Effective Annual Rate
True annual rate accounting for compounding.
Formula
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Example
12% compounded monthly gives an effective annual rate of (1.01)¹² − 1 ≈ 12.68%.
How it works
The effective annual rate tells you the single-year growth rate of an interest-bearing account or loan after compounding is taken into account. It works by first finding the value after one compounding period, 1 + r/n, and then repeating that growth n times over a full year, which gives the total annual multiplier; subtracting 1 converts that multiplier into a rate. This is why two products with the same nominal rate can have different true annual costs or returns if they compound at different frequencies.
When to use it
Use this formula when comparing savings accounts, loans, bonds, or any interest product that compounds more than once per year. It is especially useful when you need to compare offers with different compounding schedules on a like-for-like annual basis.
Worked example, step by step
Suppose a credit card advertises a nominal interest rate of 18% compounded daily. Step 1: Identify the variables: r = 0.18 and n = 365. Step 2: Substitute into the formula: EAR = (1 + 0.18/365)^365 - 1. Step 3: Compute the daily rate: 0.18/365 ≈ 0.00049315. Step 4: Add 1: 1 + 0.00049315 ≈ 1.00049315. Step 5: Raise to the 365th power: 1.00049315^365 ≈ 1.19717. Step 6: Subtract 1: EAR ≈ 1.19717 - 1 = 0.19717. Step 7: Convert to a percentage: EAR ≈ 19.72% per year.
Common mistakes to avoid
- Using the nominal rate r as if it were already the effective annual rate
- Forgetting that n is the number of compounding periods per year, not the number of years
- Mixing percentages and decimals, such as entering 12 instead of 0.12
Did You Know?
A 12% rate compounded monthly actually costs about 12.68% a year — the effective rate reveals the truth.
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