Effective Annual Rate (EAR / APY)

Why EAR exists

Two loans with the same nominal rate can cost different amounts if they compound at different frequencies. The effective annual rate (EAR, a.k.a. APY) rolls compounding frequency into a single apples-to-apples annual number.

The formula

$\text{EAR} = \left(1 + \dfrac{r_{\text{nom}}}{n}\right)^n - 1$

where $r_{\text{nom}}$ is the nominal annual rate (APR) and $n$ is the number of compounding periods per year.

Continuous compounding limit: $\text{EAR}_{\text{cont}} = e^{r_{\text{nom}}} - 1$

Step-by-step solution

Setup: $r_{\text{nom}} = 0.12$, $n = 12$.

Step 1 — Per-period rate: $\dfrac{0.12}{12} = 0.01$

Step 2 — Growth factor over one year: $(1.01)^{12} \approx 1.12683$

Step 3 — Subtract 1: $\text{EAR} = 1.12683 - 1 = 0.12683 = \boxed{12.68\%}$

Compounding monthly turns a 12% nominal rate into a 12.68% effective yield.

Comparing frequencies at 12% nominal

• Annually ($n=1$): EAR = 12.00%
• Semiannually ($n=2$): EAR = 12.36%
• Quarterly ($n=4$): EAR = 12.55%
• Monthly ($n=12$): EAR = 12.68%
• Weekly ($n=52$): EAR = 12.73%
• Daily ($n=365$): EAR = 12.747%
• Continuously: EAR = $e^{0.12} - 1 = 12.750\%$

Notice the diminishing returns — daily to continuous adds only 0.003%.

When EAR matters

• Shopping for savings accounts: the bank may quote APR but what you earn is APY.
• Comparing credit cards: daily compounding makes a "19.99% APR" card cost notably more than 19.99% a year.
• Corporate finance: converting between monthly, quarterly, and continuous-compounding conventions for valuation models.

Going the other way — nominal from EAR

Given a target EAR and a desired compounding frequency $n$: $r_{\text{nom}} = n \left[(1 + \text{EAR})^{1/n} - 1\right]$

So a 5% EAR expressed as a monthly compounded nominal: $12 \cdot (1.05^{1/12} - 1) \approx 0.04889$ ⟹ 4.89%.

Common mistakes

• Treating APR and APY as the same when they're not. For credit products, APR typically excludes compounding; APY includes it.
• Forgetting to subtract 1. $(1 + r/n)^n$ is the growth factor (like 1.1268), not the rate (0.1268).
• Adding EARs. If two accounts both have 5% EAR, combining them doesn't give 10% EAR — effective rates don't add.

Try it in the visualization

Slide $n$ from 1 to 365 and watch the EAR climb along an asymptotic curve toward $e^{r} - 1$. A side panel shows dollar-growth differences on $10,000 over one year.