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Proof of Formula for the Present Value of an Annuity

*Copyright 1995 by Campbell R. Harvey and Stephen Gray.
All rights reserved. No part of this lecture may be reproduced without
the permission of the author.*

Latest Revision: July 1996.

Consider an annuity of $1 payments, ** n** times per year for

We could find the present value of each of these individual cash flows.
The present value of the first cash flow is simply ** Z**.

The present value of the second cash flow is the value of $1 discounted
back two periods. This is equal to ** Z^{2}**.

In general, the present value of the ** k**th cash flow will
be

If we add all of these cash flows together, we get the value of the annuity:

*A _{n} = Z + Z^{2} + Z^{3}+ . .
. + Z^{n} *

or:

You may recognize this, from Calculus classes, as a finite geometric series. The formula for the sum of such a series is:

To prove this, a trick is used. Starting with our original equation:

*A _{n} = Z + Z^{2} + Z^{3}+ . .
. + Z^{n} *

we can multiply both sides of the equation by ** Z** to get
another equation:

*ZA _{n} = Z^{2} + Z^{3} + Z^{4}+
. . . + Z^{n+1} *

If we subtract the second equation from the first, several of the terms cancel out. We are left with:

*A _{n}-ZA_{n} = Z-Z^{n+1} = Z(1-Z^{n})
*

Divide both sides of the equation by ** (1-Z)** and we get:

If the annuity is ** a** dollars per period rather than $1
per period, we could use the same logic to discover that the value of the
annuity is: