The Mathematics of Money

156 Chapter 4 Annuities

Example 4.2.8 On New Year’s Day 2004, Mariano resolved to deposit $2,500 at the

start of each year into a retirement savings account. Assuming that he sticks to this

resolution, and that his account earns 8¼% compounded annually, how much will he

have after 40 years?

The payments are equal and made at the start of each year, so this is an annuity due. Thus:

FV
PMT s _n (^) | (^) i(1  i)
FV
$3,000 s 40 | (^) .0825 (1  0.0825)
Before going any further, we must fi nd the annuity factor. Using Formula 4.2.2 we get:
sn (^) | (^) i

(1  i)n  1
____i

(1  0.0825)^40  1

_____0.0825
276.72205752

Returning to the FV formula by plugging in this annuity factor, we get:

FV 
 $3,000(276.72205752)(1.0825)

FV 
 $898,654.88

So if Mariano does stick with the program, he will have $898,654.88, or nearly $900,000

after 40 years.

Summing Up

We’ve covered a lot of ground in this section. Before wrapping up with discussion of some

optional topics, it’s worthwhile to briefly summarize the formulas of this section together

in one place.

Future Value of Annuity Formulas Summary

Ordinary annuities: FV
PMT s n (^) |i
Annuities due: FV
PMT s n (^) |i (1  i)
For both formulas, the annuity factor is given by the formula:
s _n|i

(1  i)n  1
_i
When Compounding and Payment Frequencies Differ (Optional)
As noted previously, we normally assume that interest compounds at the same frequency
as payments are made to an annuity. When interest compounds more frequently than pay-
ments are made, the future value can be found by first finding an equivalent interest rate that
compounds at the same frequency as the payments. The following example will illustrate.
Example 4.2.9 Suppose that $300 is deposited each quarter into an account paying
6% interest compounded monthly. Find the future value of the account in 5 years.
There are 3 months in each quarter, and so the interest compounded each quarter will
amount to multiplying the balance by (1.005)^3. Thus, we use this factor in place of the
(1  i), and in place of i we use (1.005)^3  1 in the s n (^) | (^) i formula to get:
sn (^) | (^) i
(1^  i)
n  1
_i

 (1.005)^3 ^

20

 1

_______________

(1.005)^3  1

23.1407801

Thus the future value would be FV 
 $300(23.1407801) 
 $6,942.23.

Note that if we had simply ignored the monthly compounding and just used 6% compounded

quarterly, the future value would have been $2,312.37. While the monthly compounding

does make a difference, that difference is not large.

In cases where the payments are made more frequently than interest compounds, every-

thing depends on how the between-compoundings payments are treated. It is likely the case

(though not necessarily so) that payments made between compoundings will earn interest

for the portion of the period for which they are in the account. Assuming this, the approach

is basically the same as in Example 4.2.8, though the exponents are uglier.

cf