# Loan financing algebra question

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1. Mar 13, 2013

### jelrod45

Hey guys. Thanks for reading. I haven't done any regular algebra in a long time and feel like I am just too rusty to see something that I should be seeing. I am weighing the option of buying a house vs. renting for a year and then buying (giving me a larger down payment and a better apr on the loan).

The formula for the monthly payment for a fixed-rate house loan is given by:

(pr)/(1-(1+r)^(-n))
where p is the initial amount borrowed,
r is the monthly interest rate (apr/12)
and n is the number of payments for the loan.

What I am doing is setting the payment for a loan that I could get with apr 3.5% with a higher mortgage insurance equal to the payment for a loan with lower mortgage insurance and an undetermined value for r. Solving for r will give me the apr for the loan type with lower insurance that will result in the same amount of total payment in the long run. I am having trouble factoring r completely out of the right side of the equation to get the necessary apr. Any help is greatly appreciated!

This is the equation I have set up.

[ (p(.035/12)) / (1 - (1+.035/12)^(-360)) + 81 ] = [ (pr)/(1 - (1+r)^(-360)) ]

The trailing constant in the left term is the difference in monthly payment due to the increased mortgage insurance.

I am going to write a computer program to solve this for me, so that I can compare different total costs of houses based on the loans I qualify for rather than just comparing the market value of the property, which is why I am trying to solve for r in the general form.

2. Mar 14, 2013

### MrAnchovy

You cannot do this by rearranging the equation, you need to solve for r iteratively.

This functionality is built into modern spreadsheets (e.g. Excel's RATE function) which would be a better way to answer your underlying question than writing a program.

3. Mar 14, 2013

### jelrod45

Oh ok. Cool. I didn't know there was a RATE function in excel. I knew about PMT, but didn't know they had others. So I could have some values that are inputs to the PMT function, and use the result of that in a RATE function to get the APR of the loan where the total payment for the entire loan would be equal, right?