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
Notice that given any starting amount, the amount after 1 year would be A = P0(1 +
Annual Percentage Yield |
The annual percentage yield is the actual percent a quantity increases in one year. It can be calculated as: APY = = = |
The above formula should look very similar to the one we used to find the Relative Change in Section 2.2.
Example 1 |
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+ Bank B: APY = ( 1+ 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 |
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%