Gordon Thomson 1882-1950 I ^ y Similarly (1+i)" being the amount of 1 in 7i years, 1 is the present value of (l+^)^ due at the end of n years; and , .^,. ^ or by equation (12), i;"-, is the present value of 1 due at the end of n years ; whence, S being a sum due and P its present value, F=8v^ (13) Equation (13) is equation (9) in different symbols. It shows us that Principal and Present Value, and Amount and Sum Due, re- spectively are synonymous, and leads us to the definition that the Present Value is that sum which, invested now, will, at the end of the period, have accumulated exactly to the Sum Due. 11. Because (l-f*)^ consists of the unit which was originally invested, together with its accumulations of interest during the n years, therefore the accumulated interest alone on 1 in ?^ years is {(H-»)»-l}. 12. We have seen, Art. 9, that when the unit of time is a year, and when % is the interest on 1 in a year, then the amount of 1 in a year is (1+i), and in n years (1 + i)'*. The same principles hold if we change the unit of time. Suppose, then, we call the m** part of a year the unit of time, and the interest on 1 in such unit -, so that i becomes the nominal annual rate of interest. The amount 1 + — I , and in (% \™^ \-\ — j , and the interest actually realised on 1 in a vear, or the effective rate of interest will be CHAP. I.] On Interest, 185 Interest under these circumstances is said to be convertible m times in a year. An illustration of the difference between the effective and the nominal rates of interest, between ^("') and i, is afforded in the case of consols. The nominal rate, i, is 3 per cent., or -03; but the interest is convertible twice a year, and the effective rate, i^™), — in this case i(2) — is {(1-015)2-1}, or -030225. The higher the rate of interest, and also the more frequently interest is convertible, the greater is the difference between the effective and the nominal rates. Thus, if interest be at the rate of ten per cent., and, consequently, i = -1, then i^^) _ .1025,- tW = . 103813, and i(8) = . 104486. 13. In theory there is no limit to the magnitude of m. If m become infinite, and interest be convertible momently, we have still tlie interest on 1 in a year J (14-—) — 1 ^- ^^ ^^^^ case the nominal rate of interest has the special symbol 5 assigned to it, and we write, where i is the effective rate, i. {(,+!)■-,) ..... m But by the theory of logarithms, when m increases without limit has e^ for its limit, where e is the base of the Napierian (■+0 system of logarithms, and is equal to 2-7182818. . . . We there- fore have i={e^-l) (16) andS=log.(l+i)J-^f^M .... (17) From this point of view 8 is called the Force of Interest^ and is the nominal yearly rate of interest when the effective rat^is i. 14. The following Tables A and B afford numerical illustration of the difference between the nominal and the effective rates of interest. Table A shows for various nominal rates the correspond- ing effective rates when interest is convertible half-yearly, quarterly, or momently, and, conversely. Table B gives for the efi'ective rates the corresponding nominal rates. ft 86 Theory of Finance. TABLE A. [chap. I. Effective Rate Nominal Interest convertible Kate. Half-Yearly. Quarterly. Momently. •03 •030225 •030339 •030454 •035 •035306 •035462 •035620 •04 •040400 •040604 •040811 -•045 •045506 •045765 •046028 •05 •050625 •050945 •051271 -•06 •060900f •061364 •061837 •07 •071225 •071859 •072508 •08 •081600 •082432 •083287 -•09 •092025 •093083 •094174 '•1 •102500 •103813 •105171 TABLE B. Effective Rate. Nominal Rate. Interest convertible Half-Yearly. Quarterly. Momently. •03 •035 •04 •045 •05 •06 •07 •08 •09 •1 •029778 •034698 •039608 •044504 •049390 •059126 •068816 •078462 •088062 •097618 •029668 •034552 •039412 •044260 •049088 •058696 •068232 •077708 •087112 •096456 •029559 •034401 •039221 •044017 •048790 •058269 •067659 •076961 •086178 •095310 15. It should be observed that in Arts. 9, 10, and 11, we have used the symbol i in a general sense, for the total interest earned on 1 in a year, and that we have not restricted it to the special case where interest is convertible yearly. From the definitions of the 1+—) =(lH-i("»))'S and unless it be desired to specially emphasize the fact that interest is convertible in some particular manner, the afiix (m) may be omitted, and % may CHAP. I.] On Interest. 187 be taken to represent the interest actually realized at the end of a year by the investment of 1. Looked at broadly from this point of view, all the equations in Arts. 9, 10, and 11 remain true : but 80 far we have discussed only integral values o-f n. 1 6. If i be the interest on 1 for one year, what is the interest for the m'* part of a year % This question at one time created a great deal of warm controversy among actuaries ; see /. /. A. vols. 3 and 4. Some writers maintained that the answer must be - , and the chief m argument they urged in support of their view was that - correctly represents the interest in the wi'* part of a year on the supposition of simple interest, and that under no circumstances should com- poimd interest yield less than simple. On the other hand, rival authorities asserted that the correct expression is {(1+i)^— 1}, and that although this gives a value smaller than -, it is only rieht that it should do so. The interest m is not due till the end of the year, and if the lender receive it sooner he must be content with less, because, compound interest being supposed, he can invest his interest for the remaining portion of the year and realize interest thereon. Also, the same authorities submitted that — is palpably wrong, because if 1 at the end of the nil w** part of a year amount to f 1 +— I, it must at the end of two such parts amount to f 1 H — J , and at the end of a year to \\-\ — J , or to ( \-\-i-\-—^ — i2_|_etc. J, which is contrary to the fundamental axiom that % is the interest on 1 for a year. They therefore advocated the principle that in theoretical investigations the equation ^=P(l-f-i)" must be held to be universally true whether n be integral or fractional ; and in recent years the majority of mathematicians have adopted their view. If — be considered to be the fractional interest, great complica- tions must sometimes be introduced into formulas. See for instance the Treatise on Annuities by the late Griffith Davies, page 81, and Milne on Life Annuities, pp. 13 to 15, where not only are the expres- sions very complex, but three cases must be studied in the solution 1 88 Theory of Finance. [chap. i. of one problem. If on the contrary we adopt {(l-|-^)'«~ — 1}, these difficulties vanish. Our expressions become elegant and compact, and only one case presents itself for examination instead of three. See Baily's Doctrine of Interest and Annuities, articles 78 and 79, where the same example is discussed as we have already referred to in the works of Davies and Milne. It must be remembered, however, that in commercial transactions the interest on 1 for the m*^ part of a year is taken as — ; and even in actuarial formulas it is sometimes found convenient for purposes of numerical calculation to do the same. The effect of so doing is to omit the powers of i above the first in the expansion of (1+i) "» ; and, as the quantity i is always very small, it is evident that there is but little practical difference between the two ways of dealing with fractional interest. That way should be adopted which may be the more convenient for the purposes in hand. 17. By definition, the Discount is the difference between a sum due and its present value ; that is d=l-v (18) =Wi <^^> =«;i . . . . . . . . (20) 18. Equation (20) shows us that the discount is the value at the beginning of a period of the interest to be received at the end ; but this fact could have been ascertained by reasoning from first principles, without the aid of algebraical transformations. By Art. 10, 1 is the present value of (1+^) to be received at the end of a year ; that is, 1 is the present value of the original unit, together with the present value of the interest upon it ; and therefore the difference between the unit and its present value is equal to the present value of the interest. Looked at from this point of view then, the discount is the interest paid in advance. This result is of considerable importance in connection with life annuities and premiums. Equation (20) also tells us that 6^ is a year's interest on v. 19. From equation (18) we immediately deduce v=l-d ....... (21) I+i= ' l-d Therefore if the rate of discount as distinguished from the rate f CHAP. I.] On Interest. 189 of interest be named, we find the present value of a unit due at the end of a year by subtracting from the unit the rate of discount. 20. In this connection it should be noticed that, when a merchant seeks to discount a bill, his banker quotes to him the rate of dis- cmmfj not the rate of interest; and when the Bank of England Directors fix their rate, it is the rate of discount they determine, not the rate of interest. Through failure to keep this distinction in view, confusion has sometimes arisen. It has been usual to say that commercial discount diff'ers from theoretical discount, in that when the rate is, say, 5 per cent., the banker deducts 5 from his 5 customer's bill of 100, instead of j^, as his critics say he ought to do ; and some writers have even insinuated dishonesty on the part of the banker. Baily, for instance — Doctrine of Interest, etc., chap. iii. — remarks that the course "is neither correct nor just." But if the banker says that his rate of discount is 5 per cent., the merchant cannot grumble. The banker merely assigns a value to d from which i may be found. It is true that, at the same nominal rate, money improves faster under the operation of discount than of interest. If a banker can employ his funds in discounting at 5 per cent., it will not be to his profit to grant advances at 5 per cent, interest ; and if a merchant sell his bill to the banker at 5 per cent, discount, he must remember that he is paying more than 6 per cent, for the accommodation, but with his eyes open to this fact he suffers no wrong. 21. It is not often in business transactions that discount has to be calculated for more than a year. In fact, the great majority of commercial bills have only a fractional part of a year to run. If that fraction be denoted by — , and if d be the yearly rate of dis- count, it is customary for the banker to give for. each unit of the bill ( 1 ), and not (l— f^)"^, thus using that which by analogy may be called "simple discount." If ''simple discount" be em- ployed for periods greater than a year, erroneous and anomalous results are produced. Thus, if the bill to- be discounted have n years to run, its value at simple discount will be {l—7id), and it may very well happen that nd is greater than unity, and gives the bill a negative value. This is not because " commercial discount " differs from "theoretical discount," but because we have used " simple discount," and to simple discount the objections men- tioned in Article 4 apply equally as to simple interest. The V (>-0 190 Theory of Finance, [chap. i. correct formula is (1—^)", by the use of which the anomaly dis- appears. 22. The operation of discount is similar in its results to the operation of interest, and just as we have "compound interest," we may have, by repeating discount operations, " compound discount." W{ien a bill matures, the banker may at once employ the proceeds in discounting a new bill. If d be the nominal yearly rate of discount, and if the process of discounting be repeated m times in a year, the discount on 1 in the m!^ part of a year will be — , ai\d the^ value of 1 due at the end of the m*^ part of a year will be Repeating the operation, we have the present value of / ^\^ 1 due at the end of two m''* parts II j , and the present value 1 J . In this expression we may make m as great as we please; but by the exponential 1 I has e-^ for its ?«/ limit. Also, when discounting is performed momently, d, the nominal rate of discount, is written 8. We therefore have v=e-^ . (23) From this point of view 5 is called the Force of Discount. The term " force of discount " is more commonly employed than "force of interest." 23. By a very simple application of the differential calculus we can form a clear idea of the meaning of the force of discount or the force of interest. The differential coefficient of a function is the measure of the velocity of change in the function consequent on change in the variable. Now, v^ may be considered as the function of discount, dif^ and, taking its differential coefficient, we have -j-—v^ loge^. If now we divide by ^, we have the measure of the velocity of change 1 dv^ in the function for each unit of the function, -^-r- = log^v = -log,(l+i)=-8. In the same way (l+^)^ may be considered as the function of interest, and ^±^=(l+i)- log, (1+i), and ^^~i^^ ll^lll! =log, (1+0 = 5. CHAP. I.] On Interest. 191 The numerical value is the same, but the sign is opposite. The difference in sign shows that the force of discount is a force of decrement, and the force of interest a force of increment. We can define that force as the annual rate per unit at which a sum is increasing by interest or diminishing by discount at any moment of time. The term " force " is a misnomer ; it should rather be " velocity." But "force" having come into general use, a change would be inconvenient 24. The quantities i, v, fZ, and S, are all mutually dependent, and can be conveniently expressed in terms of each other by means of series. Thus, since by equation (22) (i+^)=(i-^)- i^A^^^ - i=(Z-|-(Z2^c?3^etc. ^ (24) l-(^^ Also, since by equation (16), i=e* — 1 ; by the exponential theorem < ^ f^ ^ *=S-f-|2*4-|3+etc. . . ... . (25)^ ^ ., /. . Again, by equation (22), (1 — (/) = (l-f i)-i. Therefore d=i-i^^i^-etc (26) and also (/=l—v ^ ( 8-^ 8^ ) =^-\2+\r''' (27) Since v=\ — d t?=l— i+i2_i3-j-etc (28) and also «;= 1—8+—— -^+etc (29) Finally, because 8 = log„ (1 -j- i), therefore, by the theory of logarithms 8=i-i-+^-etc (30) and since S=— log«2;=— log, {l—d\ therefore 8=(ZH--+-+etc (31) All the series given in this article are rapidly convergent, and when the numerical value of one of the functions is given, they offer great facilities for computing the values of the others. The student will find it an excellent exercise to set himself examples under this head. 192 Theory of Finance. [chap. I. The following Table C gives the values of the several quantities at the rates of interest most commonly in use. TABLE C. i. V. d. 5. •02 •980392 •019608 •019803 •025 •975610 •024390 •024692 •03 •970874 •029126 •029559 •035 •966184 •033816 •034401 •04 •961538 •038462 •039221 •045 •956938 •043062 •044017 •05 •952381 •047619 •048790 •06 •943396 •056604 •058269 •07 •934579 •065421 •067659 •08 •925926 •074074 •076961 •09 •917431 •082569 •086178 •1 •909091 •090909 •095310 25. In how many years will money double itself at compound interest? The answer to this question follows directly from equation (10). Writing 2 for S and 1 for P, we have, using Napierian loo* ^ n2t no logarithms, w=,- — °^'^ .v . But loo-e(l+i) = i— 7^+-^ — etc., and ° ' loge(l+'i) Ce\ 1 / 2 3 ' if we neglect the second and higher powers of i, and write for 69 loge 2 its near value ^69, we have approximately 71=-^. Whence % the common rule : — To find the number of years in which money will double itself, divide 69 by the rate of interest per cent. Mr. G. F. Hardy {Insurance Record^ March 31, 1882) has pointed out that the correction to the approximation given by this rule is very nearly a constant quantity : that is, the error involved is practically the same whatever the rate of interest. Thus log,2 ^693 •693,. _. ...... ^693 log,(l+i) i-li^ + ii3_etc. = ^-(l+i*-Ty^+etc.)=-^+-35 very nearly. With this correction the rule will give a result true usually to two places of decimals. 26. It frequently happens that there are various sums of money due at different times from one merchant to another, and which it is desired to pay all at one time. That time at which they may J L , j^€Aaji<' aJCt^/^'i'f^^ ^C^f-w"'*^^ ~to u^i^ otyT^ aA^f^^-/'''^-t>%n C^Ayttyt^ W^Kn^t.,^^^ -UJ /''^^*[(/^i tn '^'^)- i//^<. V ^ / 1 1 ■■• ..^^ . ....4. . 1 4-^4,-4.,. ^r- -, 1 ! u ■ 1 K-. 1 i 4- CHAP. I.] On Interest, 193 be paid without injustice to either party is called the equated time of payment, or the average due date. 27. If the various sums be S^, *S^„ /S^„ etc., and the times at which they respectively fall due Wi, Wj, n,, etc., to find the equated time, a-, at which the total amount (*S^i+>S^8+'S^8+etc.) may be paid, so that the parties may be on an equality as regards interest. It is evident that justice will be secured if the present value of the aggregate of the sums due at the time, re, be equal to the aggre- gate of the present values of the individual sums, and that to find X we have the equation y ''"- which may be symbolically written ^^ .^ty^^ . . . . . . (33) . (i+i)« (1+0^ If we expand each side of equation (32) by the binomial theorem and neglect all powers of i above the first, we have, after reduction, and division by the coefficient of a:, '^—'Sr+ST+Ss-f^tcr' .... (^4M^ 28. The last equation gives us the usual rule, which is approxi- mately correct : — To find the equated time of , payment of various amounts, multiply each amount by the time to elapse until it will fall due, and divide the sum of the products by the sum of the amounts. 29. If simple interest be assumed, the subject becomes intricate, and, as explained in Art. 4, yields contradictory results if regarded from different points of view. The reader may refer to Todhunter's [\, Algehraf chapter "Equation of Payments." 61^ _ -OV V .^^'^'^\r^^' THEOEY OF FINANCE. CHAPTER 11. On Annuities-Certain. 