**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})