# 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*as a decimal, and the number of compounding periods per year,*

**r***. The general formula is APY = (1 + r/N)*

**N**^{N}. When the interest is compounded monthly,

*= 12, thus*

**N**APY

_{MONTHLY}= (1 + r/12)

^{12}- 1.

When the interest is compounded biweekly,

*= 26, thus*

**N**APY

_{BIWEEKLY}= (1 + r/26)

^{26}- 1.

When the interest is compounded weekly,

*= 52, thus*

**N**APY

_{WEEKLY}= (1 + r/52)

^{52}- 1.

And when the interest is compounded daily,

*= 365, thus*

**N**APY

_{DAILY}= (1 + r/365)

^{365}- 1.

Keeping

*constant, as the value of*

**r***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*

**N**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,

*approaches infinity. APY for continuous compounding is calculated by*

**N**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
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