The Mathematics of Money

170 Chapter 4 Annuities

For convenience’s sake, when a table is used in situations such as these, the columns will

often be labeled with the nominal annual interest rates instead of the rates per period. This

saves having to divide the rate by 12 to get the rate per month. So the table we are using

might instead look like this:

SAMPLE TABLE OF ANNUITY FACTORS

Number of

Payments

(n)

INTEREST RATE

3% 6% 9% 12% 15% 18%

24 23.2659796 22.5628662 21.8891461 21.2433873 20.6242345 20.0304054

36 34.3864651 32.8710162 31.4468053 30.1075050 28.8472674 27.6606843

48 45.1786946 42.5803178 40.1847819 37.9739595 35.9314809 34.0425536

60 55.6523577 51.7255608 48.1733735 44.9550384 42.0345918 39.3802689

Be careful to make sure that you know whether it is the rate per period or the annual interest

rate that is shown in the column headers, so that you get the correct factor.

Finding Annuity Factors Efficiently—Calculators and Computers

Just as with future value factors, some calculators and computer software have the built-in

ability to calculate present value factors. The same comments made for future values apply

for present values. If you are using this approach to get your future value factors, you will

probably want to also use the approach for present value factors. You should consult your

instructor and/or owner’s manual for details of how to obtain these factors on the specific

hardware you are using.

Finding a Formula for the Present Value Factors

It is not necessary to have a table or special calculator to find present value factors any

more than it was necessary for future value factors. We can develop a formula for a _n (^) |i just
as we did for s _n (^) |i. To get things started, consider the following example, similar to some of
the problems we considered in Section 4.3:
Example 4.4.3 Suppose that Jon borrows $8,000 from his uncle to buy a car. Rather
than deal with the hassle of monthly payments to his uncle, they agree that Jon will
repay the loan in full all at once at the end of 3 years. They agree that the loan will
carry an interest rate of 4.2%, compounded monthly. To make sure that he has the
money to pay off the loan when it comes due, Jon decides to set up a sinking fund at
his company’s credit union, into which he will make monthly payments. Coincidentally,
the sinking fund also carries an interest rate of 4.2% compounded monthly. How much
should Jon deposit each month?
Remember that before we can determine the sinking fund payments, we must fi rst determine
how much Jon will actually need to pay his uncle:
FV
PV(1  i)n
FV
($8,000)(1  .0035)^36
FV
$9,072.26
Knowing that, we can determine the sinking fund payment:
FV
PMT s _n (^) | (^) i
$9,072.26
PMT s __ 36 | (^) .0035
$9,072.26
PMT(38.29504816)
PMT
$236.90
So Jon’s monthly deposits should be $236.90.