Calc FYI

APR to APY Conversion

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

Variables

Symbol Name Unit Description
$APY$ Annual Percentage Yield decimal Effective annual rate accounting for compounding.
$APR$ Annual Percentage Rate decimal Nominal annual rate before compounding effect.
$n$ Compounding frequency times/year How many times interest compounds per year.

APR vs APY

APR (Annual Percentage Rate) is the nominal interest rate — the rate quoted by banks before accounting for the compounding effect. APY (Annual Percentage Yield, also called EAR — Effective Annual Rate) is what you actually earn or pay per year after compounding is applied:

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

Why APR and APY Differ

A bank advertising 6% APR compounded monthly is not actually giving you 6% per year. Each month you earn 0.5% on the growing balance:

Month 1: 1,000 × 1.005 = 1,005.00 Month 2: 1,005 × 1.005 = 1,010.03 ... Month 12: final amount = 1,061.68

APY = (1,061.68 − 1,000) / 1,000 = 6.168% — higher than 6% APR.

Common Compounding Frequencies and APY

APR n APY
6% 1 (annual) 6.000%
6% 2 (semi-annual) 6.090%
6% 12 (monthly) 6.168%
6% 365 (daily) 6.183%
6% ∞ (continuous) 6.184%

Regulatory Importance

For savings accounts: Banks must disclose APY (Truth in Savings Act, USA). APY is what savers actually earn — a higher APY is better.

For loans: APR must be disclosed (Truth in Lending Act, USA). For loans, APR typically includes fees in addition to interest, making it a more comprehensive cost measure than the nominal rate alone.

For credit cards: Daily compounding (n = 365) means quoted APR understates the effective cost. A 20% APR credit card actually charges 22.13% APY: (1 + 0.20/365)^365 − 1 = 22.13%.

Derivation & History

The APY formula is a direct application of the compound interest formula. If you deposit $1 at APR rate r compounded n times per year, after one year you have (1 + r/n)^n. The net yield (excluding the original $1) is (1 + r/n)^n − 1, which is APY.

The distinction between nominal and effective rates was formalised in actuarial science in the 18th and 19th centuries. Regulatory requirements to disclose APY on savings products emerged in the late 20th century as consumer protection legislation recognised that compounding frequency materially affects the actual return received by depositors.

Worked Examples

Monthly compounding savings account

  1. APY = (1 + 0.05/12)^12 − 1
  2. APY = (1.004167)^12 − 1
  3. APY = 1.05116 − 1
  4. APY = 0.05116 = 5.116%

Result: APY = 5.116% (you earn 5.116%, not 5%, per year)

Daily compounding credit card

  1. APY = (1 + 0.18/365)^365 − 1
  2. APY = (1.000493)^365 − 1
  3. APY = 1.19722 − 1
  4. APY = 0.19722 = 19.72%

Result: APY = 19.72% — actual cost is 1.72 percentage points above the advertised 18%

Edge Cases & Limitations

n = 1 (annual compounding): APY = APR exactly — no compounding effect.

Fees in APR: For mortgages and loans, APR includes origination fees, making it higher than the nominal rate; APY only captures the compounding effect, not fees.

Continuous limit: As n → ∞, APY → e^APR − 1 (continuous compounding APY).

Real-World Applications

Banks in the US, EU, and Korea are legally required to publish APY on all deposit products. Consumers comparing savings accounts across banks must use APY for a fair comparison. Cryptocurrency exchanges publish APY on staking and liquidity pool yields. Fintech apps (e.g., Robinhood, Revolut) display APY prominently because it is the most consumer-friendly metric. Analysts compare bond yields using APY-equivalent calculations to normalise across different coupon frequencies.

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