-10
29
8-11
7-12
13
6-14
15
16
56
5-17
36
59.65 5:
12.52 33 548
108.74 34 55.37 53
IC5.11 35 53.32
101.66
128 81
66.58
3451
124.54 30 64.20 44 30.15
120.41 31 61.89 50
116.40 32
67
68
11.36
10 59
28.89
69
9.84
27.61
70
9.10
52
25.16 72
2637 71
8.38
7.67
54
24.00
73
6.98
18 9.13 37
19
94.81
51.34
49.41
38 47.53
55
22.85 74
6.31
56 21.75
75
$720.67
76
4-20
91 58
39
45.72
5
19.62 77
2 I
8844
40
43.95 59 18.61 78
22
23
3-24
85.41 4I 42.24
82.47 42 40.56
79.51 43 38.96
6%
17.62
79
6 16.66
62
5.72
25
76 85
44 37.39
6.
1480
26
74.17
45
35.87
64
27
28
7x
7156
46
3438
65
13.92
13.06
6904
17
32.02 66 1225
2 х х 60 00 00
80
56 1x ao
5.05
5.02
4.39
3.79
3.20
2.63
81
2.08
82
1.54
83
* 4
85
34 La
1.ΟΙ
0.50
0.00
In the conftruction of this table, which hath
not been attempted by any author,
Let the fum of the following feries, viz.
ON LIVES.
#33
I
1+R
1 + R + R ²
I + R + R² + R³
1 + R + R² + R³ + R
Their fum= n−1 +”
be divided by
&c.
− 2 R + n − 3 R² .... + R”−2
n
= the
amount of L. 1 annuity upon a ſingle life, whoſe
complement is n, at a given rate of intereſt.
Hence, the fum of n I terms, in Table IV.
of annuities certain, divided by n, will give the
amount of L. I annuity upon a fingle life whofe
complement is n.
Ex. Sought, the amount of L. I annuity during
a life aged 80 years, at 4 per cent. Here n=6, and
the five firſt terms of Table IV.=15.8244
divided by
the amount fought.
=2.64=
6
Or thus: Let s denote the amount of L. 1 an-
nuity upon a life of a given age, at a given rate,
a
Ъ
a
the probability that a life will continue during
one year preceding to that correfponding to s;
then XSR+1=the amount of L. 1 annuity upon
a fingle life, one year younger than that corre-
fponding to s; hence the foregoing table.
As the difference of the terms in Table IV. of
annuities certain, doth conſtantly increaſe; and
although the n-1 number of terms is divided
by n, in finding the amount of L. 1 annuity upon
a life whofe complement is ", yet in all
ages be-
low
134
OF ANNUITIES
{
low 68, that amount, at 4 per cent. will be great-
er than that of their expectation, confidered as fo
many years certain.
By this table, the amount of p, a fum of mo-
whoſe intereſt, at 4 per cent. remains unpaid
during a ſingle life of a given age, = 1+rs xp.
ney,
TABLE V. Shewing the true Probabilities of
Life in London, for all ages, formed by Dr
Price, from the Bills of Mortality for 10 Years
preceding Anno 1769.
Ages. Living., Dead. (Ages. Living. Dead. Ages. Living. Dd. Ages. Liv. Dd.
ΤΟ
OHN3+no no ag
|
1518 486 24
24 463
I
1032 200
2
832 85
747
59
688
42
N N N N
25 455
26
447
∞ ∞ ∞
8
8
50
48 242
49 233
224 9
9
9
72 64 6
73 585
74 53
27 439 8 51
215
28
646
23 29 422
623
20 30 413
9 54
603
14 31 404
589
IZ 32 395
9
577
569
IO
33
9
34
+34
386
57
377
9
58
16 700
431 9 52
53
190
9 55 183
183 7 79
56
169
206
1988
7
76 7 80
162
II
12
13
14
15
558
17
18
~ 3 + LO 70
549
541
98 7
35 368
9
59
155
8
36 359
9
60
147
8
37 350
9
61
139
7
534
6
38341
62
9
132
7
528
6 39 332 IO
63 125
7
522
40
7
IO
322
64 118
7
515
7 41
ΙΟ
312
65 III
7
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞~~~~~
75
48
76
77
78
no 7∞ a o
43
38
33
29
25
81
22
82
19
16
83
84 13
85
I I
86
34 9
87 7
88
6555unt & en en MMN~~~-
mx moo ma
3
3
3
3
2
2
2
2
I
89 4
I
508
7 42
302
10
66 : 104
7
90
3
I
19
501
7
292
43
10
67
97
7
91
20
494
7
44
282
10
68
90 7
92
21
487
~ N
22 479
23 471
∞ ∞ 20
8
45 272
10
69 83 7
93
133
N HO
2 1
I
8
46
262
10
70 76 6
8
47
252
10
71
70
6
}
By
ON LIVE S:
35
By Table V. the number of births or burials,
which are fuppofed to be equal, is to that of
inhabitants as 1518: 27935-759; that is, as
1:17.90; one out of forty who are born in
London arrives at 80 years of age, and the ex-
pectation of life is as follows:
Ages. Expectat. Ages Expectat.||Ages. | Expectat Ages. Expect↓
0 17.90 25 26.67
535.23 30 24.12
10 34.92 35 21.77
15 32.32 40 19.51
2029.37 45 17.64 70
50 15.90
75 6.33
80 4.98
853.32
90 1.50
8.77
55 13.93
60 11.71
65 9.71
TABLE
136
OF ANNUITIES
TABLE VI. Shewing the Value of L. 1 Annuity
upon a fingle Life, according to the true Pro-
babilities of Life in London.
Ages. 13p.cent.4p.cent. 5p.cent. Ages 3p.cent.14p.cent.[5 per c.
8--9
7--10
19.13 16.39 14.27 JI
II 00
10.07 9.26
19.05 16.35 14.25
52
1083
9.93 9.15
II
18.94
16.28 14.21
53
10.60
9.75
8.97
12
18.83
16.21 14.16 54
10.40
9.56 8.82
6-13
18.69 16.10
14.09 55
10.13
9.32
8.61
14
18.50
15.97
13.99
56
5 -15 18.27
15.80
13·85
57
16 18.04
15.62 13.71
58
17
17.83
15.46.13.59
59
4-18
21
17.61
19 17.39 15.14
20 17.17
14.96
16.94 14.79 13.06
15.30 13.47
60
26 7∞ a o
9.85
9.08 8.40
9.56
9.28
8.94
8.83 8.19
8.58 7.97
8.33
7.75
8.76
8.13 7.58
13.34
61
8.54
7.94 7.45
13.20
62
8.26
7.69
7.20
63
7.98
7.45
6.98
22
16.74 14.63 12.94
64
7.71
7.21 6.76
23
16.53 14.48
12.82
65
7.44
6.97
6.55
3--24 16.32 14.32
12.69
66
7.18
6.74 6.34
25
16.11 14.15
12.56
67
6.93
6.52 6.14
26
15.89 13.98
12.43 68
6.69
6.31 5.95
27
•
28
15.43 13.62 12.14
15.66 13.80 12.29 6)
6.48
6.11 5.77
70 6.29
5.94 5.62
29
15.24 13.47
12.02
7I 6.03
5.71 5.40
2.-30
15.03 13.31
11.89 72
31
32
14.83 13.16 11.77 73
14.62 12.99 11.64
33 14.41 12.83 11.50 75
5.29 5.04 4.79
5.02 4.79 4.56
34 14.20 12.66 11.37 76 4.77 4.56 4.34
74
♡ +
5.79 5.50 5.20
5.58 *5.31 5.04
333 ♡♡
35
13.98 I2.49 II.23 77
36 13.76
12.31
II.08
37
38
13.54 12.13 10.94
13.32 11.95 10.79 80
39 13.12 II.77 10.63
40 12.93 11.62
4.23 4.03
4.17 4.00 3.82
3.98 3.83 3.65
3.66 3.54 3.42
78
777∞∞
4.56
4.37 4.16
78
4.41
79
81
10.51 82 3.37 3.26
3.15
4I
42
43
44
12.21
12.75 II.47 10.39 83 3.12 3.02
12.56 11.32 10.27
12.38 II.17 10.15
11.03 10.04
2.93
45
46
84 2.95 2.87
85 2.60
86 2.37 2.21
12.04 10.90 9.93 87 2.CO 1.96 1.92
11.87 10.77 9.82 88 1.89 1.85 1.82
47 11.72 10.64 9.72 89 1.43
1.41 1.38
2.79
2.53
2.46
2.16
48 11.57 10.53 9.61
0.96
0.95 0.94
49 11.37 10.37 9.50
0.48
0.48 0.48
50
11.19 IO.22 9.38
Tables VI. and IX. are conſtructed by the theo-
a
rem RI+N, which expreffes the value of a
fingle life, one year younger than that correfpond-
ing to N.
TABLE
ON LIVES.
137
TABLE VII. Shewing the Value of L. 1 Annuity
upon two joint Lives, according to the true
Probabilities of Life in London.
Age. Age.3p.cent. 4p.cent. 5p.cent. Age. Age 13p.cent.4p cent. 5p.cent.;
ΙΟ 14.39 12.79 II.29
15 14.09 12.54 11.10
30 10.98 9.97 9.04
35
10.54 9.58 8.75
20 13.54
12.03 10.79
40
10.04
9.13 8.40
25
12.97
11.56 10.44
45
9.57 8.74
8.10
30
12.32 ΙΙ.ΟΙ 10.02 30 50
9.11
8.36
7.80
35
11.68 10.50 9.60
55
8.44
7.76
7.32
IO
40 10.96 9.91 9.10
60
7.52
6.96
6.59
45
10.39 9.41
8.70
65
6.58 6.10
5.81
50
9.85
8.93
8.31
70
5.65
5.25
5.04
55
9.01
8.26 7.74
60
7.91 7.32
6.90
35
10.14
9.30
8.38
65
6.86
6.37
6.04
40
9.70
8.86
8.55
70
5.85
5.46
45
9.28
5.22
8.50
7.88
50
8.88
8.15
7.61
15
13.71 12.22 10.89
35 55
8.26
7.63
7.18
20
13.06 II.79 10.57
60
7.38
6.84
6.48
25 12.55 11.34 10.24
65
6.47
6.00
5.72
30 11.98 10.83 9.85
70
5.58
5.18
4.97
35
11.38 10.32
9.44
15
40 10.73
9.76
8.93
40
8.54
9.19
7.84
45
10.16
9.29
8.60
45 8.86
8.23
7.61
50
9.60
8.83
8.22
50
8.53
7.92
7.38
55
8.34
8.18
40
7.67
55
7.96
7.42
6.98
60
7.81
7.26 6.84
60
7.17 6.69
6.34
65
6.83 6.32
6.00
65
6.33 5.89
5.62
70
5.31 5.43
5.19
5.50 5.09
4.90
20
12.76 II.47 10.20
45
8.61
7.94
7.33
25 12.27 II.04 9.95
50
8.30
7.67
7.15
30
11.76 10.58 9.61
55
7.85
7.23
6.81
45 60
35
11.19 10.10 9.24
7.05
6.53
6.20
40
10.58
9.57
8.81
65
6.4+
5.78 5.52
20
45
10.05
9.14
8.47
5.41
5.00 4.81
50
9.50 8.69 8.10
50
55
65
70
8.75
8.06 7.56
60 7.75 7.17 6.76
6.73 6.25 5.93
5.76 5.38
7.96 7.39
6.89
55
7.53 7:00 6.59
65
6.39
5.15
70
50 60 6.85 6.36
6.04
5.65
5.40
5.29 4.90 472
25
11.87 10.74 9.66
55
30 II.40 10.30 9.36
7.16 6.67 6.23
60
35
10.89 9.86 9.03
40 10.33 9.36 8.63
65 5.88 5.46 5.22
5.12 4.75 4.59
6.57 6.TI 5.30
55
45
9.82 8.94 8.29
25
50 9.34
8.53
7.95
60 6.14 5-72
5.44
55 8.62
7.94 7.46
3
60
5.56
5.18
4.97
60
7.64
7.07 6.68
4.36
4.52
65 6.66
4.39
6.18
5:88
70
5.71
5.32
5.10
65
5.22 4.81
Sr
4.64
65
4.5%
4.23
4.II
201701 4.33
3.94
رح
TABLE
138
OF ANNUITIES
TABLE VIII. Shewing the Probabilities of Life,
according to the Bills of Mortality in Kettle,
for 20 Years, preceding Anno 1790.
Ages Living Dead. Age: Living. Dd. Ages Living. Dd. ||Ages. Living. Dd.
O
625
20
I
605
40
2
~ 3+56 5∞
565 36
529 20
4
509
16
~ ~ ~ ~ ~
4567∞
24
392
6
48 295
25
26
3866
49
5 72
290 6 73
104
380
6
50
284 7 74
27
28
374
368
493
481
1 2
29
363
6
30
358
cow oo A
554
6
51
277 7
52
270 7
53
263 7
54
256 7
222 J
75
76
77
78
475
4
31
354 4
55 249
7
79
8
471
32
350
55242
7 80
9467
33
3464
57 235
10
463
34
342
58227
II
459
35
338
59
219
12 455
4
36 334
15
16
17
18
∞v answ
13 451
14446
37 330
60 211 9
61 202
9
38
5
326 3
441
5
39
323
435
40 320
431 5
41
317 3
W W W W
62 193
∞ ∞ ∞ ∞ ∞ ∞
81
9
3
63
184
3 64 175
65
166 9
∞ ∞ ∞
333 ~ ~ ~
3456
N∞
82
83
84
85
86
87
88
34
31
28
25
89
426
5
42
314 3
66
157
90
19
421
5
43
311 3
67
148
91
20
416
44
308
3
68
139
92
∞ ∞ ∞ ∞ ∞ ∞ on +33 3 3 ♡ ♡ ♡ MMN N
1∞ ∞
∞ O N 46
16
no nao
8
968
88
80
72
64
8
8
8
56
7
49
6
43
5
38 4
22
3
19
3
16
3
13
3
ΙΟ 3
7 2
5
2
21
410
45
305 3
69
130
93
22 404
6
46
302
3
70
121
94
32
I
I
23
398 6
47
200
71
I 12
95
1
By Table VIII. the number of births or burials
is, to that of inhabitants, as 625: 25368 -312.5;
that is, as 1:40.1; 2 out of 29 arrive at the age of
80 years, and the expectation of life is as follows:
757.23
80 6.41
Ages. Expectat. Ages | Expectat. Ages. Expectat. Ages. Expect.
040.10 25 34.99 50 18.39
5 45.21 30 32.54 55 15.63
10 43.01 35 29.31
60 12.98
15 40.03 40 25.83 65 10.82
2037.29 4521.98
85 4.42
90 2.30
70 8.91
950.50
Ex
ON LIVE S.
139
Ex. Let there be two lives, whofe refpective ages
and 30 years; then
are 40
N=25.83
M=32.54S
21
the double of which
$51.66=n.
265.08=m.
N— N=19.00 = { NM, expectation of their
3m
joint continuance.
the oldeft.
N-NM- 6.83=the expectation of
M—NM=13.54 furvivorship on the the youngeſt.
both. And
N+M—2NM≈20.37=
N+M-NM
part of
= M + ZN
=39.37,
3772
Sthe expectation of the
longeft liver.
TABLE IX. Shewing the Value of L. 1 Annuity
upon a fingle Life, according to the Probabi-
lities of Life at Kettle, computed at 4 per Cent.
| Ages. Values. Ages. Values. Ages. Values.
18.40 ९० 16.41 53 10.73
18.29 31 16.26 54
Ages.(Values.
6-7
65.19
5-8
10.49
77
5.07
9 18.19 32
16.II 55
10
18.08
4-11
17.97
CA) CA
33
15.94
56
34
12 17.85 35
15.78 57
15.60
13
14
15
3-16
17
on t in o
17.73
18
70
17.65
17.56
17.47
17.38
17.29
19 17.15
f f f CA) CN CO UN
36 15.42 59
O ~~ an
10.2 I
9.93
9.63
9.37
9.10
37
38
100
15.23
бо
8.82
15.03
6
8.59
39 14.82 62
8.35
40 14.55
63
8.10
41 14.28
64
7.86
42
13.99
7.52
20
21 17.00 ff
17.05 43 13.69
13.38
65
7.38
770 00 00 00 00 00 00 00 00 00
78
∞ a o
5.03
794.98
80
4.90
81
4.73
8 2
83
4.54
4.18
843.81
85
85
87
3.44
3.05
2.68
2.32
1.96
N N N N
22
23
24
+ W N
16.94
16.88
16.82
25
16.77
444
100 700
45 13.05 68
67 7.14 до
6.91
1.65
46
12.71 69
6.68
1.46
92 1.13
47
12.35
48
12.03
6.47 93 0.95
626 94 0:48
26 15.72 49 11.72 72
6.32
95
0.00
2-27
1665 50 16.45
73
5.78
28 16.61
5 1
II.21 7+
5-55
29 16.51
52
10.05 75
5.35
S 2
TABLE
140
OF ANNUITIES
#1
TABLE X. Shewing the Value of L. 1 Annuity
on the joint Continuance of two Lives, accor-
ding to the Probabilities of Life at Kettle, com-
puted at 4 per Cent.
Values at 4 per Cent.
10 14.38
15 14.16 13.86
20 13.93 13.67 13.42
25 13.80 13.53 13.34 13.12
30 13.62 13.42 13.25 12.98 12.73
35 13.26 13.06 12.85 12.60 12.39 11.97
40 12.54 12.39 | 12.22 | 12.04 11.83 11.49 11.03
Ages 10
15 20
45 11.44 11.32 11.19 11.06 10.89 10.62
50 10.22 10.II 10.01 9.91 9.79 9.60
55 9.24 9.17 9.09 9.02 8.91 8.77
60 8.09 8.04 7.98 7.92 7.85 7.74
65
7.07 7.04 7.00 6.95 6.90 6.82
70 6.08 6.05.6.02 6.00 5.95
25 30
35
5.89
10.27 9.72
9.33 8.92 8.38
8.56 8.257.84 7.39
7.60 7.37 7.07 6.75 6.29
•
6.716.55 6.34 6.10 5.75 5.38
5.82 5.70 5.55 5.38 5.154.88 4.53
40 45 50 55 60 165 170
Tables VII. and X. are conftructed by the
Theorem A-A, which expreffes the number
312
of years certain, correfponding to the value of
the joint continuance of two lives, whofe com-
plements are n and m.
+
•
:
TABLE
ON LIVES.
141
TABLE XI. Shewing the Probabilities of Life de-
rived from the Bills of Mortality in Edinburgh,
Anno 1780.
Ages. Living. Dd. Ages. Living. Dd. Ages. Living.|Dd. ||Ages.| Liv. Dd.
о
140
753
24
345
I
613 74
25
341
IO
23+nO N∞ a o
539
40
26
337
3 499
4 471
28
27
20
70
~
333
329
44
44
48
میں
233
6
72
8
6
49
227
6
73 75
6
50
22 I 6
74
69
6
51
215
6
75
52
209
6
76
36
63
57
6
451
14
29
325
53
203
5
77 51
6
437 12 30
321
7
425 10 31
317
55
415
32
313
9407
401
6
I I
397 4
12
14
15
3+no noo
393
13389
4
395
♡♡♡ ♡ ♡ ♡
33 309
57
34
305
ииии
av awe of
54
6
99%
198
78
45 6
193
79
39 5
188
183
58
177
35
301
59
171
36
297
60
165
37
38
293 5
288
391
4
39283
369
ao
365
361
357
16 377 4 40
17
18
19
20
21
373 4 41
2785
273
In In In In In
61 158
62
151 7
63
144
64
137
7
4 42
268
65 130 7
66
123
7
4
43
263
67
I
116
7
со оо оо со оо со со со со со аа
ဝ
34 5
81
29
82
25
4
83
21
84
18 3
!
85
15
3
86
12 3
87
9
2
88
7
2
89
5
2
90
3
I
91
2.
44
257
6
68
109 7
92
I
45
251
6
69
102 7
93
22
353
46 245
70
95
7
ΤΙ
88 7
27 349 4 4.7 239
By Table XI. the number of births or burials
is to that of inhabitants as 753:21968-376.5,
that is as 1 28.67; 1 out of 22 arrives at 80 years
of age, and the expectation and value of life (at 4
per cent.) are as follow.
Ages. Expedat. Values. Ages Expectat. Values. Ages. Expect. Value.
O 28.67 11.58 35
5 41.83 17.26 40
IO 41.79 1798 45
15 | 38.85 | 17:14
20 35.87 16.46
25
32.82 16.00 60
30 29.71 15.12
26.52 || 14.46
23.50 13 43
20.74
12.43
|
70 8.38 6.19
75 6.42 4.85
8a 4.82 3.66
11.45 85 3.10 2.34
10.23 90 1.50 0.95
8.74 92 0.50 C.CO
50
18.22
55
15.54
12.71
65
10.45 7.47
It
142
OF ANNUITIES
1
po-
It is required to find the amount of the
pulation in the city of Edinburgh and its en-
virons, anno 1792, exclufive of Leith. The an-
nual number of burials = 1800; which, making
allowance for adventurers, and an increaſing po-
pulation, may be, to the births and fettlers, (at
15 years of age), as 2: 3; and thefe laft to one
another, as 3: 1. Hence,
The annual number of births
2000
of burials
1800
of fettlers
700
their fum=4500
Therefore half this fum
<=2250
=31.22
multiplied by the mean expectation of life,
viz. 28.67 x3+38.85.
4
will produce, with the addition of 755 in
the caſtle, the number of inhabitants 71,000
Table XI. is a ſtriking evidence of the im-
provements in favour of life in the city of Edin-
burgh, during the period of half a century, from
anno 1740, at which time Dr Price conſtructed a
table of obſervations adapted to that city, where-
by he found, agreeable to the London tables, that
only 1 in 42 of all who died reached 80 years, and
that the expectation of an infant's life did not ex-
ceed 18; which difference in favour of human
life may ariſe from the encouragement given to
trade in all its branches fince the year 1745,
whereby the bulk of the inhabitants are better
fed, clothed, and lodged, have more elbow-room,
and freſher air, than at any former period.
The
ON LIVE S.
143
The Breſlaw table, (Table I.), conſtructed by
Dr Halley, as alfo the Norwich and Northampton
tables, publiſhed by Dr Price, agree, nearly exact
in point of expectation and value, with the hypo-
thefis, and are fuppofed to be adapted to Europe
in general; the London table (Table V.) corre-
fponds to a large and populous capital; and Table
VIII. and XI. contain probabilities of life, which
will be found to be adapted, either to Scotland
in general, or to Edinburgh in particular.
