A Formula for Calculating APR and APY

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In elementary finance courses taught in universities, students are often given easy situations to calculate examples in which the interest rate coincides with the payments or cash flows from an investment. For example, a student may be asked to calculate the future value of a yearly cash flow at the rate of 10% per year. However, investing is much more complicated when payments and interest rate quotes are not equal.

To make matters easy for investors, interest rates are almost always stated as a yearly rate. Even investments which are expected to expire in less than a year such as notes, treasury bills, and certificates of deposit (CD) have their interest rates expressed as a yearly figure. However, there are two different yearly rates that banks and other financial institutions give even though they represent the same return on investment. This article explains two common bank terms - APR and APY.

The Annual Percentage Rate (APR) is calculated as the periodic rate times the number of periods the interest will be compounded in a year. The basic formula for an APR given as:

APR = n * r

where n represents the number of periods and r represents the periodic rate compounded at each period. Suppose that an investment will pay a 2% return each month for a year. The APR of this investment would be:

APR = 0.02 * 12

= 0.24

or 24% APR. Usually, interest is compounded annually, quarterly, monthly, or continuously. The APR is considered a nominal rate (from the Latin nomen meaning name) because the APR represents the return on investment “in name only.” The real rate of return may be different because of the compounding frequency.

When interest is compounded more than once a year, the compounding frequency makes the two periods of interest applied to the principle worth more than the sum of two periods individually. This is because subsequent periods include the interest from previous periods.

An Annual Percentage Yield (APY) represents the expected return of an investment. This is because the APY takes into account the compounding frequency of an investment and the effect of interest applied to previous periods’ interest earned. To find the APY from an APR, the following formula may be used:

APY = [1 + (APR / n)]n – 1

Suppose that an investment will pay 24% APR for a year and that the interest will be compounded monthly. What is the expected return (APY) of the investment?

APY = [1 + (0.24 / 12)]12 – 1

= 0.2682

or approximately 26.82%. Notice that the APY of 26.82% is more than the APR of 24% because of the monthly compounding of interest. Notice also that no investment amount was given. Regardless of the amount invested, both the APR and APY stay the same.

It is a part of business and a part of human nature to make things seem better than they are when trying to convince someone to hand over their money. This is why sometimes an APR is quoted and sometimes an APY is quoted for different investment opportunities.

For example, when a car dealership is offering to finance the purchase of a new car, the managers of the dealership will often quote an APR interest rate because it seems lower than the real rate. When banks offer Certificates of Deposit they often quote an APY to make the investment seem more attractive. A wise investor knows that the real expected return is stated by the APY and calculates it him/herself when an APR is quoted. Now that you know what APR and APY mean, you can make better financial decisions.