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_nk = 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_nk = 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_nk = 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)