Question

A botanist secures a 30-year mortgage for $513,000 at an annual interest rate of 4.175% with 1.4 points. The loan origination fee is 0.65 points of the loan amount. Calculate the APR (in percent) for the loan. (Round your answer to the nearest thousandth of a percent.)

Answer 1

4.346%

Explanation:

The points and origination fee are assumed to be capitalized, so the loan payment amount is based on the total of the original loan amount and those fees. The APR is calculated as the rate that would be charged on the original loan value to produce that payment.

Loaned amount

The loaned amount is the sum of the original loan value and the added points.

P = $513,000×(1 +1.4% +0.65%) = $523,516.50

Payment

The monthly payment on this amount is given by the amortization formula ...

A = P(r/12)/(1 -(1 +r/12)^(-12t))

where r=0.04175, the annual interest rate, t=30, the period in years

A = $523,516.50(0.04175/12)/(1 -(1 +0.04175/12)^-360)) ≈ $2552.45

APR

There is no formula for calculating the APR. It must be developed iteratively or graphically. The attached calculator images show a couple of different ways the value can be found.

The payment of $2552.45 on a loan value of $513,000 represents an APR of 4.346%.