### Equivalent interest rates

Equivalent interest rates are those that, when applied to the same capital, in the same period, produce equal values.

A principal (P) applied at a rate (i_{1}) for a period (n) produces an amount (S) . If the
same capital applied to a different rate (i_{2}) to produce the same amount, we say that the rates are equivalents.

See, how to get the generic formula.

By the definition of equivalent rates, as above, we have:

S = S_{1}.

r = i / 100 (Interest rate in decimal form)

S = P (1 + r_{a}) (annual rate)

S_{1} = P (1 + r_{m}) ^{ 12} (monthly rate)

As we know, by definition, S = S_{1}

P (1 + r_{a}) = P (1 + r_{m}) ^{ 12},

1 + r_{a} = (1 + r_{m})^{ 12}

r_{a} = (1 + r_{m})^{ 12} -1

Example:

Be the monthly interest rate of 2%, what is the equivalent annual rate?

r_{a} = (1 + r_{m})^{ 12} - 1

r_{a} = (1 +0.02)^{ 12} -1

r_{a} = (1.02)^{ 12} -1

r_{a} = 1.2682 -1

r_{a} = 0.2682 or 26.82%

Finally, the generic formula for calculating the equivalent rates:

** r _{1} = {(1+r_{2})^{ (n1/n2)}} -1**

r

_{1}= requested interest rate;

r

_{2}= known interest rate;

n

_{1}= period corresponding to the desired rate;

n

_{2}= period corresponding to the known rate.

How to calculate

**Equivalent interest rates.**

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