Question

Derivation of FV of Ordinary Annuity (FV=PMT[(1+i)^n -1)]/i)

Answer 1

Future Value (FV) of ordinary annuity is the sum of futures values of all future cash flows or PMTs

Therefore,

Future Value (FV) of ordinary annuity = F0 + F1 + F2 +…… + Fn ……………………. (1)

Where, F0, F1, F2, Fn are future values of period 1, 2, 3, ... n respectively

Suppose our periodic cash flow is PMT and interest rate is i; therefore, equation (1) can be written in following manner –

Future Value (FV) of ordinary annuity = PMT *(1+i) ^0 + PMT *(1+i) ^1 + PMT *(1+i) ^2 +……………..+ PMT *(1+i) ^n

Or

Future Value (FV) of ordinary annuity = PMT * 1 + PMT *(1+i) + PMT *(1+i) ^2 +……. + PMT *(1+i) ^n

Where PMT is common factor, therefore

Future Value (FV) of ordinary annuity = PMT *{1 + (1+i) + (1+i) ^2 + …. (1+i)^n} ………….. (2)

Now {1 + (1+i) + (1+i) ^2 + …. (1+i)^n} forms a geometric progression. The formula for the sum of geometric progression is

{1 + (1+i) + (1+i) ^2 + …. (1+i)^n} = [(1+i) ^n -1] / [(1+i) – 1] = (1+i) ^n -1] / i

Now putting this value in equation (2); we get

Future Value (FV) of ordinary annuity = PMT *(1+i) ^n -1] / i