As annuities on lives are commonly bought or
fold at fo many years purchaſe, the values affign-
ed in Tables II. III. &c. may alfo be fo reckoned;
at the fame time, they who fell annuities have
generally 1 or 2 years purchaſe of more value
than ſpecified in the tables, from purchaſers
whoſe age is 20 years and upwards.
Ex. A perfon refiding in London, aged 30
years, wiſhes to purchaſe an annuity for life of
L. 100 per annum; what muſt he pay for it, rec-
koning intereſt at 4 per cent. ?
Here 13.31× 100 L. 1331, the purchaſe-money.
An annuity begins to run from the date of the
event which intitles it: and in the cafe of a ſimple
annuity paid in money, the annuity ceafes at the
term prior to the annuitant's death; but in the
caſe of an annuity fecured by land, the heirs
of the annuitant are intitled to payment to
the laſt moment of his life. In this cafe, by
increaſing the numbers in Table II. by .2 of a
years purchaſe, to 54 years of age, by .33, from
54 to 70, and by .44 from 70 and upwards, ve
fhall obtain the prefent values nearly of an en-
nuity (fecured by land) of L, 1 per annum cn a
ingle
144
OF ANNUITIES
fingle life, fuppofing the decrements of life to be
equal.
As the real law, according to which human
life waftes, approaches much nearer to an arith-
metical than to a geometrical progreffion,, all
thofe theorems of Mr de Moivre are rejected
which are built upon the principle, "That the
"decrements of life are in a conftant Ratio," fee-
ing they give the value of life too ſmall.
PROBLEM
IV.
Suppoſing the decrements of life to be equal, it
is required, to find the value of L. I annuity upon
the longeſt of two lives, whoſe reſpective comple-
ments are n and m, (m being the greater number),
at a given rate of intereft.
72-
64. 1 is the probability that a life, whofe
1
I
complement is ", will exift one year, and the
11
chance of its failing in that time; but the fum
of theſe two fractions is equal to unity: Therefore
having one of them given, we may, by fubtrac-
tion, find the other. Now the probabilities of
the continuance of thofe lives for one year, are
reſpectively" and "; and the probabilities
71
1
212
I
of their feverally failing are 1-2 and 1-"!;
22
712
confequently the probability of their both failing
in that time, will be I-XI-"-", which,
72
772
fubtracted from unity, leaves the probability of
there
ON LIVE S.
145
there being one of them, at leaſt, exifting at the
year's end. But,
22
η
I
I -
X I
772
1
N - I
772
I
I -
+
n
in
12
772
-1Xm-1, which, taken from unity, leaves
nIn
12
I
+
772
T n — 1 X 1 — Į
m
I
"
the firſt payment of
772
nin
2
111 -
2
12 2 X m
2 is
7172
the annuity in the parts of L. 1.
n
In like manner, "="+"
n
712
the fecond payment of the annuity, &c.; hence,
the required annuity may be expreffed in the fol-
lowing manner :
12
n R
71
nR2
I
2
+
+
772
m R
mR2
772
•
I
n — IX m
nm R
I
72
2 X 172
2
nm R-
72
3
n R³
+
3
21
3 X m
mis
nin R³
3
>
3. &c.
But the feries in the firft and fecond columns,
are the values of annuities on the fingle lives,
whoſe complements are n and m; and the third
column is the value of an annuity upon the joint
lives of the two perfons; therefore,
If, from the ſum of the values of the fingle lives
of two perfons, the value of their joint lives be
taken, the remainder will be the value of an an-
nuity on the longeſt of them.
Or thus: Let x and y reprefent the refpective pro-
babilities of the two lives continuing for one year;
then,
I—x × I—y = 1−x-y+xy, the probability of the
two lives failing in that time, fubtracted from
T
unity
146
OF ANNUITIES
unity, leaves x+y-xy, the probability of one,
at leaſt, of the two lives furviving the year; and
by fubftituting the values of the lives in place of
the probabilities thereof, (N and M being the va-
lues of the fingle lives, and NM, that of the two
joint ones,) then N+M-NM will be the value
of an annuity upon the longeſt of them.
I
Ex. It is required, to find the value of an annu-
ity of L. 1 on the longeſt life of two perfons, whoſe
reſpective ages are 35 and 25 years, allowing in-
tereft at 4 per cent. Here,
N+M=29.297
--NM=11.095
their difference-L. 18.202 is the value re-
quired.
If the two lives are equal, (N "being the value
of the two joint ones,) then 2 N-N" will be the
value of an annuity on the longeſt of them.
PROBLEM
V.
Suppofing the decrements of life to be equal,
it is required to find the value of L. 1 annuity up-
on the longeſt of three lives, (the values of the
ſingle ones being given), at a given rate of com-
pound intereft.
65. By following a proceſs fimilar to that ob-
ferved in the laſt problem, it might be fhewn, that
the value of an annuity upon the longeſt of three
lives is equal to
+
the values of the three fingle ones,
the thrce values of the joint lives, taken two
and two,
+the value of the thrs: joint lives.
Or thus:
Let
ON LIVF S.
147
Let x, y, and a reprefent the refpective proba-
bilities of the continuance of the three lives for
one year; then,
I—XX I—y XII-X-y-x+xy+xz+yz
I-
-xyz, will reprefent the probability of the three
lives failing in one year, which fubtracted from
unity, leaves,
x+y+z—xy—xz-yz+xyz, the probability of one,
at leaſt, of the three lives outliving the year; and
by ſubſtituting the values of the lives in place of
the probabilities thereof, (NMF being the value
of the three joint lives,) then N+M+F-NM-
NF_MF+NMF=value of the annuity.
Ex. Let the three fingle lives be worth 13, 14,
and I
15 years purchaſe; fought, the value of an
annuity upon the longeſt of them. Here,
13+14+15+8.750.7
-10.1-10.5-10.9-31.5
the value required = 19.2
PROBLEM VI.
To determine the value of an annuity of L. 1,
granted for a given term of years, denoted by 11,
on the contingency of its ceafing upon the extinc-
tion of an affigned life, whofe complement is m,
at a given rate of compound intereft.
66. Since every payment of the annuity depends
upon the continuance of the affigned life, whofe
complement is m, it will be worth n terms of the
feries, viz.
71
MR
I
211-2
mR2 MR3
+ + (1) which may be divided
into two other feries, viz.
T2
148
OF ANNUITIES
in
X
I
172 R
1
A₂+ (n) =P.
n)=P. (47)
R2 R3
2
x++(n
R + 1/3 (n) = = x
I
I
272
Hence P-
R =
RP - P.
nh
mr
Fr
RP_mp_ Prob. 1ſt.
↑
r
RP—" is the value required.
mr
Ex. A, aged 40 years, engages to pay to B or
his heirs, an annuity of L. 1 for 15 years, if he
(A) fhall live fo long. Sought, the value of B's an-
nuity, reckoning intereft at four per cent. Here,
n=15,m=46,P=11.118, and p=.55526. Hence,
11.118 3.234 = 9.36, is the value of B's annuity.
1.84
TABLE XII. Shewing the Value of L. 1 Annuity
for a certain Number of Years, depending up-
on the contingency of a given Life.
Ages, computing intereft at 4 per cent.
Yrs.
5
10
ΙΟ
20
25 30 35 40 45 50 55 60 65 70
4.24
5.49
7.27
IO 15
4.28 4.27 4.25
4.22 4.20 4.17 4.14 4.09 4.03 3.95 3.83 3.64
7.56 7.52 7.47 7.42 7.36 7.29 7.20 7.09 6.94 6.76 6.50 6.11
15 10.05 9.97 9.89 9.79 9.67 9.53 9.36 9.14 8.87 8.50 8.00
20 11.95 11.83 11.70 11.54 11.36 11.14 10.87 10.54 10.12 9.56 8.78
|
| |
25 12.36 13.2C 13.02 12.81 12.56 12.26 11.89 11.43 10.85 10.08
30 14.42 14.22 13.98 13.71 13.39 13.01 12.55 11.97 11.23
14.66 14.34 13.96 13.50 12.93 12.24
35 15.20 14.95
40 15.76 15.4
45 16.27 15.8
15.15 14.77 14-32 13.79 13.13
|
15.56 15.0514.54 13.92
50 16.45 16.0 15.68 15.21 14.65
11.97|11.23
Mr Simpſon, in his felect exerciſes, p. 262, is
very defective in the conftruction of this table;
which yet may eafily be calculated from the rela-
tion which the above theorem hath to P-
RP—"PN, the value of the fingle life, whoſe
complement is 7, Prob. I. Thus, for
72 1
Ex.
ང་
本
​ON LIVES.
149
Ex. A, aged 25, enjoys an annuity for the
ſpace of 20 years, if he fhall live fo long; it is
required, to find the preſent value of ſaid annuity,
reckoning intereft at 4 per cent. Here n=20,
and m=61, the complement of the life; therefore,
From the value of an annuity for 20 years = 13.590
Subtract the value of a life aged 66 years = 7.333
their difference =
multiplied by n
6.257
20
their product divided by m = 61) 125.140
(13.59
2.05
There remains the anſwer fought =
11.54
A Question of Infurance on Lives.
A, aged 16 years, is affured of the fum of
L. 500, in cafe he fhall live to be 21 years of age
complete; it is required, to find what premium
he ought to pay to infure his life during thefe 5
n years, upon that fum.
Let
b
a 65
70A's chance of
living 5 years.
71
b
11
5
70.
dying in that time.
a
b
But A's intereſt in L. 500X 500 =L. 464.3,
their difference, viz. L. 35-7, is the premi-
um of infurance; or A may lay his account
71
to loſe X 500-L. 35.7, the fame premium.
b
If this premium be paid per advance, at a gi-
ven rate; then 35-7
L. 29.34, its prefent value at
4 per cent.
Rn
If A fells his intereft in this L. 500, at a gi-
ven
150
OF ANNUITIES
ven rate;
464.3
then =L. 382.45, the value at 4
RɅ
per cent. See Prob. XV. Ex. 1ft and 2d.
Supplement to Problem I. II. III. IV. V. VI. Chap. VIII.
ART. I. To find, by a geometrical figure, the
value or amount of L. I, or of L. I annuity, up-
on a fingle life, whofe complement is n, at a gi-
ven rate of intereſt.
R
I
20/1
P
R
I
O
201m
R
I
ཚུ|"
I
O
R
O
R
I
O
Pol-
I
О
-| -|
R
I
R2
R
I
V
Ꮓ
R
I
I
R”- - 1 || Rn--2
R
R
I
I
Rn=-3
R
I
0
RN~~ I R1--2
M
R2
R
I
O
Ι
O
Ist,
ON LIVES.
151
ift, Seeing, that out of n perſons alive, at a
given age, whofe complement is n, one is fup-
pofed to fail yearly, till they are all exhaufted at
the age of 86; the two triangles V and Z, which
contain the value or amount of L. 1 annuity for
1+2+3...+n-1 years, will reprefent the va-
lue or amount of L. I annuity upon n lives,
whofe common complement is n, at a given rate;
and the two bottom columns P and M, which con-
tain the value or amount of L. 1 for 1+2+3
+n years, will fhow the value or amount of L. I
annuity for n years, at the fame rate; hence,
I
let or Z = val. or am. of L. 1 annuity for 1+2+3 . . . +n—1 years,
Por M— or of
for n years; then,
V or Z
Por M
==
val. or am. of L. I annuity (upon a fingle life.
or
of L. I
whofe complet. is n.
I
-r N
is alfo
value, and 1
R
2dly, But feeing that
upon a fingle life, whoſe
+rs=amount of L. 1
complement is n, at a given rate :
n RP
rn
M-n
Hence N =
S=
r n
= value of L. 1 annuity up-
on a life, whofe
amount complement is n.
Ex. It is required, to find the value or amount
of L. 1, or of L. I annuity, upon a life aged 36
years, at 4 per cent. Here n=50, P=21.482,
M=152.667, R=1.04, and r=.04. Hence,
n-RP
M—n
P
·M
=rn=
÷ n =
S13.829-N, the va. S of L. 1 an-
251.34s, the am. [nuity.
S.4296 =va. S of L. I, upon a life,
3.053 am. age 36 years.
ART.
152
OF ANNUITIES
I
ART. 2. To find the value or amount of L. r
or L. I annuity, upon the joint continuance of
2, 3, 4.... m lives, at a given rate of intereſt.
Here,
ift, Let n reprefent the mean complement of all
the lives,
X
the expectation of their joint
continuance,
X ➖➖➖ the expectation of the longeſt
Por M
Qor A
p or a
D
liver,
the value or amount of L. 1
annuity for x years,
the value or amount of L. I
annuity upon a life whofe
complement is x, and
I
the value or amount of L. 1
for x years; then,
K
Fig. 2.
.,
P, P, P, or M, M, M.
A
P
C
Þ, p, p, or a, a, a.
B
12,
In Fig. 2. put AB=n, AC or CD=x, of con-
fequence CB or DK-n-x=X, and complete
the rectangle BD, and triangle 2. Now, feeing
that out of n lives, whofe mean complement is
one will fail yearly during x years, DB+2=
2x P+value of a life whofe complement is x
multiplied
22
!
ON LIVES.
153
multiplied by x, will reprefent the value of L. i
annuity upon n lives for x years; and the bottom
column, viz. n-xxp+P, will fhew the value
of L. 1 on n lives for x years; hence,
j
=X
XP or M+xx Qor A
÷÷÷n=
Xpora+P or M
Val.`or am. of
L. I annuity.
Val. or am. of
L. I during the joint continuance of the lives.
Ex. It is required, to find the value or amount
of L. 1, or of L. 1 annuity, upon the joint conti-
nuance of three lives, whofe refpective ages are
40, 36, and 26 years, reckoning intereſt at 4 per
cent. Here,
M≈
n=52, x = 12.88, X=39.12, P=9.913, M-
16.434, Q=5, A=6.9, p=.6035, and a 1.657.
Hence,
387.796+64.4
642.898+88.87
23.609+ 9.913
64.822+16.434)
am.S
÷ 52=
8.696
val. 2ofL.1
14.072
ann.
1.563am. du-
.6446 val. 2 of L.1
ring the joint continuance of thefe three lives.
2d, In the cafe of mequal lives, whofe common com
plement is n, x =
n
712+ i
and 2, viz.."-- x
72
+
}
mx Por M+2Qor A
Xp or a+
I
Por M
72
x
71
of L. 1 upon the joint
and the co-efficients of P
77272
mn + n
+
÷÷m+1=
72
mn + n
Hence,
val. or am. of L. 1 ann.
m+1
Ivalue or amount
va
continuance of theſe lives.
In the cafe of 4, 5,... m unequal lives, find
their mean age, and confider it as the age of ſo
many equal lives.
U
ART.
154
OF ANNUITIES
ART. 3d, To find the prefent value or amount
of L. 1, or of L. I annuity, during the longeſt li-
ver of two or more lives, at a given rate of in-
tereft.
ift, Let / repreſent the mean complement of all
the lives,
X
X
the expectation of their joint
continuance, and
the expectation of the longeſt
liver; then
In Fig. 3. make AB or BD=2X, AC or CH=
2x, of confequence, CB or HK=2X-2x=twice
Fig. 3.
Z
2X
2X-2x
H
K
2
2x
C
2X-2x
B
2X
the
ON LIVES.
!
155
the expectation of furvivorſhip, and complete the
triangle DAB. Here,
The triangle DAB
val. or am.
of a life
2
Z
X
<
2X val. or am.
of longest liver.
[2x val. or am. of
whofe com-joint continuance.
plement is (2X-2x-value or
I
R
amount of furvi-
vorſhip, computed at the diffolution of the joint
lives, which value, x1, and added to that
of 2, will alſo give the value of the longeſt liver,
nearly exact. And the amount of 2, XI+rs,
and added to that of Z, will likewife thew the a-
mount of the longeſt liver, nearly.
Ex. It is required, to find the value and amount
of L. 1 annuity upon the longeft liver of three lives,
whofe refpective ages are 40, 36, and 26 years,
at 4 per cent. Here,
n=52, x=12.88, X=39.12; hence,
The triangle DAB2 = 17.076
by Art. 1. $ = 138.9
am.
val. S of L. 1 an-
nuity upon
the longeſt liver of thefe lives.
2d, To the value of 2. =
8.962
A
add that of Z, X =
Their fum value of the longeft liver 17.775
8.813
And to the amount of Z=s =
add that of 2, xi+is=
56.43
56.78
t
Their fum amount of the longest liver 113.20
U Q
The
156
OF ANNUITIES
The value or amount of L. I annuity upon the
longeft liver, found thus, would be exact, were it
not for this circumftance, that the triangle DAB
ftretcheth with its vertex into the regions of im-
probability, where human life cannot reach, and
of confequence gives a value rather fmall, and an
amount too great; to remedy this, obferve the
following Method.
3d, Let п,
Put P or M
x, and X reprefent as above,
value or amount of an annuity for
n years,
Qor A value or amount of a life whoſe
complement is n, and
p or a value or amount of L. 1 for n years;
then
Fig. 4.
A
21_2X
1
E
E
n
}
27-21
K/2X_n
Ger A
IV/201-21
E
zX_n
177
Per a
71
zł
ON LIVE S.
157
In Fig. 4. put AB=2n, EB or BD=2X, FB
or BC=n, and complete the fquare HB, the rect-
angle KB, and the triangles DEB, and CFB, and
join KF. Now it is evident, that the value or a-
mount of L. 1 annuity upon the longeſt liver of
any number of lives, where the mean comple-
ment of all the lives is n, will vibrate betwixt
the triangle and fquare HB; and that the line
KF, and point K, will limit thefe values; hence,
2X-
=X-x} × Por M + { 2272 X } × 2 or A
1" } XP
=2x
Xp or a + 2 XP or M
72
fval. or am. of
L.I annuity.
72
val, or am. of
L.1 upon the
longeft liver of thefe lives.
Ex. In the example to this article, n=52, x
=12.88, X=39.12, P=21.748, M=167.165, 2
=14.126, A=55.37, p.1301 and a= =7.687.
Hence,
570.667+ 353.686
385.41 +1426.33
52=
3.414+ 10.774
201707+ 82.811
I
17.968 val of L. 1 annui
111.783 am. ty.
.2728=val. Sof L. 1, du-
5-471 am ring the longeſt
liver of three lives, whofe refpective ages are 40, 36, and 26 years,
at 4 per cent.
equal lives, whofe com-
N
>
X=
112 72
4th, In the cafe of m
mon complement is n; x =
71+12
P-value of L. I annuity for n years, &c, and
the co-efficients of P and Q viz.
2.X
72
+
271
11
•
- 2 X
72
minn
2n
+
Hence,
mn+n
nin + n
m—1XP or M+2× Qor A
2
Xpora + XP or M{÷m+1=
n
upon the longeſt
Sva
value or amount of L. I
annuity.
value or amount of L. 1
liver of theſe equal lives.
Ex.
158
OF ANNUITIES
Ex. Sought, the value of the longeſt liver of 4
equal lives, whoſe common age is 30 years, at 4
Here,
per cent.
m=4, P=22.2198, and 2=14.684. Hence,
3×22.2198+2 × 14.684
5
=
=19.205 value required.
In the cafe of 4, 5...m unequal lives, find
their mean age, and confider it as the age of fo
many equal lives.
5th, If the value, &c. of an annuity be requi-
red, which fhall continue during the longeſt liver
of q out of m lives of a given age and at a given
rate; here X-
here X==expect. of the longeſt liver.
P-value of an annuity for n years, &c. and
2
m+¹× P or M+m+1—q× Qor A
Al
972 f
Xp or a +m+-1—9 × P or M
2
12
value or am.
of L. 1 ann.
Ivalue or
Imount ofL.I
a-
lives.
during the longeft liver of q out of
Ex. It is required to find the value of L. 1 an-
nuity upon the longeft liver of 10, out of 15
equal lives, whofe common age is 22 years, at
4 per cent. Here,
m=15,9=10, P=22.968 and 2=15.669; hence,
2 X 22.968+6X15.669
४
=
17.494, the value required.
m+¹,
When q is lefs than the firft term is negative.
2
ART. 4. To find the value or amount of L. 1,
or of L. i annuity, for a term of years denoted
by n, but fubject to failure on the contingency
I
of
!
ON LIVE S.
159
of an affigned life, whofe complement is m, at a
given rate of intereſt.
In fig. 2. fuppofe AB=m, AC or CD=n, and
let the rectangle DB and triangle 2be completed;
now, ſeeing that out of m lives, whoſe comple-
ment is m, one is fuppofed to fail yearly during n
years, let
P or M value or amount of L. 1 annuity for "
Qor A = -
or
bor a
then
+”×
or
11-12 x P or M + n× Qor A
xp or a + P or M
772
I
years.
of an annuity upon a
life whofe comple-
inent is n.
of L. 1 for n years;
val. or am of L. 1 annuity.
{val.
or
of L. I for a
term of years denoted by n, on the contingency
of an affigned life whofe complement is m.
Ex. It is required, to find the value or amount
of L. 1, or of L. 1 annuity, during the term of
15 years, yet liable to be demanded, or fubject to
failure, on the contingency of a life aged 36
years, at 4 per cent.
Here n=15, m=50, P=11.1184, M=20.0236,
2=5.728, A= 8.38, p.5553, and a = 1.8.
Hence,
389.144+ 85.92
700.825+125.70
19.435+ 1118 50:
63.000 + 20.024
9.501 value
16.53
amount
} of
of L. annuity.
.611
1.66
value
{{
Sof L1 during the
amount term of 15 years,
on the contingency of a life aged 36.
Σ
REMARKS,
160
OF ANNUITIES
}
REMARK S.
ft, The values, &c. acquired by Art. i. and
4. are perfectly accurate, and agree exactly with
the algebraic inveftigations, Prob. I. and VÍ.
2d, The value, &c. of the longest liver, found
by Art. 3. is difcovered with much lefs labour,
and is nearly as accurate as the refults of Prob.
IV. and V.
3d, The value, &c. of 3, 4 ...m m joint lives,
Art. 2. is as perfect as the nature of the ſcience.
will admit. The value of 2 joint lives is nearly
correct at the early periods of life; but it declines
as life advances, in fuch a manner, that, at the
mean age of 40, it would require 1 year's pur-
chaſe to be added to 2, at 4 per cent. to make it
correſpond with Prob. II. and III.
