Calculating Interest Rates for Loan of $735,000

[Shakattack12]

Homework Statement

The problem ask to calculate the interest rate of a loan. The principal is $735 000, term is 25 years (300 months), and the repayments due is $4656.17/month.

i = interest rate, PV = present value, R = repayments

Homework Equations

PV = R x [(1+i/12)^300 - 1]/(i/12)

The Attempt at a Solution

I managed to calculate the total interest to pay = $661 851
735000 = 4656.17 x [(1+i/12)^300 - 1]/(i/12)
13.155 = [(1+i/12)^300 - 1]/i
13.155i = [(1+i/12)^300 - 1]
13.155i + 1 = (1+i/12)^300
I realized that this may not be correct but I have no idea of what to do. I put this into an online calculator and it gave a value of 5.823% but I don't know how it got to it. Please help.

 

[issacnewton]

Hi I think you have written the formula of the future value. The present value would be $$\text{PV} = \frac{R}{(i/12)}\left[ 1 - \frac{1}{\left(1+\frac{i}{12}\right)^{300}} \right] $$ Now we can't solve this analytically, so use RATE function from Excel. I just checked the calculations, and I think your APR seems correct to me

 

[Shakattack12]

Thanks so much. I thought as much after a few hours that this was impossible to solve manually.

 

[Chestermiller]

Thanks so much. I thought as much after a few hours that this was impossible to solve manually.

Are you familiar with Newton's method for solving non-linear algebraic equations?

 

[Chestermiller]

Thanks so much. I thought as much after a few hours that this was impossible to solve manually.

What do you get if you expand $\left(1+\frac{i}{12}\right)^{-300}$ using the binomial expansion and retaining only the first three terms?