Nominal vs effective interest rate

A nominal interest rate is a stated annual rate that doesn’t account for how often interest compounds within the year. An effective interest rate is the actual rate after accounting for compounding within the period. The two are equal only when compounding happens exactly once per period.

Notation: $r$ is the nominal annual rate, $m$ is the number of compounding sub-periods per year, $i_s=r/m$ is the per-sub-period rate, and $i_e$ is the effective annual rate.

The future value of $P$ after one year of compounding at $m$ sub-periods is

$F=P{\left(1+\frac{r}{m}\right)}^m,$

so the effective annual rate that gives the same one-year growth is

$i_e={\left(1+\frac{r}{m}\right)}^m-1.$

A worked example. A credit card advertises 18% APR compounded monthly. The nominal rate is $r=0.18$, monthly compounding means $m=12$, the per-month rate is $0.18/12=0.015$ (1.5%), and the effective annual rate is

$i_e=(1.015{)}^{12}-1=1.1956-1=0.1956\approx 19.56\%.$

So borrowers actually pay 19.56%, not 18%, on a balance held all year. The more frequent the compounding, the bigger the gap between nominal and effective.

In the continuous-compounding limit ($m\to \infty$), the effective rate becomes $i_e=e^r-1$. For $r=0.18$ this is $e^{0.18}-1\approx 0.1972$, only slightly higher than monthly compounding.

The reason for the distinction: nominal rates are convenient for advertising and contracts (round numbers, straightforward to add up across periods), but they understate the true cost of borrowing or true return on investing whenever compounding is more frequent than the stated period. For any honest financial comparison — comparing two loan offers, two savings accounts, two investment products — convert both to effective annual rates first, then compare.

The same principle extends to other period mismatches. If your compounding period is monthly but your cash flows happen quarterly, you have to convert the rate to a quarterly effective rate before applying annuity factors or comparing PW.

In Canadian regulation, mortgages are quoted with a special “semi-annually compounded” convention that’s slightly different from the US APR convention — same underlying mechanism, different conversion. Always check what the quoted rate means.

For the basic compounding formula, see Compound interest. For comparison purposes after inflation, see Real interest rate.