1. An Annuity is a periodical payment, lasting during a fixed term of years, or depending on the continuance of a given life or combination of lives. The annuity may be payable either yearly or at more frequent intervals, but it is measured by the total amount payable in one year, which is sometimes conveniently called the annual rent. Thus if a person be entitled to receive £25 every three months, he is said to be in possession of an annuity of £100, payable by quarterly instalments. When an annuity is to last during a fixed term of years it is called an annuity-certain. When it depends on the continuance of a given life or combination of lives, it is called a life annuity, or simply an annuity. 2. The word status is conveniently used to denote the period during which an annuity is payable, whether that period be a fixed term of years, or depend for its duration on the contingencies of life. We may therefore define an annuity, generally, as a periodical payment lasting during a given status. 3. A perpetuity is an annuity that is to last for ever. Such are the dividends on consols. 4. Unless otherwise stated the first payment of an annuity is supposed to be made at the end of the first year for which the annuity is to run, or, in the case of annuities payable at shorter intervals, at the end of the first interval. Thus, if we speak of an annuity for n years, we mean an annuity consisting of n yearly payments, the first of which is to be made a year hence : or if we speak of an annuity for n years deferred m years, we mean an annuity consisting of n yearly payments, the first of which is to be made at the end of (m+1) years. This last annuity is said to be CHAP. II.] On AmiMities-Certam. 195 entered m at the end of m years, although the first payment will not be made until the expiration of (/// + 1) years. If the first payment of the annuity be made at the beginning instead of at the end of the first interval, the annuity is called an amiuity-diie. 5. For purposes of investigation we shall always consider the annual rent of an annuity to be 1. Our results will be made avail- able for other annuities by merely multiplying by the annual rent. 6. Let crW s^ =the amount of an annuitv for n Ygars. aijj=the present value of an annuity^ for n years payable, yearly. ' ' rt-^i = do. half-yearly. rt^l= do. quarterly. a^j= do. m times in a year, tt^ = do. momently, that is, a continuous annuity, a^i =the present value of an annuity-due for ?i years so that a,i| = l-f«;;r-i|. ^^=the present value of an annuity for n years, deferred m years. a^=the present value of a perpetuity. Note. — The symbols a,-;^, ^|ft^, and a^, may be qualified by the affixes (2), (4), (m), in the same way as an ordinary annuity. We may also write ^ai;;\ and d^. At Simple Interest. 7. Tofindsni When the payments of an annuity are not taken as they fall due, but are allowed to remain to accumulate at interest, the annuity is sometimes said to be forhoi'ne, and the sum to which the payments accumulate is called the amount of the annuity. Thus the amount of an annuity is the sum of the amounts of its several payments. Therefore, the last pajrment having just been made, the last but one having been made a year ago, and so on, we have, commencing with the last payment, .^„-=i+(i-fi)+(i+2i)+ . . . +(i+77:rii) = y|2 + 7^1i| (1) 8. To find a;;]. The present value of an annuity is the sum of the present values 196 Theory of Finable e, [chap. 11. of its several payments. We therefore have, beginning with the first payment, ''~^=T+i+TT2i+r+3i+ • • • +r+ri • ^^> There is no direct means of summing this series, and to obtain exactitude, each term must be calculated separately. A very close approximation may however be obtained from the formula of the differential and integral calculus given in Art. 10. 9. Several other so-called approximations have been proposed, based on plausible reasoning which however fails when simple interest is assumed. Some of these are as follows : — a. It is evident that the present value of an annuity and the present value of the amount of the same annuity should be equal, for, as regards present value, it should be a matter of indifference whether a person is to receive the annuity during the n years, or the equivalent of the annuity at the end of the n years : therefore a-^ = ^ . , and, substituting for s^ its value found above, /?. Again, the value of the annuity should be equal to the difference between the values of a perpetuity to be entered on at once and another to be entered on at the end of n years. But 1 invested now will yield % at the end of each year for ever : that is, 1 is the value of a perpetuity of %, and therefore the value of a perpetuity of 1 is — , and the value of a perpetuity of 1 deferred n «il =i/l_ M . . . . . (4) y. Again, from the nature of the case, each payment of the. annuity must contain interest on the purchase money, together with a return of part of that purchase money. The purchase money being a^i the interest on it is m^^i, and the return of capital is the balance of the yearly payment, namely (1 — m^). This last quantity constitutes the rent of an annuity which must be accumu- lated at interest in order to replace the capital at the end of the period. That is a^=s^\ (1-iaii): or q '■'-wa |»> ^L CHAP. II.] On AnnuitieS'Certai7i, i97 { K). ilhe formula of the calculus referred to in Art. 8 is as follows : — >V...K=/V^4,..H.r,-ij(g-(f).} +730{(&'^H£).}-"' where V is any function of x and 2^, its finite integral. It is skilfully used by Mr. Woolhouse, J.I. A., vol. xv. p. 100, in the solution of problems in life contingencies, and there he also demon- strates the formula. In the present case Fx= ^-, - ., and the for- mula becomes ^,_log.(l+ni) 1/ \\ +T2(^"(T+myv i2o(^"(iT^)^j • ^^^ Remembering that the modulus M of the common system of logarithms is -4342945, and that log, (l+m)==-^^^^jii^\ the formula is very easily applied, (tr^^-v'^ \% A. ^utju /»^>4^^y » 11. As an illustration of the foregoing formulas, we give, at 5 per cent, interest, the values they bring out for an annuity for twenty years. rt^o", == 13*616068, by direct summation (true value). = 14-750000, by formula (3). = 10-000000, „ „ (4). .' = 11-919192, „ „ (5). ^': = 13-616068, „ „ (6). It will be seen that formula (6) gives the result correct to six l)laces of decimals, but that the others are very wide of the mark. Annuities at simple interest have no practical importance, and the analysis is inserted here purely as a matter of curiosity. Passing to Compound Interest 12. To find s^ The amounts of the several payments form a geometrical pro- gression, with common ratio (1+*), and we have, beginning as before with the last payment of the annuity, 5;^ = i+(i+i)+(i+*y4- . . 4- (1+0^-^ ^ (l+0"-l (7) 13. This result can be easily obtained without having recourse to series. A unit invested produces an ^annuity of i per annum. The amount of 1 in ti years consists of the^ofiginal unit and the M igS Theory of Finance. [chap. ii. amount of the annuity of i for n years which the unit produced ; and therefore the amount of the annuity of * is {(l+^)"— 1}, and, by simple proportion, the amount of an annuity of 1 is ^ "*" : * as before. 14. If the annuity be payable^ times in a year, and the interest be convertible ^ times, to find the amount of the annuity. The. amount of 1 in a year is 1 1 +— j , (Chapter i., Art. 12), and in the jp** part of a year \\-\ — j"^, (Chapter i., Art. 16). The amount of the annuity, beginning with the last payment, is therefore a geo- metrical progression of -pn terms, with common ratio \\-\ — y ^ and we have, each payment of the annuity being — , when ^=^ this becomes . ' . .... . (9) 15. We see that in all cases the amount of an annuity is the fotajj^ fmteres^on 1 during the whole currency of the annuity, divided by the product of the number of instalments per annum into the interest on 1 during the interval between two instalments of the annuity. 16. To find a^ The present value of the annuity consists of a geometrical pro- gression of n terms, the first term and the common ratio both being V, We therefore have a^=?^+v^+v3+ etc+r" 1— «;" (■^ir- l-v (10) 17. By reasoning similar to that in Art. 13 we can arrive at equation (10). A unit paid down is the value of an annuity of % for n years CHAP. II.] On Annuities- Certain. 199 (produced by the investment of the unit for the ternj), and of the original unit to be returned at the end of the term : that is (1 — t;") is the value of an annuity of % for n years, and consequently the value of an annuity of 1 is "T^ . % 18. Also, by means of perpetuities, we can obtain the same result. A unit invested will yield an annuity of i for ever. Therefore the value of a perpetuity of 1 is «»4- ••••••• (11) Now, an annuity for n years is a perpetuity entered on at once, less a perpetuity deferred n years, and its value is a^ (1— 'y**).or i 19. If the annuity be payable p times in a year, and interest be convertible q times, we have (■4)-- 1 and when ^=2 this becomes (12) (13) 20. In all cases the value of an annuity is the flotal d iscount on 1 during the whole currency of the annuity, divided by the product , of the number of instalments per annum and the interest on 1 during the interval between two instalments of the annuity. 21. If the intervals between the payments of the annuity become indefinitely small, so that the payments are made momently, the annuity is said to be continuous. 22. To find the amount and the present value of a continuous annuity, interest convertible momently, we must in formulas (9) and (13) make q infinite,. when i becomes S. We then have, agree- ably with Chapter I. Art. 13, ■ -s^=^ (14) 0* = 8 (15) 200 Theory of Finance. [chap. ii. 23. If, however, the annuity be not continuous, but be payable 'p times in a year, while interest is convertible momently, we must in formulas (8) and (12) make^ infinite, when i becomes 5. We then obtain ^ 6^—1 When the annuity is payable yearly these become ^"^-1 ...... (18) e«-l l-e-w5 (19) 24. Again, if the annuity be continuous while interest is con- vertible g times a year, 'p becomes infinite in the formulas s= In the denominators, p being infinite, there appears the indeter- minate form pX (IH — j^— 1 I, the value of which is not at once evident, but it may be ascertained easily as follows : — 'U';l)--'}-'{['+fT-%*(l)"+-]-;} Now, when^ is infinite, — =0, and the above becomes '{T-«i)'4(ij-«} the value of which, by the theory of logarithms, is, q^ log« ( 1^-| — Y orlog^^l+iy CHAP. II.] On Annuities-Certain. 201 We then finally have, where the annuity is payable con- tinuously, and interest is convertible g times a year (■Hr- .o.(.+i)- (20) a=A^l (21) .og.(l+|) 25. The following two tables illustrate the preceding articles. Table A gives algebraical expressions for the value of an annuity payable yearly, half-yearly, quarterly, m times a year, and momently, with interest convertible at like intervals ; and Table B gives corresponding numerical values where the term is twenty-five years and the rate of interest is 4 per cent. Table A applies also to perpetuities. For a perpetuity, the algebraical term of the numerator will in each case vanish. Thus the value of a yearly perpetuity, interest convertible m times a year, is i- The student will find it a useful exercise to re-calculate the second table for himself. Where logarithms to the base e appear in the formulas, the calculations should be made as if all the logarithms were to the base 10, and the final result should be multipUed by the modulus 434294482. 26. It will be seen from Table A that if the times at which interest is convertible remain unchanged, we can find from the value of an annuity payable yearly the value of one payable at any other intervals, by multiplying by a quantity which is constant for all values of n. Thus if i represent the effective rate of interest, no matter how often it may be convertible, we have aia, — — ^ : ' -, , («) 1— (14-i)-" (m) i and air| = ^^ — ^^ : whence a-;r|=a^X i m{(l-f«)^-l} m{(l-fip-l} where the co-efiicient of a^ is independent of the term of the annuity. It should be noted that m{(l+i)^ — 1} is the nominal rate of interest convertible m times a year when the effective rate is i. If therefore we write j for that nominal rate, we have Q!~-\-=a^ x — Also a^=za^Xj. 202 Theory of Finance. [chap. II. s i s l-H 1 •'^ -l§ S^_^ + + + r— 1 1 1 1— ( f— 1 I— 1 "^ -^1 + a 1 + I-H 1 f— 1 d^ i^ yi •«s>|oq + •s. |oq + -H + I-H 1 I-H 1 I-H 'VI k 1 f 1 <§ •c=> \is\ Hls-I •^ |(M Hs .^|(M + •c>> joq + •cs Y^ + r-i 1 I-H + r— 1 I-H 1 T-H + I-H I-H 1 I-H '^ - S f. Tt* <« A ^T3 a A ^ 8 ^ o I-H cS ^ oq (U ^ CO ^ -M • o rt C! !^ a O fis" G o «1 a. ^^ y '^ O » v>4 1: ,\vvv>v^ ^ F=f^-^ ' . •• • . • . . (26) The same principles apply where the number of future fines is limited. It will be noticed that fines for the renewal of leases are only rent under another name, paid down in lump sums instead of year by year. 31. We have already remarked, in investigating formula (5), that each payment of an annuity must contain interest on the purchase - money, together with a return of part of that purchase-money. The buyer of an annuity expects to receive interest on his invest- ment at the agreed rate, and he would not be content unless, at the end of the term, his capital were still intact. In fact, the rentu of the annuity consists of two portions ; first, interest on the \ purchase-money ; and, secondly, a repayment of capital, called the /| sinking fund ; and while the interest may be treated as income by the aiijluitant, he must scrupulously set aside annually the sinking fund, and reserve it, with all accumulations of interest upon it at the agreed rate, in order to replace his capital at the end of the period when his annuity will expire. 2o6 Theory of Finance, [chap. ii. The capital invested being a^ , the interest on it is ia^ , and the sinking fund is the balance of the annual rent, or (1— m^). By the conditions of the case, the sinking fund accumulated must amount to the capital, or in symbols a^ = (l— m^) 5^, or ' " .. It is thus seen that the sinking fund is of the nature of an annuity which must accumulate at interest till it amounts to the original capital. 32. But we may look at the matter in another way. In the last article we have considered the whole advance to remain outstanding during all the currency of the annuity, and the sinking fund to be separately invested, to accumulate so as suddenly to extinguish the debt at the end of the period. We may now imagine each portion' of capital in the successive payments of the annuity to be at once applied towards liquidating the debt, which will thus gradually diminish until it finally vanishes. As the debt is being paid off, a less and less proportion of the annuity will be required for interest, and a greater and greater proportion will be available to refund the capital. In reality, however, the two ways of viewing the transaction are the same. In Art. 31 we have supposed the sinking fund to be invested in separate securities till it amounts to the debt, while in the present article we have practically supposed the sinking fund to be invested in the security of the debt itself 33. To separate the successive payments of an annuity into their component parts of principal and interest. The value of the annuity, or, in other words, the capital invested, is — ; — , which is the z amount unpaid at the beginning of the first year. During the first year the debt will increase by the operation of interest to (1 -^-^) ~^ , and at the end of the year a payment of 1 will be i made. The amount outstanding at the end of the first year will (1 + i) — ; I > : — and similarly for any other year. Thus, generally, if to the amount unpaid at the end of any year, just after that year's rent of the annuity has been paid, we add one year's interest and deduct 1, the remainder will be the capital unpaid at the end of the next year. We therefore have the outstanding capital at the end of each year as follows : — CHAP. U.] On Annuities-Certain. 207 •4*AV^. Ybar. Capital Outstanding. First, '-■ +i) }■ We therefore see that the capital outstanding at the end of the m'* year is the value of an annuity for the remainder of the period, (n—m) years, and this is evidently as it ought to be. 34. By the last article, the capital ouljstanding at the end of l_-yn-m+i tr^ ^•-^"^■''' (m— 1) years is : . A year's mterest thereon is (l—t;"-"*+^), which is the interest included in the m*^ payment of the annuity. The capital included in the same payment must therefore be i;«-»»+i. The capital contained in the m*'^ payment of the annuity being t;"-'»+i, and in the (m+1)'*, or (l+i)?;"~"*+i, we see that the successive instalments of capital form a geometrical progression with common ratio (1+?). This is as it should be, for if we repre- sent by Cm the capital contained in the m*^ payment, then that amount of capital being paid off, the interest on it, iC^, is no longer re- quired, and may be applied, along with Cm, at the end of next year to liquidate the debt, when the amount available for that purpose will be Cm+iCm, or C^l+e) as before. Thus Cm+i = Cm(l-^iy 35. The redemption of a loan by means of a sinking fund is fre- quently spoken of as the amortisation of the loan. 36. It very often happens that loans are granted in considera- tion of a terminable annuity. Municipal corporations borrow in this way on the security of the rates, and limited owners of real property are empowered by certain Acts of Parliament to raise money in exchange for a rent-charge for the purpose of improving the estate. In order to escape the income-tax which would other- wise be demanded on the whole annual payment, it is usual to in- sert in the deed creating the charge a schedule showing the amount of capital and interest respectively contained in each payment of the annuity. / 2o8 Theory of Finance. [chap. ii. 37. If ir be the sum to be advanced, the equivalent rent-charge K . ■ for n years will be — . The sinking fund, that is the capital to be repaid at the end of the first year, will be KI i\ or — , and, in accordance with Art. 34, the sinking fund multiplied con- tinuously by (1-f ^) will give the successive repayments of capital. In preparing the schedule, the accuracy of our work may be periodi- cally checked, say at every tenth value, by means of the principles IT laid down in Art. 33. The annual rent -charge being — , the capital contained in the m'^ payment will be , which is our verification formula. If each tenth value be correct, the inter- mediate values must be correct too, because our calculations are done by a continuous process. Logarithms will materially expe- 77" dite the work, as, starting with log — , we shall merely have to add continuously log(l+i). If the rent-charge is to be paid half-yearly, the same principles apply. We can consider the annuity to be for 2n years at rate of interest — . 