Effective Annual Rate (EAR) Calculator

In 2026, a first-time mortgage shopper comparing a 6.50% APR (monthly compounding) against a 6.45% APR (daily compounding) needs to know which is actually cheaper. The widget converts both to EAR (6.6972% vs 6.6614%) instantly - revealing the daily-compounded loan is the better deal despite the slightly higher headline APR. This is the kind of mathematical sleight-of-hand EAR exists to expose.

Jacob Bernoulli's 1683 study of compound interest discovered the mathematical constant e ≈ 2.71828, defined as the limit of (1 + 1/n)^n as n → ∞. By extension, e^r is the limit of (1 + r/n)^n. This is the foundation of the EAR formula - the discrete compounding case (1 + r/n)^n − 1 smoothly approaches the continuous limit e^r − 1 as n grows. The widget's "continuous" dial position uses this exact closed-form rather than a large-n approximation.

The first US legislation requiring transparent rate disclosure was the Truth in Lending Act (TILA, 1968), implemented by Federal Reserve Regulation Z (12 CFR 1026). TILA created APR as a standardized disclosure for consumer loans - though APR is still a nominal rate, not the effective rate. For loans, the gap between APR and EAR can be 10-30 basis points on monthly- compounded products and 200-400 basis points on daily-compounded credit cards.

The Truth in Savings Act (TISA, 1991), implemented by Regulation DD (12 CFR 230), went further for deposit products - requiring APY disclosure, which equals EAR. This means every modern US savings account ad shows the effective rate prominently. Loan ads still show APR by convention because the lower number is more marketing-friendly. The asymmetry is the source of consumer confusion the EAR calculator is designed to resolve.

International conventions vary. UK retail products typically quote AER (Annual Equivalent Rate), which is identical to EAR/APY. Canadian mortgages use semi-annual compounding (n=2) by federal law (Interest Act 1985), which is why Canadian mortgages at 6% are mathematically cheaper than US mortgages at 6% - the Canadian EAR is 6.090% vs the US 6.168%. The 8-basis-point gap is structural, not negotiable.

EAR also plays a central role in derivative pricing and bond mathematics. The Black-Scholes option pricing model uses continuous compounding (e^rt) because it simplifies the partial differential equations. Bond yields are typically quoted in "bond-equivalent yield" (BEY), which is the semi-annual nominal rate - convertable to EAR via (1 + BEY/2)^2 − 1. Failing to convert correctly between BEY and EAR can introduce 5-15 basis points of pricing error.

For US retail consumers in 2026, the practical rule is: always compare EAR (= APY) not nominal APR when shopping savings products, and always compare EAR (not APR) when shopping loans across different compounding frequencies. The widget exists to make this conversion frictionless - a single dial spin reveals the true cost or yield of any product.