Question

A borrower is offered a mortgage loan for $100,000 with an interest rate of 10% and a 30-year amortization period with monthly payments. The origination fee is 1% of the loan and the lender charges two discount points. What is the effective interest rate?

10%, 9%, 10.37%, or 10.24%?

Answer 1

monthly interest rate , i = 10%/12 = 10/12 = 0.83333% = 0.0083333

total fee charged = 3% of loan = 0.03*100,000 = 3000

Present value (PV) of loan = 100,000 - total fees charged = 100,000 - 300,000 = 97,000

period of loan = 30 years

no. of months , n= period of loan*12 = 30*12 = 360

PVIFA =

= present value interest rate factor of annuity

= [((1+i)³⁶⁰ - 1)/((1+i)³⁶⁰*i)] =  [((1.0083333)³⁶⁰ - 1)/((1.0083333)³⁶⁰*0.0083333)] = 113.9512038

let Monthly payments = m

100,000 = m*PVIFA

m = 100,000/PVIFA = 100,000/113.9512038 = 877.5686142 or $877.57 ( rounding off to 2 decimal places)

Let effective interest = r

then we have to find R which satisfies the following equation

97000 = 877.5686142* [((1+R)³⁶⁰ - 1)/((1+R)³⁶⁰*R)]

[((1+R)³⁶⁰ - 1)/((1+R)³⁶⁰*R)] = 97000/877.5686142 = 110.5326677

By trial and error we find That , R = 0.00863804 or 0.863804%

Then effective rate of interest , r = 12*R = 12*0.00863804 = 0.103656 or 10.3656% or 10.37%