2 38. From the last article it may be gathered that -^z-i=l- (29) Writ ^1 that is, the annuity for n years which a sum will purchase, less a year's interest on the purchase money, is equal to the sinking fund which will accumulate to the sum in the n years. In interest tables it is therefore unnecessary to display the values of both — and — , as, if one of these quantities be given, the other can be im- mediately found by inspection. 39. A numerical example will enable us to understand more clearly the principles laid down in the preceding articles. A sum of £5000 is to be advanced at 5 per cent, interest, and to be repaid in fifteen years by equal annual instalments, including principal and interest. It is required to separate each year's payment into principal and interest. Here, by Art. 37, the yearly charge is -^n-, or £481-711, and «15| CHAP. II.] On A nnuities- Certain, 209 the sinking fund is (481-711 -ix 5000) or 5000 or £231-711. In the following schedule the transaction is worked out at length. The amount of principal opposite any year in column 3 is obtained by multiplying by (l + 0> or 105, the amount opposite the previous year, and it will further be found that the amount opposite any year, the m'*, is equal to 481-711?;»^-»»+S or 481-711?;i«-'». Thus, for the 10'* year, ?;« = -7462154, and 2;«x 481 -71 1 = 359 -460. The principal repaid to date given in column 4 is the sum up to date of column 3. It will also be found to be the amount of an annuity of the sinking fund for the period elapsed. Thus, the amount of an annuity for 10 years is 12-578, and the sinking fund being as above 231*711, the product is 2914*440, which is also the amount opposite year 10 in column 4 9f the table. Also the principal outstanding as given in column 5 is obtained by deducting all the previous repaj-Tnents of principal as given in column 4 from the original .£5000 ; but also that opposite the m"^ year is equal to rt—^x 48 1*711. Thus, opposite year 10, the amount is 2085-560, which is 5000-2914*440. But a;;z^ or as\ is 4*3295 and ag, x 481 -71 1 = 2085*560 as before. Schedule illustrating the Eedemption of £5000 in 15 years by Equal Annual Instalments of £481*711, including Principal, and Interest at 5 per cent. 1. 2. 3. 4. 5. Year (m). Interest con- tained in m«* payment. Principal con- tained in w«* payment. Principal repaid to date. Principal still outstanding. 1 250*000 231*711 231-711 4768*289 2 238*414 243*297 475-008 4524*992 3 226*249 255*462" 730-470 4269*530 4 213*476 268*235 998-705 4001*295 5 200*064 281*647 1280-352 3719*648 6 185-982 295*729 1576-081 3423-919 7 171-196 310*515 1886*596 3113-404 8 155-670 326*041 2212-637 2787-363 9 139-368 342*343 2554-980 2445-020 10 122-251 359*460 2914-440 2085-560 11 104-277 377-434 3291-874 1708*126 12 85*406 396-305 3688-179 1311*821 13 65*590 416*121 4104-300 895*700 14 44-784 436-927 4541-227 458*773 15 22-938 458-773 5000-000 2IO Theory of Finance. [chap. ii. 40. It sometimes happens, especially in transactions connected with mining property, that, in consideration of a terminable annuity, a lender grants an advance at a higher rate of interest than he can secure from other investments, and that he wishes to realize the higher rate on the ^ whole of the capital during the entire term of the annuity, although, as we have seen, part of his capital is repaid to him year by year. He must therefore so fix the price he pays for the annuity that he may realize interest at the higher rate, called the remunerative rate, and accumulate the sink- ing fund at a lower rate, called the accumulative rate. To eff'ect this object we must apply the principles of formula (27). We have only to take i at the remunerative rate, and s at the accumulative rate. Writing A for the value of the annuity under these special conditions, and s' for the amount of an annuity at the accumulative rate, we have ^ = T^' (30) These terms are very onerous to the borrower, as he has not only to pay the higher rate of interest on the capital which remains in his possession, but also to make up to that highej" rate,- during the whole period for which the annuity has to run, the interest on the accumulations of the sinking fund which are in the hands of the lender. 41. As an example of the last article, let it be required to ascertain the annual sum to be paid in order to redeem a debt in n years, interest at the rate i to be realised on the whole debt throughout the whole term, and the instalments of capital to be invested at another rate, j. We have seen that the value of an annuity of 1 per annum to pay the purchaser interest at the rate i on his purchase -money and g' to refund that purchase-money at another rate, j, is - — r-r. For each unit of the debt the annual payment is therefore — -, or ( — |-i ) A \s' J or —+(*—;), (since — — ] — — i [ ), where a' is taken at the ratej. ; A^^ . . 4^. This last formula enables us to use a table of ordinary annuities in questions of this more complex character. If, for instance, we have a table giving the ordinary annuities which 1 CHAP. II.] On Annuities-Certain, 2ti will purchase, and we wish to find the annuity which 1 will purchase to pay i to the purchaser and replace the capital at rate ;', we have only to take from the table the annuity which 1 will pur- chase at rate 7, and add to it (*— y). 43. A numerical illustration will be useful, and it will be con- venient to repeat with the needful modifications the one already given in Art. 39. By this means a comparison can be instituted between the ordinary annuity and the special annuity now under consideration. Suppose a lender to advance £5000, and we are asked to find the annuity for 15 years which will pay him 7 per cent, on his capital and replace that capital at 5 per cent. The annuity for 15 years which 1 will purchase at 5 per cent, is •0963423. Adding to this {i—j) or -02, and multiplying by 5000, we have £581*711 the annual payment required. This is greater than in the example in Art. 39 by £100, or 2 per cent, on £5000. The sinking fund is to be taken simply at 5 per cent., and must therefore be the same as in Art. 39, viz. £231*711. The annual rent of the annuity is, as above, £581711, which is equal to the sinking fund, together with £350 or 7 per cent, on £5000. The lender may therefore treat the £350 as income during the whole of the 15 years, if he invest annually the sinking fund at 5 per cent, to replace his capital. But — as in Article 32 — we may consider that part of the capital is repaid year by year, and that on the repaid portions of the capital the lender only realizes 5 per cent, interest. The annual rent of the annuity must then supply an instalment of capital, interest at T per cent, on the capital still outstanding, and also interest at 2 per cent, on the capital already repaid, so as to raise to 7 per cent, the interest on that portion of capital also. The operation of the fund is shown in the following schedule, by which it is seen that all these requirements are fulfilled. Columns 3, 4, and 5 are identical with the columns bearing the same head- ings in Art. 39 ; but the interest in column 6 of the present schedule is always 2 per cent, more than that in the corresponding column, column 2, of the former schedule, while we have added column 8, showing the balance of interest on the principal already repaid. It will be seen that the sum of the amounts opposite any year in columns 3, 6, and 8, is always equal to the annual rent of the annuity, and that the sum of the amounts in columns 6, 7, and 8 is always £350, or 7 per cent, on the total capital advanced, X 212 Theory of Finance, [chap. II. Schedule showing the operation of an Annuity to repay £5000 in 15 years at 7 per cent, interest, Sinking Fund accumulating at 5 per cent. 1. 2. 3. 4. 5. 6. . 7. 8. Balance of Year Annual Principal included Principal Principal Interest at 7 per cent, on out- standing Interest at 5 per cent, earned on Interest on repaid Prin- cipal at end of previous year, being difference between 7 per cent, and 5 per cent, (pro- vided by Rent of repaid to still out- Principal repaid Prin- Annuity. Paymeiit. date. standing. at end of previous year (pro- vided by Annuity). cipal at end of previous year. ■^ Annuity). 1 581'711 231-711 231-711 4768-289 350000 0-000 0-000 2 243-297 475-008 4524-992 333-780 11-586 4-634 3 255-462 730-470 4269-530 316-749 23-751 9-500 4 268-235 998-705 4001-295 298-867 36-524 14-609 5 281-647 1280-352 3719-648 280-090 49-936 19-974 6 295-729 1576-081 3423-919 260-375 64-018 25-607 7 310-515 1886-596 3113-404 239-674 78-804 31-522 8 326-041 2212-637 2787-363 217-938 94 330 37-732 9 342 343 2554-980 2445-020 195-115 110-632 44-253 10 359-460 2914-440 2085-560 171151 127-749 51-100 11 377-434 3291-874 1708-126 145-989 145-723 58-288 12 396-305 3688-179 1311-821 119-569 164-594 65-837 13 416-121 4104-300 895-700 91-827 184-410 73-763 14 436-927 4541-227 458-773 62-699 205 216 82-085 15 458-773 5000-000 0-000 32-114 227-062 90-824 44. In connection with annuities, various quantities come under our notice, namely, the term, the present value, the amount, the annual rent, and the rate of interest, and in the majority of cases simple relations subsist between these quantities, so that, certain of them being given, the others can at once be found by the ordinary operations of algebra. Thus we have seen that a^v^-=.- ~^ .' — , so that, having given n and % we can find a. 45. There is one case, however, which presents difficulties. When the term, and the amount or the present value, of an annuity are given, and it is required to ascertain the rate of interest, we have to solve an equation of the %'^ degree, and an approximate solution is alone possible. This case we now proceed to investigate. 46. Having given 5, the amount of an annuity, and ?^, the term, to find i, the rate of interest. Using Table 3, which shows the CHAP. II.] On Annuities-Certain. 213 amounts of annuities at various rates of interest, we find, in the table opposite the given number of years, the value nearest to s. Let that value be denoted by s', and let the rate of interest under which it is found be denoted by j. If s', the amount found in the table, be exactly equal to s, then the rate under which s' is found must be the rate sought; that is, i=.j. If, however, s' differ from s, then j differs from i. Let i=j-{-p^ and the problem resolves itself into finding /o, which is necessarily a very small quantity. We have _ (l+t)"— I ~ i Therefore s 0'+p) = {(l+i)+p}"-l. If we expand the right-hand member of the equation by the binomial theorem, and neglect all powers of p above the first, we have . s (j^p) = {l+jf-{.n (l+y)n-ip_i ; whence, remembering that (1+^)"— 1=;V, we have By means of formula (31) the value of p, and thence of i, can be found, generally with sufficient accuracy; but should a closer approximation be desired, the process may be repeated. The amount of the annuity may be calculated at the rate of interest found by the first application of the formula, and inserted in the equation instead of s', by which means a result very accurate indeed may be obtained. 47. Other formulas for the rate of interest in s^ may be deduced, analogous to those given in the succeeding articles for the rate of interest in a^ ; but, as the problem is not one of great importance, we do not prolong the investigation hei'e. 48. Having given a, the value of an annuity,- and n, the term, to find i, the rate of interest. By the help of Table 4, giving the values of annuities, we must find a value of a near to the given value. Let that near value be denoted by a', and let the rate of interest under which it is found be denoted by;, and let the rate sought, i=j-{-p. We have _ 1-(1+^)-^ ft— _ l-{(l+j)-f-p}-" whence a (;4-p) = l--{(H-;')+f>}-". • • • (32) and a=[l^{{l-\-j)-\-p}-^](j+p)-^ , . (33) 214 Theory of Finance. [chap.il From these last two equations we can deduce as follows the four approximate formulas A, B, C, and D, for p. 49. If in equation (32) we expand by the binomial theorem, neglect the powers of p above the second, and collect the terms, we have, — writing v for (1 +i)~S and remembering that (1 —v''^)=ja\ — the quadratic equation in /o, .^-J—Li;n+2pi^^a,-nv^+^)p-j{a'-a) = . (34) If we neglect the second power of p, we have merely a simple equation to solve, from which we obtain j(a'-a) (35) 50. The value just found in A may be used in conjunction with equation (34) to obtain a closer approximation. If in the equation we write p^=zpx— — — rr, and then solve for p, we have >,4., . n(n-{-l)j(a'-a)v^+^ ' ^ ' 2 a—n'v^+^ 61. Eeturning now to equation (33), if we expand both the factors of the right-hand member by the binomial theorem, and retain only the first and second powers of p, we have another equation \ j a'-nv^+^- '^^^'^^\ ^+'j \ p^^ l-{a'-nv^+')p-\-{a'-a) = (37) and if in that equation we solve for p, neglecting the second power, we have C. Jici'-a) (3g) I 52. Lastly, if in equation (34) we write p^=px^r^ — ;^, and solve for p, we have 53. A numerical example will help us to gain a clearer idea of the formulas given in the four preceding articles. Suppose that 14-94390 is the value of an annuity-certain for 30 years, and we require to find the rate of interest. If in Table 4 we look against 30 years, we find that at 5 per cent. ai^| = 15-37245, and at 6 per cent., ai^| = 13-76483. We therefore conclude that i, the rate of interest sought, lies between -05 and •06, being nearer to -05, and we assume that j= -05. Then, using formula A, CHAP. II.] On Annuities- Certain. 215 a'=15-37245 also ?;"'= -220359 a= 14-94390 30i;3' = 6-61077 (a'-a)= -42855 j(p:-a)=, -021428 _ a- 302r»i= 8-33312 log -021428 = 2-33098 log 8-33312 = 0-92081 log/)=3-41017 p= -0025714 Formula B has the appearance of being lengthy and intricate, but it is not so. It should be observed that the second term of the denominator consists merely of the product of -^—^ — ^^'^+" and the value of p found by formula A, and that in finding the value of p by formula A we have already calculated the values of all the other terms of the expression. In the example in hand we have log (J X 30x31) = 2-66745 Wt;^ =r32194 logi!^ =^«017 T-39956 whence the second term of the denominator is -25093. Adding to this the value of a— 30^;^^ already found, 8-33312, we have the de- nominator =8 -58405. The numerator is '021428 as before, whence /3= -0024963. Formula C is the same as A, except that 0! occurs instead of a in the denominator. From it /o= -0024457. Formula D is the same as B, except that 0! occurs instead of a in the second term of the denominator. From it /o= -0024998. We therefore have by formula A. i= -0525714 B. i= -0524963 C. i= -0524457 D. i= -0524998 the true value of i being -0525. 54. Formula C gives a better value than formula A, and, if we examine the processes by which the formulas were deduced, the reason becomes apparent. Equation (32) is obtained from equation (33) by multiplying by (i+p)> """it^ ^^^ result of a loss of accuracy. If in the expansion 2 1 6 Theory of Finance, [chap. it. of 1 — {(l+y)+/)}~*^ we denote the coefficients of the ascending powers of p by ^, /, and w, so that^= {l-(l+J)-''},Z=?i(l4-7)-^^^+'\ and ^^ ^^+l) ^^_|.j^-(n+2)^ equation (32) will take the form 'mp^-{-{a—r)p—{k—aj) = 0, or say xp'-^yp-z=0 (40) where x—m, y=(a^l), and z=(k—aj), and equation (33) will take the form or [m+i-j,y+(^a-l + ^-^)p-(h-aj)=0 which is seen to be the same as ).)f%-^fy+{y^j}-^-o ..... (41) ^Equations (40) and (41) are simply equations (34) and (37) so displayed that they may be compared, and they show us that in equation (34) we neglect part of each power of p. -^niact, when, to obtain equation (34), we multiply up by (j-\-p), the result is to diminish the coefficient of each power of p by —th part of the co- efficient of the next lower power. Seeing then that formulas A and C are identical in form, and therefore equally easily applied, while C is the more accurate of the two, that is the one which should be used. 55. Formula A was first published by Mr. Francis Baily in the appendix, page 129, of his work on the Doctrine of Interest and Annuities. In the same place he investigates other formulas which do not require the aid of interest tables for their applica- tion, but at the present day these formulas have no practical im- portance, owing to the number and extent of the interest tables which are available. Formula C is due to Mr. George Barrett, who, in writing to Mr. Baily, suggests it as an improvement on formula A. (See the extract of a letter from Barrett to Baily, /. /. A. vol. iv. p. 189.) To Professor De Morgan we owe formula B. It is given by him in a paper, /. /. A. vol. viii. p. 67; and Mr. M'Lauchlan has supplied formula D, /. /. A. vol. xviii. p. 295. 56. The principles by which we reduced equation (34) from a CHAP. II.] On Afinuities-Certain. 