# Calculate Compounding Interest Rate From Total Interest Rate?

I would like to be able to calculate the interest rate that is compounded for a given total interest rate, and number of compounding events.

TotalInterestRate = ((1+CompoundingRate/#CompoundingEvents)^#CompoundingEvents)-1

Basically would like to solve this equation for Compounding Rate.

For a more concrete example, calculating the Monthly Interest Rate from a given APY. The formula for APY given a Monthly Interest Rate would then be.

APY = ((1+InterestRate/12)^12)-1

What would be a good way to go about solving this equation?

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## 1 Answer

If we refer to the effective interest rate as $r$ (what you call the total interest rate) and we refer to the nominal interest rate as $n$ (what you call the compounding rate) and the compounding frequency as $N$, the formula you have is $$r=\left(1+\frac{n}{N}\right)^N-1$$ To solve for $n$:

\begin{align} r+1 &=\left(1+\frac{n}{N}\right)^N\\ \sqrt[N]{r+1}&=1+\frac{n}{N}\\ \sqrt[N]{r+1}-1&=\frac{n}{N}\\ N\left(\sqrt[N]{r+1}-1\right)&=n\\ \end{align}

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Same, but using the symbols in the question: $\mbox{InterestRate} = 12(\sqrt[12]{\mbox{APY}+1}-1)$. – copper.hat Jul 6 '12 at 6:10
@copper.hat I believe that is for a more specific example than the OP firsts asks about. – alex.jordan Jul 6 '12 at 6:14
You are correct, I missed that... – copper.hat Jul 6 '12 at 6:16
+1 for unintentionally reminding me that "effective" and "real" interest rates are not the same thing. – John Joy Dec 3 '14 at 14:28