ANNUITIES 93 As the variable p does not occur in the numerator, we have only to consider the limit of the denominator. But we have already found that i^ limp [(1 + I)P -1] = log,,(l + 0 """ = 8 (See (3) and (4), § 24.) where 8 is the force of interest. Using these values for the limit in the denominator, we have ,_^^^(l+0-^(l+^)°-l. (i) ^-.pT^ iog. (i+o s v/ From the expression for (/P), we obtain, in exactly the same manner, -^«-^^- » ILLUSTRATIVE EXAMPLE. Find the amount of a continuous annuity of $1000 per annum for a years at 4%. „ , .. (1.04)6-! (1.04)5-1 solutlon- ^='^W=~^~7^7^~. _ .2166529 ' 2 3 - .0392207 = 5.62394. The amount of the continuous annuity of $1000 per annum is $5523.94. 36. The annuity that will amount to 1. PROBLEM. To find the annuity that will amount to 1 in n years. The problem requires the determination of the annual rent of the annuity that will amount to 1 in n years. Let R be the annual rent. The amount of an annuity with annual rent R is, by § 31, Rs^, or Rs^, as the case may be. For an annuity payable annually and amounting to 1 after n years, „ 1 J Rs^l; whence Jt = ^- = —-———. (1) °\ (!+») —l If the annuity is payable p times a year, clearly j;_ 1 _j>[(l+^-ll. ^ ^p) (l+t)»-l "7 n} 94 MATHEMATICAL THEORY OF INVESTMENT The formula for the annual rent of an annuity that will amount to K is found by multiplying equation (1), or equa- tion (2), as the case may be, by K. If the annual rent of an annuity that will amount to K be denoted by B,', where S'=KB, (3) then ]S'=X.^-=——Kt——. (-4) ^ (i+o»-i w A similar formula is derived from (2) when the annuity is pay- able oftener than once a year. When interest is convertible m times a year, the appropriate formula is found by replacing i and 1 + z, in formula (1), (2), or (4), by their values as given by the fundamental relation / i'\1" {=^1+L} -1. Y m) For reasons that will appear later, the problem of the present section may be called the sinking-fund problem, and equation (1), or its equivalent, (4), is called the sinking-fund equation. The importance of the problem consists in the fact that it is equivalent to the problem of determining the amount that must be set aside. annually to meet a given obligation at the end of a given time. For example, if a city bonds itself for $200,000 for twenty years, in order to erect a new high school, some provision must be made for the payment of the bonds when they fall due. The most convenient method is to set aside the same amount each year until the debt becomes due. The amount to be set aside each year is found by the formula (1). Substituting the values K= 200,000, i'= .04, n= 20, we have, as the required amount, $200,000 x .————= $200,000 x .03358175 = $6716.35. The subject will be taken up later, in the sections .on sinking funds. The symbol — should be looked upon as the symbol for " the s»i annuity that will amount to 1" rather than as the reciprocal ot«„-!