# Interest Calculation Problem

I want to calculate how much I would pay monthly.

$X$ = Amount borrowed. $I$= Interest Rate. $Y$ = number of years

This is what I have so far, but it's wrong; I'm getting a larger value.

$$\frac{X\left(1 + \frac{I}{100}\right)^{Y}\left(\frac{I}{Y}\right)}{12\left(\left(1 + \frac{I}{100}\right)^Y - 1\right)}.$$

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I think the question is not very clear, but it sounds like a problem that could be solved very quickly if you know about annuities. In particular, is $I$ the effective rate of interest per year? Without your previous steps, it is hard to tell where you "went wrong" to get the equation you now have. –  Tyler Mar 17 '11 at 18:32
I see Ross Has edited the equation (Thank you) is it clear now? –  user8385 Mar 17 '11 at 18:45
Actually, Ross edited the tags. –  Arturo Magidin Mar 17 '11 at 18:54

Well, we know the amount borrowed, the interest (I'm going to assume it is the effective rate per year), and the amount of time we have to pay it off. Simplifying some things first off, we must be able to find the effective rate of interest per month, $i$, which can be gleaned from the equation

$(1+i)^{12} = 1+I$

Now, in the language of annuities, we are trying to solve an equation for the present value of $X$, which is given by

$X = Pa_{\overline{12Y}|i}$

where $a_{\overline{12Y}|i} = \left[\frac{1-\left(1+i\right)^{-12Y}}{i}\right]$

Now, it is quite simple to solve for the amount of each payment, $P$.

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The present value is $X$. The future value is $0$. We know $I/Y$. So you can compute the payment (which would be negative assuming that positive amounts imply money is coming to you and negative amounts mean that you are paying money). In particular, suppose $$a_{\overline{12Y}|i^{(12)}} = \frac{1-v^{12y}}{i^{(12)}}$$ where $v = \frac{1}{1+i^{(12)}}$ and $i^{(12)}$ is the monthly effective interest rate. So $$1+i = \left(1+i^{(12)} \right)^{12}$$ Then the level monthly payment would be $$P = \frac{X}{a_{\overline{12Y}|i^{(12)}}}$$

You can do this easily on a BA-II-PLUS calculator as well.

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Shouldn't $i$ be the effective rate of interest per payment period (month)? –  Tyler Mar 17 '11 at 19:08