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

+1 vote
151 views
asked Apr 6, 2016 in BUS 2016H - Financial Mathematics by simon_rigby (4,220 points)
recategorized Apr 6, 2016 by simon_rigby

How does one prove that:

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

1 Answer

+4 votes
answered Apr 11, 2016 by Richard van Gysen (3,180 points)
edited Apr 11, 2016 by simon_rigby
 
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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