4th, The theorems contained in thefe articles,
(excepting cafe 2d, Art. 1.), may with equal pro-
priety be applied to life-annuities at fimple inte-
reſt, by making ufe of tables of fimple, in place
of thofe of compound intereft. They may alſo be
uſed in finding the value of life from a table of
obfervations.
Ex. 1ft, To find, from the Fife Table of Obfer-
vations, the value of the joint continuance of
two lives, whofe refpective ages are 30 and 25
years, at 4 per cent. Here,
N=
1
ON LIVE S.
161
N=32.54x=N° failing in NM years 15.84
M=34.99n-x
=51.69
ann.forNMyrs=14.63
Sum=n=67.53 P-val. of ann. for NMyrs 14.63
NM=22.452=val. of a life
8.05 and
51.59 x 14.63 +15.84 × 8.05 13.08, value fought;
67.53
which agrees nearly exact with the refult of the
theorm 4.
Ex. 2d, To find the value of the longeſt liver
of two lives, whofe refpective ages are 40 and 30
years, at 4 per cent. from the fame table. Here,
N=25.83/2X
M=32.54 21
N
2X
=20.37
38.00
Sum=n=58.37 P= val. of ann. for n years=22.466
NM=19.00 2=val. of a life aged 35 =15.60
Diff.=X=39.37 Hence,
20.37 X 22.466 +38 × 15.6 — 18.00, value
58.37
18.00, value; which is the
fame with the reſult of the theorem X-Xd,
Prob. III. Cafe 2.
N
:
X
of
162
OF ANNUITIES
1
Of REVERSIONS.
PROBLEM VII.
67. To find the value of the reverſion of an eſtate
in fee fimple, after one or more lives of given ages,
at a given rate of compound intereſt. Here,
I
Or,
*
I
N
=the reverfion, in per-
petuity, of L. I per an-
I
RP
RP
1
↑
nr
nr
num, after a life whofe
value is N, at a given
rate of intereſt.
Ex. Sought, the value of a reverfion, in per-
petuity, of L. I per annum, after the life of a per-
fon aged 43 years, allowing intereſt at 4 per cent.
Here,
1
—— N==25—12.683=12.317, the reverſion.
to
१
nr
In like manner,
1
NM
Ι - NMF
T
a reverfion in perpetuity after
(2 joint lives.
L3jc
3 joint lives.
If the purchaſe is difcharged in yearly pay-
ments during the continuance of the life or lives in
queftion; in this cafe, the reverſion, divided by
the value of the life or lives, will quot the an-
nual payments.
Ex.
ON LIVES.
163
Ex. A and B, whofe refpective ages are 35 and
25 years, fell the reverfion of an eſtate whoſe rent
is L. 150 per annum, for, yearly payments during
the longeſt of their lives; fought, theſe payments,
reckoning intereft at 4 per cent. Here,
25-18.2026.798
multiplied by 150
And the product divided by 18.2) 1019.7(L. 56, the
annual payment.
per
If the firſt of the annual payments be made
advance, the above divifor, viz. 18.2, increáfed
by unity, will quote L.52.1, the annual pay-
ments.
Cor. Ift, To find the preſent value of L. I, due
at the deceaſe of a perſon of a given age, at a gi-
ven rate of intereſt.
L. I, or any other fum of money, may be con-
ceived to be the prefent worth of a perpetuity,
whofe rent is equal to the intereft of that fum for
one year; hence, the reverfion of a fee-fimple
equal to that intereft, after the given life, will be
the value required. Thus,
1
*
RP
RP
-Nxr=1-rNxr would be
nr
=
the value required, were it not for one circum-
ſtance, viz. that, from the nature of the calcula-
tion, the time in which the firſt yearly payment
of a reverfionary annuity becomes due, is the end
of the year in which the event happens that en-
titles it. This is alfo the time when a reverfionary
fum becomes due. But an annuity, the firft pay-
ment of which is to be made at the fame time with
another
X 2
164
OF ANNUITIES
another payment of a fum in hand fufficient to
buy an equal annuity, is worth one year's purchaſe
more than the fum. Hence,
I
N p
=
R
72
the value of L. I due at the de-
ceaſe of a perſon whofe complement of life is n.
Or thus:
I
22
the prefent probability, that a life whofe
complement is n will fail in any one affign-
able year of its duration; (fee Prob. XIX. cafe 1.)
Hence,
I
n R
I
I
+ + (n) == the preſent value of
n R2
n R3
72
L. 1, payable at the failure of a life whofe com-
plement is n; which, multiplied by s, a fum due
at the cloſe of a given life, will produce the pre-
fent value thereof.
Ex. Let there be a legacy of L. 100=s, due at
the deceaſe of a perfon aged 60 years; fought, its
preſent value, at 4 per cent. Here,
sx 2 =100X.6147=L.61.47, the value required.
X 72
Cor. 2d, To find what fum ought to be paid, on
the deceafe of a perfon of a given age, in con-
fideration of L. I now received. It is evident,
that the reciprocal of the former corollary will
be the anſwer in this. Thus,
=
n
R
I—rN p
the fum which ought to be paid at
the deceaſe of a perfon whofe complement of life
is n, for L. I now received, which, multiplied by
t
ON LIVE S.
165
p, any prefent fum, will produce s, the amount
thereof at the cloſe of a given life. Or thus:
I
R R³
M
Seeing ++ (») ==, the amount of
n
12
72
12
L. I annuity for n years certain divided by n, and
expreffes the amount of L. 1, payable at the fail-
ure of a life whofe complement is n; therefore,
px
M
77
=S.
Ex. Let p L. 100, the age 60 years, and the
rate of intereft 4 per cent.: Here, n=26, M=
ΛΙ
44.3117, and px =100X 1.7043=L. 170.43=
s, which is the fame with the refult from the
72
theorem p×1+rs of Table IV.
PROBLEM VIII.
1
68. Suppofing the decrements of life to be equal,
it is required, to find the value of the reverfion of
an annuity, which is to continue during the life of
a perfon whofe complement is m, after the de-
ceafe of the prefent poffeffor, whofe complement
of life is n, at a given rate of intereft.
I
71
, 1-2, &c. are the probabilities
I-
72
of the poffeffor's dying in the firft, fecond, &c.
years; and
I 7112
,
,
&c are the probabilities of
772
712
the expectant's living to the end of the firſt, fe-
cond, &c. years. Hence
11 — ]
I
9
73
tod
771
# 2
X I
7- I
X
712
72
172
&c. are the probabilities that the expec-
tant
166
OF ANNUITIES
tant will receive the firft, fecond, &c. yearly
payment, which probabilities being expanded by
multiplication, and their prefent values found,
will give the value of the reverſion, viz.
M-I
1
m R
M-2
mR2
I
m n R
m—2 × n—2
2
m n R ²
&c.
But the feries in the firft column is the value
of the expectant's life, whofe complement is m;
and the fecond column contains the value of the
joint lives of the poffeffor and expectant. There-
fore, if, from the value of the expectant's life,
the value of the joint lives be taken, the remain-
der will be the value of the reverfion.
Or thus: Let xxy=y-xy be the probability
that the expectant will outlive the poffeffor the
first year; then, by fubftituting the values of the
lives in place of the probabilities thereof, M-
NM will be the prefent value of the expectant's
life after the death of the poffeffor, or the value
of the reverſion required.
Ex. Let the refpective ages of the expectant
and poffeffor be 25 and 30 years, and the rate of
intereft 4 per cent. then M=15.318, N=14.684
and NM 11.468; hence, M-NM-3.85, the
value of the reverſion.
Of
•
ON LIVES.
157
Of a Widows Scheme.
69. THE refolution of the ſeveral queſtions
reſpecting a reverfionary annuity to one life after
two, is derived from the above theorem, viz. M
-NM-the value of the reverfion. There are
two cafes of fuch an annuity, which differ from
or agree with one another in a few particulars.
Cafe 1. A and B, whofe refpective ages are 30
and 25 years, agree to pay a certain fum in hand,
or a certain annual rate during the continuance
of their joint lives, in order to purchaſe an an-
nuity to B after A's deceaſe. Sought thefe, rec-
koning intereft at 4 per cent.
M-NM
M-NM=3.85=fum in hand, or prefent valuc.
=L.0.3357 annual rate payable during
their joint lives to fecure L. 1 annuity to B.
NM
=
In this cafe, the payment of the rates ceafes at
the cloſe of the joint lives, whether A be alive or
not, and B is fuppofed to be the furvivor.
Had this cafe been conditional, viz. "if B fhall
furvive A," the fum in hand and annual rate muſt
be diminiſhed by
41.57
61.00
=.681, the probability of
B's furviving the diffolution of marriage.
Had part of the prefent value of the annuity,
fuch as L. 1.5 been paid in ready money, the
remainder, divided by NM, quots the annual
rate; and were the rates payable per advancé,
NM+I will be the divifor.
Cafe 2d, Is that of a widows fcheme, where
either
"
OF ANNUITIES
168
either a certain fum in hand, or an annual rate
during the life of A, is paid, in order to fecure a
provifion for B, after A's deceafe, or, failing B,
fo many years purchaſe of her annuity to their
children.
This cafe differs in toto from the conditional
one mentioned above; and in part from Cafe 1ſt,
in this, that the annual rate is paid during the
life of A; but agrees with it in fuppofing that
either a widow or children are left a burden
upon the ſcheme. Therefore,
Let the refpective ages of A and B be 30 and
25 years, and the rate of intereft 4 per cent; then,
MNM L. 3.85fum in hand, or prefent va-
lue.
M-NM
N
= L.0.2622 = annual rate, exclufive of
the expence of management, payable during the
life of A, to fecure L. 1 annuity to B, after A's
deceaſe, or, failing B, 10 years purchaſe of her
annuity to their children.
This, I acknowledge, is different from the u-
fual method of conftructing a widows fcheme,
and in particular from that recommended by Dr
Price, whereby, to find the annual rate, the pre-
fent value of the annuity must be divided by
that of the joint lives, as in Cafe 1ft; making
the full value of the annuity, in the prefent in-
ftance, to be paid in 19, whereas A hath a chance
of living 28 years: Beſides,
L. 3.85 val.
L..2622 ann.
L..3357 ann.
accumulated during
| A's life, by Tab.IV. =
val, of B's ann. at widowity, perfaidtab.
L. 13.74.
L. 16.83
SL. 21.55 and
L. 14.20
The above rate or value, multiplied by any
รับ
annuity
1
ON LIVE S.
169
annuity, will, with the addition of the expence
of management, fhew the rate or value correfpond-
ing to that annuity. the ages being 30
and 25.
Thus,
Exp.
Ann. Rate.
Value.
.25 +
|10=2.872 || L. 38.5
Exp.
L.25p.an.
•375+
15=4.308 57.75
•375
.5 +.2622x20=5.744
77.0 with
.5
.625+
·75 +
25=7.180
96.25
.625
30=8.616
115.5
.75
Again, let the reſpective ages of A and B be
40 and 30 years, the annuity L. 20, and the rate
of intereft 4 per cent; then 4.256×20=L.85.12
value of the annuity; and, .5+.3225x20
L. 6.95 annual rate. But the number at the
former ages may be to that at the latter, as 3: 1;
thence 5.74+3+6.95 L. 6.05, the mean annual rate
=
4
correſponding to L. 20 annuity.
Had thefe rates been conftructed upon the prin-
ciples of the expectation of life, as in the Church's
Scheme; in this cafe, the prefent value of an
annuity for 20 years, at 4 per cent.
13.590
divided by the amount thereof for 30 years 56.085
quots the annual rate = L..2423, which is lefs
than the above. In fuch a fcheme, in order to
afſiſt their operation, and to balance the advan-
tage of the marriage-tax in the Church's Scheme,
the rates might be paid per advance ;-the wi-
dow's annuity may begin to run from the firſt
term of Whitfunday or Martinmas after the de-
ceafe of the contributor ;-where there is no wi-
dow, 10 years purchaſe of her annuity might
be granted to the children of a contributor, with-
Y
in
170
OF ANNUITIES
in a twelvemonth after his deceafe;-if a contri-
butor, before his death, hath not paid 10 annual
rates, a deduction might be made out of the pro-
vifion of the widow or children, till the balance
be recovered;-and for each 500 contributors,
the annual expence of management ought not to
exceed L. 110.
Suppofing then, that there were 500 married
contributors in this fcheme, of 30 years of age.
and upwards, whofe number being constantly
the fame,
Their expectation of life = 32.5 years
The number dying yearly = 15.0
of whom, I will leave widows, 3 of them, fa-
milies of children without a widow, and 1, neither
a widow nor children.
The annual number of marriage taxes, each
at L. 9,
The expectation of widowity,
The maximum of annuitants,
3.0
19.5
= 214.0
of whom 203 will draw full, and 11, including
heirs, half annuities; and the families of chil-
dren,
Deductions out of the proviſion of wi-
dows or children,
=
The medium of annuities L. 20, and
of rates,
3.0
L. 82.6
L. 6.05
L. 110
L. 41,390
The annual expence of management
And the capital neceffary to fupport the
fcheme, when the annuitants fhall arrive
at a maximum,
7༠
Of a Marriage-Tax.
A marriage-tax may be confidered as the dif-
ference
ON LIVE S.
171
ference in point of value, of the two Widows An-
nuities, computed at the commencement of the
fecond marriage: Thus,
Suppofing the above contributor becomes a
widower, and at the age of 40 marries a fecond
wife, whofe age is 30 years; fought, his marriage-
tax, at 4 per cent.
From the value of two joint lives aged 40
and 30 years,
Subtract that of two joint lives aged 40
and 35 years,
The remainder,
L. 4.256
Multiplied by 20, will produce the tax for
that annuity,
3.822
•434
L. 8.68
Queſt. A and B, whofe refpective ages are 60
and 40 years, are fubjected to the burdens and
enjoy the advantages of a widows fcheme, where-
in B's annuity, thould fhe furvive A, is L. 25;
B, defigning to fell her annuity, defires to know
the prefent value thereof, reckoning intereft at
4 per cent.
The expectation of the joint continuance of
thefe two lives,
And B's probability of furviving the diffolu-
tion of marriage is of confequence, 35.45
40.00
10.55
•77
But M-NM-5.686 would be the years pur-
chafe of the annuity, were it certain that B would
be the fuvivor. Hence,
•77X5.686×25-L. 109.455, is the value required.
Or thus, A's expectation of life = 13, and B's
probability of furviving A =
Y 2
33
171
45
= .7173
hence,
172
OF ANNUITIES
1
hence, B's age at A's death would be 53 years,
whofe value, viz. 10.702, reduced for 13 years,
at 4 per cent, gives 6.43, the years purchaſe of the
annuity, were it certain that fhe would furvive
A; hence, 717X 6.43X 25 L. 115.25, is the
value required.
PROBLEM
IX.
It is required, to find the value of the reverfion
of an annuity, which is to continue during the
joint lives of two perfons, after the deceafe of the
prefent poffeffor, at a given rate of compound in-
tereft.
71. Let x, y, and ≈ be the reſpective probabili-
ties of the continuance of the three lives for one
year; then I will be the probability of the
poffeffor's life failing the first year; which, mul-
tiplied by xy, gives xy-xyz, the probability that
the two expectants will outlive the poffeffor the
firſt year. Hence, by fubftituting the values of
the lives in place of the probabilities thereof,
NM NAIF the value of the reverfion.
Therefore, if, from the value of the two joint
lives in expectation, the value of the three joint
lives be fubtracted, the remainder will be the
reverfion of two joint lives after one.
Ex. Let the refpective ages of the two expec-
tants and poffeffor be 43, 54, and 66 years, and
the rate of intereft 4 per cent. then,
From
ON LIVES.
73
From the value of the two joint lives
in expectation
Subtract that of the three joint lives
= 8.364
<= 5.152
The remainder =
3.212 is
the value of the reverfion of two joint lives after
one.
/
PROBLEM X.
To find the value of the reverfion of an annui-
ty, which is to continue during one life, after
two joint ones, at a given rate of compound in-
tereſt.
72. Let xy reprefent the probability of the
two lives in poffeffion outliving the first year;
then —xyX=-xyz will be the probability of
the life in expectation outliving the two lives in
poffeffion the first year; hence,
i
F-NMF will be the value of the reverfion;
Therefore,
If, from the value of the expectant's life, the
value of the three joint lives be taken, the re-
mainder will be the value of the reverfion of one
life, after two joint lives.
Ex. Let the age of the life in expectation be 43
years, and the ages of the two lives in poffeffion
54 and 66 years, allowing compound intereft at
4 per cent. Then,
=
From the value of the expectant's life = 12.683
Subtract the value of the three joint lives = 5.152
There remains the value of the reverfion= 7.531
•
PROBLEM
174
OF ANNUITIES
PROBLEM XI.
To find the value of the reverfion of an annu-
ity, which is to continue during the life of a gi-
ven age, after the longeſt of two lives, whoſe a-
ges are alfo given, at a given rate of compound
intereft.
73. Let x, y, and z repreſent the reſpective pro-
babilities of the three lives continuing one year.
Then XI-XI—y=&−x8—12+xyz will be
the probability of the life in expectation outliving
the other two the first year; and F—FR
NF
NMF will be the value of the reverfion of the
life in expectation after the other two.
But if the refpective values of the longeſt of
two and three lives, Prob. IV. and V. be compa-
red with this, it will appear to be their difference;
Therefore,
If, from the value of an annuity on the long-
cft of three lives, the value of an annuity on the
longeſt of the two in poffeflion be taken, the re-
mainder will be the value of the reverfion on the
third life, after the other two.
Ex. A, who is 43 years of age, is intitled to
the reverfion of an eftate for his life, after the
deceafe of his father and of his mother-in-law,
whofe refpective ages are 66 and 54 years;
fought, the value of A's intereft in the eſtate, al-
lowing 4 per cent,
From
ON LIV E S.
173
From the value of the longeſt of the
three lives
=
Subtract the value of the longeſt of
the two poffeffors lives =
15.190
11.904
Remains the value of the reverfion,
3.286
1
PROBLEM XII.
To find the value of the reverfion of an an-
nuity, which is to continue during the longeſt of
two lives after one, at a given rate of compound
intereft.
74. Let T-XXπ-y=I—xy+xy reprefent
the probability of the two lives failing in one
year, which, being fubtracted from unity, leaves
x+y-xy, the probability of one, at leaft, of the
two lives outliving the year; and this laſt expref-
fion, being multiplied by 1-2, gives x+y—x}
y+xyz, the probability that the longeſt
of the two lives will furvive the third the firft
year. Hence N+M-NM-NF-MF+NMF
will be the value of the reverfion of an annuity
upon the longeft of two lives after one; but this
value is the difference of the refults of Prob. I.
and V. Therefore,
XZ
If, from the longeſt of the three lives of a pof-
for and two expectants, be taken the value of the
poffeffor's life, the remainder will be the rever-
fion of the longeſt of two lives after one.
Ex. Let the refpective ages of the two expec-
tants be 43 and 54, and that of the poffeffor 66
years, and the rate 4 per cent.
From
176
OF ANNUITIES
From the value of the longeſt of the
three lives =
Take the value of the poffeffor's life =
15.190
7.333
Remains the value of the reverfion =
7.857
PROBLEM XIII.
To find the value of the reverfion of an an-
nuity, where a given term of years is concerned.
75. Ift, To find the value of the reverfion of
an aſſigned life after a given term of years.
Ex. A, aged 15 years, expects to enter upon
an eftate of L. 100 per annum, after the expira-
tion of 10 years, which he is to hold during life;
fought, the value of his expectation, reckoning in-
tereſt at 4 per cent.
Here,
From the value of the propofed life =
Subtract the value of an annuity for the
given term of years, on the contingen-
cy
of its ceafing upon the extinction
of the propoſed life, found by Prob.
VI. =
The remainder multiplied by 100 =
L. 887, is the value of A's expecta-
tion,
16.410
7.540
8.870
2d, To find the value of the reverſion of an
annuity, for the remainder of a given term of
years after an affigned life.
Ex. A, aged 25 years, who has the right of an
annuity
ON LIVE S.
177
annuity for 31 years certain, makes over the re-
verfion thereof to B and his heirs, to enjoy the
fame after his deceafe for the remainder of the
faid term; fought, the value of B's expectation,
reckoning intereft at 4 per cent.
From the value of the annuity certain
for 31 years, =
Subtract the value of the annuity for
the fame term, on the contingency of
its failing on the extinction of A's
life, found by Prob. VI. =
Remains the value of B's expectation,
11
17.588
13.858
3.730
3d, To find the value of the reverfion of an
annuity for a certain number of years, after an
affigned life.
Ex. Let the given age be 50, the term of years
12, and the rate of intereft 4 per cent. Then,
The value of the life is 11.344, the nearest to
which in Tab. V. of annuities certain is 11.118,
correſponding to 15 years;
But 12 + 15 = 27 ==
of the life and reverfion;
16.496 the value
Subtract the value of the life 11.344
There remains,
5.152 the value
of the reverfion of 12 years, after a life of 50.
4th, A perfon, 35 years of age, wants to buy
an annuity for what may happen to remain of
his life after 50 years of age; fought the value
of fuch an annuity in ready money, and alfo in
annual payments for 15 years, till he attains to
Z
i
the
178
OF ANNUITIES
the faid age, fubject in the mean time to failure,
fhould his life fail.
The prefent value of fuch an annuity is equal
to the value of a life at 50, multiplied by the
prefent value of L. 1 to be received at the end of
15 years, and alſo by the probability that the gi-
ven life will continue fo long. That is,
346
11.344 X.5553X =4.44, the number of years
490
purchaſe that ought to be given for the annuity,
reckoning intereft at 4 per cent.
Now, fuppofing the annuity L. 50, its value in
prefent money is L. 222,
Or thus: From the value of a life of 35
years,
Subtract its value for 15 years, ſubject
to failure, =
13.979
9.533
Remains the years purchafe of the annuity,=4.446
And L. 222, the value of the annuity in ready
money, divided by 9.53, its value for 15 years,
gives L. 23.3, the annual payments till the life
attains to 50 years.
5th, To find the prefent value of an annuity
to be enjoyed by one life, for what may happen
to remain of it beyond another life of equal age,
after a given term; that is, provided both lives
continue to the end of a given term of years.
Here,
Multiply the value of the annuity for the two
joint lives, plus the given term of
years, difcoun-
ted for the given term, by the probability that
the two given lives fhall both continue the given
term; the product will be the anfwer.
Ex.
ON LIVE S.
179
Ex. Let the two lives be each 30, the term 7
years, the annuity L. 10, and the intereft 4 per
cent.