2 1 7 quadratic into a simple equation may obviously be extended. We may retain all terms of our original expansion which involve powers of p not greater than the third, and write p'^=p{ ^^ ~~ / > , and p3=p J J\^ ~'^' I and then solve for p. By this means Professor De Morgan (/. I. A. vol. viii. p. 67) deduces a formula which produces very accurate results, but it is too complicated for common use. 57. A very frequent instance of the application of the foregoing formulas is the case of foreign government loans. The English Government has usually borrowed in exchange for perpetual annuities, — that is, in consideration of a sum advanced a promise has been given to pay a certain yearly interest, but without stipulation as to repayment of the principal. 58. Foreign Governments, however, generally find it more con- venient to undertake to repay the principal within a limited period. The arrangement is to issue numbered bonds, with coupons attached representing the annual interest, and these coupons are cut off by the lender and presented for payment as the interest falls due. Besides providing for the interest yearly, the Govern- ment also sets aside a fixed sum, called the sinking fund, to redeem the bonds ; and bonds to the amount of the sinking fund are drawn by lot and cancelled. The yearly interest, however, on the cancelled bonds is still provided by the Government, and applied, in addition to the sinking fund, to liquidate the loan. In fact the Government simply contracts to pay a fixed yearly instalment, including prin- cipal and interest, for a limited number of years till the whole loan is paid off, or, in other words, it raises money in exchange for a terminable annuity. The sinking fund under these conditions is called an accumulative sinking fund. 59. If such a rate of interest were offered that the public would take the bonds at par, the transaction would not be complex. The rate of interest paid by the Government would merely be the nominal rate of interest borne by the bonds. But loans of this class generally are issued at a discount, while the bonds on being drawn are paid off at par, so that, in addition to the yearly interest, the lending public are offered the inducement of a bonus on repayment. Under these conditions we must make use of one of our approximate formulas to find the true rate of interest paid by the Government. 2i8 Theory of Finance. [chap. ii. 60. Let the price at which the bonds are issued be h per unit : let the nominal rate of interest paid by the Government be i' ; and let the sinking fund per unit of the loan, that is the amount to be repaid at the end of the first year, be z :— then h is simply the present value of an annuity of {i'-rz) per annum. We therefore havey^=(i'+^)l-:ill+it! k ^ l-(l+i)-n ^^2) (i'+z) i where * is the actual, as distinguished from the nominal, rate of interest paid by the borrower, and ^i is the number of years for which the loan will run. 61. In the above equation both i and n are unknown quantities, but n is easily found. The sinking fund, being employed to pay off the bonds, accumulates at the nominal rate of interest i', and at the end of n years amounts to the total loan. We therefore have the equation (1 +»:')"-! whence iog(i^+i) and n= , \^ .J (43) log (1+0 Having found n we can then apply one of our formulas A, B, C, or D, to find i. Usually n will not be an integer, but it will be sufficient for practical purposes if we take the nearest integral value which equation (43) brings out. Instead of using equation (43), it will most often be convenient to refer to an interest table to find the value of n. 62. As an example, let us examine the Russian 1864 loan of £6,000,000. The issue price was 85 per cent., the nominal rate of interest 5 per cent., and the accumulative sinking fund 1 per cent. That is, in equation (42), Jc=-S5, i'=-05, and ^='01; therefore ^=14-1667. Also in equation (43) ^=i^f^= ^^'^^ = 3^ nearly: whence asfi = 14-1667. If we apply formula C, using 6 J per cent, for the approximate ratej, we find i= -063297. It need hardly be remarked that we can use interest tables to give an approximate rate even when the tables do not contain minute chap.il] On A^muities-Certain. 219 subdivisions of the rate of interest For instance, in the present case, if the table employed contain annuities at only integral rates of interest, we shall find that at 5 per cent. a3fi = 16-71129, and at 6 per cent. a37"| = 14*73678. We see therefore that our required rate must lie between 6 per cent, and 7 per cent., and must be about 6 J per cent., and we then calculate a^\ and 1^ at 6 J per cent., and insert the results in our formula C. 63. An interesting case occurs where a loan, redeemable by an accumulative sinking fund, is quoted in the market at other than par some years after issue, and we require to ascertain the rate of interest which it yields. Let us take the following illustration. A Government 5 per cent, loan was issued eight years ago, repayable by an accumulative sinking fund of 1 per cent. The eighth annual payment is just due but not paid, and the market price of the loan is .102 per cent. What rate of interest does it yield? The full period of the loan was, as we have seen by the last article, 37 years, and the annual payment made by the Government is 6 per cent, of the original amount of the loan. The capital at present outstand- ing is that which was outstanding when the seventh payment was just made, that is, by formula (28), 6 x «3o^^, or 92-2347 for each 100 of the original loan (the annuity being taken of course at 5 per cent, interest). The market value of the capital still outstanding is, at 102 per cent., 94*07939, and this is the value of an annuity of 6 per annum of thirty payments, first payment due at once, that is of 6(l+a29"|), whence (129^1 = 14*6799 ; and applying formula C with 5J per cent, for y, we find the rate of interest to be 5*28008 per cent. 64. We must be careful not to confound the rate of interest incurred by the Government with the rate or rather rates of interest realized by the lenders. If the loan stand at ^a discount and be repayable at par, it is evident that the holders, of the bonds which haj^pen to be drawn early for repayment will realize a higher rate of interest than will the holders of the bonds that happen to remain undrawn till a late period. Thus, in the case of the loan discussed in Art. 62, a £100 bond will cost a purchaser £85, and if the bond happen to be drawn at the end of the first year, the holder will then receive £5 for interest and £100 for principal, that is for an advance of £85 for a year, he will receive in principal and interest £105, and thus realize interest at the rate of more than 23 J per cent. K, however, the bond be not drawn till the end of the thirty- seven years, the holder will realize interest at the rate of only very little more than 5| per cent. 2 20 Theory of Finance. [chap. ii. In Art. 63 we spoke of the rate of interest yielded by a loan, but that must be understood to mean the rate yielded to a purchaser provided he take up the whole loan, or at least a sufficiently large part of it to ensure a fair average of bonds drawn. It cannot be held to refer to the rate yielded by any particular bond. 65. It will be convenient to close this chapter with one or two examples. a. What is the value of an annuity-certain, taking interest at % for the first n years and j thereafter % Let us assume that the annuity has to run for {n-\-rri) years in all. The value of the annuity for the first n years is evidently ^ . ' . When n years have expired, the value of the remammg portion of the annuity will, by the conditions of the question, be ^ :•'' — . But at the present time this latter portion is deferred n years, during which period interest is at rate *, and its present value is therefore (l+i)-"" ~' '^^' . The whole value of the annuity is thus l-(\+^)~"^(i+i)-nWLhi)L"'. /5. A person holds a lease at a rent of £20 per annum, with the option of renewing it every seven years by paying a fine of £100. What is the equivalent uniform annual rent % Interest 5 per cent. This question is likely to occur to the tenant at one of the periods for renewal, when he is debating in his mind whether or not he should continue his lease. We may therefore assume that one of the fines is just due. By Art. 30 the value of the future fines is 100 x^- •> and this we must convert into a perpetual annuity 1 — v^ by dividing by a^, or by multiplying by %. The annual rent equivalent to the fines is therefore — , which we shall find to 1— -y^ be £17 "282, making the total uniform annual rent £37*282, or £37, 5s. 8d. If the next fine be not due for seven years, we shall find that the total annual rent will be £32, 5s. 8d. y. A loan of £P is to be discharged by {i^-\-q_-\-T) annual instal- ments compounded of principal and interest. The ^ instalments are to be of £a each, the ^ of £/:^, and the r of £y. What is the rate of interest, %% We can only approximate to this rate of interest, and it will be sufficient if in our approximation we neglect all powers of i above the first. CHAP. II.] On Annuities- Certain. 221 We have +y{l_i(p+j)}|r_'ll±l)i| If now we multiply out, still neglecting all powers of % above the first, and arrange the terms, we have ^a4-^/:^+ry— P %=. a^(i>+i) +/^{^+i^.}+7{^-^t- 2 8. The following very interesting question appeared in the exami- nation paper set to the candidates at the intermediate examination of the Institute of Actuaries held at Christmas 1874 : — A Parochial Union has obtained certain loans upon the under- mentioned terms, and it now desires to consolidate the debts and redeem them by a single terminable annuity running from 31st December next — 1st Jany. 1856, £5000, by 60 equal half-yearly payments, int. 6% 1st July 1870, 3000, „ 80 do. do. „ 5% 1st Jany. 1872, 2000, „ 60 do. do. „ 4J% Kequired, 1st, the terminable equal annuity payable half-yearly for thirty years from 31st December next; and, 2nd, the rate of interest (approximate) returned upon the debt when consolidated as above. Here it will be proper to assume that the half-yearly instalment of each of the three original annuities which falls due on 31st December 1874 will be paid, and that the consolidation of the debt will take place immediately subsequent to such payment. The first step is to ascertain the annuities at present payable by the Parochial Union. By Art. 37 we find the half-yearly charge for the first debt to be 522? at 3 %, or 180-665. „ second „ ?222 at 2^%, or 87-078. ^80 1 „ third „ ?222at2J%, or 61-071. Immediately after 31st December 1874, of 1st debt 38 payments will have been made, and 22 will remain. „ 2nd ,,9 do. do. 71 do. ,, 3rd ,, 6 do. da 54 do. 2 2 2 Theory of Finance. [chap. ii. There will therefore remain outstanding, by Art. 33, Of 1st debt, 180-665 xa22-| at 3 7^, or 2879-240 „ 2nd do., 87-078 xanl at 21% or 2879-765 „ 3rd do., 61-071 XftMi at 2J7^, or 1897-995 Total amount outstanding, 7657-000 We must now take each debt separately, and find — at the rate of interest proper to that debt — the half-yearly annuity of sixty payments which the amount outstanding wiU purchase. This we shall find to be, by Art. 37, For 1st debt, . . "^^^^'^^^ at 3 7„, or 104-035 „ 2nd do., . . ^51^1?^ at 21X, or 93-170 „ 3rd do., . . IM:^ at 2i7o, or 57-956 Total consolidated half-yearly payment, 255*161 The Parochial Union has now to pay in each year, for thirty years, £510-322, in order to liquidate a debt of £7657, and we have to find the rate of interest. Using formula C, Art. 51, and 5 J per cent, for our approximate rate of interest, we have a'= 14-9439 a'= 14-9439 a =15-0043 30?;8i= 6-1410 ;(a'-a) = - -00317 — -0604 8-8029 525 ' log -00317 = 3-50106 302 log 8-8029 =0-94463 12 _Z 3 log(-/o) = 4-55643 ^=-•00036 i= 5-2147„ It will be noticed that in solving this question we have treated the annuity as one of £510-332, payable annually, and so found the annual rate of interest. This we may consider to be the nominal annual rate convertible half-yearly. It would have come practically to the same thing had we taken the case as it really stands in the question, viz., an annuity of 60 payments of 255-161 f< J^\f^ -fTi tr ^^' chap.il] On Annuities-Certain. 223 each, and assumed an approximate rate of interest 2f per cent. AVe should thus have obtained the half-yearly rate of interest. The course we have adopted is the more convenient as being better adapted to the majority of published interest tables. Another method of finding the rate of interest in the foregoing question might suggest itself; but though at first sight it seems correct, it nevertheless gives an erroneous result. Of the whole debt of £7657, there is 2879-240 at 6%, 2879-765 at 5%, and 1897-995 at 4J7o> and therefore the average for the whole debt is 5-252%, a higher rate than that already found. To take the average would be the correct course if the three portions of the debt were repayable in the same proportions at the same times. But this is not so. Those portions of the debt at the higher rates of interest are repaid by the sinking fund more slowly in the earlier years and more rapidly in the later years than those at the lower rates, and thus, as time advances, the average is destroyed. The only correct way to look at the question is to treat the consolidated debt as the present value of an annuity of the total annual sum payable by the Parochial Union. THEORY OF FINANCE. CHAPTER III. On Variable Annuities. 1. In the direct process of the calculus of Finite Differences, we form one set of functions from another by the operation called differencing, that is, if we have a series whose terms are u^, u^, u^, etc., we form another series whose terms are {u^—u-i), (u^—u^), etc., and which we represent by Aw,, A^j? etc., the successive terms of the second series being the differences between the successive terms of the first series. The operation of differencing may be repeated, the series of differences being itself differenced. 2. It is evident that this process may be inverted. Instead of forming a series of differences, we may form a series of sums. We may pass from our original series to another series, of the terms of which the terms of the original series are the differences. Thus if we have the series Uy, u^, u^, etc., we can form another series, ^1, ^2, ^3, etc., such that Fz—Fy=Ui, Vi—V^=U2, etc., or, put- ting the same operation in another form, such that ^2=^1+^*1, ^3=^2 4-W2j etc. Further, like the process of differencing, the inverse process of summation may be repeated. We may pass to another series from F,, Fg, F3, etc., in the same way that we passed to F"i, F2, Vz, etc., from u-^, u^, Ug, etc., and so on without limit. 3. The successive orders of the figurate numbers are series con- nected with each other in the way described in Art. 2, and they are represented in the following scheme : — Term 1st 2nd 3rd 4th 5th 6th 7th etc. (m) order. order. order. order. order. order. order. 1 a etc. 2 1 etc. 3 2 1 etc. 4 3 3 .1 etc. 5 4 6 4 1 etc. 6 5 W 10 5 1 etc. 7 6 15 20 15 6 1 etc. 8 7 21 35 35 21 7 etc. 9 8 28 56 70 56 28 etc. 10 9 36 84 126 126 84 etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. CHAP. III.] On Variable Annuities^ 225 4. The first order is a series of constants, which, for the sake of convenience, we may assume to be unity. Each term of the second order is the sum of all the 'preceding terms of the first order, so that the m'* term of the second order is the sum of the first (w— 1) terms of the first order, the value of which is therefore (m— 1). In the same way the w'* term of the third order is the sum of the first (m— 1) terms of the second order; and generally the m'* term of the r*^^ order is the sum of the first (m— 1) terms of the (r— 1)'* order. 5. If by /;;r| r\ we denote the m'* termjf of the.r'* order, and use 2;" as a symbol of summation of terms from 1 to m inclusive, the fundamental relation between the orders may be represented by the equation <^li;^=2r(^,frii^ ...... (1) 6. Since by the last equation, /^| f] =2r"^^iirif-ii, and ^^M^ii t\ =-^t^\ fTii, it follows that ^ijrrrifi=^iri Fi+fe"!f^i (2)' and /;;i:pi-if]— ^1 fl=/;jri rTT| (3) Equations (2) and (3) display the fundamental relation between the orders in another light from that supplied by equation (1), and show that the principles of construction laid down in Art. 2 have been carried out. The terms of the (r— 1)'* order are respectively the differences of the corresponding terms of the r** order. 7. From the method of construction of the successive orders it will be seen that always ^rm=i (4) and that, of the r** order, all the terms below the r** are equal to zero. 8. Carrying out the analogy of equation (4), we may assume that /q] 5] = 1, and also t^\ fi=0. These conventional symbols (like the symbol a:" with which the reader in his algebraical studies has already become familiar), although they have no meaning in them- selves, will be found useful in our subsequent investigations, as they will enable us to make perfectly general the formulas which we shall deduce. 9. It is easy to show that the m'* term of the /* order is equal ^ (m-l)(m-2). . . (m-r+1) ^ , ^ u ^x, to ^^ ^— j-^-^i — ^ ■ — -. Let Ui, u^, Ui, etc., be the succes- |r-l sive terms of the r** order. Then, by the ordinary formula of Finite Differences, 2 26 Theory of Finance, [chap. hi. 7.^=,,,+ (m-l)A^. + ^'^"^|^^'^~^W + etc. ( m-l)~(m-2) . . . (m-r+lU _ But the first term of each of the orders except the first, is equal to zero, and the first term of the first order is equal to unity ; and these first terms constitute the successive orders of diff'erences of the /^ order of numbers. That is, u-^ and all its differences, up to and including the (r— 2)** diff'erence, vanish, and the (r— 1)''^ difference is equal to unity. Therefore _ (m-l)(m-2) . . . (m-r+1) ,. *m I r\ — - r-r~i • * \^/ 10. The principles so far investigated can now be applied to the solution of questions connected with variable annuities. Let an annuity of the f^ order be an annuity the successive payments of which are the corresponding terms of the r*^ order of figurate numbers. Thus an annuity of the first order will be an annuity whose payments are all unity ; an annuity of the second order will be an annuity whose payments are 0, 1, 2, 3, etc. ; an annuity of the third order will be an annuity whose payments are 0, 0, 1, 3, 6, etc. ; and, generally, an annuity of the r*^ order will be an annuity whose m*'^ payment is, by formula (5), (m-l)(m-2) . . . (m— r+l) Let the amount of an annuity for n years of the r"^ order be de- noted by s^i r\ , and the present value by a^ y\ . 