Table 21.2 on page 873 shows that at an annual interest rate of 10% (a nominal rate) compounded daily for one year, $1000 yields $105.16 in interest, which is 10.516% of the principal. If instead you had $1000 just at simple interest of 10.516% for one year, you would earn exactly the same amount. Thus, 10% compounded daily is effectively the same as 10.516% simple interest, and we say that 10.516% is the effective rate.
Since there is no compounding done inside a compounding period, the effective rate for a compounding period is the nominal rate.
Effective Rate and APY DEFINITION
The effective rate is the rate of simple interest that would realize exactly the same amount of interest over the same length of time. For a year, the effective rate is called the annual percentage yield (APY).
For an interest rate per compounding period, a principal of $1 grows to in periods, so the interest earned on that $1 is the final principal minus the original principal of $1, or . Hence, we have the following formula for the effective rate:
Formula for Effective Rate RULE
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Mostly, we will be interested in the effective rate on an annual basis. For a nominal annual interest rate compounded times per year, the interest rate per compounding period is , and an amount of $1 grows in one year to
The effective annual rate of interest (the APY) is the amount of interest earned
divided by the original principal. Since that principal is $1, we have the following.
Formula for APY RULE
where
The only difference between the effective rate and the APY is that the term APY is used only for an annual rate (the “A” in APY). The effective rate could be over any length of time.
EXAMPLE 5 Finding the APY
For a nominal annual rate of 10% compounded monthly, what is the APY?
What is the APY for an account that earns 5% per year, compounded quarterly?
In some cases, you know the principal, the current balance, and the interval of time, and you want to learn the interest rate. For example, money market funds typically report earnings to investors each month, based on interest rates that vary from day to day, but often do not report the average interest rate. We can find the equivalent average effective daily rate, from which we can calculate the APY.
The compound interest formula gives the end-of-month balance as , where is the balance at the beginning of the month, is the average daily interest rate, and is the number of days that the statement covers. So we have
Taking the th root (the power) gives
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EXAMPLE 6 Daily Interest Rate on a Money Market Account
Suppose that the monthly statement from the fund reports a beginning balance () of $7373.93 and a closing balance () of $7382.59 for 28 days (). What is the effective daily rate?
We have
Thus, the average effective daily rate is 0.00419194%. Compounding daily for a (non-leap) year, we would have , for an APY of 1.54%.
For an account that is compounded daily and earns 10% APY, what is the effective daily rate? Hint: For any initial principal in such an account, at the end of 365 days we have .