Effective Annual Rate (EAR) Calculator

What is effective annual rate (EAR)?

Effective annual rate (EAR), also known as annual percentage yield (APY) or effective annual yield, is the actual interest rate you earn or pay on an investment or loan when compounding is taken into account. It reflects the true cost of borrowing or the real return on an investment.

While the nominal rate (APR) is the stated annual rate, EAR accounts for the effect of compounding — earning interest on interest — throughout the year. The more frequently interest compounds, the higher the effective rate.

The EAR formula

For periodic compounding:

$$EAR = \left(1 + \frac{r}{n}\right)^n - 1$$

Where:

• $r$ = Nominal interest rate (as a decimal)
• $n$ = Number of compounding periods per year

For continuous compounding:

$$EAR = e^r - 1$$

Where $e$ ≈ 2.71828 (Euler's number).

Example calculations

Monthly compounding

For a 6% nominal rate compounded monthly:

$$\begin{aligned} EAR &= \left(1 + \frac{0.06}{12}\right)^{12} - 1 \\[0.5em] &= (1.005)^{12} - 1 \\[0.5em] &= 1.0617 - 1 \\[0.5em] &= 6.17\% \end{aligned}$$

Daily compounding

For a 6% nominal rate compounded daily:

$$\begin{aligned} EAR &= \left(1 + \frac{0.06}{365}\right)^{365} - 1 \\[0.5em] &= 6.18\% \end{aligned}$$

Continuous compounding

For a 6% nominal rate compounded continuously:

$$\begin{aligned} EAR &= e^{0.06} - 1 \\[0.5em] &= 6.18\% \end{aligned}$$

Impact of compounding frequency

Compounding	5% APR	10% APR	15% APR
Annual	5.00%	10.00%	15.00%
Semi-annual	5.06%	10.25%	15.56%
Quarterly	5.09%	10.38%	15.87%
Monthly	5.12%	10.47%	16.08%
Daily	5.13%	10.52%	16.18%
Continuous	5.13%	10.52%	16.18%

Key observations:

• Higher nominal rates show larger differences between compounding frequencies
• The difference between daily and continuous compounding is negligible
• Monthly vs. annual compounding makes a meaningful difference

EAR vs APR

Feature	APR (Nominal Rate)	EAR (Effective Rate)
Compounding	Ignores	Includes
Purpose	Stated rate	True cost/return
Use case	Loan advertisements	Accurate comparison
Calculation	Simple	Requires formula

When to use each

Use APR when:

• Calculating simple interest
• Understanding stated loan terms
• Legal disclosures require it

Use EAR when:

• Comparing products with different compounding
• Calculating actual returns or costs
• Making investment decisions
• Evaluating savings accounts

Practical applications

Comparing savings accounts

Two accounts offer:

• Bank A: 4.85% APY (already EAR)
• Bank B: 4.80% APR compounded daily

Bank B's EAR: (1 + 0.048/365)^365 - 1 = 4.92%

Bank B actually offers the better rate despite the lower stated number.

Comparing credit cards

Card A: 18% APR compounded monthly Card B: 17.5% APR compounded daily

Card A EAR: (1 + 0.18/12)^12 - 1 = 19.56% Card B EAR: (1 + 0.175/365)^365 - 1 = 19.12%

Card B costs less despite having the same type of APR.

Mortgage analysis

A mortgage at 6.5% APR compounded monthly: EAR = (1 + 0.065/12)^12 - 1 = 6.70%

Over a 30-year, $300,000 mortgage, this 0.20% difference represents thousands of dollars.

Converting between APR and EAR

APR to EAR

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

EAR to APR

$$APR = n \times \left[(1 + EAR)^{1/n} - 1\right]$$

Example: Convert 6.17% EAR to APR (monthly compounding):

$$APR = 12 \times [(1.0617)^{1/12} - 1] = 6.00\%$$

Why compounding frequency matters

The mathematical insight is that more frequent compounding means interest starts earning interest sooner. Consider $1,000 at 12%:

Compounding	After 1 year	Difference
Annual	$1,120.00	—
Semi-annual	$1,123.60	+$3.60
Quarterly	$1,125.51	+$5.51
Monthly	$1,126.83	+$6.83
Daily	$1,127.47	+$7.47
Continuous	$1,127.50	+$7.50

While differences seem small for one year, they compound dramatically over time.

EAR over multiple years

The power of EAR becomes clear over longer periods. For $10,000 at 6% APR compounded monthly:

Years	Simple interest	With compounding	Extra earnings
1	$10,600	$10,617	$17
5	$13,000	$13,489	$489
10	$16,000	$18,194	$2,194
20	$22,000	$33,102	$11,102
30	$28,000	$60,226	$32,226

Continuous compounding

Continuous compounding is the mathematical limit as compounding periods approach infinity. While no real financial product compounds truly continuously, it's useful for:

• Theoretical calculations
• Options pricing models
• Upper bound estimates
• Academic finance

The formula uses Euler's number (e ≈ 2.71828):

$$FV = PV \times e^{rt}$$

Tips for using EAR

1. Always compare EAR to EAR — Don't compare APR to EAR
2. Check the compounding frequency — It matters, especially for loans
3. Consider fees — EAR doesn't include account fees or closing costs
4. Use for all time periods — EAR works for any investment horizon
5. Verify advertised rates — Some ads use APR, others use APY/EAR
6. Higher rates mean bigger differences — Compounding matters more at higher rates