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Finance maths

Effective annual rate

Converts a nominal rate compounded n times a year into the single yearly rate that grows money the same amount.

reff=(1+rn)n−1

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

What the symbols mean

reff
the effective annual rate (also called AER or APY), as a decimal
r
the nominal (stated) annual rate as a decimal (12% = 0.12)
n
the number of compounding periods per year

When to use it

Use it to compare offers with different compounding: 12% compounded monthly is really about 12.68% a year, which beats 12.5% compounded yearly. Savings products often state this figure (as AER or APY); loan rates are often quoted differently, so check which rate a document gives.

Worked example

A card charges 12% a year, compounded monthly (n = 12). What is the effective annual rate?

  1. Start from the formula.

    reff=(1+rn)n−1

  2. Substitute r = 0.12, n = 12 and work it out.

    reff=(1+0.1212)12−1
    =(1+0.01)12−1
    =1.0112−1
    ≈0.126825

  3. As a percentage:

    reff×100=0.126825·100
    ≈12.6825%

  4. Compounding more often than once a year always gives an effective rate at least as high as the nominal rate; the limit for continuous compounding is er−1=0.127497.

Answer r_eff ≈ 0.1268, which is about 12.68% a year

Why it works

One year of compounding multiplies money by (1+r/n)n. A single yearly rate that does the same satisfies 1+reff=(1+r/n)n.

Common mistakes

  • Assuming the effective rate equals the nominal rate whenever n > 1; it is always higher (for r > 0).
  • Comparing a nominal rate with an effective one; convert both to effective rates first.
  • Forgetting to subtract 1 at the end, which gives the growth factor 1.1268 instead of the rate.