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in BUS 2016H - Financial Mathematics by (4.2k points)
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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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Best answer

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, \(PV = \frac{1-v^n}{d} \)

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