Interest rates are quoted in several different ways, and understanding the differences is essential for comparing borrowing costs. The nominal interest rate is the stated rate without accounting for compounding or fees. APR, or Annual Percentage Rate, represents the yearly cost of borrowing including the interest rate plus certain fees such as origination or closing costs, expressed as a standardized annual rate. APY, or Annual Percentage Yield, is the effective annual rate after accounting for intra-year compounding, and it is mathematically equivalent to the Effective Annual Rate (EAR) calculated as \[\text{APY} = \left(1 + \frac{r}{n}\right)^n - 1\] where \(r\) is the nominal rate and \(n\) is the number of compounding periods per year.
Because APR ignores intra-year compounding while APY reflects it, APY is always equal to or higher than APR whenever interest compounds more than once per year. This distinction matters most when comparing loans that have the same APR but different compounding schedules, such as daily-compounding credit cards versus monthly-compounding mortgages. More frequent compounding adds interest to the principal more often, so daily compounding costs more than monthly, which in turn costs more than annual compounding even with an identical stated APR. Borrowers should rely on APY when comparing products that compound at different frequencies, and on APR when comparing loan offers with different fees and structures, because APR includes closing costs and other charges in a standardized way.
Beyond nominal and effective rates, the real interest rate reflects the true economic cost of borrowing after accounting for inflation. The relationship is captured by the Fisher equation, \((1 + i) = (1 + r)(1 + \pi)\), where \(i\) is the nominal rate, \(r\) is the real rate, and \(\pi\) is inflation. Approximately, the nominal rate equals the real rate plus inflation, so a 5% mortgage during 3% inflation yields only about 2% in real terms. Continuous compounding represents the theoretical maximum compounding frequency and is computed as \(A = Pe^{rt}\); it is sometimes used for precise APR-to-APY comparisons. US regulations require institutions to quote APY uniformly under the Truth in Savings Act so consumers can compare deposit products across banks, which is why standardized rate displays are used throughout the industry.