From the value of a ſingle life at 30+7=
Subtract the value of two joint lives, each 37
Remains, the value of an annuity for the
life of a perfon of 37 years of age, after
another of the fame age, which, being
diſcounted for 7 years, = 2.6.
49 49
13.67
10.25
3.42
fhall con-
The probability that two lives at 30
tinue 7 years is (by Mr De Moivre's hypothefis)
12×10=.765; and 2.6 ×.765=1.989, the num-
ber of years purchaſe which cught to be given for
an annuity, to be enjoyed by a life now 30, after
a life of the fame age, provided both continue 7
years; and the annuity being L. 10, its prefent
value is L. 19.89.
In like manner, fuppofing the term one year,
and the ages and the rate of intereft the fame, the
preſent value of the fame reverfionary annuity is
L. 32.4; and if the term is 15 years, the value is
L. 9.7.
Hence, fuppofing the fcheme of a fociety for
granting annuities to widows to be, that if a
member lives 1, 7, or 15 years after admiffion,
his widow, of the fame age with himſelf, fhall be
intitled to an annuity of L. 20, L. 30, or L. 40;
we may find what ought to be the annual pay-
ment of each member of a given age, and at a
given rate of compound intereft.
By the above, the value of L. ro per annum, to
a life of 30 after another of 30, provided the joint
lives do not fail in one year, is L. 32.4, at 4 per
cent,
Z 2
The
180
OF ANNUITIES
The value of L. 20 per annum, in the fame
circumstances, is, therefore, =
The value of L. 10, after 7 years, is =
And of L. 10, after 15 years, is =
64.80
19.89
9.70
Their fum = 94·39
is the value of the expectation in a fingle prefent
payment; which, being divided by 11.182, (the
value of two joint lives at 30), gives L. 8.44, the
annual payment during the joint lives. In like
manner, were the ages 40 and 50, the annual
payments would be L. 8.69 and L. 9.05.
PROBLEM
XIV.
To find the value of an annuity, or the rever-
fion thereof, depending on furvivorſhip, and the
expectation of life,
76. 1ſt, A and B, whofe refpective ages are 25
and 40 years, enjoy an annuity equally betwixt
them; which, after the deceaſe of either of them,
is to belong to the furvivor for life; fought, the
value of the right of each in that annuity, reckon-
ing intereft at 4 per cent.
کو
From the value of A's life
15.318
Subtract half the value of the joint lives 5.327
Remains, the value of the right of A = 9.991
In like manner B's right =
7.869
2d, An annuity, after the deceaſe of A, aged 66,
is to be divided equally between B and C, whofe
refpective ages are 43 and 54 years, during their
joint
ON LIVES.
181
joint lives, and then is to go entirely to the laſt
furvivor for life; fought, the value of B's expec-
tation, reckoning intereſt at 4 per cent.
From the value of the reverfion of the life B
after the life A, found by Problem VIII. 6.433
Subtract half the value of the reverfion of the
joint lives of B and C, after the life A,
Problem IX. =
Remains, the value of B's expectation
1.582
4.851
3d, A, aged 40, expects to come to the poffef-
fion of an eſtate, fhould he furvive B, aged like-
wife 40, and offers to give fecurity for L. 40 per
annum out of the eſtate at his death, provided he
ſhould get into poffeflion: What ſum ought now
to be advanced to him, in confideration of fuch
fecurity, reckoning intereſt at 4 per cent.?
From the value of the perpetuity =
Subtract the value of the longeſt of two
lives, each 40 =
Remains, the value of the reverfion of ditto
25.000
16.566
8.434
which, multiplied by 40, gives L. 337.36, the
value of the given eftate, were it certainly to be
enjoyed, after the extinction of the longeſt of two
lives, both 40; but that A's life, in particular,
fhould fail laft, rather than B's, is an even chance:
The true value of the reverfion, therefore, is half
the laft value, or L. 168.68.
4th, B, aged 40, will, if he lives till the de-
ceafe of A, whofe age is 30, become poffeffed of
an eftate of L. 40 a-year; fought, the worth of
f
his
182
OF ANNUITIES
his expectation in prefent money, reckoning inte-
reſt at 4 per cent. Here affume two lives, whoſe
common age is equal to that of the older of the
two propofed lives, A and B, and find the expec-
tations of life in A and B, viz. 28 and 23. Then,
From the value of the perpetuity
Subtract the value of two joint lives, each
40 =
Remains, the reverfion of the two joint
lives =
25.000
9.826
15.174
the half of which, viz. 7.587 would be the value
of B's expectation, were their lives each 40:
Therefore fay,
As 28: 237.587: 6.232, which, multiplied
by 40, gives L. 249.28, the value of B's expecta-
tion.
If B's age be 30, and that of A 40 years; then,
From the value of the perpetuity
Subtract the value of the joint lives B and A,
viz. 10.428 +6.232, found above, =
25.00
16.66
Remains, the value of B's expectation
8.34
5th, C and his heirs are intitled to an eftate of
L. 200 per annum, upon the deceafe of B, aged
40, provided B furvives A, whofe age is 30 years;
fought, the value of their expectation in preſent
moncy, reckoning intereft at 4 per cent. Aflume
two equal lives, whofe common age is that of the
older of the two propofed lives, B and A; then,
From
J
ON LIVES.
183
From the value of the perpetuity=
Subtract that of the longeſt of two equal
lives, each 40 =
Remains, the reverfion of ditto =
25.000
16.566
8.434
the half of which, viz. 4.217, would be the years
purchaſe required, if the ages of A and B were
each 40 years: Therefore,
28: 23: 4.217 3.464, the number of years
purchaſe required, anfwering to L. 692.8.
If B's age had been 30, and A's 40 years, in
this cafe,
=
From the value of the perpetuity
Subtract that of the longeſt of the two lives.
B and A, viz. 17.452+3.464, found a-
bove =
Remains, the value of C's expectation
25.000
20.916
4.084
Hence, in a widows fcheme, when the refpec-
tive ages of hufband and wife are 30 and 25 years,
and the proviſion for a family of children, where
there is no widow, L. 200; we may find the pre-
fent value of faid provifion, reckoning intereft at
4 per cent.
By the above, the years purchaſe of the chil-
dren's provifion would be 3.128; but the intereſt
of L. 200, for one year, at 4 per cent. is L. S :
Therefore 3.128 × 8 = L. 25.024, is the expecta-
tion of a family of children, without a widow.
PRO-
184
OF ANNUITIES
1
PROBLEM XV.
Of Infurance on Lives.
77. 1ft, IF A, aged 30 years, is willing to advance
P
money in hand, or to pay a certain fum yearly du-
ring life, to fecure L. 100 to his heirs; or, in other
words, to infure his life upon L. 100, allowing
intereft at 4 per cent. In this cafe, N=14.684,
and (Problem VII. cor. 1.) =40X 100=L. 40,
the fum payable in hand; which, divided by
14.684, gives L. 2.73, the annual payments du-
ring life.
72
2d, A, aged 40 years, agrees to advance a cer-
tain fum in hand, or to make annual payments
during the ſpace of 15 years, in cafe he lives fo
long, in order to fecure L. 300 to him or his heirs;
fought, the prefent value of faid fum, or annual
payments during thefe 15 years, reckoning intereſt
at 4 per cent. Here, (Problem VI.), N=8.817, and
=.6224X300=L. 186.7, the prefent value;
which, divided by 8.817, gives L. 21.20, the an-
nual payments during 15 years, in cafè A lives
fo long.
I-- rN
R
3d, A, aged 40 years, wants to make provifion
for B, aged 30, in cafe the latter thould happen
to furvive: What ought the former to give in a
fingle payment, and alfo in annual payments du-
ring their joint lives, to fecure L. 500, payable at
his death to the latter, reckoning intereft at 4 per
cent.?
In
ON LIVE S.
· 185
In this cafe, the value of the joint lives of A
and B10.428, L. 500, at 4 per cent, is equal to
an eſtate of L. 20 per annum, and (Problem XIV.
4.) the value of B's expectation of furviving A is
8.34; therefore, 8.34× 20 L. 166.8, is the fingle
payment, and 166.8, divided by 10.428, gives
L. 16, the annual payments during the joint lives
of A and B.
4th, An eftate of L. 20 per annum will be loft
to the heirs of a perfon now 40, fhould his life
fail in 12 years: What ought he to give for the
affurance of it for that term, reckoning intereſt
at 4 per cent.? Or, What is the prefent value of
the eſtate, to be entered upon at the failure of fuch
a life, fhould that happen in 12 years?
From the value of an annuity certain for 13
years =
Subtract the value of the life fubject to fail-
ure in 12 years =
Remains, the value of the eftate for 12
years, upon the chance of the life's
failing in that time.
9.385
8.154
1.231
Again, 25, the perpetuity, X .6246, the value
of L. 1, diſcounted for 12 years X
T2I
445
,
the chance
of the life's failing in that time, gives 4.246, the
value of the eſtate after 12 years, upon the chance
of the life's failing in that time; and thefe two
values, added together, give 5.477, the value of
infurance, correfponding to L. 109.54, which, di-
vided by 8.154, the value of the life, quots
L. 13.43, the annual payments.
A a
5th,
186
OF ANNUITIES
5th, There is an eftate, which, if A, aged 7
years, happens to die before he attains to the age
of 21, is, after his deceaſe, to go to B and his
heirs for ever; fought, the value of B's expecta-
tion, reckoning intereft at 4 per cent.
If the reverſion of the eſtate, after A's life had
belonged to B and his heirs; then, in this cafe,
From the perpetuity =
Subtract the value of a life 7 years old
There would remain B's expectation =
25.000
16.698
8.302
But B's expectation is limited to 14 years of A's
life; therefore,
From the perpetuity =
Subtract the value of a life 21 years old =
25.000
15.781
Remains, the reverfion of a life of that age = 9.219
and
9.219×.5775, the value of L. 1, diſcounted
the chance of A's living 14
for 14 years X
592
6922
,
692
years, gives 4.554; which, being fubtracted from
the former value, leaves 3.748, the value of B's
expectation.
531
Again, had the reverfion of the cftate been li-
mited to B, aged, for inſtance, 30 years; in this
cafe, 3.748x427, the chance of B's living 14 years,
gives 2.873, the value of B's expectation; which,
multiplied by 4, the intereft of L. 100 for one
year, produces L. 11.6, the prefent value of L. 100
to be paid by B, in cafe the furvivorſhip fhould
take place.
•
PRO-
ON LIVES.
187
78.
PROBLEM XVI.
To find the Value of Succeffive Lives.
LET N and M reprefent the values of
an annuity of L. 1, to continue n and m years, at
a given rate of compound intereft;
I
I
then N=
Rn
RR (47)
I
=1—rN,
Rr
r
which gives
I
and M=
Rm
I
=1-rM.
RTL
1
Hence, = 1−rÑ× 1—rM=ı—rN—rM+
2
Rn+m
r² NM. Now,
let V be the value of an annuity, which is to con-
I
tinuen+m years; then, I—I—rÑ×
1-rM; which gives,
V
Rn+m
=
I-rNXI-rM
=N+M-rNM, the value
of an annuity to continue during two fucceffive
lives, whofe fingle values are N and M.
Ex. 1ft, A, aged 40 years, enjoys an annuity
for his life; and, at his deceafe, has the nomina-
tion of a fucceffor, aged 18 years, who is alfo to
enjoy the annuity during his life; fought, the
prefent
A a 2
188
OF ANNUITIES
prefent value of the two fucceffive lives, reckon-
ing intereſt at 4 per cent. Here N=13.196, and
M=16.105.
N+M-rNM=29.301-8.5 20.810,
Hence,
the value required.
In like manner, if there be three fucceffive
lives, whofe fingle values are N, M, and F, then,
Ꮴ
Ꮴ -
I—I—rNX1--rMX1~ go.
rF
ceffive lives.
the value of three fuc-
Again, if the two fucceffive lives are of the
fame age, which is moſt likely, then n+m2n,
and
T:
1
R2n
and V-
2
=1—rV=1—rÑ, which gives
i-r N
five lives.
2
the value of two equal fucceffive lives.
1-1-N
3
the value of three equal fuccef-
Ex. 2d. It is required, to find the value of four
equal lives, following one another in fucceffion,
whofe common age, at admiffion, is 24 years,
reckoning intereft at 4 per cent. Here N 15.437
and
4
.9786
24.465, the value.
.04
Univerfally, if N ftands for the value of an
annuity to continue a certain number of years,
then
ON LIVES.
189
then
I—I—N
r
will reprefent the value of an an-
nuity to continue n times as long. If n is inde-
n
finite, then 1-rNo, and v, the value of
the perpetuity.
v=
PROBLEM XVII,
Of a Copy-bold, and the renewing of Leafes.
SUPPOSIN
79. UPPOSING any number of equal lives
of a given age, and that' upon the failing of any
one, or all of them, they fhall be immediately re-
placed, and I fhall then receive a fum of money
S, agreed upon; fought, the value of that expec-
tation, and at what intervals of time I may ex-
pect to receive the faid fum.
Let n denote the number of years during the
intervals betwixt the payments of the fum S; and
let N repreſent the value of the life or lives du-
ring theſe intervals; then,
N=
I
I
Rn
(47.) which gives R"
T
1-rN.
Now feeing a fum of money, S, is to be received,
ad infinitum, at the equal intervals of time denoted
by n; its preſent value may be found from the
following feries, viz.
S
S
Ri R211
S
S
t
+ +
&c. =
R$ 119
R-I
payments.
But
(40.) the prefent value of all the
190
OF ANNUITIES
But in place of R", ſubſtitute its value found
I
above, viz.
1-rN
then R-1=
rN
I-N'
and confequently
S
I-rN
R-I
XS, the value required.
r N
I
Again, let —_——~=2, then R=2, and
Log. 2
Ir N
n= the intervals of time at which I may ex-
Log. R
pect the fum S.
Ex. ft. Let there be three perfons, each 20
years of
age, at the deceaſe of any one of which,
I fhall receive L. 250, which life is to be imme-
diately replaced, and whenever a life becomes va-
cant, I or my heirs are to receive a like fum;
fought, the value of that expectation, and at what
equal intervals of time I may expect the faid fum,
allowing intereft at 4 per cent.
I-N
rN
Here, N=9.192 by Prob. III. and
XS=1.72X250=L. 409, the value required.
Again, n=
*.1990631
0170333
11.68 years of interval.
Ex. 2d. A purchaſes a leafe-hold eftate, upon
three equal lives, aged 15 years, for the fum of
L. 2000, on condition that his heirs fill up the
leafe continually, whenever all the lives become
vacant, paying a fine of L. 1000; fought, the
fent value of the whole purchaſe allowed for the
eftate, with the annualrent anfwering thereto,
and at what equal intervals of time the heirs of
pre-
A
ON LIVE S.
19t
A may expect to pay the faid fine, allowing in-
tereft at 4 per cent.
I-N
rN
Here N 20.25, by Prob. III. and
XS=.2346x1000 L. 234.6 the value of
the fines;
To which add,
Their fum is,
2000.0
2234.6the value of the
the purchaſe, whofe annualrent, at 4 per cent.
is L. 89.384; and
2 =
.7212464
.0170333
=42.3 years of interval.
If only one life is concerned, which, at the de-
mife of the incumbent, is to be replaced with a
fucceffor, as in the cafe of a benefice, or of a
copy-hold; in which, let N denote the value of
the life, and s the fine, if there is any, to be paid
on admiffion; then,
I-N
rN
pa-
Xs=value of the copy-hold or right of
tronage, to which add the ſum s, if the purchaſe
is made during the vacancy; then the value
of the patronage.
N
=
Ex. 3d, Let an intrant's age be 30 years, the
fine on admiffion L. 150; fought, the value of
the patronage at 4 per cent. Here N=14.684,
and 150. Hence,
s=
IrN
N
Xs=.7025×150=L.105.375, the value
fought. Or, if the purchaſe be made during the
vacancy, then,
$
rN
-L. 255.375, the value of the
patronage.
PRO-
192
OF ANNUITIES
PROBLEM XVIII.
Of the Expectation of Life.
80. THE expectation of life is that period of
time, during which a perfon of a given age may
juſtly expect to continue in being, and is always
equal to the years purchaſe of an annuity of L. I
during life, upon the fuppofition that money
bears no intereft.
Cafe ift, It is required, to find the expectation
of a fingle life of a given age, fuppofing the de-
crements of life to be equal.
In eſtimating the expectation of life, we muſt
reckon from the prefent inftant, and fuppofing n
to be the complement of life, make = I,
1
n
n
72
I, the
firſt term of the feries, + +2, &c. expref-
12
12
72
fing the expectation of life; which being a feries
decreaſing every inftant, the laft term thereof
viz. =0, and the number of terms, though in-
η
72
n
definite, is expreffed by n; therefore, x= (8)
12
n
X
2
2
the fum of the feries is equal to the expectation of
that fingle life whofe complement is n.
Ex. A perfon of 24 years of age, may expect
to live"
62
=31 years longer.
2
2
2d,
ON LIVES.
193
2d, It is required, to find the expectation of
two joint lives, of given ages.
Let the complement of the elder life be n, and
that of the younger m; then the fum of the
feries>
n
n2
In
m
+ X
n
771
12
m
n
772
n
+ X (1) will
2
772
be the expectation of the joint lives. Now, the
terms of this feries are the products of the two
arithmetical progreffions, which exprefs the ex-
pectations of the fingle lives, carried on to a num-
ber of terms equal to n; but the fum of the
firft is and, fince the greateſt term of the fe-
cond is
12
2
,
772
m
12
the leaft
and the number of
m
TH
terms n, the fum thereof will be
772
172 -72
n
n
2m. n
n
+ X
X
=n
:n—= (7, Equ. 3d.)
m2
m
2
772
2
2711
hence, the fum of n terms of the feries of products
will be =
พท
22
1212
-xn.
2
72
2772
X n
+
I
Τ
n + 1 x n Xn --I
›
--IX-X which being
2.2.3
77
1/2
reduced, becomes
ท
7171
+
2
4172
I
n
7212
, it will become
12112
2
611"
་་
I
and, neglecting
12772
the expectation required.
Ex. Two perfons, whofe refpective ages are 30
and 25 years, may expect to live together 28-
8.568=19.43 years.
If the two ages are equal, the expectation of
72
7272
21
71
12
their joint lives will be
2 On
2
6 3
3d, It is required, to find the expectation of any
number of equal joint lives, of a given age.
Let there be, for inftance, two equal lives;
Bb
then
194
OF ANNUITIES
then the first term of the feries, expreffing the
expectation of their joint continuance, will be
12; and the other terms will alſo be ſquares,
n
n
X
n
n
n XI
n
fo that the whole will be a ſeries of fquares (whofe
roots are in arithmetical progreffion) beginning
with I, and ending in a cypher, and n the
number of terms; therefore, calling m the index
of the power in the feveral terms,
will be
m+1 3
the fum of the feries, or expectation required:
See the following note *. Or let there be three e-
qual lives: then the firſt term of the ſeries expref-
fing the expectation of their joint continuance
will be
n
n
n
xx=1³; and
n
n
n
n X I n
m+I 4
their ex-
pectation required; and univerfally calling any
number of equal lives m, and their complement of
n
m+1
life n, then will expreſs the expectation of
their joint continuance in being; fee the follow-
ing note *.
* The ARITHMETIC of INFINITIES.
Ex.
If there be any ferics of quantities, all of the fame power, whoſe
roots are in arithmetical progreffion, 0, 1, 2, 3, 4, &c. and let m
denote the index of the power; N the number of terms; L the laſt
term; and S the fum of the ſeries; then,
NL
1, 2, 3, 4,
02, 12, 22, 32, 42,
2
0³, 1³, 23, 33,
43,
3
m=
04, 14, 24, 34,
44,
4
NO,
12, 135
kee
,3/3 ,2 ,1 ,1 ܢ o, a
3
m+I
S. Thus,
NL
2
2
NL
3
3
NL
+
4
NL
I
4
m + I
NL
5
S
2 NL
mojet
3
WA
3 NL
4
Thi
ON LIVES.
195
Ex. Fifty perfons, whoſe age at a medium is 30
years, may expect to continue all in being
year 36 days.
56
51
= 1
: I
4th, It is required to find the expectation of
three joint lives, whofe ages are given. Let the
complements of the eldeft, fecond, and youngeſt,
be feverally denoted by n, m, and t; then by fol-
lowing a proceſs fimilar to that obſerved in cafe
2d, the expectation of their joint continuance
would be found to be,
n
12
1
I
223
2
6 7/2
t
12 mt
See the note in cafe 9th.
Ex. Three perfons, whofe refpective ages are
50, 45, and 35 years, may expect to continue all
in being, 18-9.59+1.86=10.27 years.
5th, It is required, to find the expectation of
the longeſt of two lives, of given ages. Here, by
reafoning as in Prob. IV, and calling N and M the
expectations of the fingle lives, and NM that of
their joint continuance; then,
N+M-NM will be the expectation of the long-
eſt of theſe two lives.
NL
Ex. Sought, the expectation of the longeſt of
This may be proven by induction, whereby it is fhewn, that the
greater the number of terms is, the nearer will the expreffion
approach to the fum of the feries, and is only equal to it when
the number of the terms is indefinite.
1+144
B b 2
two
195
OF ANNUITIES
two lives, whoſe reſpective ages are 30 and 25
years. Here,
28+30.5-19.43=39, the expectation of the long-
eft liver.
6th, It is required, to find the expectation of the
longeft of three lives of given ages. Let N, M,
and F repreſent the expectations of the three fingle
lives, and NMF that of their joint continuance;
then, by reaſoning as in Prob. V. N+M+F—
NM-NF-MF+NMF will be the expectation
of the longeſt of theſe three lives.
7th, If the ages are equal, then the fum of the
72
772
n
772
feries 1- +1- &c. will be the expectation
of the longeſt liver: but this feries is equal to a
number of units whofe fum is n, minus a feries
of m powers, whofe roots are in arithmetical pro-
greffion, beginning with 1, and ending in o;
therefore n
Hence,
272
72
77272
77
112 +1
m+I
is the fum of the feries.
3
4-
=The expect. of the longeſt of
4.12
5
77172
971
2
equal lives
3 whofe com-
plement of
4 life is n.
m
Ex. Let there be 100 perfons, whofe age, at a
medium, is 30 years, fome of them may expect
to live
5600
ΙΟΙ
=55.44 years,
1
8th,
ON LIVES.
197
8th, It is required, to find the expectation of
widowhood in married perfons, whofe ages are
given.