11. To find s^ f|,the amount of an annuity for n years of the r*'^ order, We have s^ fi=i?^ f]+(l-f i)/^5:i:]] r| + (l+«)^^^r:2| fi+etc. s^l ;rz]l=^^ ^^|+(l+*);^rrril^ + (l+^)'^;^=:2|;^l+etc. -\-{l-^if-Hi] ,-=11 +(1 +*)%-] ^1 Adding the two equations together by the help of formula (2), =^M^i| t\ +(1 +i)^ f| -f (1 +^)'^7r:il rl +etc. + (1 4-0""'^2l ^ + (l+irh\f\ Therefore, after arranging the terms, \V* *^n— ^ • * • • • v/ By means of this formula we can find s^ rl having given s^ irii|. CHAP. III.] On Variable Annuities, 227 12. We have seen, Art. 8, that an annuity of the 0'* order con- sists of but one payment, or rather that the annuity is really only a unit in possession at the beginning of the period. We may therefore infer that the amount of such an annuity is (l+i)**, and its present value unity, and we may use the conventional symbols 55i,5|=(l+i)« (8) flii|ffl=l (9) 13. By formula (7), g^ ^ = ^ <>1 "^^E lLQ. But ^^|T| = 1, and by formula (8), ^ o] = (1 +*)". Therefore Si| i^ = (lil^tlll. This % result agrees with formula (7) of Chapter II., and shows that we have correctly interpreted the symbol 5„I oi 14. From formula (7) we have 57il0|=(l+i)" HA HA. n{n—\){n—1) 6 etc. = etc. And, to take a numerical example, when w=5, and t=*05 g5]ol= 1-2762815625 •2762815625x20 = ^1'^^^ 55] 11= 5-52563125 •52563125 55|jl = 10-5126250 ~X20 7x20 7x20 71 = 5 ' ^ ^ 2 •6126250 55131 = 10-252500 6 •252500 .*j5^4|= 5-05000 «(«-l)(»-2)(«-3)_, 24 •05000 X 20 «6l5]= 1* 228 Theory of Finance. [chap. hi. The last result proves the correctness of our work, because an annuity for 5 years of the b*^ order consists of but one payment of unity, made at the end of the fifth year. 15. To find a^ ^, the present value of an annuity foi- n years of the r'* order. Following the method adopted in Art. 11, a^ Fl ='v%\ T\ -{■'vk\ r\ ■\-vH2\ r| +etc. +«;"/^' f | Xn-r+2) to ^«+i X (^+1M^^-1) . ♦ . (7^-r+3) |r— 1 ' jr—i ' which we may write N'. That is, to obtain N' we multiply N by o 230 Theory of Finance. [chap. hi. X i-j-7- The first factor of this expression may be written ri-r + 2 1+i 1_ in the form -. ?*— 1, which shows that although it will always be ~n-\-\ greater than unity, yet it will diminish with the increase of w, and tend to approach unity as a limit. N' will be greater than N as long as ^"^ X T^j-r-r, > 1 ; that is, as long as ^"^, ^ >(1+^) ; 7i— r+2 (1+*) n—r-{-2 or, in other words, iV'=iV when — — — -=(! + *): that is, when n—r-\-2 n= ~ — . — — =B say, a finite quantity, and when n still in- creases iV begins to diminish. If ilf represent the value — which is finite V — of JVwhen 11= B, the value of iV becomes successively, as n still increase, .^±^M, .._^J|+_J)(|±^Jf, etc. , that is, M is continuously multiplied by a series of factors of the form «;— — ^— S^JL each of which is less than the preceding one, and less than v-^p- — ^ ^, which is less than unity. Therefore when n R-r+2' ^ ( B,-X-\ ) »» becomes (R-\-m), iV becomes less than M{ v ^ — ^ \ - But the ( E—r-\-2 ) coefficient of M being less than unity, diminishes as m increases, and may be made as small as we please by sufficiently increasing m. When therefore m, and consequently n, becomes infinite, N - vanishes, and we have a^l,-|=^-^l4El (11) 1 The value of the perpetuity of the r*^ order is therefore finite when that of the (r— 1)"^ order is finite. But the value of a perpetuity of the first order being finite, so will be that of the second order, and of the third order, etc. ; and generally the value of a perpetuity of the r^^ order will be finite as long as r is finite. 22. From formula (11) we have tt-oo"| 1 = 1 _ 1 ftc»|ll=J 1 1 CHAP, iii.l On Variable Annuities, 231 and generally / Jb'^ 23. The reasoning of Art. 21 may appear to the student to be somewhat laboured. That is only because we have avoided the use of the Differential Calculus. If we call in the aid of the Differential Calculus, we can very easily prove that t^^/^r+i] r\ vanishes when n becomes infinite. The term which we are in- vestigating is ^(^-1) ^ ' • ^''r^'_^\ \ViY\ the numerator and denominator of which both increase without limit with the in- crease of n. To determine the limit of the value of such a fraction, we must substitute for the numerator and denominator their respective differential coefficients, repeating the process, if neces- sary, until a fraction is obtained of a form exhibiting its ultimate or limiting value. In the case in hand we shall find that the (r— 1)'* differential coefficients will answer our purpose. Sub- stituting these differential coefficients for the original numerator and denominator, we have -^. -rr-r-.. r i/ i ■ -ix^* which evidently {log,(l+*)}'^-Hl + l)" vanishes when n becomes infinite, and we therefore conclude that ^^''^tJTiI r\ also vanishes. 24. In Art. 17 of Chapter II., we found the value of an annuity of the first order by means of reasoning without having recourse to series, and we may now extend the reasoning to annuities of higher orders. The m** payment of an annuity of the (r— 1)''^ order, if it be in- vested immediately on its receipt, will produce at the end of each succeeding year the sum of i X ^^| fHii , and it will itself still remain in hand at the end of the ti years for which the original annuity was granted. If, then, each of the payments of the original annuity be thus forborne, at the end of the first year there will be entered on an annuity, arising from interest, for (^—1) years of i x ^r| r=i!> at the end of the second year another annuity for (w— 2) years of i x t^ ^rryi, and so on, each of these annuities, which w^e may call secondary an- nuities, running concurrently with all those that have preceded it. Thus, at the end of the m'* year the payments of all these secondary annuities will aggregate i (/i] ^nzij +^21 -^—\\ + etc. H-^^^Tal r=:i|), which by formula (1) is equal to ix/,ir|f|. We therefore see that an annuity of the (r— 1)'* order forborne, will produce for n years an annuity of x of the r^ order, and in addition there will of course remain in hand at the end of the n years the aggregate of all the 232 Theory of Finance. [chap. iii. n payments of the original annuity, the aggregate of these payments being, by formula (1), t;{;^\\ ^i, and the present value of the aggregate being «;»* ^^i| ^| . Thus we reach the result, a^ r=i| =ia-^ r\ +^" t^W r\ \ whence, as in formula (10), "71I r\' ^ '■ — Similarly for perpetuities. If the payments of a perpetuity of the (r— 1)"^ order be forborne, each of them will produce another perpetuity, the mF" payment producing a perpetuity of i/^| ^rrji, and the aggregate payment of these secondary perpetuities to be received at the end of the m"^ year will be ^ (^r| 7^\\ -k-k\ ^li + etc. '\-t^^\\ r-\\), or by formula (1), ^^^nj f|. Therefore a~\ fziii ='iti^ r\ ', whence, as in formula (11), a-\ r\ =&EII. % It will be noticed that in the argument in this article we have escaped the difficulty which occupied us in Arts. 21 and 23. 25. To find the amount and the present value of an annuity whose successive payments are Ui^ u^, u^j etc. If for u^, Us, etc., we substitute their equivalents in terms of u^ and its diff'erences, we have lh = Ui "'^^X^illV, _^ . U2=Ui-\-Aui ■"- -' ' '■'■ Us=Ui-\-2Aui-\- A^u^ u^=u^-{-3Au^ + 3A^u^ + A^Ui ; and generally u^=u,-]- (m-l)A2^,+ ^'^"'"^li'^~^ W+ etc. Therefore the given annuity is equivalent to an annuity of Ui of the first order, together with an annuity of Auj of the second order, and an annuity of A^u^ of the third order, etc. Let s and a be the amount and the present value respectively of the annuity u-i, u^, u^, etc., and let the series u-^, u^, u^, etc., have (r— 1) orders of differences : then 5=5^11^*1 + Sn\ 2 1 Aui + s^ 3| A^^u^ + etc. + 5i! r\ A''-% (13) T- <^=S ri ^1+^^ 2 1 ^^: 4-«Hi 3] A^Ui-\-etc. +a^ fj A'-% (U>^ If the given annuity be a perpetuity, then ^=^^1 r| Ui+a-^ 2 1 ^^1+ etc. +a— i ^ A^'-^u^ -T+12-+ • • • +-ir- .... (15) 26. The great power of the formulas demonstrated in the last article may best be illustrated by some examples. i CHAP. III.] On Variable Annuities, 233 a. Let it be required to find the value of an annuity certain for n years whose several payments are 1, 2, 3, etc., n. Here ?*i = l, A2(i = l, and all the higher orders of differences of u^ vanish. Therefore a=a^ ri +a^ 2] _ (14-t)fliin-^^ i p. Similarly, if it be required to find the amount of the same annuity, i y. If the annuity be a perpetuity increasing 1 per annum for ever, 6. Let it be required to find the value of an annuity for forty <^ years, whose several payments are 4, 7, 12, 19, etc. ; interest 5 per ^ cent. Here 2*1 = 4, A2ii = 3, and A%i = 2. Therefore ft=4a4cn 1] +3a4(r| 21 +2aio'l 3l Taking the values of the annuities as given in Art. 18, a=5507*253. ^;j €. Find the value of an annuity for fifteen years whose several ^ payments are 21, 39, 54, 66, etc. Here2*i = 21, Aw, = 18, ^hi^ = — ^. Therefore ft = 2 1 ai5-| i] + 1 8ai5-| 2 1 — 3fti5"| 31 This is an annuity which increases till the maximum payment is 84, and which then decreases till the fifteenth payment v;^ishes. 27. A lease is granted for ten years at an ann^d rent, with power to the tenant to renew ten times for a Ijk^ period on pay- ment of a fine of 1 for each year of the lease^ expired. What is the value of the fines % Interest 5 per cenfe"^ Here the fines are respectively 10, 20, etc., 100, payable at the end of 10, 20, etc., years. A sum payable periodically at the end of 10, 20, etc., years may, in accordance with Chapter II. Art. 30, be looked upon as an annuity at rate of interest {(14-*)^^ — 1}=J say. If the annuity at rate j be denoted by a'io"| , we have the value of the fines lOft'io"! n + lOft'iol 2^^. Calculating the values of these annuities when i=*05, and consequently y= '6288946, we shall find the value of the fines to be 39*66. 28. The following example is an instructive one : — An annuity-certain deferred twenty years, and after that to run twenty years, is to be paid for by an annual premium, the first payment, ir, to be paid down now, and afterwards, at the beginning 234 Theory of Finance, [chap. hi. of each year, a regularjy^diminishin^ amount, the last premium being paid at the beginning of the twentieth year, after which the premium becomes extinct. Find tt, having given ^''=•456387 at 4 per cent, interest. Here the benefit to be received is an annuity for twenty years de- ferred twenty years. Its present value is therefore v^'^a-^\. The con- sideration for this benefit is an annuity for twenty years, commenc- ing at TT, and decreasing — - each year. If this annuity were pay- able at the end of each year, its value would be i^{a-^\ \\ —-^oj^\ 21) but, as it is payable at the beginning of each year, its value is ''■(1+0(^201 r|— n7;^2o'| 2 1 ). The benefit and the consideration for the benefit must be equal in present value : whence ■(1 Y%) U20-J n -20^20"! 21 j=«'''fl^2o'| n and TT 'o^o^-^\ n uXAOV ^'"= '456387 13-5903 783654 OSf^ ^ a2o-|r|= 13-5903 ^^^^^ 20^20= 9-1277 6795 '7 .f 4-4626 ^-04 - ^^^ _^^\^ a2o-|2-i = lM-565 ^ ^^ ^^r^ «2olr|= 13-5903 2 -^^^' ]^ V- i-a2oi.^= ^'5^83 ^20^(20-11-1= 6 -2023 20 8-3325) 6-2023 (-744.35 . "'>' 8-0120x1.04 3695 lxr;:v^" ^ 3205 362 (1 +^) («^2o"! n "20^20^1 2| j=8-3325 29 .4 7r= -74435 29. In former days, when De Moivre's hypothesis as to the law of life was of greater importance in the science of life contingencies than it now is, attention was much directed to the annuities-certain the payments of which are the powers of the natural numbers. Mr. Baily devoted two chapters of his work to this subject. CHAP. III.] On Variable Annuities. 235 By means of the formulas of this chapter, the amounts and pre- sent values of such annuities can be readily found. Thus, let it be required to find the value of an annuity, the payments of which are 1*, 2S 3* ,etc. Here 1* = 1 2* = 16 3* = 81 4* = 256 6* = 625 A A» A3 A* , 15 50 60 24 1 65 110 84 24 1 175 194 108 369 302 671 A« 6* = 1296 Therefore the value of the annuity is a^ 1] +16a^ 2| +50a^ ^ +60^^^ 4| +24ai| 51 To find the amount of an annuity for n years, the successive payments of which are ?i% (71— 1)^ (71— 2)% etc. Hereof 1=71' A7fl=— (371^ — 37i + l) A«Wi = (67i— 6) Ax=--6 and s=7i35^ n— (3712^371+1) s^ 2|+(67i-6)sr^ 3| -^H i\ 30. Formulas (13), (14), and (15) apply only to annuities where the differences of the payments vanish after a finite number of orders, and it is not sufficient if the differences merely rapidly diminish but do not vanish. The {r—Xf" difference is multiplied into an annuity of the r*^ order, and we have seen that the amounts and values of annuities of the higher orders are very large. There- fore, although the (r— 1)"^ difference may be very small, its co- efficient will be very large if r be large, and the product will be a quantity which cannot safely be neglected. 31. Let us take the following case : — A A2 1st payment. 1-000000 -010000 -000100 2d „ 1-010000 •010100 •000101 3d „ 1-020100 •010201 etc. 4th '„ 1-030301 etc. etc. ' etc; A3 A* -000001 000000 These terms are in geometrical progression, the ratio being 1*01, and the differences diminish with considerable rapidity, the fourth having no significant figure in the sixth decimal place. Let it be required to fiiid the value of the perpetuity at 5 per cent by means of formula (15). 236 Theory of Finance. [chap. iii. A%i= -000001 -f -05 - ^^- •000020 I A22^ = -000100 •000120^-05 •002400 'i . .1 Awi= -010000 •012400 -r •OS ^^:.^^-^' •248000 u,=: 1-000000 1 -248000 -r -05 24-960000 The true value of the perpetuity is 25-0, so that although the first difference neglected is only -0000001, there is a considerable error in our result. This is accounted for by the fact that the coefficient of the first neglected difference is — or 3,200,000. Of course theoretically any required degree of accuracy could be obtained by computing a sufficient number of terms to a sufficient number of decimal places, provided that the differences of the series of payments diminish faster than the increase in the coefficients ; but more convenient formulas for such cases we now proceed to find. 32. From the example in the last article, we infer that formulas (13), (14), and (15), are not applicable to annuities which increase or decrease in geometrical progression ; but it is easy to give a general demonstration of the fact. Let there be a geometrical series, the first term of which is K 1)' and common ratio R. and its differences : — Term A The following scheme shows the A2 A3 A^ K K{R-\), KR KR{R-\) KR^ KR\R^\) KR^ KR\R-l) KR^ etc. K{R-iy. K(R-\f K{R- KR{R-IY KR{R-lf etc. KR\R-\f etc. etc. etc. It therefore appears that if the common ratio of the original series be R, the successive orders of differences form another geometrical series with the common ratio (-K— 1) and the diff'erences CHAP, m.] O71 Variable Annuities, 237 can never vanish. Therefore, by Art. 30, formulas (13) to (15) are inapplicable. 33. To find the value of an annuity, the payments of which are in geometrical progression with common ratio 11. If a be the value of the annuity, we have rt=(l+i)-i+7^(l+j)-«4./i:3(l-|-i)-3+ etc. +^^'-^1+*)"" i {7^(l+i)-^+i?«(l+0-H^'(l+0-'+etc 4-^'(l+0- -{M' .. -"^ _ U+M (16) . , 11 1 ,, , . (l+*)-i^ J i Let^^^=^^-^,sothat;=^— , ^ Then i.wi±zt! . . . :- . (17) M J If the annuity be a perpetuity, we can find its present value only when — -!l^<1, that is, when E<(l-\-i). In other cases the l"r* present value will be infinite. When B<{l-{-i) and the annuity is a perpetuity, then i^.n^f^ ^= 1_ (18) U ^^ (l+*)-i^ / v>»^ jjj ^]^Q example in Art. 31, it=r01, and the value of the perpetuity at 5 per cent, is therefore, by formula (18), or 25. Thus the value of the increasing perpetuity is equal to the value of a fixed perpetuity of 1 calculated upon the assumption that the rate of interest is diminished by the rate of increase of the pay- ments. 34. From formula (17) we see that by changing the rate of interest, we can substitute for an annuity, the payments of which are in geometrical progression, another annuity, the payments of which are uniform. The changed rate of interest may, however, present anomalous features. If R be greater than unity, then j\ the substituted rate, will be less than ? ; and if 7/ be greater than 238 Theory of Finance. [chap. hi. (1 +i), then/ will be negative. If E be less than unity, then J will be greater than % ; and there is no limit to the magnitude which j may assume when E is diminished. 35. It is to Mr. William M. Makeham that we owe those formulas in this chapter, by which, with the aid of the figurate numbers, we can deal so effectively with variable annuities. Previous writers had investigated particular cases, but Mr. Make- ham has furnished the general theory. He published it in a re- markable paper in /. /. A. vol. xiv. p. 189. THEORY OF FINANCE. CHAPTER IV. On Loans Repayable by Instalments. 1. In Chapter II. we gave full consideration to loans which are repayable^within a limited term by equal periodic instalments, including principal and interest. It is the object of the present chapter to extend our investigation to cases in which the capital is repayable in any other manner whatever. The formulas of Chapter III. will be of great value to us in our inquiry. The analysis divides itself naturally into two branches which are of equal importance ; namely, first, knowing the conditions of the loan, to find that value for it which will secure to an investor a given rate of interest \ and secondly, to ascertain the rate of interest which the loan yields at a given market price. 