The Mathematics of Money

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

Installment Plan Interest Rates: The Approximation Formula

An alternative means of calculating the APR is an approximate calculation of the actual

rate by using a formula such as the one given below.

FORMULA 10.3.2

APR Approximation Formula

APR 
 ______2nr

n 
 1

where

n represents the NUMBER OF PAYMENTS

and

r represents the NOMINAL SIMPLE INTEREST RATE

The symbol 
 used above indicates “is approximately equal to.”

Example 10.3.6 Calculate the APR interest rate for Tonya’s piano loan by the

approximation formula.

APR 
 ______n 2nr 1 
 2(24)(0.08)___________

25

15.36%

This is not the same as the 14.65% rate we found by using a spreadsheet or table. However,

even though these two rates are not exactly the same, they are “in the same ballpark.” The

approximation formula is claimed to give only an approximation, not the exact value. Note

also that the APR interest rate is considerably higher than 8%. Either of these rates makes

that quite apparent!

We could also find the APR for Bill’s dining room furniture by both of these methods as

well. Since we did not have a simple interest rate for Bill’s plan, though, we would first

have to calculate it.

Example 10.3.7 Bill is paying off a $1,400 purchase with nine monthly payments

of $171.11. Calculate the APR for this installment plan, using amortization tables and

using the approximation formula.

Using an amortization table with guess and check or solving for the annuity factor and fi nd-

ing it in an annuity factor table we come up with a rate of 23.40%.

By spreadsheet:

Rows Omitted

11 9 $171.11 $3.27 $167.84 $0.02

10 8 $171.11 $6.48 $164.63 $167.86

1 Rate: 23.40% Initial Balance: $1,400

2

3

A B C D

Month Payment To Principal Ending Balance

1 $171.11 $143.81 $1,256.19

To Interest

$27.30

4 2 $171.11 $24.50 $146.61 $1,109.58

5 3 $171.11 $21.64 $149.47 $960.11

E

By table:

We can solve for the annuity factor:

$1,400
$171.11a _n (^) |i
a _n (^) |i
8.181871311
10.3 Installment Plans 453