# Equivalent Fees

Two fees i1 Hey2 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 iThe.
• 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 im.
• The amount M ' at the end of the 12 month period will be equal to M '= P (1 + im)12 .

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

Therefore, P (1 + iThe) = P (1 + im)12
Hence we conclude that 1 + iThe = (1 + im)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 + iThe = (1 + is)2
1 + iThe = 1,082
iThe = 0.1664 = 16.64% a.a.

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

1 + iThe = (1 + im)12
1 + iThe = (1,005)12
iThe = 0.0617 = 6.17% a.a.

Next: Nominal Fees