Effective Annual Interest Rate

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Effective Annual Interest Rate (EAR)

The effective annual interest rate (EAR) is the actual annual rate of return on an investment or the true annual cost of borrowing after accounting for compounding. It shows what you effectively earn or pay over one year, unlike the nominal (stated) rate which ignores compounding frequency.

Formula

EAR = (1 + i/n)^n − 1
where:
– i = nominal annual interest rate (as a decimal)
– n = number of compounding periods per year

For continuous compounding: EAR = e^i − 1

Why EAR matters

  • More frequent compounding (larger n) increases the EAR for the same nominal rate.
  • EAR lets you compare different financial products on an apples-to-apples basis, regardless of how often interest is compounded.
  • EAR does not include fees, taxes, or changes in the nominal rate over time.

Example

Two investment offers:
– Investment A: 10% nominal, compounded monthly (n = 12)
EAR = (1 + 0.10/12)^12 − 1 ≈ 0.10471 → 10.471%

  • Investment B: 10.1% nominal, compounded semiannually (n = 2)
    EAR = (1 + 0.101/2)^2 − 1 ≈ 0.10360 → 10.360%

Although Investment B has a higher nominal rate, Investment A delivers the higher effective annual return. On $5,000,000 principal, the annual interest difference is:
– A: $5,000,000 × 10.471% = $523,550
– B: $5,000,000 × 10.360% = $518,000
Difference ≈ $5,550 in favor of Investment A.

Effect of compounding frequency (for 10% nominal)

  • Semiannual: 10.250%
  • Quarterly: 10.381%
  • Monthly: 10.471%
  • Daily: 10.516%
  • Continuous: e^0.10 − 1 ≈ 10.517%

EAR vs. Nominal Rate (APR)

  • Nominal rate (often called APR for loans) is the quoted rate and ignores compounding within the year.
  • EAR reflects compounding and is therefore the true annual rate. Use EAR to compare yields or costs across products.

Practical uses

  • Comparing savings accounts, CDs, bonds, and loan offers.
  • Calculating the real annual cost of credit cards and mortgages (when periodic rates are given).
  • Converting a periodic rate to an annualized effective rate: if r is the periodic rate and there are m periods per year, EAR = (1 + r)^m − 1.

Limitations

  • Assumes the nominal rate is constant over the year (rates can change).
  • Ignores fees, taxes, and non-interest costs that affect net return or cost.
  • Does not convey risk, liquidity, or other qualitative differences between products.
  • Less informative for very short-term instruments where compounding has limited effect.

Quick tips

  • When comparing offers, convert all quoted rates to EAR first.
  • For loans, check both the EAR and any fees to understand true cost.
  • For credit cards, use the monthly periodic rate to compute EAR: EAR = (1 + monthly rate)^12 − 1.

Bottom line

EAR is the most accurate single-number measure of the annual return or cost when compounding is involved. Use it to compare financial products, but remember to also account for fees, taxes, and risks when making decisions.