2. When a corporation or a foreign government contracts a loan, it generally undertakes obligations of a twofold nature. It agrees to pay the lender interest at a fixed rate aa long as his advance remains outstanding, and it promises to repay at stated periods the capital itself. Frequently the borrower holds out to investors inducements beyond the stipulated rate of interest, by issuing the loan at a discount and repaying it at par, or by issuing it at par and giving a bonus on repayment. Sometimes, on the other hand, where the credit of the borrower is good, he may find it to his advantage to raise the loan at a higher rate of interest than the public demands from him, and therefore to issue his bonds at a premium. These complications, however, will not render our investigations much more intricate. 3. Let C=the capital repayable by the borrower. y=the nominal rate of interest thereon paid by the borrower. i=the actual rate of interest realized by the lender, whom we may also call the investor, or purchaser. A'=the present value of the capital at rate ?. CJ^vM'- y^ = the purchase-money, or the value of the loan. 240 Theory of Finance. [chap. iv. To illustrate the symbols, let us suppose a loan to be contracted for £10,000,000 at 3 per cent., by ten thousand bonds of £1000 each, the bonds to be paid off at maturity with a bonus of 25 per cent. Here the capital repayable by the borrower is really £12,500,000, which sum we therefore represent by (7, and the nominal rate of interest is not -03, as would appear from the stated conditions, but ^f ^ , or -024, which we represent by j. The loan being nominally issued at par, we have ^ = 10,000,000. The points to be noted in connection with our symbols are, that C represents the capital repayable by the borrower, including any bonus he may contract to pay along with it, and that j represents the ratio between the annual interest contracted for and the capital repayable as defined above. We must therefore be careful to dis- tinguish, as in the foregoing example, between the nominal rate of interest as stated in the conditions of the loan, and the nominal rate j which concerns us in our investigations. These rates will in many cases be identical, but they are not necessarily so. 4. To find the value of a loan, repayable by instalments at stated periods of time, with interest in the meantime at ratej, so as to yield the purchaser a given rate of interest, i. The value of the loan consists of two parts, the value of the capital and the value of the interest. The value of the capital at rate i being K^ the value of the interest is evidently (A—K). Had the borrower contracted to pay interest at rate i, then the loan would have stood at par, and A would have been equal to C ; and there- fore the value at rate i of the interest in this particular case would have been (C—K). The annual interest on the loan would, on the same supposition, have been iC, and the value of each annual unit of interest payable by the borrower is therefore -^7^—. But the borrower has actually contracted to pay interest at rate j instead of i, or jC annually, and the value of this interest is ^— (C—K), or ^{C—K). Adding to this the value of the capital, we have the value of the whole loan, A=K+l(C-K) (1) V 5. It will perhaps enable us better to understand the demon- stration of the last article if we confine our attention for a moment to a single unit of the loan. Suppose that unit to be represented l)y Ci, and to be payable at the end of 11^ years, and let its value be CHAP. IV.] On Loans Repayable by Instalments. 241 /f,, so that ir, = (l+i)-»*i. The interest payable on the unit is an annuity of; per annum for n^ years ; but the value of 1 per annum for 7i, years is - ""^ .^' — ^, or — "7 ^ and therefore the value of; per annum is -i(Ci— ^1). Adding to this the value of the unit of the capital itself, we have | ifi+i (Ci— /iTi) i , the total value of that particular loan-unit under consideration. If now we take into' account the other units C2, C's, etc., due at the end of ?ia» w,, etc., years, we have corresponding formulas [k.-^-L {0,-K,)\ , ji^s+i (^3-^3) I , etc., and, adding to- gether the values of all these portions of the loan, we have for the value of the entire loan (A^i+ir,+-^s+etc.)-f4|((7i+C2+C3+etc.)-(^i4-^2+^3+etc.)| or I K-^ 1-{C-K) I as before. 6. In Art. 4 we have treated the capital of the loan as a whole. For purposes of calculation it will often be convenient to represent the entire capital by unity, so that we write C=\. Under these circumstances formula (1) takes the elegant form A=K+i(l-K) = l-(l-/0(l-j) (2) 7. In formulas (1) and (2) we do not limit the repayments of capital to any particular conditions. AVe simply find the present value, K, of the capital, under whatsoever arrangements it may be repayable, and insert it in the formulas. The formulas will take various shapes according to the various methods by which loans may be liquidated. Thus, if the loan of unity is to be repaid in one sum at the end of n years, we have A^\-(\-if')U-i\ = l-a^(i-j) ^ (3) where a^ is taken at rate ?'. Again, if the capital of 1 be repayable by n equal annual instal- ments of — each, then / y 242 Theory of Fifiance. ^ [cuAr. IV. Also, if the capital of C be repayable by annual instalments, u^ at the end of the first year, 2*2 at the end of the second year, and so on, then in formula (1), K='m^-\-v'^u^-\-xi^u^-\- etc., and if the payments i/j, w^, etc., form a series the differences of which vanish after a finite number of orders, then by Chapter III., formula (14), i^=o^^ n Wi+a^i 2 1 Awi+0^^ 3-1 A2m,+ etc. . . (5) 8. The following examples will elucidate the subject : — (a.) A bond for £1000 is to be sold. It bears interest at 3 per cent., and will be repaid at par in twenty years. An intending purchaser desires to make 5 per cent, on his investment. What price can he afford to give for the bond? Here formula (3) is applicable, writing a-^-=a'^\ at 5 per cent., and i=-05 andy=-03. ^ = 1000{l-«2o-:('05--03)} = 1000{l-12-462x-02} = 750-76. (/?.) A loan of £1000 is to be repaid as follows, with a bonus of 25 per cent. : — One twenty-seventh at the end of four years, five years, and so on. Finally, one twenty-seventh at the end of thirty years, interest being payable in the meantime at the rate of 6 per cent. What price must a purchaser give so as to realize 5 per cent, on his outlay % Here in formula (1) we must write C=1250 and y=^-— ^, or -048; also, ii:=!^!||lx 1250, at 5 per cent. logv« =1-93643 C -i<:= 664-39 logffi2f| =1-16563 840 log 1250 =3-09691 i,c- "26576 4-19897 5315 log 27 =1-43136 -05)31-891 .ir) = 637-82 log^ =2-76761 K = 585-61 i(C-K)= 637-82 ^ = 1223-43 y. A loan of £10,000 is repayable as follows : — £94 at the end of the first year, 102 „ second year, 110 „ third year, CHAP. IV.] Oji Loans Repayable by Instalments. 243 and so on till the whole loan is liquidated, interest at the rate of ^ 3 per cent, being^allowed in the nieantime^on^theoutstaaiimg j| capitat~ "Keijuired the"value of thoToan at 5 per cent. We must » fireTTind the term of the loan. By the well-known formula for summation in the calculus of Finite Differences, if there be a series of n terms, the first of which is Wj, the sum of the series is ^i^j ^.^i!?^ JAmi -f etc. In the present case the sum of the series is 10,000, Ux is 94, and A?^i is 8. Making use of these values, we have a quadratic equation in ?i, namely, 47i' + 90^^—10,000=0 ; whence w=40. The capital repayments are therefore of the nature of an increasing annuity for forty years — first payment 94, second payment 102, third payment 110, etc., and by formula (5) we have iir= 94(240] ri+8a4o'| %. In Chapter III., Art. 18, we have already calculated the value of ai^\ g] , and employing those figures, we have A'= 3449*316. Making use of this value in formula (1), ^ = 3449'316+4x 6550-684 o = 7379-726, or £73, 15s. lljd. per cent. 9. In the foregoing investigation we have divided the loan into two parts, the capital and the interest, and we have expressed the value of the interest in terms of the value of the capital. We might have pursued a different, and apparently a more direct course, and valued the capital and interest separately, but the effect of the method of solution which we have followed, and which is due to Mr. W. M. Makeham, is generally to reduce the series to be valued by one order of differences. Thus, suppose we take the case of the capital of unity repayable by n equal annual instalments of — each, — formula (4) — and value the capital and interest independently. The value of the capital is ^. The interest payable at the end of the first year is 7', at the end of the second year jl W, at the end of the third year [ 1 V, and so on. The value of the interest is therefore /a^ij r|— -^ril 2l> and the value of the whole loan A—a^X — {-/)— •^-^^. This formula, although it involves an \n j n annuity of the second order, is really identical with formula (4). i Theory of Finance. \ [chap, i v. Thus, ^«^^(i+-'-i)+(i+^)-"47 n\ I j I = 1-^ _^Vl_i\ as in formula (4). 10. AVe pass now to the converse problem: — Having given the value of the loan, to determine the rate of interest. 11. We have by formula (1), A=K-^L{C-K\ whence, by simple algebraical transformation, _i=i^-| ....... (6) In this equation, seeing that i is the unknown quantity, and that K is calculated at rate %, we cannot assign the true value to /f, but if we find an approximate value for Z, and insert it in the formula, we shall obtain an approximate value for i; and the nearer the assumed value is to the true value of K, the nearer will the resulting approximate value be to the true value of '%. 12. As an illustration of the formula, let us take the following example. What rate of interest does a Government loan issued at 73 per cent, yield when it is redeemable at par by uniform annual drawings of 2 per cent. % Here the 3 per cent, paid by the Government yields a little over 4 per cent, on the issue price of 73, and, in addition, the lender will on repayment get a bonus of 27 on each 73 invested, and this is'^ evidently; equal to fully 1 per cent, per annum additional in- terest. We may therefore assume, to begin with, a rate of 5 per cent. The formula becomes i = -03 x ^^ ~n — •> where ^501 is taken as a trial at 5 per cent. The result is, i = -03 x -^ir-r^ ^ = "05349. The assumed rate turns out to be too low, and, to get a closer ap- proximation, we might insert in the formula the value just found for i, and work it out again. We can, however, proceed in another CHAP. IV.] On Loans Repayable by Instalments, 245 way, which will often, by the help of interest tables, be more easily applied. Assuming a higher rate, say 5 J per cent., we find i to be -05070, and from the two approximate rates for i we can find a third more . ^ar than either. Thus 5 gives 5*349 per cent. 5-5 gives 5-070 per cent. Diff. -5 gives Diff. - -279 •279 whence 5-|-ic=5-349— -—a:, nearly, whence a;=-224, and the rate which we seek is 5*224 per cent, very nearly. 13. The rationale of the method of approximation above illus- trated is easily seen. If the trial rate, which we may denote by /„ were the true one, then that rate itself would be the result of making use of it in the formula, but, seeing that I^ is only approximately true, our result, which may be written J^, is also only approximate. If now we assume another trial rate, /j, differing from /i by A, say, we get another approximate result, J^, diff'ering from J^ by 8, say. Since a change of A in the trial rate produces a change of S in the resulting rate, therefore a change of x in the trial rate will produce a change of -y-x in the resulting rate. We seek a rate which, when used in the formula, will reproduce itself, and, if that rate be /i -fa*, we must therefore have I^-\-x=Jx-{-^x^ whence x= — x^ — ^^^ This method of approximation may often be advantageously resorted to with other formulas than that to which we have just now applied it. / 14. It is very easy to show that formula (6) of the present (chapter is an extension and generalization of Baily's, No. 35 of Chapter II., which applies only where the loan is repayable by equjd annual instalments, including i)rincipal and interest. In the case of such a loan we have, by Chapter II., Art. 34, the capital included in the m"" payment equal to <;"-»»+». The present value of the capital in the w'* payment is v*^v^-'^+\ or i^+\ or (l + O'^*^*) ; and there being n such payments, the present value of the whole of 246 Theory of Finance. [chap. iv. the capital is %(l+i)-('*+i). Now, if an appropriate rate / be obtained by inspection of the tables, we have in formula (6) C = i^=^^i±^, and ir=?i(l+ /)-("+!), while A is the value of the loan. Therefore l_ (!+/)-« ^ , , -{ l-(H-/)-- ^ 1+ ^ ] whence 1—1= Ix ^^(^+^) -A In Chapter II., formula (35), we have denoted {i—I)hjp,I by J, ^7^ ' by a', ^ by a, and ?z(l +/)-("+!) by 7i?;«+^ Making these substitutions in the ^bove formula (8), we at once have formula (35) of Chapter II. ] 15. Just as Barrett's formula — No. (38) of Chapter II. — is an im- provement on Baily's, so we may obtain an improvement on No. (6) of this chapter. We have seen, Art. 4, that {C—K) is the value of the interest when it is payable at rate i. Therefore —{C—K) is the value of the interest for each unit of the rate, and ^ is the {C—K) annual rate of interest for which a payment of 1 down will pro- •C-A vide. Hence i— — is the extra annual rate of interest for which L> — K the discount {C—A) will provide, and this extra rate, which we may denote by A, must be added to j, the rate actually payable, in order to get i, the rate realized by the lender. Seeing that i is unknown, we may take a near value, and denoting that by /, we have approximately h=I ^~^ (9) C-K ^ ^ 16. We have shown that formula (6) is a generalization of Baily's No. (35) of Chapter II., and now we can similarly show that formula (9) is a generalization of Barrett's No. (38) of Chapter II. As in Art. 14, we have C equivalent to a', A equivalent to a, and K equivalent to %«;"+\ while we have now denoted by h that which we formerly wrote p, and by / that which CHAP. IV.] On Loans Repayable by Instalments. 247 a —a we formerly wrote/ Formula (9) therefore becomes p=j— :;^.^ which is identical with Barrett's formula. 1 7. Following precedent, we shall close this chapter with a few examples. (a.) A loan of £2,000,000 is issued, repayable as follows : — 50,000 at the end of 5 years. 60,000 „ „ 6 „ 70,000 „ „ 7 „ and so on till all is repaid, interest on the outstanding amount being allowed at the rate of 5 per cent. The issue price of the loan is 93 per cent. What does the loan cost the borrower % Here, by a process similar to that followed in example y of Art. 8, we find that the loan will all be paid off in 16 years after the repayments commence. The repayments are in the form of an annuity, and as the first payment of that annuity will take place at the end of five years, the annuity is deferred four years. We therefore have for the value of the capital, after dividing by 10,000 for the sake of brevity, K=v^{^a\^\ •\-a^\ 21), while C=200, and ^ = 186. If we use formula (9) and take for a trial rate 5 J per cent., we have ai6-| =10-46216 16«;^«= 6-79330 -055)3-66886(66- 706 = =ai6|2-i 368 388 360 5ai6|= 52-311 ai6^^^i= 66-706 ._.055x 200-186 ^-^^^^200-96-0725 119-017 reversed 712708 -•^^^^103-9275 952136 log -055 = 2-7404 8331 „ 14 =1-1461 238 i-8865 12 log 103-9275 = 20166 8 log A= 3-8699 ^^= 96-0725 yi= -007401 ; =-05 I = -057401 = 5 -7401 per cent. 248 Theory of Finance, [chap. iv. The trial rate of 5| per cent, is evidently too low. If we try again at 6 per cent, we have a'^\ 21 = 63*459 and ^=90*2899, and the rate brought out by the calculation will be 5*7656. Applying now formula (7) we have /i = 5*5 per cent., /i = 5*7401 per cent., A=-5, and S=*0255, whence the final rate i = 5*753 per cent. (|8.) Colonial 5 per cent. Government bonds repayable at par in 19 years a^e quoted in the market at 107f per cent., after making allowance for the interest accrued since the last payment of dividend. What rate of interest do they yield % and, to yield the same return, what should be the price of 4 per cent, bonds repayable at par in 25 years'? For the first part of the question, using again formula (9) and trying 4 per cent., we have (iir being equal to 47*464), i= 4*410 per cent. Trying again i\ per cent, we have (Z" being equal to 43*330), 2* =4*385 per cent. Interpolating by means of formula (7) we finally have for the rate yielded by the bonds 4*390 per cent. For the second part of the question we may conveniently employ formula (2) where y=*04 and i=*0439, and where A" is iP taken at rate i, or -34161. The formula becomes ^ = 1 — -65839 X ( 1 - jtoT) ) or 94-151 per cent. 18. In practically applying the formulas of this chapter and of those that precede, circumstances may render necessary various modifications ; and it may also frequently happen that there is no formula given which will directly meet the case in hand. Tlie principles will however remain constant, and the actuary who has fully mastered the principles will find no difficulty in adapting the formulas to special conditions. In the aff'airs of life mathematical rules cannot be made rigidly to apply, and the actuary, having made himself thoroughly acquainted with the mathematical rules, must never fail to exercise a sound judgment when he comes to make use of them. • i ' "'~ -^-^-~r-r--i---~-- ■'*'■'■■' ■ ........„„ . 1 :.^ 1 1 1 1 1 • "'■ ( i. ■ ^25*^^ THEORY OF FINANCE. CHAPTER V. On Interest Tables. 