What is APR?

How to Calculate APR from the Annual Interest Rate

When you apply for a home loan or auto loan, you will pay interest on the loan according to the annual interest rate that the lender offers you. However, if closing fees are added to the amount borrowed, the actual interest rate will be higher than the quoted annual interest rate. The actual interest rate is called the Annual Percentage Rate, or APR for short.

APR represents the true interest rate that you would pay on a loan if you borrowed just the principal (without fees) but made the same monthly payments (including fees) for the same length of time. Another way to think about APR is that it represents the actual interest rate on a loan when you consider the closing fees to be a form of interest. This is best illustrated by example.

Suppose you take out a loan for $10,000 plus $1000 in fees at a 5% annual interest rate for 10 years. The total amount you borrow is thus $11,000. If you use a standard mortgage calculator, you will find the monthly payments are $116.67. (Use 11000 for the principal, 120 for the months, and 0.05 for the annual interest rate.)

Now, suppose you borrowed just $10,000 for 10 years. What interest rate would make your monthly payments come out to $116.67? The answer turns out to be 7.11%. Even though the quoted annual interest rate is 5%, the APR is 7.11%.

By law, banks and lending institutions are required to report the APR along with the loan terms. If the loan comes with no extra closing fees, then the annual interest rate and APR are equal. In fact, most many loan advertisements list only the APR and not the annual interest rate, so you won't have to make extra calculations.

If you don't know the Annual Percentage Rate, you can compute it using the APR calculator above. Just enter the number of months, the original amount borrowed (principal), the fees, and the quoted annual interest rate. You do not have to convert the interest rate to a decimal; the calculator will do the math for you. The calculator will then output the true Annual Percentage Rate. You can also read below to learn how APR is computed mathematically.

Relation Between APR and Annual Interest Rate

Banks determine your monthly payments M based on the annual interest rate I, the principal P, the fees F, and the number of months in the loan term N. The equation for monthly mortgage and car payments is

M = (P+F)(I/12)(1+I/12)^N/[(1+I/12)^N - 1].

In this equation, I is a decimal. That is, 6% is treated as 0.06. Keeping M, N, and P the same but ignoring F, The Annual Percentage Rate is the number R such that

M = (P)(R/12)(1+R/12)^N/[(1+R/12)^N - 1].

In this equation, R is the only unknown quantity. This equation is too complex to be solved for R using standard techniques in algebra, but the calculator can compute the APR instantly. It uses an iterative procedure similar to "guess and check" to pinpoint the value of R that solves the equation.

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