Calculating an APR

Date: 10/01/2002 at 22:57:08
From: Yoshi Menna
Subject: Calculating the APR

Compound interest seems like a black art to me...

A company made a loan for 12 months. The total amount was $790.60 and 
they charged $94.83 interest. How can I calculate the APR?

I have tried to reverse: (1+r)^2*P-(1+r)R-R but that really gets ugly.

Date: 10/02/2002 at 08:57:05
From: Doctor Mitteldorf
Subject: Re: Calculating the APR

Yoshi-

The APR depends on the exact terms of repayment. You can't calculate 
it just knowing the amount of the principal and the amount of the 
interest.

Here's a way to think about these problems: a dollar is not constant 
in value in an interest-bearing world. A dollar tomorrow is worth less 
than a dollar today. Many problems involving loans and present values 
can be solved with the paradigm that there is a single number r that 
represents an interest rate, such that the value of a dollar goes down 
as e^(-rt). This is to say: the value of a dollar t years from now is 
e^(-rt) times its value today (t can be a fraction, like 1/12 of a 
year).

Problems in APR amount to taking a loan (a dollar amount today) and
equating it to the value of a given repayment schedule. The single
parameter you adjust is r, and by trial-and-error, find the r that
makes the total value of all payments at their respective times equal
to the value of the loan amount today.

(There is one caveat in this simple prescription: The r that I define 
here is a "true rate." Legally and conventionally, the number that is 
reported as the APR is a "monthly rate," which is defined as 12 times 
the amount of interest that this true rate r would generate if it were 
compounded over a month. As a formula, the monthly rate R can be 
obtained from the true rate r with  R = 12*(e^(r/12)-1). For example, 
if the true rate r is 12%, then r/12 is 1% per month, so that in a month 
the value increases by a factor of e^0.01. This is about 1.01005, or 
1 + 1.005/100, so the amount of interest generated in a month is 
about 1.005%.) So the APR is quoted as 12 times this, which is 12.06%.)

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/