From the expectation of the longest liver
Subtract that of their joint continuance
==
NM-NM
+NM
Remains the expectation of widowhood ≈ N+ M—2NM
Ex. In married perfons whofe refpective ages
are 30 and 25 years, the expectation of widow-
hood is 58.5-38.86=19.64.
In married perfons of equal ages, the expecta-
tion of widowhood is equal to that of marriage,
feeing
2n
༢
72
n
3
3
which expreffes either of them.
9th, If the expectation of life, for any number
of years not exceeding n the complement thereof,
be required; in this cafe calling
years, the fum of the feries
*
the number of
72
+
+
72
72
n I
12
-X
will exprefs this expectation; but the fum of the
271-X
firft and laft terms =
and x the number of
72
217--X
272
terms; hence, х × x (7. Equ. 3.) is the expecta-
tion required. When x=n, the expreffion be-
12
comes and gives the expectation of the affign-
2
,
ed life for its whole poffible duration.
Ex. The expectation of life for 15 years, in a
perfon aged 30, is
97
1 12
X15=12.99 years.
Here it may be obferved, that the above expref-
fion alfo difcovers the number of perfons alive, to
which one perfon added annually to a fociety, or
left annually in widowhood, at a given age, will
increafe
198
OF ANNUITIES
1
increaſe in x years. Thus, calling N=
H
2
the expec-
tation of a widow's life; and fuppofing one to come
upon a fociety every year, the number of annuitants
alive, deduced from hence, will, in x years, be
4N-x
4N
Xx. And calling W the number of widows
age, then
4N—x
left annually, at the fame
X Wx
4N
will be the number of annuitants alive at the end
of x years. The maximum, therefore, is WN,
which will always happen when xn, the com-
plement of the widow's life.
Ex. Let there come yearly upon a fund, 19.3
widows, whofe expectation of life is 19.64, (cafe
8th ;) it is required, to find the number to which
they will increaſe in 15 years; and what will be
their maximum. Here,
4N~x
4N
x=15, W=19.3, N=19.64, and,
63.56
xWx= × 289.5=234.2, the number of
78.56
widows at the end of 15 years. And their maxi-
mum is WN=379. In like manner, the fum of
!
N-1
3
the feries
12--X
+
+
n
2
n
3712 3122
+
X3
3722
3
X
n
not only expreffes the expectation of two e-
qual joint lives, for the time x, but alfo gives the
number of marriages in being together, that will,
in x years, grow out of one yearly marriage be-
tween perfons of equal ages, whofe complement of
life is n, which expreffion, when x=n, becomes
n
3
the expectation of two equal joint lives; or,
calling M the number of yearly marriages, their
maximum
ON LIVES
199
nM
maximum will be
at the end of n years. If the
3
perfons are of unequal ages, let the complement of
the oldeſt life be n, and that of the youngeſt m,
n+mXx²
the expreffion will be x- + ; and the num-
2nm
3nm
ber of marriages will come to a maximum when
x=n.
Ex. Let there be a fociety, wherein there are
32 yearly marriages, between perfons whofe re-
ſpective ages are 33 and 25 years; it is required, to
find the greateft poffible number of married per-
fons, as alfo the maximum of widows and widow-
ers in fuch a fociety. Here, x=n=53, m=61
and M= 32; hence, 53-49.5+3.5=19; which,
multiplied by 32, gives 608, the greateſt number
of married perfons; which number being equal
to all the widows and widowers in life at one time,
and the proportion of thefe to one another being
as 5 to 3; hence, 8: 5:: 608: 380, the maximum
608:380,
of widows, and 228, the greateſt number of wi-
dowers.
The EXPECTATION of LIFE difcovered by FLUXIONS.
ift, Let
-X
repreſent the probability that a life, whofe comple-
n
#X-XX
ment is 7, will continue to the end of x time. Then
will be
2
n
the fluxion of the fum of the probabilities, whofe fluent, (to find
which, feeing the fluxion of x is xx+xx=2xx, divide the whole by
x, add unity to the exponent of the power of x in each term, and di-
vide by the exponent fo increaſed), viz. x-
x2
271
27-X
Xx, is the
211
22
becomes and gives
2
fum itſelf for the time x; this, when x=1,
the expectation of the affigned life for its whole poffible duration.
2d, Let
200
OF ANNUITIES
Supplement to Prob. XVIII.
Let N, M, P, and T repreſent the reſpective ex-
pectations of 4 lives, whofe complements are n,
, p, and t; and c, the arithmetical mean com-
',
plement
be the probability that two unequal joint
n-x X 112-X
2d, Let
1212
n-xxm x
ves will continue to the end of x time;
ทาน
Xx, will be the
uxion of the fum of the probabilities; the fluent of this expreffion,
iz. x
n+mx²
+
2nm 3nin
x3
12
when xn, becomes
122
the expec-
2 6m²
ition of two unequal joint lives, whofe complements are n and m.
3d, Since
72-X
72
222
72-X
122
is the probability that two equal joint lives will
ontinue to the end of x time,
im of the probabilities, whofe fluent,
72
Xx will be the fluxion of the
x2
viz. x—
+3
+
72
3722
when x
ecomes the expectation of two equal joint lives.
3
he number of equal lives be m, then
heir joint continuance.
n
m+I
Univerſally let
will be the expectation of
n-xx m~xxt-x
4th, Let
nmt
be the probability that three unequal
oint lives will continue to the end of x time; then this expreffion,
aultiplied by x, produces
mtxx + tx^x
mtx
ntxx + mx²x
x
nmxxnx²x
nmt
the fluxion of the fum of the proba-
ilities; the fluent of which expreffion, being reduced and x puti,
71
ecomes
122
X
I
I
ޖމ 3
3
2 6
+-+
การ
12 mt
the expectation of three joint
ives whofe refpective complements are, n, m, and t.
5th, Let
ON LIVÉ S.
201
plement of all the lives wanting n, the comple
ment of the oldeſt; then,
x
x
5th, Let-x=
n
12
| *
1A,
be the probability that two equal lives,
x²
n
whofe complement is n, will fail in the time x, then 1 - xx
Xx will
be the fluxion of the fum of the probabilities that one of them at leaft
will ſurvive the time x; the fluent of this expreffion, viz. x
212
3
3x²
when x, becomes 7- , the expectation of the longeſt li-
3 3
ver. Univerfally let the number of equal lives bem, then is the
expectation of the longest liver.
7-x
2X
6th, Let
X
72
n
mn
m+i
be the probability that there will be a furvivor
of two equal joint lives at the end of x time; then
2x
2x2
X x is
n
222
the fluxion of the fum of the probabilities, whofe fluent, viz.
x2 2x3
n
32223
when xn, becomes n
2n n
-- the expectation of fur-
3 3
vivorſhip between two equal lives, and is the fame with that of their
joint continuance.
x
x
7th, Let I
X
7
m
be the probability that the oldeſt of two lives
will furvive the other at the end of x time; then,
2.2
Xx
xx will
772
nm
be the fluxion of the fum of the probabilities, whofe fluent
2m
when xn, becomes
882
6m
Hence the expectation of
2
3nm
furvivorſhip on the part of the oldeſt
бль
of the youngeſt
Expectation of widowity
C c
272
Add
2
2
* | N
m
Add
11
212
210
n
=/ M
n
+
6m
6m
12
+
2
317%
202
OF ANNUITIES
, The expectation of the joint continuance of 2
=N- N
n2
n²
unequal lives
2
6т
n
212
of 3
+
2
6c
21
of 1*
322 2
n
3m
NM.
NMP.
224
of m equal lives
of m unequal lives:
2
=
n
6c
mti
- 272
+
n
12mp
3223
1202
20mpt
¨n" +1
172 +1
I
mix nmpt
NMPT.
n being the
mean comple.
of all the lives.
Expectation of widowity, as on p. 201.
Add
21 ~
"
+
2
2
37N.
2 | ~
n2
1
6m
of the longest liver
2
m
2
+
n
6m
8th, Let there be three joint lives, whofe refpective complements.
are n, m, and t; the expectation of furvivorship on the part of
the oldeſt
the ſecond
783
12mt
1113
the youngeſt
771
12nt
t3
I 27}}
*To abridge the labour of calculating the expectation either of
the joint continuance, or of the longeft liver of me unequal lives, whoſe
mean complement is ʼn ;
In the fluent
m + 1 x nmpt
I
which expreffes the fum of the pro-
babilities that all the lives will fail in the time, x; let x=n, the
mean complement of all the lives, and fubtracting that fluent from
222
I
212
>
m+ we fhall have
77+1
m+I
m + 1 x nmpt
expectation of their
n --
joint continuance; and expectation of their joint continuance —
that of the longeft liver.
2d,
ON LIVES.
203
2d, The expectation of the longeſt liver
of 2 unequal lives N+M—NM=M+—N.
3m
n+m+p
of 3-
of 4
NMP.
3
n+m+p+t
NMPT.
4
of m equal lives
of m unequal lives
mn
M+I
n-Expectation whofe mean
is n.
is
of their joint continuance complement
of
9 lives out of m=
gn
m+1
Or in the cafe of 4, 5, 6... m unequal lives, find
their mean age, and confider it as the age of ſo
many equal lives.
Ex. Sought, the expectation of the joint continu-
ance and of the longeft liver of three lives, whofe
reſpective complements are 40, 50, and 60. Here,
20—9.77+1.7712, Expect. of their joint contin.
of the longeſt liver.
50 12
d
= 38,-
3d, Ift, the time, be fought, in whichp, a part of m
equal lives, will fail, whofe complement is ; here,
prz =t, the time fought.
mti
77
12
な
​m+I
71
xt=p; and making t=1,
=p, the number which fail in one year; hence,
In any number of lives, their expectation of life,
and the number which die yearly, are reciprocal of,
and diſcover one another.
4th, Let there be any confiderable number of
lives, in their natural order, extending from a
Cc 2
given
204
OF ANNUITIES
given age, whofe complement is n, to the utmoſt
period of human life; then,
272
t, = their mean complement of life.
3
n
The given age+. the mean age; and,
3
their mean expectation of life.
n
3
11
Thus in a table of equal decrements, where the
complements of life form a feries of fquares,
n² Xn 271 the mean complement, &c.
3
3
Ex. 1ft, In any approved table of obfervations,
ſuch as that of Dr Halley, where 90 is the utmoſt
extent of human life, 30 will be the mean age of
all the perfons in the table, and its expectation, viz.
28, will be that of human life, or of infancy by
the table.
Moreover, 28x2+30=86, is with propriety
put as the utmoſt extent of human life by Mr De
Moivre, in his Hypothefis of equal decrements,
which is adapted to Halley's table.
Ex. 2d, The marriages of minifters wives, in
the eſtabliſhed Church, take place from the age
of 18 to 45 years; hence, +18=27, the
mean age of marriage, which is alſo the mean age
45-18
3
of the youngest widows; hence,
86-27
3
+27=4633
the mean age of widowity, whofe expectation of
life, viz. 193, multiplied by 19.3, the number
eft yearly, will produce 379.58 the maximum of
annuitants.
3d, In fuch a number of lives, the expectation
Qf
ON LIVES.
205
of the longeſt liver is nearly equal to the comple
ment of life in the youngeſt.
4th, In fuch a number of lives, which are con-
ſtantly the fame, being kept up by the addition
of new lives as the old drop off, their expecta-
tion of life is the fame with that of the youngeſt.
5th, To find the period of doubling the number
of inhabitants in any province. When the num-
ber of births, with the addition of the annual
fettlers, does fo far exceed the burials, that their
difference bears a confiderable proportion to the
whole number of inhabitants, and is equal, fup-
poſe, to a 36th part thereof; the number of inha-
bitants, if this part continues conftantly the fame,
will be doubled in 36 years. But, as every ad-
dition to the number of inhabitants, from the
births and fettlers, produces a proportionably
greater number of births; if we fuppofe the excefs
to increaſe annually in proportion to the number
of inhabitants, fo as to preferve the ratio of theſe
to one another always the fame; upon this fuppo-
fition, the population of a province, till it doubles
itſelf, will exactly refemble the accumulation of
money at compound intereſt: therefore, calling
this ratio of the excefs of the births and fettlers
I
above the burials, to the inhabitants, —;
I
in the population| r,
of a province, is
equivalent to
R,
in compound intereſt.
hence,
i
206
OF ANNUITIES
hence, by remark 2d upon compound intereft,
L. 2
L. +1
*
the period of doubling.
I
Ex. Let
36"
period of doubling.
.30103
then
=25.3 years, the
.0119
If M, the amount of the population P, in the
time t, be fought, when the ratio of the exceſs of
the births above the burials to the inhabitants is
I
ز
—; in this caſe, PX
r
=M, the amount.
XIX.
PROBLEM
Of the Probability of Survivorship.
Cafe 1ft, Two lives being given, to find the
probability of one of them fixed upon ſurviving
the other.
82. A
B C D
F G
-|-|-
S
Let AS, the complement of life, be divided in-
to an indefinite number of equal parts, AB, BC,
CD, &c. reprefenting moments of time; then
will the probabilities of living from A to B, from
A to C, from A to D, &c. be repreſented by the
, &c. and the probability
BS CS DS
AS AS AS
fractions
of life's failing in any interval of time AF is mea-
AF
AS
fured by the fraction Again, when the in-
terval AF is once paft, the probability of life's
GS
continuing from F to G, is and of its failing
FS'
FG
in that interval, is
Hence,
FS
The
ON LIVES.
207
The probability of life's continuing from A to F,
and then failing from F to G, is X=78; of
FS FG FG
AS FS AS
conſequence, the probabilities of life's failing in
any two or more equal intervals of time betwixt
A and S, are exactly the fame, the eſtimation being
made at A confidered as the preſent time.
Theſe things being premiſed, let the comple-
ments of the two
A
B
С с
H
D d
---
S
S
lives be AS=n, and BS=m; upon which take the
two intervals AC, BD=x, as alfo the two mo-
is the pro-
ments Cc, Dd=x; then
N-X
n
X
72
bability of the firft life's continuing to the end of
x
772
the time x, or beyond it; and is that of the ſe-
cond's continuing from B to
in the interval Dd; hence, I
D, and then failing
X
x
X
XX
X
is
72
772
71772
772
the probability of theſe two events happening to-
gether at the end of the time x, or it is the fluxion
of the fum of the probabilities, whofe fluent, viz.
x
m
x²
2nm
is the fum itſelf for the time x. Let now n
be put=x, then the probability of the first life's
n
210
furviving the ſecond will be, which fubtrac-
ted from unity, leaves the probability that the ſe-
cond life will furvive the firſt.
Hence the probability of furvivorſhip on the
of the oldeſt life, whofe complement is "
of the youngeſt,
part
n
2772
is 772=1-
m
2772
Ex.
208
OF ANNUITIES
Ex. Let n=46 and m= 56; then,
22
2m
72
='41
•59
2m
The chance of furvivor. on the part of
the oldeft.
S the
the youngeſt.
When the lives are equal, is this probability on
the part of either of them.
2
A, aged 60, whilſt in health, bequeaths to B, aged
30 years, the fum of L. 500, in cafe he (B) fhall
be the furvivor; fought, the prefent value of B's ex-
pectation, at 4 per cent. Here,
n=26, m=56, and P-15.983. Hence,
P
500X /
22
L. 307.365
72
multiplied by I
.767
2m2
produces B's expectation
L. 235.75
Cafe 2d, Let there be three lives, A, B, and C,
whoſe reſpective complements are n, m, and t; it is
required, to find the probability that any one of
them fixed upon will furvive the other two.
x
X
The fluent of the expreffion, I—-X-X
being taken, viz.
72
3
2mt 6nmt,
n
ኣ.
192 1,
and x put=n, we ſhall
have the probability that the oldeſt life A
3mt
will furvive B and C.
Again, the probability,
that the fecond life B will furvive the other two,
is,
17?
2t
72
Gut
; and theſe two, fubtracted from unity,
leave the probability of the youngest life C furvi-
ving A and B. Hence the probability of furvivor-
ſhip on the part of
A,
ON LIVES.
209
A, after B and C =
n²
3nit
m
B,
A and C =
2t
6mt
C,
A and B =
m
222
I
21
6mt
Ex. Let n=40, m=50, and t=60. Then,
•177 = A's
•32
•327
B's
chance of furviving
•494 = C's
{
B and C.
A and C.
A and B.
3
When the lives are equal, will be the probabi-
lity that any one of them fixed upon will furvive
the other two.
Cafe 3d, Let there be three lives, A, B, and C.
It is required, to find the probability of furvivor-
fhip on the part of any two of them after the
third.
The fluent of the expreffion 1—
x
x
I -XI- X
n
m
being taken, and x put = n, will give the proba-
bility that the two oldeſt will furvive the young-
eſt, =
n
21
2122
भूर
6mt
Hence the probability of furvi-
vorſhip on the part of
A and B after C =
n
n²
21
6mt
A and C
B =
72
n2
2771
6mt
B and C.
- A
A = 1-
2771
2/2
712
+
2t
3Fut
Ex. Let n=40, m=50, and t=60. Then,
.244 A and B's
31A and C's
.444 B and C's
C.
chance of furviving B.
A.
When the lives are equal, will be the probabi-
D d
lity
210
OF ANNUITIES
lity that any two of them fixed upon will furvive
the third.
Cafe 4th, It is required, to find the order of
furvivorſhip betwixt the three lives, A, B, and C,
whoſe reſpective complements are n, m, and t; or
the probability that A fhall furvive B, and B fur-
vive C, &c.
ift, A, B, C, =
2d, A, C, B, =
3d, B, A, C,
6mt
12
6mt
51
n²
2t
3mt
772
n
n
A
4th, B, C, A, =
2t
2t
6mt
n
n²
5th, C, A, B,
=
2 m
3mt
172
13
6th, C, B, A,
= 1
I -
2t
2m
+
12
6mt
Here it is manifeft, that thefe fix orders, taken
two and two, fuch as the 1ft and 2d, or the 1ft and
3d, are reſpectively equal to the firſt equations in
the two foregoing cafes, &c.
Ex. Let n=40, m=50,
Order 1st, .083
2d,=.088
<=
3d, .158
and t=60. Then,
ठ
Order 4th, .177
5th, = .227
6th,=.277
When the lives are equal, will exprefs the pro-
bability of furvivorſhip betwixt any order of the
three fixed upon.
A, aged 70 years, whilft in health, makes a
will, whereby he bequeaths L. 300 to B, aged 45,
and L. 500 to C, aged 36 years, with this condi-
tion, that if either of them die before him, the
whole is to go to the furvivor of the two: Sought,
the values of the expectations of B and C, eftima-
ted
ÖN LIVE S.
211
ted from the time that the will was writ, and rec-
koning intereſt at 5 per cent. Here,
n=16, m=41, t=50, and P, correfponding to
n, 10.838.
Was it certain that A would die before either
of them,
P
B's expectation would be 300XL. 203.21
C's
n
P
500×==L. 338.69
Which expectation of C, multi-
plied by the probability that
he and B fhall furvive A,
Produces C's expectation, arifing
n
.6865
from the profpect of A's dying L. 216.916
before either of them,
But C hath a further expectation,
arifing from the profpect that he
fhall furvive A, and that A fhall
furvive B, in which cafe he thall
obtain
Which expectation, multiplied by
the 5th order,
203.21
.1535
Produces his chance of B's legacy, =
31.193
To which add
216.916
Their fum is the whole of C's exp.-L. 248.11
In like manner, B's expectation, arifing from the
profpect that they will both fur-
vive A,
To which add his chance of C's le-
gacy,
= L.
L. 139.504
40.10 £
ex-}
= L. 179.605
D d 2
PRO-
Their fum is the whole of B's ex-
pectation,
212
OF ANNUITIES
PROBLEM XX.
Of Half-yearly and Quarterly Payments of an Annuity.
83. 1ſt, IN cafe of an annuity a, paid every
half-year, during ʼn years, at a given rate, let P be
the value of the annuity a for n years, V, the va-
lue of the half-yearly annuity for the fame num-
ber of years, and d=R-1, the intereſt of L. I
for half a year, (Table in 46th); then
a
a
a- Rn
P- (47.) V=
a
r
-, hence P: V::
Rn
If quarterly, d=R =
a
a
La — Rn
I
d
I
:
r
zd
(49.) divide both by
which gives
1, and 2=V.
4d
rP
: V.
2d
Ex. Let there be an annuity of L. 30 per annum,
paid every half-year, by equal portions, for the
ſpace of 20 years; fought, the value thereof, at 4 per
cent. Here a=15, P=L.407.7, and d=.01984.
+P
Hence =L.410.96=V, the value fought.
4
2d
Or thus: The annuitant, in cafe of half-yearly
payments, has the advantage above the annual
payments of the intereſt of a for half a year, that
is, of a yearly, during the continuance of the
annuity; which intereft, at 4 or 5 per cent. is e-
qual to or part of the annuity; therefore
to the value of the annuity paid yearly, add its
99th or 79th part, the fums will be the values of
the half-years payments.
I
2d,
ON LIVES.
213
?
*
of a
2d, If an annuity is paid for life, in this caſe,
half-yearly payments are of a year's purchaſe
better than yearly ones.
The annuitant, by the laſt, hath the advantage
of the intereſt of of the annuity during life, be-
fides an equal chance of receiving one half-year's
value more than if he had been paid yearly;
which chance is therefore, at his death, worth 4
part of the yearly annuity. Now theſe two ad-
vantages, it is manifeft, will always amount to
of a year's purchaſe in preſent value.
Again, half-yearly payments, which begin im-
mediately, are half a year's purchaſe better than
thoſe that begin at the end of half a year; that is,
by the above, they are of a year's purchaſe bet-
ter than yearly payments which begin at the end
of the year. But yearly payments which begin
immediately are a whole year's purchafe better
than the fame payments to begin at the end of the
year: therefore the difference of value between
yearly and half-yearly payments, fuppofing both
to begin immediately, is of a year's purchaſe in
favour of the former.
4
3d, To find how many years poffeffion will re-
imburſe an annuitant for the price paid for his
annuity.
Calling n the number of years required, a pur-
chafer will be reimburſed for his money, when the
years-purchaſe of his annuity for n years certain
is equal to the value of his life; that is, when
I
I
Rn
L.
I
=N; hence R” -
——, and n =
I-N
I-rN'
L. R
which is always lefs than the expectation of life
confidered
i
214
THE BRITISH
confidered as ſo many years certain. (Conſtruc-
tion of Table II.)
Ex. A perfon, aged 30 years, purchaſes an an-
nuity for life: Sought, how long he muſt live be-
fore he can be reimburſed for the purchaſe-mo-
ney, at 4 per cent.
