Loan financing algebra question

[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.

 

[pbuk]

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.

 

[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?

Thanks for your help!

 

[pbuk]

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?

Thanks for your help!

Exactly that. Note also that to convert APR to a monthly rate you need to use the following formula: 1 + i_monthly = (1 + i_APR)^1/12. The formula you have used converts an annual rate of interest applied monthly, which is not the same as an APR.

 

[CellCoree]

I understand the importance of using mathematical equations and formulas to make informed decisions. In this case, it seems like you are trying to determine the best option for financing a house, taking into account the different loan types and their associated costs. Your approach of setting the monthly payment for both loan types equal to each other and solving for the interest rate is a valid method.

To factor out r completely from the equation, you can use the distributive property to expand the terms within the parentheses on the right side of the equation. This will allow you to combine like terms and isolate r on one side of the equation. Alternatively, you can also use logarithms to solve for r.

It is great that you are using a computer program to help you compare different total costs for houses based on different loan options. This will allow you to make a more informed decision and potentially save money in the long run. Keep in mind that in addition to the monthly payment, you should also consider other factors such as closing costs, interest rates, and the length of the loan when making your decision.

Overall, your approach to solving this algebra problem is sound and I encourage you to continue using mathematical equations and formulas to make informed decisions in your personal and professional life.