Annuity (finance Theory) - Proof of Annuity Formula

Proof of Annuity Formula

To calculate present value, the k-th payment must be discounted to the present by dividing by the interest, compounded by k terms. Hence the contribution of the k-th payment R would be R/(1+i)^k. Just considering R to be one, then:

$a_{n|i} = sum_{k=1}^n frac{1}{(1+i)^k} = left( frac{1}{1+i} - frac{1}{(1+i)^{n+1}}right) sum_{k=0}^infty frac{1}{(1+i)^k}$

We notice that the second factor is an infinite geometric progression of the form,

$sum_{k=0}^infty kappa^k = frac{1}{1-kappa}$

therefore,

$a_{n|i} = left( frac{(1+i)^n - 1}{(1+i)^{n+1}}right )left( frac{1}{1-1/(1+i)}right) = left( frac{1-(1+i)^{-n}}{1+i}right )left( frac{1+i}{i}right) = frac{1-(1+i)^{-n}}{i}.$

Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (n−1) years. Therefore,

$s_{n|i} = 1 + (1+i) + (1+i)^2 + cdots + (1+i)^{n-1} = (1+i)^n a_{n|i} = frac{(1+i)^n-1}{i}$