The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

FORMULA 4.4.2

The Present Value of an Annuity Due

PV 
 PMT a _n|i (1i)

Finding Annuity Factors Efficiently—Tables

Just as tables of annuity factors exist for future values, they also exist for present values.

An example of such a table is given below:

SAMPLE TABLE OF ANNUITY FACTORS

Number of

Payments

(n)

RATE PER PERIOD (i)

0.25% 0.50% 0.75% 1.00% 1.25% 1.50%

24 23.2659796 22.5628662 21.8891461 21.2433873 20.6242345 20.0304054

36 34.3864651 32.8710162 31.4468053 30.107505 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

Just as with the future value annuity factor table, this table allows us to look up the value of

the appropriate annuity factor for any interest rates and values of n that the table includes.

For example, if we need the annuity factor for a 3-year monthly annuity at 6%, we would

look in the n 
 36 row and 0.50% column and see that the value is 32.8710162. Using our

annuity factor notation, we would write a __ 36
(^) .005
32.8710162.
The following example will illustrate the use of a table factor to find the present value
of an annuity.
Example 4.4.1 Find the present value of $275 per month for 3 years at 6%.
Following our present value formula we have:
PV
PMT a _n (^) | (^) i
PV
($275)a __ 36 | (^) .005
The annuity factor we are looking for can be found in the table; in fact, it is the factor we
used as an example in our discussion above:
PV
($275)(32.8710162)
PV
$9039.53
The use of tables presents the same issues as it did with future values. A table needs to be
quite large to cover every reasonably possible value for n and i. However, there are a number
of situations where tables may be a very useful approach for present value calculations. As
we saw when first discussing annuities, one common application of annuities is payments
on loans. The table given above might be used, for example, by the finance office of a car
dealership. Payments on car loans are almost always monthly, and cars usually are financed
only over 2, 3, 4, 5, or at most 6 years. Similarly, though almost any interest rate is possible
for a car loan, it would not be unusual to have only certain selected interest rates in use at
any given time. If there are only a limited number of possible values for n and i, a table of
annuity factors can be kept to a reasonable size.
Example 4.4.2 A customer at a car dealership says that he can afford a $275
monthly car payment. He is looking for a used car and would be taking out a 3-year
loan; on the basis of his credit rating, he would qualify for a 6% rate. How much can
he afford to pay for the car?
The monthly payments are an annuity, and the money to buy the car would be the
present value of that annuity. This question, then, is the same as the one we worked out
in Example 4.4.1. He can afford to pay $9,039.53.
4.4 Present Values of Annuities 169
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