We described an interest rate (*r*) is the rate of return that equates the value of different cash flows on different dates.

If you could pay $975 today to receive $1,000 in 1 year, you can consider the value of waiting 1 year to be $25 ($1,000 – $975.) It can be expressed as an interest rate, *r *= 25/975 = 0.02564 or 2.56%.

In many cases, interest is paid more frequently than annually. In such cases the future value formula can be expressed as:

FV_{N} = PV(1+*r*_{s}/m)^^{mN}

where m is the number of compounding periods and N is the number of years.

Interest can also be compounded continuously, in which case FV_{N} = PV_{e}^{rsN}` `

where e is the transcendental number e = 2.718… Most financial calculators have a function for e^{x}.

Consider a stated interest rate of 6% annually. For different compounding periods, the **effective annual interest rate** the investor would earn is summarized below.

m | mN | r_{s}/m | Future value per $ |

Annual | 1 | 6%/1 = 6% | 1(1.06) = 1.06 |

Semiannual | 2 | 6%/2 = 3% | 1(1.03)^{2} = 1.0609 |

Quarterly | 4 | 6%/4 = 1.5% | 1(1.015)^{4} = 1.0614 |

Monthly | 12 | 6%/12 = 0.5% | 1(1.005)^{12}` = ` 1.0616 |

Daily | 365 | 6%/365 = 0.0164% | 1(1.0000164)^{365} = 1.0618 |

Continuous | 1e^{0.06} = 1.06184 |

The more frequently a given rate is compounded, the higher the ending value for the investor.

## 2 thoughts on “The Effective Annual Interest Rate”

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