Effective annual rate and compounding: comparing rates properly
A bank offers a savings account at 12% a year, compounded monthly. Another offers 12.5%, compounded annually. Which is better? You cannot tell from the stated rates alone, because they compound at different frequencies. The effective annual rate, EAR, puts them on the same footing: it is what a pound actually earns over a year.
Stated rate and periodic rate
A stated (or nominal) annual rate is quoted per year but applied per period. The periodic rate is the stated rate divided by the number of periods in the year. A stated rate of 12% compounded monthly applies 1% a month, twelve times.
The effective annual rate
Interest earned in one month earns interest in the months after it, so a year of 1% monthly growth is more than 12%:
EAR = (1 + stated rate ÷ m)m − 1
where m is the number of compounding periods a year. For 12% stated and monthly compounding:
EAR = (1 + 0.12 ÷ 12)12 − 1 = 12.68%
So the 12% monthly account beats the 12.5% annual one, since 12.68% is more than 12.5%. Try any rate in the effective annual rate calculator.
More frequent compounding means more interest, with a limit
Keep the stated rate at 12% and change only how often it compounds:
- Annually: 12.00%
- Semiannually: 12.36%
- Quarterly: 12.55%
- Monthly: 12.68%
- Daily: 12.75%
- Continuously: 12.75%
Each step adds less than the one before. The ceiling is continuous compounding, which uses the exponential function:
EAR = estated rate − 1
At 12% this is 12.7497%, and daily compounding is already within a hair of it at 12.7475%.
Going backwards: from EAR to a stated rate
Often you need the rate for a given frequency that produces a known EAR. Take the root first:
stated rate = m × [(1 + EAR)1/m − 1]
An EAR of 10% corresponds to a stated rate of 9.76% compounded semiannually, 9.57% compounded monthly, or 9.53% compounded continuously. The continuous rate is the natural log of (1 + EAR). A common slip is to divide the EAR by m, which gives a stated rate that does not reproduce the EAR.
Using the effective rate in a time-value problem
The cleanest way to avoid errors in a time-value problem is to match the rate to the cash flows. If payments are monthly, use the monthly rate and count months. Or, if the cash flows are annual, convert to an EAR and use annual periods. For example, 1,000 at 8% compounded monthly grows to 1,000 × (1 + 0.08 ÷ 12)36 = 1,270.24 after three years, compared with 1,259.71 if compounded annually. The time value of money guide has more on matching periods.
The holding period return
A related idea is the holding period return: the total return over whatever period you held an asset, including income. It is (ending value − beginning value + income) ÷ beginning value. To compare holding period returns across different lengths, convert them to an effective annual rate: raise (1 + HPR) to the power of one over the number of years held, and subtract one.
Mistakes to avoid
- Comparing stated rates across different frequencies. Convert both to EAR first.
- Multiplying a monthly rate by 12 and calling it effective. That is the stated rate. Compound the monthly rate instead.
- Using the continuous formula for ordinary compounding, or the reverse.
- Forgetting that the effective rate depends on frequency, but the stated rate does not.
Where this is tested
Rates and compounding appear in the quantitative methods topic and run through fixed income, where yields are quoted on different bases, and derivatives, where continuous compounding is standard. The outcomes below set out the conversions.