APR vs APY Calculator — Compounding Comparator

In 2026, a consumer credit counselor in Brooklyn sits with a client who is shocked that her 21% APR credit card actually costs 23.37% per year. The widget plots the simple-vs-compound curves over 24 months, the dashed and solid lines pull apart, and the client immediately understands why minimum payments never seem to make a dent. That single visualization is often the moment a debt repayment plan becomes possible.

The mathematical foundation of compound growth dates to Jacob Bernoulli, a Swiss mathematician who in 1683 studied compound interest in a paper titled Quaestiones nonnullae de usuris. Bernoulli asked: what happens to (1 + 1/n)^n as n grows larger? He proved the expression converges to a constant — bounded between 2 and 3 — which Leonhard Euler later named e (Euler's number, ~2.71828). Continuous compounding at 100% APR yields exactly e dollars per dollar invested in one year.

Irving Fisher's 1907 The Rate of Interest integrated compound interest with time-preference theory, formalizing the equation (1 + nominal) = (1 + real) × (1 + inflation). This equation lets the modern consumer distinguish a 5% APY savings yield in a 3% inflation environment (real yield ~1.94%) from a 5% APY in a 6% inflation environment (negative real yield of −0.94%). The widget reports nominal APY; the FAQ shows how to convert to real APY using current Federal Reserve inflation data.

The US Truth in Lending Act (TILA, 1968) and Regulation Z required LENDERS to disclose APR — making consumer-credit costs comparable across institutions for the first time in US history. The Truth in Savings Act (1991) and Regulation DD required BANKS to disclose APY on deposit products. Together these two laws created the modern dual standard: you see APR on loans and APY on savings. The widget makes it trivial to translate between them.

The Consumer Financial Protection Bureau (CFPB), established in 2011 after the financial crisis, enforces TILA on lenders. The Federal Reserve enforces Regulation DD on banks. The SEC regulates yield disclosure on securities (Rule 482 for mutual funds). FINRA Rule 2210 (Communications with the Public) requires fair-and-balanced yield presentation in broker-dealer advertising. Together this framework makes APY disclosure auditable across every consumer-financial product in the US.

Modigliani-Miller's 1958 capital-structure theorem and Markowitz's 1952 mean-variance portfolio theory both use continuously-compounded returns extensively. Sharpe's 1964 CAPM is most naturally expressed in continuous-compounding terms. Black-Scholes (1973) option pricing relies on continuous compounding throughout. The widget's continuous mode is the same mathematics that prices every option, future, and structured product on global derivatives exchanges.

The six product presets in this widget reflect 2026 US consumer-finance market norms: credit card 21% APR daily (Federal Reserve G.19 data), 30-year mortgage 6.5% APR monthly (Freddie Mac weekly survey), high-yield savings 4.5% APR daily (top-quartile online bank), 12-month CD 5.0% APR monthly, 60-month auto loan 7.5% APR monthly (Experian Q1 2026 report), DeFi USDC supply yield 8% APR continuous (Aave / Compound on-chain). Each preset is anchored to a published primary source and ships with the correct compounding frequency required by the relevant disclosure regime.