The Mathematics of Money

154 Chapter 4 Annuities

We can work this out exactly as before:

Operation Result

1 .005916667
 1.005916667

^180
 2.891749861

 1 
 1.891749861

/.005916667
 319.732352909

Though the decimal makes this a bit tedious, this will work fi ne, and this is the preferred

method if you happen to be using a calculator that does not follow order of operations.

It may be more effi cient to avoid the decimal and enter i as a fraction, combining steps

as before. Make sure to put ( )’s around the denominator!

Operation Result

(1.071/12)^180
 2.891749861

 1 
 1.891749861

/(.071/12)
 319.732352909

If you have been entering the formula all at once, you can do that here as well, but between

the fractions and the added percents it gets awfully long, which invites typos.

Either of the approaches demonstrated in Example 4.2.5 will work out fine. The important

thing is to make sure that you can consistently and correctly use whichever approach suits

you and your calculator best.

Example 4.2.6 Each month, Carrie deposits $250 into a savings account that pays

4.5% interest (compounded monthly). Assuming that she keeps this up, and that

the interest rate does not change, how much will her deposits have grown to after

5 years?

Carrie’s payments are equal, and made at regular intervals, and so her deposits make up

an annuity. Since the timing of the payments is not specifi ed, we assume it is an ordinary

annuity, and so we can use Formulas 4.2.1 and 4.2.2 to fi nd its future value. We fi rst must

fi nd the annuity factor, and to do that we must fi rst fi nd:

i 
 0.045______

12

and

n 
 5(12) 
 60

Note that we have decided here to leave i as a fraction. Substituting into Formula 4.2.2,

we get:

sn _ (^) | (^) i
(^1 ^ i)
n  1
__i

(^)  1  0.045__ 12 
60
 1

0.045__
12

67.14555214

Using this in Formula 4.2.1, we get

FV
PMT s _n (^) | (^) i
FV
$250(67.14555214)
FV = $16,786.39
Notice that in all of our examples the payment and compounding frequencies are the
same. We quietly assumed this in developing our formula for s _n (^) |i. It is of course possible
for the annuity payments and compounding frequencies to differ, but this difference con-
siderably complicates matters. A way of dealing with this situation is outlined at the end
of this section, but for most purposes we would normally just assume that the frequencies
are identical.
cf