Derivation of Formulas
i) Future Value of an Annuity
Future value of an annuity is
FVA_{n} = A(1 + k)^{n -1} + A (1 + k)^{n - 2} + .......A (1 + k) + A ...............(a1)
Multiplying both sides of the equation a1 by (1 + k) gives.
(FVA_{n}) (1 + k) = A (1 + k)^{n} +A(1 +k)^{n -1} +... A (1 +k)^{2} +A (1 +k) .......(a2)
Subtracting eq. (a1) from eq. (a2) yields
FVA_{n}k = A[((1 + k)^{n} - 1)/k] ......................................(a3)
Dividing both sides of eq. (a3) by k yields
FVA_{n} = A[((1 + k)^{n} - 1)/k]
ii) Present Value of an Annuity
The present value of an annuity as:
PVA_{n}k = A (1 + k)^{-1} + A(1 + k)^{-2} + .... + A(1 + k)^{- n} ............(a 4)
Multiplying both sides of Eq (a 4) by (1+ k) provides:
PVA_{n} (1 + k) = A + A (1 + k)^{-1} + ...... + A (1 + k)^{-n} +1 .....................(a5)
Subtracting eq (a4) by (a5) yields:
PVA_{n}k = A[1 - (1 + k)^{-n}]
= A [((1 + k)]^{n} - 1)/(k (1 + k)^{n}) .....................(a6)
Dividing both the sides of Eq (a6) with k outcomes in as:
PVA_{n} = A [((1 + k)]^{n} - 1)/(k (1 + k)^{n})