1. In the preceding chapters we have given formulas by means of which all the values to be found in interest tables could be in- dependently calculated as required, but it is evident that such a process would Ije tedious, and that the convenience is great of having those values that will be commonly wanted ready prepared and presented in tabular form. The solution of many problems too-^ such, for instance, as finding the rate of interest involved in a term annuity — is rendered very much more easy by a reference to tables, and to the skilful actuary they are at all times invaluable auxili- aries in his work. It will be frequently noticed that where a novice goes through an intricate calculation in answering a question, the adept produces the same result seemingly without thought or effort ; and on inquiry it will usually be discovered that tables are the tools he uses to shorten his labour and save his time. 2. An intimate acquaintance with the nature of the tables he may find in his hands is essential to the actuary, for without that knowledge he cannot turn them to the best account. It is therefore very desirable for him to practise the construction of tables for himself, although those he requires may already be in print, as by actual experience in their manufacture he will much more readily obtain a clear knowledge of the properties of his tools, than by only theoretical study. The principal object of the pre- sent chapter is therefore to explain the best methods of constructing and verifying interest tables, and it will for the most part be left to the reader to gain for himself practical skill in using them, by studying the first four chapters of the work with the tables in his hands. Sufficient tables are given at the end to enable him to do so without going beyond the pages of the book itself. 250 Theory of Finance. [chap. 3. In interest tables the functions most commonly tabulated for each rate of interest are the following : — (i.) (l+^)", the amount of 1 in ?i intervals, (il) if-^ the present value of 1 due at the end of n intervals, (iii.) %;, the amount of an annuity of 1 for n intervals, (iv.) «^|, the present value of an annuity of 1 for n intervals. — =s~^ the sinkiner fund which will redeem a debt of 1 in n intervals ; or, in other words, the annuity which will accumulate to 1 in ti intervals. z=a~J-, the annuity for n intervals which 1 will pur- (V.) (vi.) a^ chase. The values under each heading are arranged in columnar form, commencing with the value for one interval, and finishing generally with that for one hundred intervals. 4. The following is an example of the form which the interest table sometimes assumes : — Interest 5 per cent. (i.) (ii.) (iii.) (iv.) (V.) (vi.) n (i+^r ^« H ^^ n\ 1 1-050000 -952381 1-0000 •9524 1-000000 1-050000 2 1 -102500 -907029 2-0500 1-8594 •487805 -537805 3 1-157625 -863838 3-1525 2-7232 •317209 •367209 etc. etc. etc. etc. etc. etc. etc. 5. All tables are not an*anged in the same manner. Frequently the functions, all at one rate of interest, are placed in parallel columns as in the specimen in Art. 4, so that there is a distinct table for each rate of interest. Sometimes each function is kept by itself, the rates of interest being side by side. Such are the tables which are given at the end of the book. 6. Corbaux in his work, published in 1825, Doctrine of Comj)oimd Interest, supplies very complete tables. He gives all the columns (i.) to (vi.) mentioned in Art. 4, for each rate of interest rising by J per cent, from 3 per cent, to 6 per cent. ; and not only does he do so, but for each rate of interest he also gives the values of all the CHAP, v.] On Interest Tables. 251 functions when interest is convertible either half-yearly or quarterly. Some tables, instead of supplying the values for interest con- vertible half-yearly or quarterly, very minutely subdivide the rate of interest, and also begin with very small rates. In this way they answer the same purpose as Corbaux's tables. Thus an annuity of 100 per annum payable quarterly for twenty years, at 4 J per cent., interest convertible quarterly, is equivalent to an annuity of 25 per annum for 80 years at 1| per cent., interest convertible yearly. We therefore see that in speaking of interest tables, it it more .approi)riate to name intermls, as we have done, rather than years. The tables of Colonel Oakes, published in 1877, are of the last- named description. They furnish columns (i.) to (iv.) at each rate of interest, rising I per cent, from f per cent, to 10 per cent. Such also are the tables of the late Mr. P. Hardy, F.R.S., published in 1839. They contain columns (i.) to (iv.) for rates beginning at \ per cent., and rising by J per cent, to 5 per cent., and also for 6, 7, and 8 per cent. Ranee's, published in 1852, are of the same kind, and give cols, (i.) to (iv.) for rates beginning at J per cent., and proceeding by steps of J per cent, to 10 per cent. 7. It may be noted here that although some published tables give all the six columns (i.) to (vi.), it is not really necessary that columns (v.) and (vi.) should both appear. We saw. Chapter II., formula (29), that al^— ^=s^^ If therefore a~\ the annuity which 1 will purchase, be given, we can at once, by deducting the rate of interest, find the sinking fund. We therefore, in the specimen tables at the end of the book, have not included column (v. ) of Art. 4. 8. We have said that it is usual to tabulate the functions for all values of n from 1 to 100, but for ordinary purposes a table of such extent is not essential. It is not often in practice that the values and amounts of annuities are required for so long a period as 100 years. A table, too, may be effectually used for longer terms of years than it actually includes. Thus, suppose there is a table which goes up to 50 years only, and it is required to find the values of the functions for 60 years. We have (l+i)«'=(H-i)«'x (1+^)10, ^=^xv'', aw\=a5o\-¥^aTo\,, and ,siTol =(1 +i)^°s3o'| +510^1 . It is, however, convenient to possess exten- sive tables, especially if they are of the description of those of Oakes, and not of Corbaux, and we must use them for intervals instead of years ; because, for example, an annuity for 25 years at 252 Theory of Finance, [chap. v. 5 per cent., payable quarterly, is equivalent to one at IJ per cent, for 100 intervals. 9. Insurance premiums are payable in advance, and thus are of the nature of annuities- due. It might therefore be thought useful for the purposes of life offices to tabulate the values and amounts of annuities-due instead of those of ordinary annuities : but the more usual form of tables, which is that we have adopted, provides for every object. Thus the amount of an annuity-due for n years is evi- dently s^izjiii — 1, which can be found by inspection from a table of the amounts of ordinary annuities. For instance, to find the amount of an annuity-due for 25 years at 4 per cent, interest, we enter table 3 with 26 years, and, deducting unity, the result is 43-31174. Again, to find the sinking fund payable at the beginning of each year to redeem a debt of 1 in % years, we can find the sinking fund payable at the end of each year and multiply it by v. For example, to find at 4 per cent, interest the sinking-fund payable at the beginning of each year which will redeem 1 in 25 years, we use table 5, and find first the annuity which 1 will purchase for 25 years. This is -064012. Deducting the rate of interest, -04, we have -024012, the sinking-fund payable at the end of each year to redeem 1 in 25 years. Multiplying now by -961538, the value oft* taken from table 2, we have -023088, the quantity required. This might be obtained equally easily in another way, namely, by find- ing the amount of an annuity-due for 25 years, and taking the reciprocal by means of Barlow's or Oakes's tables of reciprocals.^ 10. To construct a table, the most obvious course is to calculate independently each value that is to be tabulated ; but that course would very seldom be the best to pursue. It would usually be very laborious \ and, besides, in order to insure accuracy, the whole work would have to be done in duplicate. It is generally much l)referable to take some formula connecting the consecutive values of the function, and by means of it to compute them one from the other in succession. In this way each value is made to depend on all that go before it, with the consequence that if an error occur in one it is carried on to those that succeed, and we can therefore feel confidence that if any particular value be correct, all those that go before it are correct also. This method of constructing tables is * By means of Orchard's Tables, the sinking fund payable at the beginning of each year to redeem a debt in n years, can be found by inspection. We have only to enter the table of annual premiums with an-i|, and the result is the sinking fund required. It is beyond the scope of the present work to describe Orchard's Tables. CHAP, v.] On Interest Tables. 253 called the " continuous method," and when it is used, a periodical check, say at every tenth value, is all that is required. 11. To employ the continuous method three things are necessary. We must have a convenient working fm'mula connecting the value of the function for n years with that for (^ + 1) years : we must know the imt'ml value on which all the others are to be built : and we must have a verification formula by which to apply our periodical checks. In interest tables these three requisites are simple, and easily ob- tained. In many other tables they do not present themselves so obviously, but the computer will find it to his advantage to seek them in order to construct his tables continuously, on account of the great facilities which the continuous method gives for insuring accuracy. 12. To construct a table of (1+i)'*. This table, for the majority of the rates of interest in use, can best be formed by direct multi- plication. The values in the column are connected by the relation (l-\-i)"-^^ = {l-\-i) (l-\-iy\ and i being a small quantity not usually consisting of many digits, multiplication by (1 + *) is easy. We add to (l+O" *^^^ result of its multiplication by i, and so obtain (l-f/)""*"*. This is our working formula. The amount of 1 in one year is (1+0j ^^^ initial value. To check our work we must calculate by means of logarithms, say every tenth value ; or, if we construct the tenth value by logarithms, we can form the twentieth, thirtieth, etc., by raising the tenth to the second, third, etc., powers by ordinary contracted multiplication. 13. When i consists of but one significant figure — foi when t=-04 — the work of making the table is very easy. A type of the operation is given in the margin. The number of decimal places will go on increasing indefinitely unless the increase be checked. When we have obtained as many as we require, we must, as in the example, cease to allow them to extend, merely 2. r0816 taking account of the proper carriage from the neg- 43264 lected figures to the figures which we retain. Thus, 3^ 1*124864 in the marginal example, the result of multiplying the 44995 last two figures of (l-f*)" ^Y '^^ is 256 : we neglect , i~ifiQQKQ the 56, and as the figures neglected are greater than ' 46701 49, we increase the next figure by unity, carrying 3 instead of 2. In order to insure accuracy in the last ^ 1*216653 decimal place of the tables, we must work to two * * 254 Theory of Finance. [chap. v. places more than we mean finally to keep, and when our work is finished we must cut down the results to the required limit, taking care always, when the value of the rejected figures is greater than 49, to increase by unity the last place (1+^)" retained. Thus, if we wish to cut down 24647155 4| per cent. to five places of decimals, we should write 2*46472, whereas to cut down 2*4647145 we should write 2*46471. 1. 1*0425 41700 2606 14. Where i consists of more than one significant figure, we can generally find a short method of multi- plication. Thus, if i=-0425, we can divide by 4, instead of multiplying by 25, of course correctly placing the result as regards the decimal point. Again, if the rate be 4f per cent., we can divide by 6 the result of the multiplication by 4. A few lines of each of these examples are given in the margin. 15. Where % is such a number that multiplication becomes troublesome, recourse must be had to logar- ithms. This leads us to the next problem. 16. To construct a table of log (1+*)". Because log (l+i)" = 7i log (l+^), the table con- sists of the successive multiples of log(l+i), and these may be formed most conveniently by addition. The value of log(l-|-i) should be written at the top of the column and again at the foot of a card, which is moved down as the additions are performed. A verification is naturally obtained at every tenth value, the tenth being ten times the first, the twentieth ten times the second, etc. ; the figures of each pair, there- fore, being the same, the decimal point only being moved. 2. 1*086806 43472 2717 3. M32995 45320 2832 4. 1*181147 47246 2953 5. 1*231346 % ^ % (1+i)" 4| per cent. 1. 3. 1*046667 41867 6978 1*095512 43820 7303 1*146635 45865 7644 1*200144 48006 8001 1*256151 * * * 17. The last figure of log(l-f«) is only approxi- mately true, and the error in it is continuously multiplied as the work proceeds, so that a correction must be introduced to counteract the accumulation of error. Thus, at 4 per cent, log (1 + ^) = -01703334. If in our operation we only retain six places of decimals, the value of log (1 + 1^^ CHAP, v.] Oil Interest Tables. 255 10. will come out -170330, and of log(l+^y^ 1-703300, whereas they should be -170333 and 1703334 respec- tively. We may keep correct the last place to be retained in one or other of two ways — either by working with two places more figures than are to appear in the final table, or by apply- ing a correction as the work progresses. This second method is as follows : — The seventh and eighth figures of log(l-|-?'), at 4 per cent., are 34, or almost exactly a third of a unit in the sixth place. If, therefore, we work with only six figures, and increase by a unit in the sixth place the second, fifth, eighth, etc., values, our results will be accu- rate. The specimen in the margin shows both these methods of correction. Eight figures are there used, although only six are to be retained, and the two that are to be cut off are separated by a space from the others. When the final cutting down process is effected, the usual correction must be made when the figures neglected are greater than 49. The necessity of going over the work to correct the last figure will be obviated if, at the top of the column, hid not on the moveable card, we increase the seventh figure by 5. In the margin the operation is repeated with this adjustment, and it will be noticed that without further alteration the six figures to be retained are accurate. It will also be observed that if we had used only six places of figures in our operation, and, applying the second method of cor- rection above named, if we had, to form the second, fifth, eighth, etc., values, added 017034, instead of 017033, which is used to form the other values, the result would have been also correct. By examining the first two figures of log(l-|-i) which are to be neglected, we can always see at what intervals in our work the last place which we retain is to be increased or diminished by a unit. Thus, at 6 per cent., log(l-f?) is -02530587, or, to six figures, •025306. This last quantity is -13 in excess in the last place, and therefore when we use it for continuous addition, we must diminish our results by a unit at the fourth value (because 4x "13 = -52 > -49), and thereafter at every eighth value. log (1+0" 4 per cent. 1. 017033 34 2. 034066 68 3. 051100 02 4. 068133 36 5. 085166 70 6. 102200 04 7. 119233 38 8. 136266 72 9. 153300 06 170333 40 * * * log(l+i)« 4 per cent. 1. 017033 84 2. 034067 18 3. 051100 52 4. 068133 86 5. 085167 20 6. 102200 54 7. 119233 88 136267 22 153300 56 170333 90 8. 9. 10. 256 Theory of Finance, [chap. v. 1 8. To construct a table of v"*. Since y"+* = yx«^", we could form the table by beginning with ?;, and multiplying continuously by v; but this would not be con- venient. The quantity v has generally many significant figures, and the multiplications would therefore be lengthy. By changing the working formula into «;'*=(14-*)^"'''S we can reduce the labour, and make it no greater than that involved in preparing a table of (1+i)". Commencing then with the last value of v^ to be tabulated, we work backwards ; but in other respects we proceed exactly as if we were j^r^paring the column (l-f*)" All that we have said regarding this last function will therefore apply, and it is unnecessary to repeat illustrations here. 19. It is very useful when we have completed an entire column of a table to be able to verify by one operation the whole work, or to be able at once to check a printed table with which we are not familiar. In the case of the majority of columns of interest tables, excellent formulas for this purpose are easily found. We have 5-^-| = l + (14-i) + (l+0'+ • • • +(1+*)""^ If therefore we add up our column of {\-\-iY and increase the sum by unity, the result, if our work is correct, will be s,h:t|. Similarly, aj^\=v-\-v^-\- . . . d?", .so that the sum of our column of t'" should be rtiT. 20. To construct a table of s^. Since s»4-ii=S/7. +(1+^)", it follows that the column (1+*)" con- sists of the differences between the successive values in column ■s*^, and we can therefore, if we have already formed a table of (l+i)", construct a table of 5^1 by mere summation. Commencing with 1, the amount of 1 per annum for one year, we add succes- sively (l+i), (1 + ^)% etc., so forming ^2;, sg], etc. At any stage the work may be checked by calculating independently the amount of the annuity by means of the formula s^ = r • As the values in the table of (l-f?)" are only approximately true in the last place, the errors^ although they are in both directions, and so will in general tend to neutralize each other, may sometimes at certain points in the table of 571 fall in such a manner as to produce a sensible inaccuracy in the last place. We must therefore, to insure exactitude, work to at least one place more than we mean finally to retain. CHAr. v.] On Interest Tables. 