82 MATHEMATICAL THEOEY OF INVESTMENT 32. The present value of an annuity. PROBLEM. To find the present value of an annuity of 1 per annum, payable annually for n years. Let it be required to find the present value in terms of the effective rate i. The several annual payments will be, in effect, a series of sums due in 1, 2, 3, ..., n years. Their present values will then be, by (5), § 26, (l+z)-i=t,, (1+;)-^^ ... ^^-"^V. Denoting the sum of the present values by <^, we have a^=v+v2+vs+...+v". The right member is a geometrical progression with first term v, last term v", and ratio v. The sum is therefore ^v»+i—v v—y»+i a'^l~~^^=~l^^ If we remember that v is defined by the equation v==(l+I)-\ the expression on the right reduces easily by dividing numerator and denominator by v, so that ^-£1='-Q±^. ^, If the nominal rate./ is given, the formula (1) may be written
<^='
i-.. ^(^i)
' i V"/ i \'" 1+3-) -1(1+^) -1 \ m/\ mf
(2)
PROBLEM. To find the. present value of an annuity of 1 per annum, payable p times a year for n years. The stated payment will be -, and the times to elapse before the several payments are to be made will be, in years and frac- tions of a year, 133 -, -, -, -,' ..;, n— -, n. P P P P - ANNUITIES 83 The present values in terms of the effective rate i will be 1 -'I1! -2!2 1 1 -(l+O "^-v, -(l+O ^-v, , -(l+o-"=-^. p-'ppp p p Denoting the sum of these present values by ff^, we have -I 1 2 _1 aw=-(v''+vp+..+v' p + v'"). Bl p- The right member is a geometrical progression of np terms, with 1 L 1 1 first term —v", last term —v", and ratio v1'. Consequently, the v p sum is ^ii (P^IV'^P-V" a"1 — fi 1 p v-1 i Dividing numerator and denominator by v, and noting that v=(l+z)-1, ^efind ^ l-^; ^1-(1+^. ^ y[(l+^-l] ^[(1+y-l] If the nominal rate j, convertible m times a year, is given, we must express 1 + i in terms of 1 + - by the fundamental relation I iV l+z=(l+^-); \ ml ( A"""' sothat ^4_______=ll-^L. (4) ^(i+iy-i ^ (1+1)^1 \ ml \ ml Again, if the conversion interval for interest coincides with the payment interval for the annuity, m =p, and (4) becomes (i\-"p ,o-ll-^11" 1+^ ^l^atrate-L (5) 'n ? ^. ^ ^ ^^ ^ - ^ P The case covered by formula (5) occurs frequently in practice.