How to Calculate APY, aka Effective Annual Rate

Calculating APY from the Nominal Interest Rate

The Annual Percentage Yield (APY) is also known as the Effective Annual Interest Rate. It is the actual yield of a stated (nominal) annual interest rate depending on how frequently the interest is compounded. For example, suppose you have a $1000 bond with a nominal annual interest rate of 12%. If the interest is compounded only once per year, then the bond will be worth $1120 after one year.

However, if the interest on this bond is compounded monthly, then each month the bond's value increases by 12% ÷ 12 = 1% over the previous month. After one year the bond will be worth $1000(1.01)¹² = $1126.825. Thus, the actual yield (or the effective yield) is (1126.825-1000)/1000 = 12.6825%. In financial terms, we say that 12.6825% is the APY of 12% compounded monthly.

APY Formulas

To calculate APY for a given nominal interest rate, you need to know the nominal rate r as a decimal, and the number of compounding periods per year, N. The general formula is APY = (1 + r/N)^N. When the interest is compounded monthly, N = 12, thus

APY_MONTHLY = (1 + r/12)¹² - 1.

When the interest is compounded biweekly, N = 26, thus

APY_BIWEEKLY = (1 + r/26)²⁶ - 1.

When the interest is compounded weekly, N = 52, thus

APY_WEEKLY = (1 + r/52)⁵² - 1.

And when the interest is compounded daily, N = 365, thus

APY_DAILY = (1 + r/365)³⁶⁵ - 1.

Keeping r constant, as the value of N increases, so does the APY. For example, if we have a nominal annual interest rate of 6% and we compound it monthly, biweekly, weekly, and daily, then we get

APY_MONTHLY = (1 + 0.06/12)¹² - 1 = 0.061678 = 6.1678%
APY_BIWEEKLY = (1 + 0.06/26)²⁶ - 1 = 0.061763 = 6.1763%
APY_WEEKLY = (1 + 0.06/52)⁵² - 1 = 0.061800 = 6.18%
APY_DAILY = (1 + 0.06/365)³⁶⁵ - 1 = 0.061831 = 6.1831%

The maximum possible value for the APY is achieved with continuous compounding, that is, N approaches infinity. APY for continuous compounding is calculated by

APY_CONTINUOUS = e^r - 1,

where "e" is the mathematical constant 2.718281828459045... Using the example above, a 6% rate compounded continuously has an APY of

APY_CONTINUOUS = e^0.06 - 1 = 0.061837 = 6.1837%.

© Had2Know 2010