Here N 14.684, and
years.
I
L. IN
•395
L. R
<= 23.23
.017
When an annuity is bought at a number of
1
years purchaſe, = —, in this caſe,
22
L. 2
L. R
2r
I—ÎN
2,
=2, and
the time in which money doubles itſelf
at a given rate.
When the number of years purchaſe=-, n be-
comes indefinite, and the poffeffion muſt be a per-
petuity.
PROBLEM XXI.
1
Of the British State Lottery.
84. The State-Lottery is a fpecies of gambling
permitted by the legiſlature, for the purpoſe of ſe-
curing a certain yearly revenue to government.
The Lottery anno 1791 contains 50,000 tickets;
and the number and value of the prizes, begun
drawing February 20. 1792, are reprefented in the
following ſcheme.
SCHEME
STATE LOTTERY.
215
SCHEME of British State Lottery 1791.
|| ||
Sum.
60,000
40,000
30,000
25,000
20,000
11
||
||
||
||
Number. Prizes.
2 of 30,000
2 of 20,000
3 of 10,000
5 of 5,000
10 of
2,000
15 of
1,000
30 of
500
50 of
100
50
||
15,000
15,000
5,000
100 of
14,150 of
5,000
20 = 283,000
ift & laſt 2 of 1,000
a
b
2,000
N° of
L.
500,000=
500,000={
{
Sum of
the Prizes.
14,369= {Prizes.
11
a+b=
S
In this Scheme let
14,369 repreſent the number of Prizes.
35,631
50,000
=L. 500,000
Blanks.
Tickets.
the fum of the Prizes.
From thefe data the following particulars are
inferred.
ift, To find the value of each Ticket, &c.
S =L. 10 =v, the value of each Ticket.
a+b
S
=L. 34.79=V,
a
Prize.
put p=L. 16.8, the market-price of each Ticket,
then pvxa+b = L. 340,000, equal Profit of
government, including the expence of manage-
ment, &c.
2d, To
216
THE BRITISH
2d, To find the chance or probability of draw-
ing a Prize or a Blank. Here,
Seeing a+ba :: 1 :
ratios.
Ift,
2d,
3d,
a
a+b
1옹 ​이용
​a
b
I
3.48-
I
1.4 +
I
2.48-
a
a+b
Hence the following
= 3.48—the probability (a Prize.
of 1 Ticket
a Blank.
=1:1.4 + drawing
=1:2.48 —; that is, to each
Prize there are 2.48
Blanks.
Theſe ratios may be called the indices of the
Lottery, which, being pointed to, or multiplied
by any number of tickets, ſhow the probable
number of prizes or blanks in theſe tickets.
I
3.48
Ex. × 3.48=1:1, and ſhows that 3.48
tickets have an equal chance of drawing 1 prize
and 2.48 blanks; but the market-price of the
tickets being L. 58.36, there is therefore a lofs of
58.36—V=L. 23.57.
I
Again, × 1.4 = 1:1, and fhews that 1.4
1.4
X
I,
tickets have an equal chance of drawing 1 blank,
and .4 of a prize: there is therefore a lofs of
23.52—13.92=L. 9.6.
3d, To find how many Blanks or Prizes there
are in n Tickets.
72
12
In n tickets
prizes, and n-
blanks,
3.48
3.48
there are
n
12
blanks, and n—
prizes.
1.4
1.4
Ex. Let n=20, there will be
20
3.485.75 prizes,
and
STATE LOTTERY.
217
{
and 20—5.75=14.25 blanks. At the fame time,
the greater the number of tickets, the greater is
the probability of obtaining the proportional
number of prizes: Thus, 3.48 tickets have only
an equal chance of drawing I prize; whereas
a+b tickets are certain of drawing a prizes:
therefore the chance of drawing a proportional.
number of prizes varies from probability to cer-
tainty in proportion to the number of tickets.
4th, To find the probability which 2 tickets
have of drawing a certain number of prizes, not
exceeding 1, let p=the prizes expected in ʼn tick-
the probability required.
ets;
I
n
then --×
P 3.48
5
I
Ex. Let n=5, and p=3; then 10.44 2.09
12 the probability of 5 tickets drawing 3
prizes.
5th, To find the probability which 2 tickets
have of drawing any one particular prize in the
fcheme.
That ticket will draw a prize
11
11
14.150
of L. 20,
the pro-
a+b
219
better than L. 20,
babi-
a+b
lity=
39
of L. 1000 and upwards,
That n tickets will draw a prize
better than L. 20,
a+b
of L. 20,000,
I
3.53 •
I
228.3°
I
1282.
1?
the probability =
228.3
72
25,000
E e
6th,
218
THE BRITISH
6th, To find the probability which any 2
tickets fixed upon have of drawing 2 capital
prizes, for inftance, of L. 10,000 each. Here,
3
50,000
Ι
16666.8
= probability that 1 ticket fixed
upon, will draw a prize of L. 10,000. But in
the prefent cafe of 2 tickets drawing equal prizes,
as thefe events are independent of one another,
therefore
= 1:277,777,777-4=
I
X
I
16,666.6 16,666.8
the probability required. Hence the vaſt diſad-
vantage attending fhares of tickets drawing ca-
pital prizes. Thus the probability that 2 half
tickets will draw the half of two L. 10,000 pri-
zes, is as 1 : 277, &c. as above,
7th, To find the number of prizes and blanks
drawn daily, during the continuance of the lot-
tery, being 38 lawful days.
Each law-Prizes
ful day the Blanks = 937,
N° of
Tickets
N°
No of L. 20 Prizes
378, with 5 interfperfed.
25
1315,
30
372,
14
ac-
8th, Infurance in the Lottery is not only ex-
prefsly prohibited by acts of parliament, on
count of the bad effects it may have upon the
morals and fortunes of adventurers; but it is alfo,
upon a fair calculation, fo high, that it is not to
be fuppofed any perfon will riſk incurring the le-
gal penalty by fuch a practice. Thus,
ift, Should an adventurer infure, upon his tic-
ket, the fum P, not exceeding the leaft prize, up-
on condition that the office fhall make good that
fum,
STATE LOTTERY.
29
fum, in cafe of a blank, while he referves to him-
felf the chance of drawing a prize; fought, the
premium of infurance.
It is evident, that here only the blanks need to
be infured; therefore Pxb will be the infurance
upon the whole, a+b number of tickets, and
I
= PX - -the infurance upon each ticket.
1.4
Ex. Let P-L. 16.8, then
of infurance.
PXb
a+b
16.8
1.4
L. 12 premium
20
Let P-L. 20, the leaſt prize, whilſt the hold-
er retains the chance of a higher one, then, 1.4
L. 14.285, the premium of infurance.
Hence any fum P, not exceeding the leaft
prize, multiplied by the chance of drawing a
blank, in any lottery, will give the premium of
infurance upon that fum.
2d, Should an adventurer infure upon his tick-
et the fum of L. 50, whilſt he retains the chance
of a higher prize; fought, the premium of infu
rance. Here 5ob+the number of L. 20 prizes
multiplied by 30=the infurance upon the whole
number of tickets; therefore,
506 +14150X30
a+b
=L. 44.12, the premium fought.
Ee2
RE-
←
220
THE BRITISH
i
REMARKS.
ift, In fuch a Lottery, the true value of ea
ticket, exclufive of the expence of management
is L. 10; and it is owing to the confidence which
the credulity of the public hath in the powers of
fortune, that the market-price rifes to 16 guineas
or upwards.
2d, The fcheme of this Lottery is conftructed
with great art: There is a valuable bait of
L. 100,000 in four prizes, hung out to catch the
attention of the public; and the number of L. 20
prizes being fo great, there are not 2.5 blanks to
I prize; whereas the chance of obtaining a
L. 20 prize being ſcarce diſtinguiſhable from that
of obtaining a prize at all, the number of the
former ought in all justice to be diminished,
and the 2 prizes of L. 30,000 each rejected, fo
as to increafe the number of the intermediate
prizes.
PROBLEM XXII.
Of a Tontine.
85. A Tontine is a kind of lottery on lives, where
the capital, divided into a number of ſhares, is
formed by the voluntary fubfcriptions of thoſe
who chufe to riſk their money, in fuch a ſcheme,
upon the continuance of their lives. There are
feveral particulars refpecting a tontine which need
to be determined.
int,
OF A TONTINE.
221
ift, In a tontine, where 100 perfons, for in-
ſtance, whoſe mean age is 20 years, fubfcribe
each of them L. 100, forming a capital of L. 10,000,
which is funk to the heirs of the ſubſcribers, it is
required to find the number of furvivors at the
end of 10, 20, 30, &c. years, and what premium
will then be due to each furvivor, reckoning in-
tereft at 5 per cent. Here, n=66, m=100, and
t=10; hence, (fupplement to problem XVIII.)
111+1xt 1010
12
= 15.3, who fail in 10 years; but the
66 15.3,
intereſt of the capital being L. 500,
500
84.7
= L. 5·9,
the intereft due to each furvivor at the end of
that period; hence,
living 84.7, per cent 5.90, at the end of 10 years.
·69.4,
54.1,
7.20,
9.24,
20.
30, &c.
2d, In a tontine, where the fhares are in pro-
portion to the value of life, with a view to make
a dividend of the capital when the furvivors can
receive each of them L. 1000; in this cafe, if
the ſhare of a fubfcriber aged 10 or 12 years be
L. 100, that of a child one year old will be L. 79,
reckoning the value of life at 4 per cent; thus
16.88: 13.36:: 100:79. In like manner,
from 1 to 5 the fhares are from 79 to
5 to 10
10 to 18
18 to 27
27 to 39
1
95
95 to 100
-100 to 95
95 to
90
go to
79
In fuch a tontine, let there be 100 fubfcribers
whoſe mean age is 20 years, and the fhares at
medium L. 90 each, forming a capital of
L. 9,000 ; it is required, to find the time in which
dup
a
they
1
222
OF A TONTINE.
they will be fo far reduced, that each furvivor
can receive L. 1000. Here, m=100, n=66, and
p=91. Hence,
пр 6006 =59.46=t, the time fought.
m+I ΙΟΙ
3d, In a tontine to continue a limited time, it
is required, to find the capital of each furvivor
having a fingle fhare, at the expiration of that
period.
Ex. In the equitable and univerfal tontine,
Briſtol, commencing on the 16th day of January
1792, and continuing 7 years, wherein each
fhare is 6 s. 6 d. per quarter, with 5 d. per quarter
for agency, it is required to find,
is
Ift, Out of 100 fubfcribers, whofe mean age
25 years, the number of lives failing in 7 years,
100, n=61, and t=7; hence,
Here m
infixt
73
707
61
=11.6, the number required.
7
Here,
2d, The amount of each fhare at the end of
years, reckoning intereft at 4 per cent.
a=.325, R'—1.3159, and r=.00985; hence,
Rt-I
ax———L. 10.425, the amount of each fhare.
r
But it is evident, that, as the lives fail from year
to year, the 11.6 dying out of each 100 fubfcri-
bers will only have contributed the half of their
quota, viz. 11.6× 5.2125=L. 60.465, to be divi-
ded among 88.4 furvivors; hence,
60.465
88.4
.684, to which add L. 10.425, their fum,
viz. L. 11.11, is the capital of each furviyor, ha-
ving a ſingle ſhare, at the end of 7 years.
Theſe quarterly payments, viz. 6 s. 11d. laid
out
INSURANCE FROM FIRE.
223
out at 4 per cent. fimple intereft, where there is
no rifk, would, at the end of 7 years, amount
to L. 11.42.
SUPPLEMENT TO CHAP. VIII.
Of Infurance from Fire.
86. IN the city of Edinburgh and its environs
there are about 50,000 inhabitants; and of con-
fequence, by allowing 5 perfons to each family,
there will be 10,000 families in that city and its
neighbourhood: Now fuppofing that L. 100 was
infured upon each of theſe at 2 s. 6 d. per annum,
this, for each 1000 families, would produce an
annual revenue to the inſurance-office of L.
Ι
125.
Again, admit that I family out of 1000, or 10
out of the whole city, were expofed to the accident
of fuffering lofs by fire yearly, which is a high
proportion, and, we truft, feldom verified by ex-
perience; this would occafion an annual lofs to
the office of L. 100 upon each 1000 families; of
confequence there would be a faving of L. 25
yearly, befides intereft, to defray the expence of
management, upon each 1000 polices at 2 s. 6 d.
a-piece. Were the riſks in large and fmall pro-
perties the fame, being in proportion to their va-
lues, their polices would be equal; but from the
number of hands, fires, and lights, &c. employed,
and from the connection of the parts of a build-
ing, the riſk attending L. 1000 capital is great-
er than 10 times that attending L. 100 value.
Therefore, to find this rifk when the policy is,
for inftance, 4s. L. .2 per cent.; here, as
-
224
INSURANCE FROM FIRE,
}
.125×1000 –625, the riſk is as 1 to 625.
.2
And when the policy is 4 s.-L. .2 per cent. upon
625 capitals of L. 500 each;
The annual revenue to the office=5X.2 X 625=L.625
The annual lofs would be
And the yearly advantage
* 500
= L. 125
In like manner, when
The policy is 2 s. 6d≈.125]
[1000
3 s.
=.15
833·3
4 5.
=.2
the risk is as I to
625
5 s.
=.25
500
6 s.
=.3
416.6
The number of polices =
3375
Policy.
2s. 6d.
3 s.
Capital.
M. Revenue.
Lofs.
Gain.
L. 100,
125
100
L. 25
45
4 S.
from L. 100 to L. 500,
from L. 500 to L. 1000,
312.5
250
61.5
875
700
175
5 S.
from L. 1000 to L. 2000,
1875
1500
375
6 s.
L. 2000 and upwards.
2500
2000
500
5687.5
Add the intereft upon the mean revenue at 4 per cent.
Hence 3375 polices produce of yearly gain
1136.5
237.5
L 1374
Of
CHA P. IX.
Of the Widows Scheme in the Church of Scotland.
THE fund eſtabliſhed by law, for a pro-
87.
vifion to the widows and children of the minifters
of the eſtabliſhed church, and of the profeffors in
the univerſities, of Scotland, is a noble monument
of the knowledge, moderation, and perfeverance of
the late Reverend Dr Alexander Webſter, one of
the miniſters of Edinburgh, the principal contri-
ver and promoter of this ſcheme; at the fame
time it is a ſtriking evidence of the experience
which the minifters of the church then had, of
the ſtraitened circumftances of many widows and
orphans of minifters, who, from a competency,
were reduced, at once, to a ftate of indigence;
feeing that, out of 962 minifters and profeffors,
only 135 declined contributing to this fund; a
fund, by which, when it is complete, above 370
widows will be fupported, and a fum not lefs
than L. 1360 diftributed yearly among the or-
phans of contributors.
F f
Previous
226
OF THE WIDOWS SCHEME.
Previous to the eſtabliſhment of the ſcheme, fe-
veral attempts had been made, by the different
fynods in Scotland, to provide for the widows
and orphans of minifters; all which proving in-
effectual, from their limited nature, and for want
of a common rule and proper authority to enforce
it, Dr Webſter and others, about the year 1742,
prepared the plan of a ſcheme, which fhould com-
prehend the whole minifters of the church, and
be enforced by the authority of law. For this pur-
pofe they had propofed a fet of queries to the dif-
ferent prefbyteries; and from the lifts tranfmitted
by them, anno 1742, in anſwer to thefe queries,
founded upon an inquiry for 20 years back, the
following facts were difcovered :
1st, That the number of benefices in the church,
independent of the offices in the four univerfities,
was 970.
2d, That, taking one year with another, 27
minifters died yearly. That 18 of them left wi-
dows, whofe mean age was fuppofed to be 52
years; 5 of them children without a widow; and
4
of them neither widows nor children: Hence
the expectation of a widow's life, by Dr Halley's
table, viz. 16.66, multiplied by 18, produces 300,
the greateſt number of minifters widows fuppofed
to be alive at one time.
3d, That 3 families of children, under 16 years
of age, were left annually by widows of mini-
fters who died or married.
Theſe facts being difcovered, the falaries of the
general collector and clerk were eſtimated at
L. 200
OF THE WIDOWS SCHEME.
227
L. 200 yearly, which, together with a fum not
exceeding L. 39 allotted to defray the incidental
expences of the truſtees, made the annual expence
of management, in all L. 239; which afterwards,
upon admiffion of the profeffors in the four uni-
verfities to the privilege of the fcheme, was in-
creaſed to L. 250. And in order to deprefs the
annual rates as much as poffible, fo as to leave
room for taxes and deductions, a high rate of in-
tereft was affumed in the calculation of the rates,
namely 4 per cent. Upon theſe principles and
data, a ſcheme was prepared and laid before the
General Affembly 1743, which approved thereof;
and the parliament, upon the humble application
of the Affembly, were pleafed to eftablifh the
fame by law, and enacted that the ſcheme fhould
commence on the 25th day of March 1744, ma-
king the first year thereof, which ended on the
22d day of November thereafter, to confift of 8
months wanting 3 days. From the lifts tranfmit-
ted by preſbyteries at Candlemas 1745, it appear-
ed that the number of benefices and offices in the
church and univerfities, namely 1011; and con-
fequently the annual produce for the fupport of
the fund, was conſiderably leſs, and that the num-
ber of widows to be provided for, viz. 364, was
confiderably greater than had been fuppofed in
the calculation of the rates previous to the afore-
faid act of parliament; and upon inftituting a
new.calculation, founded upon Dr Halley's Ta-
ble of Obfervations, it further appeared, that by
reafon of thefe difadvantages, the free ftock would
become ſtationary against the 1771, and that
L. 10,000 would then be wanting to raife the ne-
ceffary capital of L. 80,000. The truſtees, for
preventing this and other difagreeable confequen-
Ff2
ces,
228
OF THE WIDOWS SCHEME.
I
ces, did, purſuant to the order of the General
Affembly 1748, petition the parliament to order
certain fums, out of the firſt and readieft of the
annual produce, to be applied from time to time
towards raifing the capital; and to authorife de-
ductions, in certain cafes, from the annuities of
widows and the provifions of children. The par-
liament were pleafed to enact accordingly.
During the ſpace of 30 years, from the year
1748, the affairs of the fcheme continued, as
they ſtill continue, to profper. The truſtees, how-
ever, about the year 1773, with the advice of Dr
Webſter, propoſed a number of alterations and
improvements refpecting the ſcheme to the Gene-
ral Affembly. In confequence of which, after ta-
king the fenfe of the church upon theſe altera-
tions, the Affembly 1778 did again petition par-
liament to confirm the fame by law; the parlia-
ment were pleaſed to comply with their requeſt,
by a new ftatute, 19 Geo. III. cap. 20.
The feveral particulars refpecting the perfons
who are liable in payment to, or receive benefit
from the fund, together with the terms of pay-
ment, the riſe of the capital, till it amounts to
L. 100,000, and the duty of the truſtees, who are
appointed to fuperintend the whole, are all accu-
rately expreffed in the late act of parliament,
which is in force from the 29th day of September
1778, a copy of which is fuppofed to be in the
poffeffion of every contributor. In the prefent
ſtate of the fund, in the year 1790, it is propoſed
to inveſtigate the 12 following articles :
ART. I. To find the connection betwixt the fe-
veral claffes of annual rates and their correſpond-
ing annuities in the widows fcheme.
88.
OF THE WIDOWS SCHEME.
229
88. According to lifts tranfmitted by prefby-
teries previous to the commencement of the
ſcheme, the benefices in the church appeared to
be, as above, 970; minifters were fuppofed to be
admitted to their benefices at the mean age of 27
years, which, by Dr Halley's table, makes their
expectation of life 30 years. The widows of mi-
nifters were fuppofed to live, at a medium, about
20 years; and the expence of management was
eſtimated at L. 239, which being divided by 970,
quots .24641. for each contributor's annual fhare
of expence. Now fuppofing an annuity of L. 5
was granted to each widow, or 10 years purchaſe
of her annuity to a minifter's children where
there is no widow, it is required to find the an-
nual rate which each contributor ought to pay,
reckoning intereft at 4 per cent. Here,
The preſent value of L. 1 an-
nuity for 20 years,
Multiplied by the annuity, =
13.00793
5
Produces,
65.03965, which
being divided by 61.00707, the amount of L. I
annuity for 30 years quots,
To which add the yearly expence of
management,
Their fum is,
1.0661
.2464
L. 1.3125=
L. 163, the annual rate of each contributor,
including his expence of management, and fup-
pofing the annuity to be L. 5. Hence, by doubling,
&c. the annual rate and annuity, when
L. s. d.
L.
L.
the rate is 2 12 6, the annuity=10, and 10 years-purchaſe=100,
3 18 9,
0,
5 5
6 11 31
15,
20,
25,
150,
200,
250.
From
56
!
230
OF THE WIDOWS SCHEME.
From the conſtruction of theſe four claffes of
annual rates, it appears that, befides bearing the
burden of its correfponding annuity and yearly
expence of manageinent,
the firſt claſs is taxed with 5 s.
the ſecond
the third,
with IOS.
with 15 s.
and the fourth,
with 20s. of annual expence
nearly.
Theſe four claffes of annual rates are admirable
for the fimplicity of their conſtruction; and be-
ing built upon the principles of the expectation
of life and a high rate of intereft, are as mode-
rate as poffible, and leave full room for taxes
which are juft, and deductions which are not op-
preffive, in order to their anſwering the purpoſes
intended by the act of parliament. Thus,
ift, The rates themſelves are taxed as above,
and the higher the rate the greater is the tax.
2d, Contributors at their marriage, and married
intrants at their admiffion, are taxed with the
payment of one year's rate of the clafs to which
they are fubjected.
3d, The half-years rates payable out of anns
and vacant ftipends may alſo be confidered as a
tax, feeing the decrement of life in the incum-
bent is then fufpended; befides the advantage
which the fund receives from 4 of the contribu-
30
tors who die yearly leaving neither a widow nor
children, and from the expectation of a miniſter's
life being 24 years greater than is fuppofed in the
calculation of the rates. Again,
4th,
7
OF THE WIDOWS SCHEME.
231
4th, There is a deduction of one years purchaſe
out of the provifion of a widow, feeing her annu-
ity begins to run from a full year after the laſt
payment of her huſband,
5th, There is a deduction of two years purchaſe
out of the provifion of children, which is fo much
lefs than that of a widow.
