How to Calculate APY, aka Effective Annual Rate

Calculating APY from the Nominal Interest Rate

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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

APYMONTHLY = (1 + r/12)12 - 1.

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

APYBIWEEKLY = (1 + r/26)26 - 1.

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

APYWEEKLY = (1 + r/52)52 - 1.

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

APYDAILY = (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

APYMONTHLY = (1 + 0.06/12)12 - 1 = 0.061678 = 6.1678%
APYBIWEEKLY = (1 + 0.06/26)26 - 1 = 0.061763 = 6.1763%
APYWEEKLY = (1 + 0.06/52)52 - 1 = 0.061800 = 6.18%
APYDAILY = (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

APYCONTINUOUS = er - 1,

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

APYCONTINUOUS = e0.06 - 1 = 0.061837 = 6.1837%.


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