APY Calculator

APR (Annual Percentage Rate) is the nominal annual rate before compounding; APY (Annual Percentage Yield) is the effective annual rate after compounding is applied. The formula APY = (1 + APR/n)^n - 1 makes the gap explicit: at the same nominal 5% APR, compounding monthly gives 5.1162% APY, daily gives 5.1267%, and continuously gives 5.1271%. Banks must quote APY on deposit accounts (so savers can compare them apples-to-apples) and APR on loans (so borrowers see the cheapest stated rate first). That single regulatory split is why a 5% savings account and a 5% loan are not symmetric: the savings account pays you APY, but the loan still charges you APR plus fees.

The APY gain from compounding more often shrinks fast as frequency rises. At 5% nominal, monthly compounding is 5.1162% APY and daily is 5.1267%, a gap of just 0.0105 percentage points. On a $10,000 deposit that is $1.05 of extra interest in year one. The mathematical reason is that the function (1 + r/n)^n converges to e^r as n grows; once you are past monthly, each step toward daily, hourly, or continuous adds only the tail of that convergence. The practical implication is that a higher headline APY at a competitor with the same compounding frequency matters far more than chasing daily-versus-monthly at the same APY.

As compounding frequency approaches infinity, the APY converges to e^r - 1, where e is Euler's number (about 2.71828). This is called continuous compounding and it represents the mathematical ceiling for any given nominal rate. At 5% nominal: monthly gives 5.1162%, daily gives 5.1267%, hourly gives 5.1270%, and continuous gives 5.1271%. Some money-market and zero-coupon instruments quote continuous rates because the math is cleaner for derivatives pricing, but no real-world deposit account actually compounds infinitely often. Use the continuous figure as a sanity check on how close to the ceiling your daily-compounding account already is.

Yes. The Truth in Savings Act of 1991, implemented by the Federal Reserve as Regulation DD (12 CFR Part 1030), requires US banks and credit unions to disclose APY (not just APR) on every savings account, money market account, CD, and interest-bearing checking account. The same APY must appear in ads, account-opening disclosures, and periodic statements, and it must be calculated using a standardized formula so consumers can compare offers directly. The rule was passed specifically because competing banks were advertising the same nominal rate with different compounding frequencies and presenting them as equivalent, which made shopping for the best yield close to impossible.

Yes, monotonically. For any positive nominal rate r, the APY (1 + r/n)^n - 1 strictly increases as n increases, with continuous compounding (e^r - 1) as the upper bound. At 6% nominal: annually = 6.000%, semiannually = 6.090%, quarterly = 6.136%, monthly = 6.168%, daily = 6.183%, continuous = 6.184%. The marginal gains shrink quickly (the jump from annual to monthly is 0.168 points, monthly to daily is just 0.015 points), but the direction never reverses. A flat or lower APY at higher frequency would violate the math, so if you see one, the nominal rate has been changed somewhere in the disclosure.