6th, There is alfo a deduction out of the pro-
vifions of the widows and children of thoſe con-
tributors who have not paid annual rates to the
amount of three years of their widows annuity,
without reckoning intereft thereon, till the ba-
lance is recovered; befides the advantage which
the fund derives from widows, not having chil-
dren under 16 years of age, who happen to marry
not-contributors, and of confequence are ftruck
off from the benefit of the fund: Whereas the
only advantage which the fcheme affords above
the calculation, is the reverſion of 10 years annu-
ity to the children under 16 years of
of age, of thoſe
widows who happen to die before they have com-
pleted the 11th year of their widowhood.-Thefe
rates, taxes, and deductions, together with the
provifion in the act of parliament to fecure the
rife of the capital to L. 100,000, by the applica-
tion of L. 200 yearly for this purpoſe, will, we
truft, give a ſtability to the fcheme, and make it
anfwer all the purpoſes for which it was intend-
ed.
I
In the conftruction of thefe rates, fuppofing
the number of benefices and offices in the church
and univerſities to be 1013, including the third
charge in Perth and the fecond in Linlithgow,
and fixing the annual expence of management, a-
greeable
!
232
OF THE WIDOWS SCHEME.
A
greeable to the two first acts of parliament, at
L. 250, we ſhall have 250L..2467 for each con-
1013
tributor's fhare of expence, which, for an annu-
ity of L. 5, will alfo produce a rate, as above, of
L. 1 6 3, nearly exact.
Had thefe rates been calculated upon the ftrict
principles of reverfionary annuities, the calcula-
tion would have been more laborious, and the
rates and marriage-taxes much higher than they
are at preſent in the widows fcheme. Thus :
Suppofe a contributor, aged 30, marries a wife
of 25 years of age, whofe annuity is L. 20;
fought, the annual rate which he ought to pay,
reckoning intereft at 4 per cent. Here,
3.85
× 20+.2464×4
14.684
required.
L. 6.23, the annual rate
Again, If the fame contributor becomes a wi-
dower, and, at the age of 40, marries a fecond
wife aged 30 years, his marriage-tax would be
L. 8.68. See Prob. VIII. Of a Widows Scheme.
Thefe annual rates are due to the fund, either
for a whole or an half year, at the fame terms
that an incumbent hath a right to his ftipend or
falary for the fame periods, and are payable to the
collector at the firft term of Candlemas thereaf-
ter; excepting in the cafe of the half-years rates
payable out of anns and vacant ftipends, the for-
mer of which is always half of the contributor's
rate, and the latter is fixed at L. 3, 2 s. for each
half-year of vacancy, thefe, though due at the
fame terms with the former, are not payable to
the collector till the firft term of Candlemas which
fhall happen a full year after the anns and vacant
ftipends fhall be due and payable. The medium
of
OF THE WIDOWS SCHEME.
233
of annual payments by contributors being within
a trifle of the third clafs of L. 5, 5 s. this third
claſs is eſteemed, in all calculations refpecting the
fcheme, as the medium of the annual rates of all
the contributors, and L. 20 the medium of all the
widows annuities.
ART. 2. To find the number of benefices in the
church, and offices in the univerſities, which are
all fubject to the payment of one or other of the
annual rates.
89. By the lifts tranfmitted to the truſtees, by
prefbyteries and univerfities, it appeared that, at
Martinmas 1744, there were,
Of Benefices in the Church,
And of Offices in the Univerſities,
Of the faid Benefices and Offices,
9427
}}
IOII in all.
69.
(full 962)
vacant 49
1011 in all.
From the beſt information which I have recei-
ved, it appears that, at Martinmas 1789, there
were,
Of Benefices in the Church,
And of Offices in the Univerfities,
943
685
}
IOII in all, as above.
It being left optional, by the firſt act of parlia-
ment, to the minifters of the church then living,
to accede to, or decline the privilege of, the
fcheme; there were, at Martinmas 1744, 135 de-
cliners, that is, perfons who declined accepting
the benefit of the fcheme, confequently there re-
mained only 827 original contributors; and as
not only their fucceffors, but the fucceffors alfo of
the 135 not-contributors are obliged to become
ſubject to the ſcheme, the greateſt number of con-
tributors alive at one time will be 962, making
G g
in
234
OF THE WIDOWS SCHEME.
in all, with the addition of 49 vacancies, 1011
benefices in the church and univerfities liable to
the payment of one or other of the annual rates.
Hence, as the above number of benefices alfo in-
cludes the anns, the greateſt ſum of annual rates
paid in any one year, exclufive of marriage-taxes,
will be 1011 X 5.25=L. 5307, 15s.
As feveral of the not-contributors, at the com-
mencement of the fcheme, were below 30 years
of age, the number of contributors will not be
complete till about the 60th year of the fund; and
there being 7 of thofe alive at Martinmas 1789,
the number of contributors may be placed in this
form in the fucceeding years of the fund, at Mar-
tinmas yearly.
Tear.
Fund. Contribut•
1790
47
955
1791
48
955
1792
49
957
1793
50
958
1794
51
959
1796
53
960
1799 56
961
1803
бо
962
;
Although there be feveral minifters of the church,
who alfo hold offices in the univerfities; yet this
diminution of the number of contributors, fuch
not being liable to double rates, is more than
counterbalanced by the number of thoſe who
have refigned their benefices or offices, and are
ftill fubject to the fund.
From the lifts tranfmitted by prefbyteries anno
1742, it appeared, as mentioned above, that out
of 897, the greateſt number of minifters in life at
one
OF THE WIDOWS SCHEME.
235
one time, 27 died yearly, &c. According to which
proportion, out of 962 minifters and profeffors,
28.956 would die yearly: of whom 19.3 would
leave widows, 5.36 children without a widow,
and 4.29 neither widows nor children.
From the truſtees report for the 46th year of
the fund, ending on the 22d day of November
1789, it appears that there have died of minifters
and profeffors, fince the commencement of the
fcheme,
Which being divided by the
number of years,
1345
quots29.455,
45.663
the number who have died yearly.
Hence the number alive at one
time,
962
Divided by the number
quots 32.66,
29.455
who die yearly,
the expectation of a miniſter's life.
From the fame report it appears, that 19.3 left
widows, 5.8 children without a widow, and 4.4
neither widows nor children, yearly. But feeing,
by Dr Halley's table, out of 962 perfons of 27
years of age and upwards, 31.79 would die year-
ly; Dr Webfter, in his calculations, fuppofes that
30 minifters and profeffors would die yearly; out
of whom 20 would leave widows, 6 of them chil-
dren without a widow, and 4 of them neither
widows nor children. He fuppofes further, that
the mean age of widows, at the death of their
hufbands, was 52, and that the expectation of
their life was 20 years.
ART. 3. To find the annual number of thofe
G g 2
contri-
236
OF THE WIDOWS SCHEME.
contributors who are liable to the payment of the
marriage-tax.
90. As 30 are fuppofed to die annually out of
the whole minifters and profeffors, of whom 26
leave widows or children; and in regard many
wives predeceaſe their huſbands, and feveral of
them leave no children; we cannot fuppofe fewer
than 28 to marry annually, who, in confequence
of their marriage, except to annuitants, are liable
in a fum equal to one year's rate of the claſs to
which they belong; the juftneſs of which infe-
rence depends upon the fuppofition, that all in-
trants are bachelors, and that there are no fecond
marriages. Therefore, independent of conjecture,
it is found in fact, that there are about 26 incum-
bents who marry yearly in the church and uni-
verſities; and that there are about 6 married per-
fons admitted yearly, for the firſt time, to bene-
fices and offices, making in all 32. Beſides, of
thoſe who are admitted into the church and uni-
verfities, there are 5 or 6 who are 40 years of age
or upwards. Now, fuppofing that the half of
thefe are either married, or afterwards do marry,
and, in confequence of their marriage, are, by the
late act, liable in a fum equal to 2 years rates of
the clafs to which they belong, this will increaſe
the annual number of marriage-taxes to about 35.
Hence the fum of marriage-taxes paid annually
by contributors will be 35X 5.25=L. 183, 15 s.
Moreover, 32, the number of yearly marriages,
being multiplied by 19.43, the expectation of
marriage between two perfons of 30 and 25 years
of age refpectively, will produce 622, the greateſt
number of married perfons in the church and u-
niverfities at one time; which number being equal
to
OF THE WIDOWS SCHEME.
237
to that of all the widows and widowers in life at
one time, and the proportion of theſe to one an-
other being as 19.4 to 12.6, we fhall have
32: 19.4 :: 622: 377, the greateſt number of wi-
dows in life at one time. See the following Article.
A marriage-tax is payable to the collector, at
the firſt term of Candlemas which fhall be one
full year
after marriage; or, in the cafe of a mar-
ried intrant, after his admiffion to a benefice or
office.
ART. 4th, To find the greateſt number of an-
nuitants which can come upon the fund. This
may be done, ift, by the annual reports of preſby-
teries, which, among other articles, contain lifts
of all the widows alive at Martinmas each year,
the medium of which lifts for a few years, could
we depend upon their accuracy, would fhew the
maximum of annuitants. Thus, during a fpace of
20 years from Martinmas, 1749 their mean num-
ber was below 380; and the number of widows,
annuitants, and others, alive at Martinmas 1789,
was 376, including 4, which are added upon ac-
count of the great number of families of children
of the first order, being 8 more than ufual, which
came upon the fund at Whitfunday 1781.
2d, By their expectation of life when they com-
mence widows. Let the refpective ages of huf-
band and wife be 30 and 25 years; then, to the
age of the wife, viz.
add the expectation of marriage, by Dr Halley's
==
25
19.5
table
their fum, increaſed by .5 on account of fecond
marriages, gives 45, the mean age of widowhood;
therefore 19.56, the expectation of a perfon's life
aged
1
238
OF THE WIDOWS SCHEME.
"
aged 45, multiplied by 19.3, the number left
yearly, will produce 377.5, the maximum of an-
nuitants; whereas 16.66, the expectation of a
perfon's life aged 52, multiplied by 20, the num-
ber of widows fuppofed to be left yearly, produces
only 334, which is the number Dr Webſter
brings out, in the third table of his calculations.
So fenfible was the Doctor of the defect of this
maximum, that, from the year 1781, he fuperadds,
to the number of annuitants by calculation, no
fewer than 66, making in all 400 in life at one
time.
3d, By the numbers already upon the fund.
Thus at Whitfunday 1779, the number of annui-
tants drawing full was 289.5, including heirs, be-
fides 9 drawing half annuities; at which term
there were 181 benefices which had caft no widows
upon the fund, but which, in the ſpace of 11
years, produced 34 additional ones, including 4
which are added for a reafon mentioned above;
at Whitsunday 1790, there were 123 benefices
which had produced no widows to the fund.
Hence,
181:34::123: 23, the number of additional wi-
23:23,
dows at Whitfunday 1801, making in all 342.5,
including heirs, drawing full, and 10 drawing
half annuities, when the number of contributors
will be complete; at which period there will ſtill
be
52
benefices unproductive of widows; hence,
962-52:342 5:52:19.5 the number of widows
ftill to come upon the fund after Whitfunday
1801. Hence the maximum of annuitants will
be 362, including heirs, drawing full, and 10
drawing half annuities; that, is there will be
362-5=357 drawing full and 10 drawing half
annuities,
OF THE WIDOWS SCHEME.
239
!
annuities, together with 10 not entered upon the
fund; in all 377, exclufive of heirs.
It is fomething remarkable, that although, in
affuming the expectation of a widow's life from
52 years and upwards, there is a defect of 2.9
years, yet Dr Webſter's calculation, Table III, a-
grees nearly exact with the fact, till about the
year 1781. To account for this I obferve,
ift, That the Doctor, in fuppofing 20 widows to
be left yearly, and that theſe are all alive at the
following Martinmas, affumes .7+.3=1 more
than the fact; which circumftance would increaſe
the number of annuitants by 16, when that
number was complete.-2d, That, from the na-
ture of the calculation, he fuppofes the number
of contributors to be complete in the year 1781;
whereas there were 15 decliners ftill alive, and,
by proportion, there were 107 original contribu-
tors and others ftill in life, who would produce
37 widows, fuppofing the maximum to be 334-
3d, That the Doctor's calculation comprehends
all the widows alive whether married the fecond
time or not; which circumftance may increaſe
the number about 10 more than the fact, when they
arrive at a maximum; now the fum of thefe, viz.
16+37—10=43, added to 334, will give 377,
the true maximum of annuitants upon the fund;
or, fubtracted from the true maximum, will leave
334, the number of widows by the Doctor's cal-
culation, Ann. 1781, fuppofing there had been 962
contributors from the beginning. The induction
of theſe particulars is a further proof that the
maximum found in this article is near the truth.
The number of annuitants cannot be complete,
not only till after the death of all the decliners,
which may happen about the 60th year of the fund,
but
240
OF THE WIDOWS SCHEME.
1
but alſo while any of their fucceffors in office are
alive, which may be till about the 120th year of
the ſcheme; during the laft 60 years of which
period, there are only about 20 additional widows
to come upon the fund. Hence the number of
annuitants, including heirs, drawing full annui-
ties, at the term of Whitfunday yearly, befides
10 drawing half annuities, may be repreſented by
the following table.
Years. [Fund. Annu.
1790 | 47 319
I 48 22
Years. Fund. Annu.
1806
63
345
7
64
46
5
8
♡ ♡ to eso no
2
49 24
9
66
47
3
50
26
1811 68
48
4
51
28
13 70
49
52
330
15
72
350
6
53
32
17
74
51
7
54 34
19
76
52
55
36
1821 78
53
9
56
38
24
81
54
1800
57 40
27 84
55
I
58
42
1830 87
56
2
59
43
33 90
57
4
61
1¦
344 1863 120 362
A widow's annuity is computed to run from the
26th day of May, or the 22d day of November,
which ſhall be one full half year after the huſband's
death; and the first year's or half year's annuity
is payable on the 26th day of May, which fhall
be a full year, or a full half year, reſpectively,
after the time from which the annuity is com-
puted to run; and it continues payable on the
26th day of May yearly, during the widow's life,
and her remaining unmarried; excepting that,
by the late act of parliament, annuitants, having
no children under 16 years s of age, and marrying
contributors,
1
↓
OF THE WIDOWS SCHEME.
241
contributors, are entitled to one half of their an-
nuities, until the capital is completed; after which
they fhall enter upon their full annuities; and as
there is nothing due to her heirs for the half year
in which an annuitant dies, the heirs only of
thoſe who die from Martinmas to Whitfunday
are entitled to half years annuities at the follow-
ing term of Whitfunday.
2
As a widow is fuppofed to live at a medium
20 years, did fhe come upon the fund immedi-
ately at her huſband's death, her annuity, rec-
koning intereft at 4 per cent. would be worth
13 years purchaſe; but as fhe hath no claim up-
on the fund till three quarters of a year at a me-
dium after the above term, and as fhe lofes one
quarter of a year at her own death, her annuity
is only worth 12 years purchaſe.
ART 5. Refpecting families of children of the
firft order.
By theſe are underſtood thofe families of chil-
dren whofe fathers at their reſpective deaths left
no widows upon the fund.
89. In calculations refpecting the fcheme, it is
ſuppoſed that 6 fuch families of children are left
yearly, who, by act of parliament, are intitled to
10 years purchaſe of their fathers widows annui-
ty, that is in all, to the fum of L. 1200. And
the proviſion of each family is divided equally a-
mong
them where there are more children than
one; which provifion of children falling due in
any one year, computed from the 22d day of No-
vember to the 22d day of November in the year
following, is payable by the collector, on the
13th day of Auguft thereafter.
Hh
ART.
242
OF THE WIDOWS SCHEME
ART. 6. To find the number of families of
children of the fecond order which come yearly
upon the fund, that is, children, who being under
16 at the death or marriage of their refpective
mothers, are intitled to the reverſion of what they
have not drawn of 10 years annuity.
90. According to lifts tranfmitted by prefbyte-
rics, anno 1742, it appeared, as mentioned above,
that 3 fuch families were left annually by widows
of minifters who died or married, having chil-
dren under 16 years of age. Now although the
fame number die, yet it is probable that fewer
marry fince, than before the commencement of the
ſcheme, on account of the lofs of their annuities;
we will fuppofe that there are 2.5 families of chil-
dren left annually by widows who died, or marri-
ed, having a child or children under 16 years of age.
But it is evident that no widow who furvives her
hufband 17 years, can leave a family in thefe cir-
cumftances, and by the act of parliament, it is
only the children under 16 years of age, of fuch
widows as have not drawn 10 years annuity, or
have not ſurvived the 11th year of their widow-
hood, who are intitled to the reverfion; there-
fore,
17: 2.5: 11: 1.6 the number of families of
children of the fecond order, who are yearly in-
titled to the reverfion of their reſpective mothers
annuity; which reverfion is worth from o to 10,
that is 5 years purchafe; Therefore, 1.6× 100=
L. 160, is the yearly fum due to families of chil-
dren of the fecond order.
In this article I differ from Dr Webfter, who
makes the number of families of children of the
fecond order to be 1.5, becaufe, from the nature
of
OF THE WIDOWS SCHEME.
243
of the calculation, the Doctor's number does not
comprehend thoſe widows who may happen to
marry, having children under 16 years of age;
whereas I include thefe, calling them .5 yearly,
feeing they are excluded from the number of wi-
dows actually upon the fund. At the fame time,
I avoid the fuppofition, that any woman, aged 69
years, can leave a child or children under 16 years
of age.
!
ART. 7. To find the yearly value of the deduc-
tions from the proviſions of the widows and chil-
dren of fuch contributors as happen to die before
they have paid a fum equal to three years annu-
ity, correfponding to the annual rate to which
they were fubjected, without computing intereft
thereon.
7
91. Such a deduction will take place, unlefs a
contributor at his death hath paid a fum, inclu-
ding his marriage-tax, amounting to 11 years
rates, or L. 60. Now as contributors, who are ad-
mitted into the church or univerfities at the
mean age of 27, cannot be fuppofed to marry
fooner, at a medium, than the third year of their
incumbency, or in the 30th year of their age,
we may conclude that the propofed deductions
will take place only with refpect to the families
of thoſe contributors who fhall happen to die in
the third, and in any fubfequent year before the
12th of their incumbency, or in the 30th, and
in any fubfequent year before the 39th of their
age. Now, by Dr Halley's table, out of 16669,
the number of perfons alive of 27 years of age
and upwards, 76 die annually in the 30th to the
38th year of their age, both included; according
Hh 2
to
244
OF THE WIDOWS SCHEME.
1.5
to which proportion, 4.386 contributors will die
annually in the 30th to the 38th year of their age,
both included, out of 962 miniſters and profef-
fors. But parts fewer of contributors die
yearly, than according to the proportion of the
above table, and _44 parts of thoſe who die, leave
29.5
29.5
neither widows nor children; which 5.9
29.5
parts of
4.386, fubtracted from the fame, leave
3.509 the
number of contributors dying yearly in the 30th
to the 38th year of their age, leaving widows or
children; each of whom have paid, including
their marriage-tax, from 3 to 11, that is 7 years
rates, making in all 24.56 L. 128.94; but they
ought to have paid 3.509 × 60L. 210.54; there-
fore, fubtracting one from the other, there re-
inain L. 81, 12 s. the yearly deductions.:
However inconfiderable theſe deductions may
appear, yet of fuch confequence were they in the
early periods of the fund, that an act of parlia-
ment was obtained, anno 1748, authorifing the
fame, in order to prevent a deficiency in the ca-
pital of L. 10,000, which was calculated to hap-
pen about the year 1771, which defect they have
prevented. At the fame time it is provided, that
every widow fhall have at leaſt one half of her an-
nuity for fubfiftence until the deficiency be made
good, and then enter on full payment.
ART. 8. Reſpecting the annual expences of ma-
nagement.
92. In the first draught of the fcheme, which
extended only to the minifters of the church,
theſe expences were limited to L. 239. In the two
acts
OF THE WIDOWS SCHEME.
245
acts of parliament eſtabliſhing the fund, where-
in the profeffors in the four univerfities were alſo
comprehended, the annual expences were fixed
at L. 250; namely L. 210 for the falaries of the
general collector and clerk, and L. 40 for the an-
nual expences of the truſtees. By the late act of
parliament anno '1778, the falaries of the collec-
tor and clerk are the fame as formerly, but the
incidental expences of the truſtees are augmented
to L. 70, with a provifo that they do not in any
one year exceed that fum. Which expences of the
truſtees, in imitation of Dr Webſter, are not
brought as a charge upon the fund, from the fup-
pofition that the intereſt received by the collec-
tor in any one year, doth fo far at leaſt overba-
lance the intereſt of the capital for that year,
computed at 4 per cent.
ART. 9. Refpecting the capital of the fund.
93. Had there been no capital, the rates being
adjuſted to the annual demands upon the fund,
it is evident that during the first 14 years of the
ſcheme the annual rates would have been too low,
and that afterwards, till the number of annuitants
arrived at a maximum, they would have rifen gra-
dually to L. 8.5 at a medium, which is out of all
proportion to the value of an annuity of L. 20. By
the act of parliament eftablishing the fund, anno
1744, when the maximum of annuitants was fup-
pofed not to exceed 321, the capital was limited to
L. 63,860, including the loans of L. 30 lent to con-
tributors; the intereft of which, together with the
annual rates and taxes, was thought fufficient to dif
charge all dentands upon the fund. By the act of
parliament, anno 1748, when it was found that the
number
:
246
OF THE WIDOWS SCHEME.
!
number of annuitants would exceed 364, the ca-
pital was extended to L. 80,000, including the
loans of L. 30, granted to contributors at 4
per cent.
And by the late act, anno 1778, when
it was fuppofed that the number of widows might
rife to 400, the capital is allowed to be raiſed to
the fum of L. 100,000, to be lent out upon he-
ritable fecurity, hitherto at 4 per cent.; and
the loaus of L. 30 to contributors are prohibi-
ted, which of confequence will all revert to the
capital about the 95th year of the fund. And in
order to fecure the rife of the capital to L. 100,000,
the trustees are directed to take care that the
fum of L. 200, together with the favings and
furpluses of intereft, annual rates, &c. that theſe,
after all demands upon the fund are diſcharged,
be applied annually for that purpofe, by lending
out the fame, with the advice and confent of the
learned judges who prefide in our courts of law,
until the capital is made up. When it is comple-
ted, there will be an overplus, reckoning intereft
at 41 per cent. after paying all demands upon the
fund, of about L. 1400 yearly; of which L. 150
muſt be referved as a provifion for thofe 18 wi-
dows who are ftill to come upon the fund: as for
the remainder, the church muft confult and de-
termine in what manner it ought to be difpofed
of, when the capital is raiſed to L. 97,000.
