ANNUITIES 97 Therefore ^-^=^_^_^_,.- ^,y_J- ^ . i i _ q+O" 1 But l-(l+z)-" (1+0--1 (1+0--1 (l+O'-l =1. Consequently, "Tp)— Trt ^p)' ^ "n} ^ as was to be proved. COROLLARY. When the annuity is payable annually, J—J-=i. (2) ^ ^ To see the truth of (2) we have only to make p =1 in (1), since „(!) _ .. »(i) _ „ onrl 1" — 1 m 4- iV —11=1 ^n-l = ^l "nl — °n1' anu3(11 — ± LV- ^ -^ J Formula (2), which is a special case of the theorem, is the form in which this fundamental relation is most used. As we shall see later, this formula enables us to dispense with a set of tables, either for — or for —, since, when one is given, the <^ «.n formula enables us to find the other at once.
EXERCISE Prove the relation (2) directly by means of the expressions for — and — 39. The term of an annuity certain. PROBLEM. To find the term of an annuity when the amount, the annual rent, and the rate of interest are given. Suppose first that the annuity is payable annually. By formula (1) of § 31, the amount of an annuity of 1 per annum payable annually is (l+^y-l t/ and consequently the amount of an annuity of -K per annum is ^^(1±D-^L (2) " z 98 MATHEMATICAL THEORY OF INVESTMENT Let -S"be the amount of the annuity whose term is sought. Then .tl±ff^. „ The required term will be found by solving the equation (3) for n. If equation (3) be divided by .R and multiplied by i, and 1 be transposed to the right member, (1+0-1+H. (4) Equation (4) is an exponential equation which may be solved by taking logarithms of both sides and solving the resulting equation for n. The solution is _Iog(l+Jz) n- log (1+0 <5) If the annuity is payable p times a year, formula (1) is re- placed by (3) of § 31, viz. ^>_ (1+Q---1 ^[(l+z-y-lj and consequently equation (3) will be replaced by s q+0"-i y ^ K —————-^——— = K. (6) ^[(i+0?-i] v rom equation (6), (1 +,)- = 1 + ^p [(1 + {y, _ 1]. (7) Finally, the solution of the exponential equation (7) is ^{i+fpCO+^-i]} n=—————————————————. /o,\ log (1+0 (0) If we write p[(l + iy _ lj ^j^ as in (3') of § 23, formula (8) becomes ' log(l+l;'».) n = ——-——————-. (Q\ log (1+z) w ANNUITIES 99 PROBLEM. To find the term of an annuity when the present value, the annual rent, and the rate of interest are given. Suppose the annuity is payable annually, and let A denote the present value. Then, by formula (6) of § 32, ^.i-a+o-^ ^ where B denotes the annual rent of the annuity. From equa- tion (10), - (1+0-=1-1. (11) The solution of the exponential equation (11) for the time n is log^l——'z) n=-————K———. (12) log (1+0 If the annuity is payable p times a year, equation (10) is replaced by the equation K ^q+O-'- ^ (13) ^[(I+OP-I] From (13), (l+z-)-=l--1 .^[(1+z^-l], (14) ti and the solution of the exponential equation (14) for the time n is ri i -> log^l-^.^[(l+,)?-l]^ n=———————————————————. (15) log (1+z) i If we write j,_ for the expression p[(l+?')''—I], according to (3') of § 23, formula (15) takes the convenient form log (l-1 ./<.,) (16) w=— —————————— log (1+z) If the relation between A, -K, and i should be such that l-Ai<0 R 100 MATHEMATICAL THEORY OF INVESTMENT (that is, if R is less than the annual interest charge), the con- ditions of the problem are inconsistent, since the expression
log(l-|..),
which occnrs in formnia (12), would then be imaginary, and the time would not be a real number. Suppose, for example, that the annual rent is $100, the rate is .04, and the present value is assumed to be $3000. The quantity ^i would be 3000 x .04 ., 100—————' and, consequently, 1 ——;=_. 2, I