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and then replace s _n (^) |i with its formula, we get:
a n (^) |i
(1 i)n 1
__i
__(1 i)n
We can simplify this formula by multiplying both top and bottom by 1/(1 + i)n:
a n (^) |i
(1 i)n 1
__i (^) __(1 (^1) i)n (^)
(1 i)n (^) _(1 ^1 i)n (^)
which simplifies to:
a _n (^) |i
(1 i)n 1
___i(1 i)n
a _n (^) |i
1 ___(1 ^1 i)n
_i
a _n (^) |i
1 (1 i)n
____i
Stating this formally, we write:
FORMULA 4.4.4
The Present Value Annuity Factor (Traditional Form)a _n (^) |i
1 (1 i)n
____i
Admittedly, this does offer some advantages over our original formula, particularly in that
is does not require calculating s n (^) |i first. There is also something to be said for following con-
vention and tradition. On the other hand, at this point plenty of practice should have us all
reasonably comfortable calculating s n (^) |i , and so finding s _n|i on the way to a _n (^) |i doesn’t impose
too heavy a burden. The conventional formula is not appreciably simpler, and it involves a
negative exponent, something with which you may or may not be comfortable, depending
on your prior algebra background and something that can easily lead to calculator errors if
you are not careful.
Rather than insist on one formula or the other, we will present both. The difference
between the two is a matter of taste and style, not mathematical necessity. Both formulas
will give correct results in all situations. Your instructor may allow you the choice of using
either, or may require you to use one or the other. If you are given the choice, you should
try a few examples calculating the factor each way so that you can find which formula
works best for you.
The following example will illustrate calculating a n|i in each way.
Example 4.4.6 Find the present value annuity factor for a 5-year annuity with
quarterly payments and a 9% interest rate.
Using Formula 4.4.3:
We fi rst fi nd the future value annuity factor just as we have been:
sn (^) | (^) i
(1 i)n 1
__i
(^) ( 1 0.09_____ 4 )
20
1
0.09_____ 4
24.91152003
Then plugging this in to Formula 4.4.3, we get:an _ (^) | (^) i
s _n (^) | (^) i
(1 _ i)^ n^ ^ ___24.91152003^
(^) 1 _____ 0.09 4
20 15.96371237
4.4 Present Values of Annuities 173