86 MATHEMATICAL THEORY OF INVESTMENT annuity at the end of its term would be the same as the amount of an ordinary annuity having the same term; i.e. im I o(P) _ <(P) //( 0—. —— 0— for all values of m. " ILLUSTRATIVE EXAMPLE. Use the formulas of the present section to solve Example 1 at the end of the previous section. Solution. If money is worth 6% effective, the present value of the six pay- jients, of which the first is cash, is $1000 a^ =$1000 (1+a^) —(^'^n = $1000 (1 + 4.21236) = $6212.38. The solution is similar when money is worth 6% nominal, payable semiannually. 34. Perpetuities and capitalization. A perpetuity is an annuity whose payments are to be continued forever. Obviously the amount of a perpetuity would be an expression without mean- ing, since, as time went on, the amount of an annuity would increase beyond all bounds. On the contrary, the present value of a perpetuity is a definite sum, viz. the limit of dy, as n increases indefinitely. The notion of a perpetuity is one of great im- portance in practical business affairs, as we shall soon see. PROBLEM.To find the present value of a perpetuity whose pay- ments are made annually. Denoting the present value of the perpetuity of 1 per annum by a», we will have, by definition, T T l——V <(„ = lim an-, = lim —:— B=«. n&m 1 Butv = —— l+i is a fraction whose value is less than unity; hence lim v" = lim ————^ = 0, n=oo n=w l-L-rl)
and, consequently, lim———=-. n=« i i We have therefore <»»=-. (1) ANNUITIES 87 If the annual rent of the perpetuity is £, the present value will be ,, -Bffl. = — (1') tf Equation (1') shows that an annual income oi-R per year T_> indefinitely continued is equivalent to a cash sum of —. For % example, if money is worth 5%, an income of $2000 per year would be equivalent to $40,000 in cash. Again, an acre of land that can be made to yield a net income of $5 per annum would be worth $100 with money at 5%. The cash equiva- lent of an annual expenditure may be found in the same way. For example, if it costs $5 a year to dig the weeds from a block of cobblestone gutter, it would pay to spend $100, in addition to the cost of the cobblestone gutter, to substitute a cement gutter needing no cleaning, when money is worth 5%. The process of determining the cash equivalent of an annual income or an annual outgo that is supposed to continue indefi- nitely is called capitalization. It is customary to use the expres- sion " to capitalize an income or an outgo " at such and such a rate. In the example relating to the net income of the acre of land, we say that " the annual income of $5 capitalized at 5% is $100." PROBLEM. To find the pressent value of a perpetuity whose pay- ments are made at intervals of k years. If the first payment be made at the end of k years, the second at the end of 2 Jc years, and so on, the sum of the present values of the payments will be ^+v'tk+vsl:+..., and if the payments be. continued for nk years, the sum may be denoted by a^y We have ,.d-^ a^-^-^^ 88 MATHEMATICAL THEORY OF INVESTMENT as the present value of the first n payments. Dividing the nu- merator and denominator of the expression on the right by v*, and remembering that ,/ — ^ (l+O*'
1 —ynk '(l+i)"-!'
we have, finally,a^,k=——————(2)
If now the number of payments be increased indefinitely, nk will be increased indefinitely, and so lim r"1 = 0. a=oo Denoting lim a^. ^ by a^ ,., we have at once n =® w fc=———I———. ('3') *' (l4.,)fc_l '-"^ If, moreover, the payments be R, the present value of the per-
petuity will be
^-(i^T<3')
The expressions on the right of (3) and (3') may be put in forms better adapted for computation by noting that if numerator and denominator be multiplied by ?', one factor in the result will be —. The equation (3') then takes the form 8ir! .Bo^ =:?.-!-. (3") 1 «tl In (3") the value of — can be found from the tables, thus avoid- «F1 ing the laborious division by the value for (l+iy'—l, which would be necessary if (3) or (3') were used. It should be noted that formulas (1) and (3) could have been obtained directly by finding the limit of the sums of the two infinite geometrical progressions v+v2+vs+... and ^+v2t+^,s<+.... ANNUITIES 89 EXAMPLES 1. If it costs $1000 a year to guard a railroad crossing, how much could the railroad company afford to expend in order to abolish the grade cross- ing, on the supposition that money could be invested at 4% ? 2. A bridge costing $25,000 must be renewed every thirty years. What sum should be set aside when the bridge is first built, to provide for an indefinite number of renewals, on the supposition that the cost of renewal will remain constant and that money will remain at 4% ? Solution. By formula {S") the amount required will be 25,000 1 .04 '^at4%' The tables give — = -0178301;