Effective Interest Rate vs Nominal Rate: What You Are Actually Paying

Nominal rate (APR / stated rate): the annual rate before compounding Effective Annual Rate (EAR / AER): the true annual rate after compounding EAR = (1 + r/n)^n - 1 r = nominal annual rate (as a decimal) n = number of compounding periods per year Example: 12% nominal, compounded monthly (n=12): EAR = (1 + 0.12/12)^12 - 1 = (1 + 0.01)^12 - 1 = (1.01)^12 - 1 = 1.12683 - 1 = 0.12683 = 12.683% The effective rate (12.68%) is higher than the nominal (12%). The gap grows as compounding frequency increases.

Nominal rate: 10% per year Compounding frequency | Effective Annual Rate Annually (n=1): 10.000% Semi-annually (n=2): 10.250% Quarterly (n=4): 10.381% Monthly (n=12): 10.471% Daily (n=365): 10.516% Continuously: 10.517% Key observation: beyond monthly compounding, the difference is small. But annual vs monthly at higher rates matters more: Nominal 24% (typical credit card), monthly vs annual: Monthly compounding: (1 + 0.24/12)^12 - 1 = 26.82% effective Annual compounding: 24.00% effective Difference: 2.82 percentage points -- significant on large balances

To compare products quoted in different ways: Nominal rate = n x ((1 + EAR)^(1/n) - 1) Product A: 5.1% AER (effective, annually) Product B: 0.42% per month (monthly, stated as monthly rate) Convert Product B to EAR: Monthly rate = 0.42% = 0.0042 EAR = (1 + 0.0042)^12 - 1 = (1.0042)^12 - 1 = 5.16% Product B (5.16% EAR) pays slightly more than Product A (5.1% AER). Without this conversion, you might incorrectly pick Product A.

Credit card with 2% monthly rate: EAR = (1.02)^12 - 1 = 26.82% effective annual rate This is higher than the "24% APR" many people assume. Payday loan example: 1% per day interest EAR = (1.01)^365 - 1 = 3,678% effective annual rate (This is why daily-rate lending is so destructive.) Mortgage with 4.5% APR, monthly repayments: Monthly rate = 4.5% / 12 = 0.375% EAR = (1.00375)^12 - 1 = 4.594% The true cost of capital is slightly above the stated APR.