Nominal vs. Effective Rates
Why 12% compounded monthly is not really 12%
One card quotes 12% compounded monthly and another quotes 12.5% compounded annually. Which actually costs more?
A quote is not a rate
A nominal annual rate $i$ compounded $m$ times per year credits $i/m$ each sub-period. Because those sub-period credits themselves compound, the true annual growth - the effective annual rate (EAR) - is higher than the quote:
More frequent compounding, higher EAR
For a fixed nominal rate, increasing $m$ raises the EAR, but with diminishing effect: the jump from annual to monthly is far larger than from monthly to daily. In the limit of continuous compounding the EAR approaches $e^{i} - 1$.
Never compare two quoted rates directly unless they share a compounding frequency. Convert both to EAR first - that is the only apples-to-apples comparison.
- 1. Inputs: $i = 0.12$, $m = 12$, so $i/m = 0.01$.
- 2. Growth over the year: $(1.01)^{12} = 1.126825$.
- 3. Subtract 1: $EAR = 1.126825 - 1 = 0.126825$.
💡 Hint
Check your understanding
- A nominal rate needs a compounding frequency to be meaningful.
- The effective annual rate is $EAR = (1 + i/m)^m - 1$.
- Convert every quote to EAR before comparing borrowing or investing options.