APY vs APR — How to Read Yield Numbers Honestly

A 5% APR savings account that compounds daily and a 5% APY savings account look identical in the marketing copy. They are not. The first one will pay you about 5.13% over a year; the second one will pay you exactly 5%. On a €10,000 balance that is €13 of difference — small, but in the opposite direction of what most savers assume. Flip the same trick onto a 24% APR credit card and the gap explodes: the effective annual cost of carrying a balance is closer to 27%. Reading APR and APY as the same number is one of the most common and most expensive mistakes in personal finance.

The formulas

The core conversion is mechanical:

• APY = (1 + APR/n)^n − 1
• APR = n × ((1 + APY)^(1/n) − 1)

Here n is the number of compounding periods per year — 12 for monthly, 52 for weekly, 365 for daily, 8,760 for hourly. APR is the nominal annual rate; APY (also called EAR, effective annual rate) is what the balance actually grows by after compounding has done its work over a full year.

Two intuitions help. First, APY is always greater than or equal to APR — they are equal only when n = 1 (annual compounding) or when the rate is zero. Second, APY grows monotonically in n: more frequent compounding always increases the effective rate, but with diminishing returns. The jump from yearly to monthly is much larger than from daily to hourly.

APR vs APY in real products

Which number a product quotes is almost always the one that flatters the seller. Banks advertise APY on savings accounts, certificates of deposit, and high-yield checking because compounding makes the headline number bigger than the underlying nominal rate. Lenders advertise APR on mortgages, auto loans, personal loans, and credit cards because skipping compounding makes the headline number smaller than what you will actually pay.

US regulation institutionalizes the asymmetry. Truth in Savings (Regulation DD) requires depository institutions to disclose APY on deposit accounts; Truth in Lending (Regulation Z) requires lenders to disclose APR on credit products. Each rule was written to protect consumers from the other side’s framing — and both are now the standard the industry hides behind. EU disclosure rules use slightly different terminology (TAN for nominal, TAE/TAEG for effective) but the mechanics are identical.

The practical takeaway: never compare a product’s APR to another product’s APY directly. Always convert to the same metric first.

A worked example

Take a 4.25% APR savings product and run it through different compounding frequencies on a €10,000 starting balance held for one year.

Compounding	n	APY	Year-end balance
Annual	1	4.2500%	€10,425.00
Quarterly	4	4.3182%	€10,431.82
Monthly	12	4.3338%	€10,433.38
Daily	365	4.3413%	€10,434.13
Continuous	∞	4.3415%	€10,434.15

Two things stand out. First, the difference between annual and daily compounding is €9.13 — meaningful but not dramatic. Second, the difference between daily and continuous compounding is two cents. Beyond daily, the curve is essentially flat, which is why no one in retail finance bothers with hourly or per-second compounding.

Continuous compounding

The mathematical limit of (1 + r/n)^n as n approaches infinity is e^r, where e ≈ 2.71828. So the continuously compounded APY for a 4.25% APR is exactly e^0.0425 − 1 ≈ 4.3415%. This is the formula derivatives traders, fixed-income quants, and academic finance papers use, because exponential rates are mathematically clean — they add cleanly across periods, integrate cleanly into Black-Scholes, and avoid the period-discretization headache.

In real consumer products it almost never matters. The gap between daily and continuous compounding at 4.25% is 0.0002 percentage points. At 10% it is 0.013 points. Even at 30% credit-card rates it is only 0.18 points. Unless you are pricing exotic derivatives or running a quantitative back-test, daily compounding is a perfectly good approximation of the continuous limit.

Where to go next

If you want to plug in real numbers, the APY calculator converts between APR and APY at any compounding frequency and shows the year-end balance. To see how the same logic plays out over multiple years, Compound Interest walks through the long-horizon math where small APY gaps turn into large dollar gaps. For yield-bearing crypto positions, Staking Rewards handles the protocol-specific cases where validator rewards, slashing risk, and lockup periods complicate the simple APY-vs-APR picture. And on the borrowing side, the Loan Calculator lets you compare quoted APRs head-to-head against the true effective rate you will pay.