A quoted rate is incomplete until you know how often it compounds. Banks and lenders quote a nominal annual rate of r, compounded m times per year. In loan contexts it's called the APR, or annual percentage rate. Each period credits r/m, and there are m·t periods:
FV = P × (1 + r/m)^(m·t)
What is the effective annual rate (EAR)?
The effective annual rate (EAR) is the single once-a-year rate that would produce the same result, and it's the only fair basis for comparing offers:
EAR = (1 + r/m)^m − 1
Twelve percent nominal, compounded at different frequencies:
| Compounding | m | EAR |
|---|---|---|
| Annual | 1 | 12.000% |
| Semi-annual | 2 | 12.360% |
| Quarterly | 4 | 12.551% |
| Monthly | 12 | 12.683% |
| Daily | 365 | 12.747% |
| Continuous | → ∞ | 12.750% |
There are two lessons in the table. First, frequency always helps the side receiving interest, since EAR rises with m. Second, the benefit saturates fast. Going from daily to continuous adds almost nothing. The limiting case, FV = P·e^(r·t), is continuous compounding. You'll meet it again in option pricing years from now. For now, just recognise it as the ceiling of the table.
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How do APR and EAR work in India and the US?
India, in practice. Indian bank fixed deposits typically compound quarterly. An FD quoted at 7% nominal actually yields (1 + 0.07/4)^4 − 1 = 7.19% EAR, which is the "annualised yield" printed beside the rate on the bank's page. Credit cards quote monthly rates of about 3–3.6% per month. 3.5%/month sounds like 42% a year, but the EAR is (1.035)^12 − 1 = 51.1%. Unpaid card debt is a doubling machine running at ln 2 / ln(1.035) = 20.1 months per double. That figure should make you wince. It's also where the Rule of 72 stops being safe, so watch what happens to it. Applied to the EAR, 72/51 = 16.9 months runs three months fast, because 51% is far outside the 4% to 15% band the rule is calibrated for. Applied to the nominal, 72/42 = 20.6 months is almost spot on, and only by luck. Using the nominal instead of the EAR pushes the estimate up by roughly what the rule's high-rate undershoot pulls it down. Two errors cancelling isn't a method. At these rates, compute exactly.
US, in practice. Deposit products advertise APY, the annual percentage yield, which is simply EAR under the Truth in Savings Act, so US savers can compare directly. Loans advertise APR. A 24% APR card compounding daily is (1 + 0.24/365)^365 − 1 = 27.1% EAR.
How should you compare rates with different compounding?
So here's the analyst's rule. Never compare quoted nominal rates across different compounding frequencies, and convert both to EAR first. An 8.6% deposit compounded quarterly, at an EAR of 8.88%, beats an 8.8% deposit compounded annually. The fine print is worth 8 bp here. On loan products it's routinely worth hundreds.