Explanation

The ** effective annual rate** is a value used to compare
different interest plans. If two plans were being compared, the interest
plan with the higher effective annual rate would be considered the better
plan. The interest plan with the higher effective annual rate would be the
better earning plan.

For every compounding interest plan there is an effective annual rate. This effective annual rate is an imagined rate of simple interest that would yield the same final value as the compounding plan over one year.

Formula symbols:

S | Final value of investment |

P | Initial value of investment |

i_{eff} |
Effective annual rate |

r | Annual percentage rate (APR) |

k | Interest periods per year |

After a term of one year the final value, S, of a
** compounded** interest investment with an initial value P compounded k times
per year at an annual percentage rate of r is given by:

S = P(1 + r/k)^{k}

After a term of one year the final value, S, of a
** simple**
interest investment with an initial value P at an annual percentage rate of
i

S = P(1 + i_{eff})

Setting these two equal we get:

P(1 + i_{eff}) = P(1 + r/k)^{k}

Dividing each side by P we get:

(1 + i_{eff}) = (1 + r/k)^{k}

And solving for i_{eff} we get:

i_{eff} = (1 + r/k)^{k} - 1

Example calculation

Which of the following plans is the better investment plan? Note that the answer is not immediately obvious. Plan 1 has a higher interest rate than Plan 2, which causes it to earn more money. However, Plan 1 is compounded less frequently than Plan 2, and this would cause it to earn less money. It is exactly in these types of situations that calculating the effective annual rate can show you which plan will earn more.

*Plan 1:*

8.0% annual percentage rate, compounded monthly

*Plan 2:*

7.9% annual percentage rate, compounded daily

*i*_{eff}* for plan 1:*

i_{eff} = (1 + r/k)^{k} - 1

i_{eff} = (1 + 0.08/12)^{12} - 1

i_{eff} = (1.0066...)^{12} - 1

i_{eff} = (1.08299...) - 1

i_{eff} = 0.08299...

i_{eff} = 8.299...%

*i*_{eff}* for plan 2:*

i_{eff} = (1 + r/k)^{k} - 1

i_{eff} = (1 + 0.079/365)^{365} - 1

i_{eff} = (1.000216...)^{365} - 1

i_{eff} = (1.08219...) - 1

i_{eff} = 0.08219...

i_{eff} = 8.219...%

Plan 1is better since 8.299...% > 8.219...%

Here is a calculator for effective annual rate.

*i*_{eff} = (1 + r/k)^{k}
- 1

i_{eff} |
Effective annual rate |

r | APR, Annual percentage rate |

k | Times per year compounded |

Enter values for the above formula:

(Example: For r enter 5.0% as 0.05, etc.)