APR vs. APY: Nominal vs. Effective Rates

APR vs. APY — definitions

• APR (Annual Percentage Rate): the nominal annual rate. It ignores compounding within the year. Regulations often require loans to disclose APR.
• APY (Annual Percentage Yield) / EAR (Effective Annual Rate): the actual yearly return once intra-year compounding is included. Regulations often require deposits to disclose APY.

They disclose the two sides of the same product in a way that flatters each: APR makes a loan look cheaper (ignores the compounding you owe); APY makes a deposit look better (includes the compounding you earn).

The conversion

$APY = \left(1 + \dfrac{APR}{n}\right)^n - 1$

where $n$ is compounding periods per year. Rearranged: $APR = n \left[(1 + APY)^{1/n} - 1\right]$

Continuous compounding limit: $APY = e^{APR} - 1$.

Step-by-step solution

Setup: $APR = 0.18$, $n = 365$ (daily).

Step 1 — Daily rate: $\dfrac{0.18}{365} \approx 0.0004932$

Step 2 — Annual growth factor: $(1 + 0.0004932)^{365} \approx 1.19716$

Step 3 — Subtract 1: $APY = 0.19716 = \boxed{19.716\%}$

So the "18%" credit card actually charges about 19.72% per year in effective yield — a 1.72 percentage-point gap.

Why APY > APR (for $n > 1$)

Every compounding period adds interest to the balance, and the next period's interest is calculated on that larger balance. The more frequent the compounding, the bigger the gap.

• $n = 1$ (annual): APY = APR exactly.
• $n = 2$ (semiannual): APY = $(1 + 0.09)^2 - 1 = 18.81\%$.
• $n = 12$ (monthly): APY $\approx 19.56\%$.
• $n = 365$ (daily): APY $\approx 19.72\%$.
• $n \to \infty$ (continuous): APY $= e^{0.18} - 1 \approx 19.72\%$.

Daily is essentially indistinguishable from continuous.

What to compare in real life

When shopping, always compare like with like:

• Deposits (savings, CDs, money market): use APY.
• Loans (mortgages, auto loans, personal loans): APR is legally required and includes most fees — but it does not include intra-year compounding effects. For the true cost, ask for the effective rate.
• Credit cards: APR compounded daily — use APY to see what you'd actually pay.

Edge case: APR with fees

For loans, legally-disclosed APR often bakes in origination fees, points, mortgage insurance, etc. That APR is already higher than the pure interest rate, even before considering compounding. APY for a loan with fees may actually be lower than APR (depending on fee amortization conventions).

Inverse problem

Bank advertises a 5% APY. What is the monthly APR? $APR = 12 \cdot (1.05^{1/12} - 1) \approx 12 \cdot 0.004074 = 0.04889 = 4.89\%$

A 5% APY corresponds to a 4.89% APR compounded monthly.

Common mistakes

• Assuming APR = APY. Only true for annual compounding. For daily compounding, they can differ by 2+ percentage points at high rates.
• Comparing a loan APR to a deposit APY directly. You're not comparing the same metric — convert both to the same basis.
• Forgetting fees. The advertised APR on a mortgage usually includes fees; the coupon rate (what's on the loan) does not.

Try it in the visualization

Slide the APR from 0% to 30%, and $n$ from 1 to 365. Watch the APY curve arch above the APR line, with the gap widening as either variable grows. Dollar-example panel shows yearly interest difference on a $5,000 balance.