Section PF.9 – Annual Percentage Yield (APY)
Because of compounding throughout the year, with compound interest the actual increase in a year is usually more than the annual percentage rate. If $1,000.00 were invested at 10%, the following table shows the value after 1 year at different compounding frequencies:
| Frequency | Value after 1 year |
| Annually | $1100.00 |
| Semiannually | $1102.50 |
| Quarterly | $1103.81 |
| Monthly | $1104.71 |
| Daily | $1105.16 |
If we were to compute the actual percentage increase for the daily compounding (last row of the table above), there was an increase of $105.16 from an original amount of $1,000.00, for a percentage increase of [latex]\frac{105.16}{1,000}[/latex] = 0.10516 = 10.516% increase. This quantity is called the annual percentage yield (APY). This quantity is also sometimes called the annual effective yield or annual effective rate.
Notice that given any starting amount, the amount after 1 year would be A = P0(1 + [latex]\frac{r}{n}[/latex])nt . To find the total change, we would subtract the original amount, then to find the percentage change we would divide that by the original amount:
| Annual Percentage Yield |
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The annual percentage yield is the actual percent a quantity increases in one year. It can be calculated as: APY = [latex]\frac{Balance~After~One~Year~-~Starting~Principal}{Starting~Principal}[/latex] = [latex]\frac{(P_0(1~+~\frac{r}{n})^n~-~P_0)}{P_0}[/latex] = [latex](1~+~\frac{r}{n})^n~-~1[/latex] |
The above formula should look very similar to the one we used to find the Relative Change in Section 2.2.
| Example 1 |
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Bank A offers an account paying 1.2% compounded quarterly. Bank B offers an account paying 1.1% compounded monthly. Which is offering a better rate? We can compare these rates using the annual percentage yield – the actual percent increase in a year. Bank A: APY = ( 1+ [latex]\frac{0.012}{4}[/latex])4 – 1 = 0.012054 = 1.2054% Bank B: APY = ( 1+ [latex]\frac{0.011}{4}[/latex])12 – 1 = 0.011056 = 1.1056% Bank B’s monthly compounding is not enough to catch up with Bank A’s better APR. Bank A offers a better rate. |
| You Try PF.9.A |
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a. Calculate the APY for an account that pays 4% compounded daily. b. Calculate the APY for an account that pays 4% compounded monthly. c. Calculate the APY for an account that pays 4% compounded annually. If needed then round your answers to three decimal places. |
The table below shows the APY for $1000 invested in an account that pays 10% at different compounding frequencies:
| Frequency | Value after 1 year | APY |
| Annually | $1100.00 | 10% |
| Semiannually | $1102.50 | 10.25% |
| Quarterly | $1103.81 | 10.381% |
| Monthly | $1104.71 | 10.471% |
| Daily | $1105.16 | 10.516% |
Notice that the APY is always greater than the APR if interest is compounded more than once a year.
Section PF.9 – Answers to You Try Problems
PF.9.A
a. 4% compounded daily: APY = 4.081%
b. 4% compounded monthly: APY = 4.074%
c. 4% compounded annually: APY = 4%
