Effective Annual Rate from a Nominal Rate

Also known as EAR · APY · effective rate from nominal

\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1

Worked example: 12% nominal compounded monthly → 12.6825% effective — press Try an example to run it live, then adjust anything.

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Effective Annual Rate from a Nominal Rate explained

A quoted rate is not a cost until you know how often it compounds. "12% a year" charged monthly is really 1% twelve times, and $(1.01)^{12} - 1 = 12.68\%$ . The extra 0.68 points is interest earned on interest, and it grows with the compounding frequency: the same 12% compounded daily comes to 12.747%, approaching the continuous limit $e^{0.12} - 1 = 12.75\%$ .

This is the only fair way to compare two offers. A card at 19.99% compounded daily and a line of credit at 20.2% compounded annually are not what they appear, and the nominal figures rank them the wrong way round. Disclosure law exists precisely because of this gap, which is why lenders must publish an effective or annualised figure alongside the headline rate.

Going backwards recovers the nominal rate a lender must be quoting to produce a given effective one. Notice that $m$ cannot be solved for: it sits in the base and the exponent at once, and no elementary rearrangement frees it.

Effective Annual Rate from a Nominal Rate formula

\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1

Where

$\mathit{EAR}$ = Effective annual rate
$r$ = Nominal annual rate
$m$ = Compounds per year

Missing one of these? Work it out first, then come back

Nominal annual rate — Real Interest Rate (Fisher Equation)
Compounds per year — Loan Payment (Amortized Loan or Mortgage), Total Interest Paid Over a Loan

Part of this formula set

The loan and interest formulas

5 formulasPayment, total interest, effective rate and doubling time — everything that decides what borrowing actually costs.