Effective Annual Rate Calculator

How Nominal Rate and EAR Differ

A 12% nominal rate compounded monthly — twelve 1% monthly slabs produce a higher annual return than the stated rate.

What Is the Effective Annual Rate (EAR)?

The effective annual rate (EAR) is the true annual return on an investment or the true annual cost of a loan, accounting for the effect of compounding within the year. It is also called the effective interest rate or — in US banking — the annual percentage yield (APY).

When interest compounds more frequently than once a year, the borrower pays — or the investor earns — more than the stated nominal rate suggests. A savings account advertising "6% APR compounded daily" actually yields 6.1831% per year. That extra 0.18% is the compounding effect captured by the EAR.

Banks and lenders are required in many jurisdictions to disclose both APR and APY (EAR) so consumers can make meaningful comparisons. Always compare EARs when evaluating two products with different compounding frequencies.

What Is APR (Nominal Interest Rate)?

The annual percentage rate (APR), also called the nominal interest rate, is the simple annualised rate without accounting for intra-year compounding. It is calculated by multiplying the periodic rate by the number of periods per year.

For example: a credit card that charges 1.5% per month has an APR of 1.5% × 12 = 18%. But because interest compounds monthly, the EAR is actually (1 + 0.18/12)¹² − 1 = 19.56%. That is the real annual cost of carrying a balance.

In the United States, the Truth in Lending Act (TILA) requires lenders to disclose APR. In Europe, the Annual Percentage Rate (APRC) includes fees and charges. Always check whether the quoted rate is the nominal APR or the effective EAR/APY.

How to Convert Nominal Rate to EAR

For a nominal rate r compounded n times per year, the EAR formula is:

For continuous compounding (n → ∞), the formula simplifies to EAR = eʳ − 1, where e is Euler's number (≈ 2.71828). At r = 12%, this gives EAR = e⁰·¹² − 1 ≈ 12.7497%.

How to Convert EAR to Nominal Rate

To find the nominal APR that produces a known EAR with n compounding periods, rearrange the EAR formula:

Example: you want a 10% EAR from a monthly-compounding account. What nominal rate must you find?

For continuous compounding: Nominal = ln(1 + EAR). At EAR = 10%, Nominal = ln(1.10) ≈ 9.5310%.

Continuous Compounding and EAR

Continuous compounding is a theoretical limit where interest is calculated and added to the principal infinitely many times per second. It is the maximum possible EAR for a given nominal rate. In practice it is used in options pricing (Black–Scholes model), central bank discount rates, and some advanced fixed-income calculations.

The difference between daily (365×) and continuous compounding is tiny but mathematically elegant. At a 12% nominal rate: daily EAR = 12.7475%, continuous EAR = 12.7497% — just 0.0022% apart.

Continuous compounding becomes more significant at high interest rates. At 100% nominal rate, daily compounding gives EAR ≈ 171.46%, while continuous compounding gives EAR = e¹ − 1 ≈ 171.83%.

Why Compounding Period Matters

The table below shows the EAR for a 12% nominal rate across different compounding frequencies. The more frequently interest compounds, the higher the true annual cost or return.

EAR vs APR: Which Should You Use?

Use APR (nominal rate) when you need to quote a rate to a lender or institution, or when comparing loans where compounding frequency is the same. APR is the number required on most mortgage applications, credit card disclosures, and loan agreements.

Use EAR (effective rate / APY) when comparing products with different compounding frequencies. A savings account offering 5.1% APR compounded daily beats one offering 5.15% APR compounded annually: their EARs are 5.2341% and 5.15% respectively.

For borrowers, a higher EAR means a higher true cost. For investors, a higher EAR means higher earnings. Always verify which rate type is being advertised before committing.

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