257 21. If we do not already possess a table of (1 +*)**> we can never- theless form our table of s^ with great facility for the majority of rates of interest, by simple multiplication. . r)er"^cent The relation to boused is Sri:+i| = (l4-0^n| + l« At , , qq each step we multiply our previous result by (1+i), * j 4 exactly as we did in Arts. 13 and 14, the only differ- ^ " o^i" once in the operation being that now we add a unit ' | g^g at the same time that we make the multiplication. vT^Tfi" In the margin we show the construction of the * i-i 24864 table at 4 per cent interest. It is needless to add . -t-z. ^ . ^ . - ^, ^ , 4. 4-246464 further examples. 1-169859 22. At each stage, for our addend we multiply „ ^ ,,^„^^ our previous result by i and add unity: that is, '' -[.216653 to form .s„.i.-i| we add to 5^; the quantity i.-f/jj + l- ^ 6-63-^976 But from the equation . to form — at least when the rate of interest is integral — the table of s^ , as by so doing we obtain two tables in one operation. 23. A very useful formula is available to check our complete column of 5^ , or to verify a printed table of this function — •*?r|H-%',+53i+etc.-fs^ ^(lH-0+(lH-i)' + (l +0'>+etc.+ (lH-y~^^ (l + 0^7ij- ^ % i If, therefore, we multiply the last value in the column by (1 + Oj deduct n, and divide by % we have as result the sum of the column. Should the equation hold we know that all our work is correct, unless indeed there be an exact balance of errors — an unlikely event; but, should the equation not hold, we may be sure that there is an error somewhere, and we must set to work to discover and eliminate it. We may divide our table into sections, and apply our formula again, and so localize the error. Thus, for example, the sum of the column under 4 per cent, in Table III. is 4687*75784, A 1 • *u * 1 ^ J *u * 1 -04 X 237-99069-60 . and, applymg the formula, we find that ~ — -.- is 258 Theory of Finance. [chap. v. 4687*75744, differing from the sum of the column by 40 in the last two places, which is due to the fact that the figures in the last place of the tabulated function are only approximately true, and that in multiplying s^\ by 1'04 we have not made allowance for the carriage from the figures beyond the fifth. Had we used six places of decimals in seo"!, ^^® result of the formula would have been 4687 '75781, or only 3 out in the fifth place. Suppose, however, that the fiftieth value had by mistake been printed 153*66708, the result of addition would have 'differed from that given by the application of the formula by -99960, thus showing that there must be a mistake somewhere. Splitting the column into three equal sections, we find the sum of the first twenty values to be 274*23005, while (A+!)^fQ "i "^^ ^ 274*23008, thus showing I that the error is not in that section. Summing the second section we have 1196*43339, which, added to the sum of the first section, gives 1470*66344, and il±!)£illli2= 1470*66352. The error is I therefore not in the second section, and it must be in the last ; and, examining the values in it, we find .s^| to be wrong. 24. To construct a table of an\ . This table may be formed from a table of «?^ exactly as a table of 5^1 may be formed from a table of (1+i)*^ If however we have not already a table of n^^ ^ we can construct that of ft^l directly by means of 4 p^^. ^^^j^ the relation «»., I =(l+iK,-l ^^^ 22-623490 We must proceed as m Art. 21, except that Q04Q40 we must begin at the end of the table and work backwards, and that we must deduct instead of ^9- 22*528430 add unity at each step. The example in the ^^^^^^ margin shows the construction at 4 per cent. 58. 22*429567 897183 25. From the relation a^=,^-J^ it follows that 57. 22*326750 * 893070 i a^l = 1 —v^K Therefore the addends that we form are the arithmetical complements of v'^, from which ^^- ^^ ggg^go v'^ can be derived very easily, almost by inspection. Therefore, if we wish to construct a table of ^^ 55. 22*108613 we may, without much additional labour, do so by first forming one of a^i, and so secure both tables at once. * # * CHAP, v.] On Interest Tables. 259 26. A very similar relation to that given in Art. 23 is available to check our column of a^ . We have a\\ +(t2i H-597 proceed. At 4 per cent, logv is '3 in excess in & 1 "___ the sixth place, and we must therefore deduct loga^, 275564 a unit every third time it is used. As before, j^^ ^ 982967 we may write logi; on a moveable card to save 7 log a ^ 460311 trouble. , "TTTIT^ logasl 443278 31. There is no continuous formula for the tt: log?; 982966 construction of the columns .s_^ and aj^. We riogaai 576928 must take the reciprocals of s^ and a^i] respec- loga^j 559894 tively. In order to insure accuracy, it will , 08296^ therefore be necessary carefully to verify emh T\ma- 665571 tabulated value. This may best be done, not ' — by performing the work in duplicate, as the lf>g^5l ^ ^8538 same error might thus be repeated, but by ♦ * ♦ again taking out the reciprocals of the values in the newly formed columns, and these should be the values of ,9^ and «,i] respectively, with which we started. 32. If both the columns .s~^ and a~^ are wanted, one may be formed from the other in such a way as to check them both. Since sZ^=aZ^—i, it follows that the two columns have the same differ- n| «| ences. If, therefore, we first form the column sz! by means of a table of reciprocals, and difference it, we shall so produce the differences (which are negative) of the column az^ Starting now with the first value of the column al\ namely, (1 +i), and adding (algebraically) continuously the differences, we complete the colunm. To check the whole work, we take the reciprocals of the values in the last column, and these should be the successive values of a^ . Q 262 Theory of Finance, [CIIAP. V. The following is an example at 4 per cent. 0-- 1-000000 " 490196 ro4oooo •490196 . 830153 •530196 •320349 r 915141 /. 949137 •360349 •235490 •275490 •184627 966135 •224627 •150762 975848 •190762 •126610 981918 •166610 •108528 985965 •148528 •094493 988798 •134493 •083291 •123291 Year 6'-' —A 1 2 4 5 6 7 8 9 10 \Vhere i consists of but one significant figure, as in the example, this method of construction does not possess any advantages ; but if there be several significant figures in % then considerable benefits are experienced by its adoption. 33. The column log sZ^ may be conveniently formed by means of the logarithms of *%i, because, omitting the characteristics of the logarithms, log 51^=1— log -s^. We have only to take the arithmetical complements of the logarithms of the successive values of .Vi|, which can be done by inspection. In the same way log aZ] may be formed from loi? ff^ . Table I. VL" \\d^4M ^ ('^ ' ^ ■'"""'"' "f ' •"""" <^ +'^" :u-) n 20/ 3 /o lY[. 4^/0 4iVc 5*^/0 6«/o n 1 I 030000 I -035000 1-040000 I -045000 1-050000 1 1-060000 1 2 1-060900 1 071225 I -081600 I -092025 1-102500 1 1-123600 2 3 I 092727 1-108718 1-124864 1-141166 1 157625 1 1-191016 a 4 1-125509 I -147523 1-169859 1-192519 1-215506 1 1-262477 4 6 1-159274 I -187686 I 216653 I -246182 1-276282 i 1-338226 6 « 1-194052 1-229255 I -265319 I -302260 1-340096 I -418519 6 ! 7 I -229874 1-272279 I -315932 I -360862 1-407100 1-503630 7 i 8 I -266770 I -316809 1-368569 I -422 10 1 1-477455 1-593848 8 ! 9 1-304773 I -362897 I -423312 1 I -486095 1-551328 1689479 9 10 1-343916 I -410599 I -480244 ; 1 -552969 1-628895 1-790848 10 1 1-384234 1-459970 1-539454 I -622853 1-710339 1-898299 1 2 1-425761 I -51 1069 1-601032 1-695881 1-795856 1 2012196 2 3 1-468534 1-563956 I -665074 I -772196 1 -885649 2-132928 3 4 I -5 1 2590 1-618695 I -731676 1-851945 I -979932 ! 2-260904 4 15 1-557967 I -675349 I -800944 1-935282 2-078928 1 2-396558 16 i 6 1 -604706 1-733986 1 1-872981 , 2-022370 2-182875 t 2-540352 6 1 7 I -652848 1-794676 I -947901 1 2-II3377 2-292018 ; 2-692773 7 8 1-702433 1-857489 2-025816 2-208479 2-406619 1 2-854339 8 9 1-753506 I -922501 2-106849 2-307860 2-526950 ! 3 025600 9 20 I -8061 1 1 1-989789 2-191123 2-411714 2-653298 I 3-207135 20 i 1 I -860295 2-059431 2-278768 2-520241 2-785963 3-399564 1 2 1-916103 2-131512 2-369919 2-633652 2-925261 3-603537 2 3 1-973587 2-206114 2-464716 2-752166 3-071524 3-819750 3 4 2-032794 2-283328 2-563304 2-876014 3225100 4-048935 4 26 2-093778 2-363245 2-665836 3-005434 3-386355 4-291871 26 6 2-156591 2-445959 2-772470 3-140679 3-555673 4-549383 6 7 2-221289 2-531567 2-883369 3-282010 3-733456 4-822346 7 8 2-287928 2-620172 2-998703 3-429700 3-920129 5-111687 8 9 2-356566 2-711878 3-118651 3-584036 4116136 5-418388 9 30 2-427262 2-806794 3-243398 3-745318 4-321942 5-743491 30 1 2-500080 2-905031 3-373133 3-913857 4-538039 6-088101 1 2 2-575083 3-006708 3-508059 4-089981 4-764941 6-453387 2 3 2-652335 3-1 11942 3-648381 4-274030 5-003189 6-840590 3 4 2731905 3-220860 3-794316 4-466362 5-253348 7-251025 4 35 2-813862 3-333590 3-946089 4-667348 5-516015 7-686087 36 6 2-898278 3-450266 4-103933 4-877378 5-791816 8-147252 6 7 2-985227 3-571025 4-268090 5-096860 6-081407 8-636087 7 8 3-074783 3-69601 1 4-438813 5-326219 6-385477 9-154252 8 9 3*167027 3-825372 4-616366 5-565899 6-704751 9-703508 9 40 3-262038 3-959260 4 -80102 1 5-816365 7-039989 10-285718 40 1 3-359899 4-097834 4-993061 6-078101 7-391988 10-902861 1 2 3-460696 4-241258 5-192784 6-351615 7-761588 11-557033 2 3 3-564517 4-389702 5-400495 6-637438 8-149667 12-250454 3 4 3-671452 4-543342 5-616515 6-936123 8-557150 12-985482 4 45 3-781596 4-702359 5-841176 7-248248 8-985008 13-764611 45 6 3-895044 4-866941 6-074823 7-574420 i 9-434258 14-590487 6 7 ; 4-01 1895 5-037284 6-317816 7-915268 9-905971 15-465917 7 8 4-132252 5-213589 6-570528 8-271456 ' 10-401270 16-393872 8 9 4-256219 4-383906 5-396065 6-833349 8-643671 1 10-921333 17-377504 9 50 5.584927 7-106683 9-032636 1 1 1 -467400 18-420154 60 1 *'I'5J23 5-780399 7-390951 9-439105 12 040770 19-525364 1 2 4650886 5-982713 6-192108 7-686589 9-863865 12-642808 20-696885 21-938698 2 8 4-790412 7-994052 10-307739 13-274949 3 4 4-934125 5-082149 6-408832 8-313814 10-771587 13-938696 23-255020 4 65 6-633141 8-646367 1 1 -256308 14-635631 24-650322 I 56 6 1 5-234613 6-865301 8-992222 1 1 -762842 15-367412 26-129341 6 7 5-391651 7-105587 9-351910 12-292170 16-135783 27-697101 7 1 8 5-553401 7-354282 9725987 12-845318 16-942572 29-358927 8 ' 9 5720003 7-611682 10-115026 13-423357 17-789701 31-120463 9 60 5-891603 7-878091 10-519627 14-027408 18-679186 32-987691 60 Table IV. Present Value of 1 j^er (iniimii : — /•/.:. a^ n 1 3Vo 3iVo 4V0 4iVo 5Vo 6% n 1 •97087 •96618 '25^54 ' -95694 •95238 -94340 C 2 I •9.1347 1-89969 I -88609 I I -87267 I -85941 i'^3339 2 ^ 3 2-82861 2-80164 1 2-77509 I 2-74896 2-72325 2*67301 3 4 371710 3-67308 i 3 -62990 3-58753 3-54595 3-46511 4 1 ^ 4-57971 4-51505 4-45182 4-38998 4-32948 4 -2 1 236 5 6 5-41719 5-32855 1 5-24214 1 5-15787 5-07569 4-91732 6 7 6 23028 6-II454 1 6-00205 1 5-89270 5-78637 5-58238 7 8 7-01969 6-87396 6-73275 i 6-59589 6-46321 6-20_279 8 : 7 7861 1 7-60769 ; 7-43533 ! 7-26879 7-10782 6-80169 9 \ 10 8 53020 8-31661 8-11090 7-91272 7-72174 7-36009 10 ! 1 9-25262 9-00155 i 8-76048 1 8-52892 8-30641 7-88688 1 1 2 9-95400 9-66333 9-38507 9-11858 8-86325 8-38384 2 8 10-63496 10-30274 9-98565 ^ 9-68285 1 9-39357 8-85268 3 4 1 1 -29607 10-92052 10-56312 10-22283 i 9-89864 9-29498 4 15 1 1 -93794 11-51741 11-11839 IO-73955 1 10-37966 9-71225 15 6 12-56110 12-09412 11-65230 1 1 -23402 i io-83""777 10-10590 6 7 13-16612 12-65132 12-16567 11-70719 I 11-27407 10-47726 7 8 •3-7535I 13-18968 12-65930 12-15999 ' 1 1 -68959 10-82760 8 9 14-32380 13-70984 13-13394 12-59329 1 12-08532 1115812 9 20 14-87748 14-21240 13-59033 13-00794 12-46221 11-46992 20 1 15-41502 14-69797 i 14-02916 13-40472 12-82115 11-76408 1 1 2 15-93692 15-16713 i 14-45112 13-78443 13-16300 12-04158 2 3 16-44361 15-62041 ! 14-85684 14-14778 13.48857 12-30338 3 4 16-93554 16-05837 15-24696 ■ 14-49548 13-79864 12-55036 4 25 17-41315 16-48152 15-62208 : 14-82821 14-09395 12-78336 25 ! 6 17-87684 16-89035 15-98277 ! 15-14661 14-37519 13-00317 6 i 7 18-32703 17-28537 16-32959 1 15-45130 14-64303 13-21053 7 8 18^76411 17-66702 16-66306 j 15-74287 14-89813 13-40616 8 19-18846 18-03577 16-98372 16-02189 15-14107 13-59072 9 30 19-60044 18-39205 17-29203 : 16-28889 15-37245 13-76483 30 " 1 20-00043 18-73628 17-58849 16-54439 15-59281 13-92909 1 2 20-38877 19-06887 17-87355 16-78889 15-80268 14-08404 2 , 3 20-76579 19-39021 18-14765 17-02286 16-00255 14-23023 3 4 21-13184 19-70068 18-41120 17-24676 16-19290 14-36814 4 1 35 21-48722 20-00066 18-66461 17 46101 16-37419 14-49825 35 i 6 21-83225 20-29049 18-90828 17-66604 16-54685 14-62099 6 7 22-16724 20-57053 19-14258 17-86224 16-71129 14-73678 7 8 22 -49246 20-84109 19-36786 18-04999 16-86789 14-84602 8 9 22-80822 21-10250 19-58449 18-22966 1 17-01704 14-94908 9 40 23-11477 21-35507 19-79277 18-40158 17-15909 15-04630 40 ! 1 23-41240 21-59910 19-99305 18-56611 17-29437 15-13802 1 2 2370136 21-83488 20-18563 18-72355 17-42321 15-22454 2 3 23-98190 22-06269 20-37080 18-87421 17-54591 15-30617 ! 3 4 24-25427 22-28279 20-54884 19-01838 17-66277 15-38318 4 45 24-51871 22-49545 20-72004 19-15635 1777407 15-45583 45 6 24-77545 22-70092 20-88465 19-28837 15-52437 (5 7 25-02471 22-89944 2 1 -04294 19-41471 , 17-98102 15-58903 7 8 25-26671 23-09124 21-19513 19-53561 18-07716 15-65003 8 9 1 25-50166 23-27656 21-34147 19-65130 18-16872 15-70757 9 50 25-72976 23-45562 21-48219 19-76201 j 18-25593 15-76186 50 1 25-95123 23-62862 21-61749 19-86795 18-33898 15-81308 1 2 26-16624 23-79577 1 21-74758 19-96933 18-41807 15-86139 2 3 26-37499 23-95726 21-87268 20-06635 18-49340 15-90697 3 4 26-57766 24-11330 21 -99296 20-15918 18-56515 1 15-94998 4 55 26-77443 24-26405 I 22-10861 20-24802 18-63347 1 15-99054 55 6 26-96546 24-40971 ' 22-21982 20-33303 18-69855 1 16-02881 6 7 27-15094 24-55045 i 22-32675 20-41439 18-76052 1 16-06492 7 8 27-33101 24-68642 22-42957 20-49224 , 18-81954 16-09898 8 9 27'S"583 24-81780 i 22-52843 20-56673 1 18-87575 16 13111 9 60 27-67556 24-94473 22-62349 20 63802 18-92929 16-16143 60 1 "■»' — ^ t/ 1 ^T«1 A:(yL:nt '^ ^ ^^^ "d^^ ^^^ , Table V. 4^ ]fi.nnuity which 1 ivill purchase : — tiz. («7,|)- ■■ 3% r — n sr/o 4"/o 4iVo 5Vo 6»/„ n I I -030000 I -035000 I •040000 I ^045000 ro5oooo ro6oooo 1 2 0-522611 0-526400 0-530196 •533998 •537805 o'545437 2 8 •353530 •356934 •360349 •363773 •367209 •3741 10 u 4 •269027 •272251 -275490 •278744 •282012 •288591 4 6 •218355 •22 148 1 •224627 •227792 •230975 •237396 5 6 •184598 •187668 •190762 •193878 •I 9701 7 •203363 6 1 7 •160506 •163544 •166610 •I 6970 1 •172820 •179135 7 8 •142456 •145477 •148528 •151610 •154722 •161036 8 9 •128434 •131446 •134493 •137574 •140690 •147022 9 10 •II723I •120241 •123291 •126379 •129505 •135868 10 1 •108077 •111092 •114149 •I 17248 •120389 •126793 1 2 •100462 •103484 •106552 •109666 •I 12825 •119277 2 3 •094030 -097062 •100 1 44 •103275 •106456 •I 12960 8 4 •088526 •091 57 1 •094669 •097820 •101024 •107585 4 15 •083767 •086825 •089941 •0931 14 •096342 •102963 15 6 •07961 1 •082685 •085820 •089015 •092270 •098952 6 7 •075953 •079043 •082199 •085418 •088699 ■095445 7 i 8 •072709 •075817 •078993 •082237 ■085546 •092357 8 9 •069814 •072940 •076139 •079407 •082745 •089621 9 20 •067216 •070361 •073582 •076876 •080243 •087185 20 ; 1 •064872 •068037 •071280 •074601 •077996 •085005 1 2 •062747 •065932 •069199 •072546 •075971 •083046 2 ! 3 •060814 •064019 •067309 •070682 •074137 •081278 3 4 •059047 •062273 •065587 •068987 •072471 •079679 4 25 •057428 •060674 •064012 •067439 •070952 •078227 25 i] •055938 •059205 •062567 •06602 1 •069564 •076904 6 7 •054564 •057852 •061239 •064719 •068292 •075697 7 8 •053293 •056603 •060013 •063521 •067123 •074593 8 9 •052II5 ■055445 •058880 •062415 •066046 •073580 9 30 •05 1 01 9 •054371 •057830 •061392 •065051 •072649 30 1 •049999 •053372 •056855 •060443 •064132 •071792 1 -2 •049047 •052442 •055949 •059563 •063280 •071002 2 3 •048156 •051572 •055104 •058745 •062490 •070273 1 3 4 •047322 •050760 •054315 •057982 •061755 •069598 4. 35 •046539 •045804 •049998 •053577 •057270 •061072 •068974 S5 G •049284 •052887 •056606 •060434 •068395 ^ 7 •0451 12 •048613 •052240 •055984 •059840 •067857 j 7 8 •044459 •047982 •051632 •055402- •059284 •067358 8 9 •043844 •047388 •051061 •054855 •058765 •066894 9 40 •043262 •046827 •050523 •054343 •058278 •066462 ; 40 1 •042712 •046298 •050017 ■053862 •057822 •066059 1 2 •042192 •045798 •049540 •053409 ■057395 •065683 2 3 •041698 •ot^878 •049090 •052982 •056993 •065333 3 4 •041230 •048665 •052581 •056616 •065006 4 45 •040785 •044453 •048262 •052202 •056262 •064701 ; 46 6 •040363 •044051 •047882 •051845 •055928 •064415 c 7 •039961 •043669 •047522 •051507 ■055614 •064148 ; 7 8 •039578 •043306 •047181 •051 189 •055318 •063898 1 8 9 •039213 •042962 •046857 ■050887 •055040 •063664 9 50 •038865 •042634 •046550 •050602 •054777 •063444 50 ' 1 •038534 •042322 •046259 •050332 •054529 •063239 1 1 2 •038217 •042024 •045982 •050077 ■054294 •063046 2 3 •037915 •04 1 74 1 "045719 •049835 •054073 •062866 8 4 •037626 •04147 1 •045469 •049605 •053864 •062696 4 55 •037349 •0412 13 •045231 •049388 •053667 •062537 55 •037084 •040967 •045005 •0491 8 1 •053480 •062388 C 7 •03()83. •040732 •044789 •048985 ■053303 •062247 7 ' 8 •036588 •040508 •044584 •048799 •053136 •062116 8 9 •036356 •040294 •044388 •048622 •052978 061992 9 : CO •036133 •040089 •044202 •048454 •652828 •061876 C0| \.' IZ- / / f 7: ' 2 2i/ o -^l 9 / ^ '^ &J. f 1^ ■%! / 11 I ^ W) ^ f- *• o r ^ 3*^ .V ,x ..^^ •/ '