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/
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