Effective annual yield from a stated APR at any compounding frequency

Why APR and APY Differ

APR (Annual Percentage Rate) is the simple annual interest rate without accounting for compounding. APY (Annual Percentage Yield) is the effective annual rate after compounding effects. The two differ because compounding adds interest on interest during the year. A 5% APR with monthly compounding gives 5.116% APY. With daily compounding, 5.127% APY. With continuous compounding, 5.127% (capped at e^r - 1). The gap widens at higher rates — 20% APR with monthly compounding is 21.94% APY, a noticeable 194 basis point difference.

When APR vs APY Actually Matters

Savings and investment products (CDs, money market, high-yield savings, dividend yields): APY is the number you earn. Banks advertise APY because it looks higher. Loans and credit cards: APR is what you pay. Lenders advertise APR because it looks lower. A 20% APR credit card effectively charges about 22% APY if unpaid, but the headline is 20%. Reading the small print tells you which number is being quoted — sometimes banks mix APR and APY strategically to make their products look more attractive.

Compounding Frequency Options

Annual: 1 compounding per year (rare in modern products). Semiannual: 2 (some bonds). Quarterly: 4 (some savings accounts). Monthly: 12 (most credit cards, many savings accounts). Daily: 365 (most savings accounts, CDs). Continuous: effectively infinite (theoretical upper bound, used in financial modeling). As frequency rises, APY approaches a ceiling — the difference between daily and continuous is typically under 1 basis point.

The Formula

APY = (1 + APR/n)^n - 1, where n is compounding periods per year. At higher APRs, small differences in n produce meaningful APY gaps. At low APRs (under 3%), APR and APY are close enough that the distinction is academic. At higher APRs (10%+), the distinction matters materially — especially on compounded debt like credit cards, where the unpaid balance keeps growing at the higher effective rate.

Worked Example

APR: 5%. Monthly compounding (n=12). APY = (1 + 0.05/12)^12 - 1 = 5.116%. Difference: 0.116 percentage points or 11.6 basis points. Same APR with daily compounding (n=365): APY = (1 + 0.05/365)^365 - 1 = 5.127%. The extra 1.1 basis points from daily versus monthly compounding is roughly the same on 10,000 balance as 1.10 per year — trivial on small balances, meaningful on institutional-scale portfolios.

Using This for Loan Shopping

Credit card issuers always quote APR. Their effective APY for a balance carried is higher by the compounding factor. A 22% APR card compounds monthly to roughly 24.36% APY — a meaningful 236 basis points higher than the headline. Similarly for variable-rate loans: compare offers on effective APY rather than headline APR to see the true cost. Savings account marketing usually does the opposite — quoting APY to make the deposit look more rewarding than the underlying daily-compounded rate would suggest.

Regulatory Requirements Around APR and APY Disclosure

Truth in Lending Act requires lenders to disclose APR on most consumer credit products. Truth in Savings Act requires banks to disclose APY on deposit products. These rules exist precisely because consumers historically confused the two. And EU have similar rules under various consumer credit directives. When comparing products within the same category (two savings accounts, two credit cards), the disclosed rate is directly comparable. Comparing across categories requires converting to a common basis, which this calculator does automatically.

Why the Difference Matters More on Debt

A 22% APR credit card with a 5,000 balance compounds daily to an effective rate of roughly 24.36%. On 5,000, the difference between 22% and 24.36% per year is about 118. Small sounding, but this compounds on top of itself for carried balances. Over 3 years of minimum-payment-only behavior on a 5,000 credit card balance, the difference between paying down at true APY versus assumed APR is roughly 400-600 of additional interest. The disclosed APR understates what you actually owe if you carry balance; always look up the effective APY for any debt you cannot pay off monthly.