# EXAMPLE QUESTION: How to derive the formula for an annuity?

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How does one prove that:

$$\ddot{a}_\overline{n|} = 1 + v + v^2 + \cdots + v^n = \frac{1-v^n}{d}$$

## 1 Answer

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Well Simon, it seems you have included an extra R1 cashflow at the end of year $$n$$. The annuity listed says that the last cashflow is received at year $$n-1$$. Nevertheless, here is a proof!

Equation  1: $$PV = 1 + v + v^2 + v^3 +....+ v^{n-1}$$

Equation  2: $$v.PV = v + v^2 + v^3 + ...+ v^n$$

Equation 1 - Equation 2:

$$PV - v \cdot PV = 1 + v + v^2 + ...+ v^{n-1} - (v + v^2 + v^3 + ... + v^{n-1} + v^n)$$

$$= PV (1-v) = 1- v^n$$

$$\implies PV = \frac{1 - v^n}{1 - v}$$

And  $$(1 - v) = d$$

Therefore, $$PV = \frac{1-v^n}{d}$$