94. ART. 10. To trace the progreſs of the capital
from L. 87751.522, in the 45th year of the fund,
till it arife to L. 100,000. Here,
A
Art.
OF THE WIDOWS SCHEME.
247
Art.
2.
3.
2.
હું
4.
5.
6.
The number of contributors, being
r
from
955 to 962.
of perfons liable to the mar-
riage tax,
of vacancies,
35.
49.
of annuitants, including heirs,draw-
ing full, befides 10 drawing half
annuities, being from 319 to 362.
of families of children of the firſt
order, being 6, = L. 1200.
of families of children of the fe-
cond order, being 1.6, L. 160.
7. Deductions from the provifions of widows and
children,
L. 81.6.
8. The falaries of the collector and clerk, L. 210.
1. The medium of annual rates, L. 5.25; and of
annuities, L. 20; and the rate of intereft up-
on the capital 4 per cent. The progrefs of
the capital may be reprefented by the fol-
lowing table.
I. II
III.
IV.
V.
VI.
VII.
1789 46
1790 47
1791 | 48
1792 49 1040
1793. 50
1041
1794 | 51
1042
1796 | 53 1043 5475.75 332 8228.4
Rates. their prod. Ann. their exp.
1039 5454.75 319 7968.4
1039 545475 322 8c28.4
5460. 324 8068.4
5465.25 326
5470.5 328 8148.4
1795 52 1043 5475.75 330 8188.4
87,751.522
88,747-932
89,724.2
90,704.768
810-4
91,689.809
92,579.5
93,674.03
94,668.34
1797 54 1044 5481.
334 8268.4
95,667.67
1798 55 1944 5481.
336 808.4
96,666.97
1799
56
1044 548!.
338 8348-4
97,666.25
1800 57
10455486.25 340 8388.4
98,670.75
1801 58
1045 5486.25
342 8428.4
99,675.43
1802 59 1045
586.25
343
343
8448.4
100,700.3
1803 60 1045
5486.25 | 343
343
8448 4
84484
101,762.16
1804 61
61 1046 5491.5
344
8468.4
102,855-75
In
248
OF THE WIDOWS SCHEME.
1
In this Table, Column VII. fhews the amount of
the free ſtock at Whitfunday for each correfpond-
ing year of the fund in Col. II. the intereſt of
which, computed at 4 per cent. together with the
fum of the rates and taxes of the following year,
Col. IV. is added to faid ftock, and their fum, di-
miniſhed by the charge upon the fund, Col. VI,
is the free flock of the following year. Col. III.
fhews the number of rates and taxes due at Can-
dlemas, for each year of the fund; and Col. V. the
number of annuitants, including heirs, drawing
full, befides 10 drawing half annuities at the term
of Whitfunday yearly.
I
Ex. To ſtock at Whitfunday 1801,
Add intereft to Whitfunday 1802,
And 1045 rates and taxes,
L.99675.43
3987.02
5486.25
Total,
L. 109148.7
Deduct annuities diminiſhed by de-
ductions, Art. 7.
= L. 6878.4
Childrens proviſions
and falaries,
11
1570.0
8448.4
Stock at Whitfunday 1802,
11
L. 100700.3
Thus, about the 61ft year of the fund, the capi-
tal will be made up, the number of contributors
will be complete, and there will be 18 widows
ftill to come upon the fund in the courfe of 60
years.
ART. 11. Refpecting
Refpecting alterations upon the
fcheme.
97. 1ft, It is required, to find what capital
would
OF THE WIDOWS SCHEME.
249
would be neceffary to fupport the fund upon its
prefent footing, when the annuitants are arrived
at a maximum, reckoning intereſt at 4 per cent.
From the widows annui-
ties,
=
Childrens provifion and fa-
laries,
Their fum, ==
L. 7340
1570
8910
5573.I
L..
83422.5, the
Deduce rates, taxes, and
deductions,
Divide by
.04) 3336.9 $3 Capital.
2d, It is required, to find what diminution
might be made upon the annual rates, fuppofing
the annuities to remain the fame as at prefent,
when the capital is raiſed to L. 100,000, reckon-
ing intereſt at 4½ per cent.
2
To 357 annuitants drawing
full,
L. 7140
Add 20, including heirs,
drawing half annuities, = 200
Childrens provifion,
And falaries,
1360
280
•
Their fum,
8980
Deduce intereſt and deduc-
tions,
4581.6
L.
annual furplus.
Divide by rates and taxes,
Here, 1046 X 5.25 — 4398.4 = L. 1093.1, is the
1046) 4398.4
(4.2, Annual
40.
Rate.
?
I i
3d,
{
250
OF THE WIDOWS SCHEME.
3d, It is required, to find what augmentation
might be made upon the annuities, the annual
rates remaining the fame as at prefent, when the
capital is raiſed to L. 100,000, and reckoning in-
tereft at 4 per cent.
From intereft, rates, and de-
ductions,
Deduce falaries,
10073. I
280.0
L.
Divide by annuitants and
children,
435
:)
9793.I
(22.51, the
Annuity.
ART. 12. Containing a general view of the
fcheine.
96. Ift, It is peculiar to this fcheme, that the
annual rate varies, not with the age of a contri-
butor, but only in proportion to the annuity;
that every intrant who is married, or afterwards
marries, is liable to a marriage-tax; that its con-
ſtruction is built upon the principles of the expec-
tation of a minifter's life, which, being above 34
years, including the vacancy, with a high rate
of intereft, depreffes the rates as much as poffible;
that the widow's annuity begins to run, from a
full twelvemonth after the laft payment of the huf-
band; that widows who happen to marry are de-
prived of the whole, or of part of their annuities;
and it is owing to this fcheme that the diftinction
of families of children of the fecond order was
ever heard of.
2d, As to the difadvantages attending this
fcheme, if there be any, they are fuch as are in
common with all laudable undertakings of this
kind.
1
OF THE WIDOWS SCHEME.
251
kind. Thus the leading principle of fuch a
ſcheme not being justice and equity, but charity,
mercy, and compaffion for the indigent; the
more you pay, the lefs is received, and the leſs
you pay, the greater is the value of the annuity.
As all intrants are obliged to contribute for the
fupport of the fund, this circumítance, though
an advantage to the fcheme, is often a hardship
upon the individual, who is thus obliged to pro-
vide for a widow, when poffibly he may have no
thoughts of taking a wife. As heirs in general
diftinct from widows and children have no inte-
4
reſt in the fund, the contributions of
30
parts of
thoſe who die yearly are loft to them and their
heirs for ever, if what is given for the fupport of
the widow and fatherlefs can be called a lofs. The
depriving a widow of her annuity in confequence
of a ſecond marriage, may be confidered as a dif-
advantage attending the fcheme, not from a prin-
ciple of juftice or charity, feeing every woman
who marries is fuppofed to be fupported by her
huſband, but in a political view, as the increaſe
of fociety is thereby fo far difcouraged. And the
capital of the fund, like all other fums of money
laid out at intereft, is liable to a diminution in
its value, from the poffible fall of the intereft of
money. But I infift, with much more pleaſure,
3d, Upon the advantages attending the ſcheme;
as hereby, in common with other inftitutions of
the fame kind, fo many widows and orphans re-
ceive the means either of fubfiftence or of educa-
tion, which muft prove a benefit to contributors
themſelves, who, in confequence of the fcheme,
are enabled to marry with greater advantage than
they
I i2
252
OF THE WIDOWS SCHEME.
they could have done had it not taken place. The
annual rates are exceeding low, on account of the
advantages which the fcheme itſelf enjoys; as, 1ſt,
All entrants being ſubject to the ſcheme, they en-
ter early, and of confequence have a chance of
continuing long contributors to the fund. 2d,
Bachelors and widowers being liable to the pay-
ment of the annual rates, the heirs of a ſeventh
part of thoſe who die yearly derive no advantage
from the fund. 3d, Anns and vacant ftipends,
where there is no wafte in the life of the incum-
bent, are fubjected to the payment of half
years
rates. 4th, There are a number of taxes and de-
ductions, fee Art. 1. which are not fuppofed in
the calculation of the rates. 5th, Should a con-
tributor fail in the payment of his rates, which
are privileged debts, there is a vifible fund, name-
ly, his ftipend or falary, out of which the ſcheme
may be reimburſed for all loffes occafioned by
him; and fhould this fail, the collector hath 26
chances out of 30, of recovering at 5 per cent.
fimple intereft, all arrears due by a contributor,
out of his widow's or childrens provifion. 6th,
The rife of the capital to a certain amount, which
is lent out upon heritable fecurity, and its re-
covery, fhould it fuftain any material lofs, are
both fecured by act of parliament. 7th, The
number of contributors is fixed and determined,
being limited to the number of benefices in the
church and offices in the four univerfities; fo that
the truſtees are enabled to forefee what demands.
are likely to be made upon the fund from year to
year. 8th, The ſcheme itſelf hath undergone fe-
veral alterations and improvements, and may fill
undergo more by favour of the Britiſh legiflature.
9th, The whole concerns of the fcheme are under
the
OF THE WIDOWS SCHEME.
253
the direction and management of truſtees, moſt
refpectable for their knowledge, experience, and
perfeverance in the affairs of the fund; namely,
the Reverend Minifters of the prefbytery, and
learned Profeffors in the univerſity of Edinburgh,
together with the minifters in the prefbytery feats,
or who are admitted to offices in the univerfities,
if they choſe to accept, and delegates from pref-
byteries and univerfities, who, did they all con-
vene, would form a general court of directors, as
refpectable for their numbers and importance as
the Venerable Affembly itſelf.
Therefore, I cannot conclude without exprefs-
ing my reſpect and gratitude to the Reverend and
learned Truſtees, who are appointed to manage and
direct the operations of the fund, and thanking
them for their fidelity and unwearied application
in conducting the concerns of a ſcheme, whereby
the widow's heart is made to fing for joy.
SUP-
SUPPLEMENT to CHAPTER IX.
A
DISSERTATION
RESPECTING
The DISPOSAL of the SURPLUS in the Fund of the WIDOWS
SCHEME, when the Capital is raifed to L. 100,000; addref
fed to the Truſtees for managing faid Fund.
Mr PRESES,
99.
THE queſtion now under confideration,
reſpecting the difpofal of the furplus in the fund
of the widows fcheme, is of a fimilar nature with,
but of a different tendency from that determined
by the church, anno 1748, and of confequence
ought to have a ſimilar though oppofite determi-
nation. In the year 1748, the Reverend Dr A-
lexander Webſter, the principal contriver and
promoter of the widows fcheme in the Church of
Scotland, diſcovered by calculation, that there
would be a deficiency in the capital of the fund, of
L. 10,000 about the year 1771. What was then
to be done to prevent this deficiency? Muft the
rates be raiſed? No; Muft the annuities be dimi-
nifhed?
OF THE WIDOWS SCHEM E.
255
nifhed? No; Touch not, would he fay, either of
thefe pillars of the fcheme, left by fapping and
undermining the foundation, you make the whole
fabric tumble about your ears. What then? Why,
the Doctor did the only thing which he ought to
have done; he did an act of juſtice, by obliging
every contributor to pay at leaſt three years pur-
chafe of the annuity correfponding to the rate to
which he was fubjected, before his widow or chil-
dren could be entitled to their refpective provi-
fions.
In the prefent cafe, when the capital is raiſed
to L. 100,000, the annual furplus may be about
L. 1473, reckoning intereft at 44 per cent.* What
must be done to prevent the farther increaſe of
the capital? Shall we diminiſh the rates in general?
or fhall we increaſe the annuities? Either, or both
of thefe would be defirable, did we not thereby
perpetuate a very great hardſhip upon aged con-
tributors, by continuing the payment of the rates
till the clofe of life; when we have it in our pow-
er to redreſs it.-But,
To find how this matter may be determined to
the fatisfaction both of contributors and annuitants,
by giving each party a fhare of the furplus, nearly
in proportion to their numbers. There may be
435 annuitants upon the fund, when they arrive
* When the capital is complete, an. 1802, there may be 338 an-
nuitants drawing full, and 20, including heirs, half annuities;
Hence to rates, taxes, and deductions,
Add the intereft of the capital at 4 per cent.
L5573
4500
Their fum
10073
To the fum of annuities paid to widows,
Add chil. prov. and expence of management
L. 696of
1640
8600
Their difference is the annual furplus
11
L. 1473
at
256
OF THE WIDOWS SCHEME.
1
at a maximum, including both orders of children,
which are equal in expence to 68 widows; and
the number of contributors alive at one time.
is 962.
Now it is obvious that annuitants of all deſcrip-
tions, whether old or young, are equally the ob-
jects of charity, and of confequence, ought to be
benefited only by the plan of augmentation; fup-
pofe by 5 per cent, or by 1 fhilling upon the pound
of their annuities; this will exhauft L. 435 of the
above furplus. Again,
With respect to contributors, in the relation
they ſtand in to the fcheme, by the payment of
the rates, &c. take the following table.
Years of minr.
From o to 12
12 to 28
74
Contributors.
3.6 fail who pa
Amount at 4 per cent.
yrs pur.
67.5 for an annuity worth 15.0
149
13:5
IC.O
;24
14.0
132
605
10.0
28 to 65 | 18.45•
o to 33 15.0
33 to 55 15.0
By this table, out of thoſe who are alive, during
the first 12 years of their miniftry, 3.6 fail yearly,
who, allowing 5s. for expence of management,
have paid at a medium, including deductions, the
fum of L. 67.5, for a L. 20 annuity worth 15
years purchaſe. It would therefore be highly
improper to grant a diminution of the rate during
this ſtage of life, not only on account of the fmall
value paid for the annuity, but alſo becauſe the
more the rate was diminifhed, the greater would
be the deduction out of the widows provifion.
In like manner, in the cafe of thofe who are alive
from the 12th to the end of the 28th year of their
miniſtry*, when the full value of the annuity is
S
252
* To find the time in which a rate of L. 5 per annum, allowing 5 s.
for expence of management, will diſcharge the augmented annuity of
L. 21, worth 12 years-purchafe, at 4 per cent.
correfponding to 28.13 years.
50.4 N
B
5
difcharged
OF THE WIDOWS SCHEME.
257
diſcharged, it would alfo be unneceflary to diminiſh
the rate during this period, becauſe contributors,
have not paid the full value of the annuity. But
in the cafe of thoſe who are alive from the 28th
year of their miniftry to the clofe of life, it is
manifeft that, during this interval, there ought
to be a total exemption from the payment of the
rate, as far as the above furplus of 1473-435=
L. 1038 will admit; feeing thefe contributors
have paid, at a medium, the fum of L. 324 more
than the value of the annuity. Therefore,
I call upon you, Mr Prefes, I call upon the
church in general, ſeeing you have it now in your
power, to do an act of juſtice to our aged brethren,
who have borne and ſtill will bear the burden in
the heat of the day, by fufpending the payment
of the rates after a certain age, or rather, by ex-
empting aged contributors after a certain num-
ber of annual payments, including the marriage-
tax, be that number what it will; which number,
could I depend upon the only table of obfervations
which approaches near to the better fort of lives
in Scotland, might be 30*, including the mar-
riage-tax; upon the fuppofition that each contri-
butor thus exempted was obliged to pay,
of vaffalage, 5 per cent. or 1 s. upon the pound,
in name
*It is required, to find, if L. 938 of annual furplus will admit of
a fufpenfion of the rates beyond 30 annual payments, including the
marriage-tax. Suppofing 30 the mean age of admiffion, there are a-
live, by the Fife table of obfervations, at the age of 30 years,
358-2-356, and at 60 years of age, 211-4.5=206.5, whoſe ex-
pectation of life is 12.98; and there are admitted into the church and
univerſities 29.455 contributors yearly. Hence,
356 206.5 29.455: 17.08, and 12.98X17.08-222, alive at 60
and upwards. Hence, 222 X 4.25 = L. 943.5, the fum remitted to
theſe contributors.
Kk
of
258
OF THE WIDOWS SCHEME.
of the annuity correfponding to the rate to
which he was fubjected, and that L. 100 of fur-
plus was reſerved, in order to co-operate with the
favings upon the rates and annuities, till the ex-
emptees and annuitants arrived at a maximum,
in forming an additional fund of L. 9000, to dif-
charge the annuities of thofe 18 widows who are
ftill to be provided for, as appears from the fol-
lowing
TABLE.
Expect. Total alive.
29.455 × 32.66 = 962.
25.85
=
X 24.29 628.
18.45 X 14.00 = 258.
17.79 X 13.50 = 240.
17.08 X 12.98 = 222.
Age.
Rate.
Living.
30
42
12
58
28
59
29
60
30
61
31
62
32
63
33
=
16.34 X 12.53 205.
15.60 X 12.10 = 189.
14.85 × 11.66 = 173.
If the capital be allowed to rife to L. 109,000, as
fuggeſted above, by the accumulation of L. 150
per annum, without intereft, during the ſpace
of 60 years, when the annuitants will arrive at a
maximum: in this cafe, the above furplus of
L. 1038, appropriated to contributors, with a tax
of 1 s. per pound of the annuity, will reduce the
years of exemption to 28, when a contributor
hath paid the full value of the augmented an-
nuity, and when the fourth part of thoſe alive at
I
one
OF THE WIDOWS SCHEME.
259
one time in the church will be exempted annu-
ally. Moreover,
If the annual furplus fhall happen to rife con--
fiderably above L. 1000, both plans of augmen-
tation and exemption may be uſed to advantage;
but ſhould it not exceed or fall below that fum,
the plan of exemption ought alone to be employ-
ed. Now, that fuch a plan ought to be ad-
opted, in preference to all others, will appear,
ift, From its being founded upon a principle
of juftice. In fuch a Widows Scheme as this, we
acknowledge that charity, mercy, or compaffion
for the indigent, are the leading principles: there-
fore, as long as it is neceffary for the exiſtence of
the ſcheme, rates ought to be paid till the cloſe of
life; but when this hardfhip, as in the prefent
cafe, can be removed, and juftice mixed with
mercy, it ought to be done. With what confi-
dence can we prefume to diminiſh the rates in ge-
neral, or augment the annuities, whilft we leave
this burden upon aged contributors unremoved?
A grievance which, remaining unredreffed, muſt
blaft the reputation of every other plan, at leaſt
in the opinion of every difcerning perfon. Befides,
every one who is exempted after 30 annual pay-
ments of the rate, will have contributed a fum,
viz. L. 295, whofe intereft, at 4 per cent. is with-
in a trifle of L. 12 per annum; whilft the other
contributors, from 30 years and under, will have
only paid, each of them, at a medium, a fum,
viz. L. 124, whofe intereft, together with the an-
nual rate, does not exceed L. 10.2 per annum:
fo that the former benefit the fcheme as much af-
ter an exemption, as the latter do with the pay-
ment of the rate.
K k 2
zd,
260 OF THE WIDOWS SCHEME
1
2d, From its poffeffing the beneficent proper-
ties of all other plans. It prevents the accumula-
tion of the capital beyond a limited fum; it aug-
ments the annuities with L. 435, and diminiſhes
the rates to the extent of L. 1038 per annum,
and that too moft effectually where there is moſt
occafion for a diminution; and as a penny faved
is fo much gained, it may be faid to augment the
annuity, in the cafe of exemptees, by the va-
lue of the rate during the life of the contribu-
tor. Moreover, was it thought adviſeable to cre-
dit the truſtees with the fmall difcretionary pow-
er of augmenting or diminiſhing the years of ex-
emption, according as it ſhould be found in fact,
that the number of annuitants or of exemptees
exceeded or fell fhort of the calculation, or that
the intereſt of money was below 4 per cent.;
this plan might keep the affairs of the ſcheme up-
on fuch an even footing, as to prevent all future
applications to parliament for redrefs of grievan-
ces, which cannot fail of being both expenfive
and troubleſome; whilft the loffes fuftained by
bad debts or otherwife are effectually guarded a-
gainft by the law now in force.
2
3d, From the fimplicity of its conftruction and
operation. Let any one attempt to diminiſh the
rates or augment the annuities, he will find more
difficulties than poilibly he expected, and will be
in danger of unhinging the whole frame of the
fcheme. But by this plan, the annuities are paid
in guincas, in place of as many pounds Sterling,
and the rates remain entire as from the com-
mencement of the fund, which ftill enjoys the full
benefit of deductions and of half-years rates out
of
1
2
OF THE WIDOWS SCHEME.
261
of anns and vacant ftipends; fo that the affairs of the
ſcheme, in general, proceed in their uſual courſe,
which cannot fail of being a great advantage, in
point of eafe, both to the truſtees and general
collectors. The only calculation necefiary will be
to determine the years of exemption; for which
purpoſe had we a table of obfervations adapted to
the lives of minifters, this might be done with ac-
curacy; if not it may be diſcovered from fact,
with this caution, that the number of exemptees
will not arrive at a maximum till about the 85th
year of the ſcheme. But let none object to this
plan,
ift, That I am in danger of lofing the concur-
rence of the younger members of the church,
who form the greater part of the fociety. No; I
will caſt myſelf upon their generofity: Dr Web-
fter did fo at the commencement of the fcheme,
and was not diſappointed. Had it not been for their
difintereſted conduct, the fcheme would never
have taken place; nor can I fuppofe that, in the
courſe of half a century, this public fpirit fhould
be degenerated in the younger members of the
fociety. But I afk lefs than the Doctor did; I
only aſk an act of juſtice towards their aged bre-
thren, whilſt the youngeſt in the church have at
leaft a chance of reaping the fame benefit with the
oldeſt, and many may there be who fulfil the
30th year of their miniftry, before which they
have no caufe to complain, not having diſcharged
the full value of the annuity. Nor,
2d, That this plan comes from an obfcure cor-
ner of the church, and from an inconfiderable
hand.
262
OF THE WIDOWS SCHEME.
[
hand. No; However inconfiderable the author
of this differtation may be in point of mental a-
bilities, he is yet one who hath been deeply enga-
ged in the ſcience of annuities of all defcriptions,
and in particular hath given fuch an illuftration
of this very ſcheme, as he trufts will ſtand the
teft of the moſt impartial examination. But how-
ever this be, he is one who will yield to none in
regard for his country, and in his good wishes
for the profperity of the Church of Scotland; in
evidence of which, he hath taken this method
of being ſerviceable to the church, in the reſolu-
tion of a queſtion which may come to be deter-
mined, when poffibly this very hand is laid in the
duft.
CULTS,
Otr 18. 1792. S
2.}
DAVID WILKIE.
JUL 12 1920
}
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