Two fees i_{1} Hey_{2} are equivalent if, applied to the same capital **P** over the same period of time, through different capitalization periods, produce the same final amount.

- Be the capital
**P**applied for one year at an annual rate i_{The}. - The amount
**M**at the end of the 1 year period will be equal to M = P (1 + i_{The}) - Consider now the same capital
**P**applied for 12 months at a monthly rate i_{m}. - The amount
**M '**at the end of the 12 month period will be equal to M '= P (1 + i_{m})^{12}.

By the definition of equivalent rates seen above, we should have **M = M '**.

Therefore, P (1 + i_{The}) = P (1 + i_{m})^{12}

Hence we conclude that **1 + i _{The} = (1 + i_{m})^{12}**

With this formula we can calculate the annual rate equivalent to a known monthly rate.

*Examples:*

**1) What is the annual rate equivalent to 8% per semester?**

In a year we have two semesters, so we will have: 1 + i_{The} = (1 + i_{s})^{2}

1 + i_{The} = 1,08^{2}

i_{The} = 0.1664 = 16.64% a.a.

**2) What is the annual rate equivalent to 0.5% per month?**

1 + i_{The} = (1 + i_{m})^{12}

1 + i_{The} = (1,005)^{12}

i_{The} = 0.0617 = 6.17% a.a.