Question

Suppose that you are purchasing a house (loan amount = $200,000) and inquire about the terms for a 30-year fixed-rate mortgage.

LOAN B:

• annual interest rate of 3.625% with monthly payments and compounding

• 1.25 discount points

• origination fee of 1%

• $18 in third-party closing costs

What is the EBC for the loan if we prepay at the end of the third year?

Answer 1

First we shall calculate the exact amount of the mortgage to be paid in installments:
Loan amount=	200000
Add: Discount points(200000*1.25%)	2500
Add:Origination fee(200000*1%)	2000
Add: Third-party closing costs	18
Total amount of the mortgage=	204518

Now, using the PV of annuity formula,

we will find the monthly payment amt. on the mortgage

ie. PV of Mortgage=Pmt.*(1-(1+r)^-n)/r

where ,

PV of mortg. = 204518

Pmt.+ the mthly pmt.--that is to be found out---??

r= given as 3.625% p.a. ie.3.625%/12=0.3021% or 0.003021 p.m

n= no.of mths.= 30 yrs.*12= 360 mths.

Plugging in these values, in the above formula,

204518=Pmt.*(1-(1+0.003021)^-360)/0.003021

Now, solving for pmt. , we get the mthly. Pmt. As

Mthly.pmt. On the mortgage=204518/((1-(1+0.003021)^-360)/0.003021)=

932.73

2..

To know EBC for the loan if we prepay at the end of the third year

We need to find the future value of the principal balance on the mortgage at end of 3 yrs. Ie.3*12= 36 months

FV of principal bal.=FV of original principal value-FV of the monthly interest annuity

ie. FV =(PV*(1+r)^n)-(Pmt.*((1+r)^n-1)/r)

where,

FV= Future value (principal) remaining of the mortgage

PV=Present value /original mortgage amt.=204518

Pmt.=mthly.pmt.= 932.73 (as found above)

r= rate / mth. ie. 0.003021

n=no.of mthly. Pmts.=36

So, plugging in the values.

ie. FV=(204518*(1+0.003021)^36)-(932.73*((1+0.003021)^36-1)/0.003021)

192561.95

Now this is the unamortised principal balance on the mortgage at end of 36 mths. Or 3 years ---which needs to be paid in full , along with the 36 th monthly pmt.

so, the effective interest rate is calculated using the PV of ordinary ( mthly.) annuity formula, as follows:

Like finding a bond's present value, for the original house value,

200000=(932.73*(1-(1+r)^-36)/r)+(192561.95/(1+r)^36)

& solving for r,ie. IRR of these cash flows=

0.3696 % p.m

ie.

Effective annual rate will be:

(1+mthy rate)^12-1

ie. (1+0.3696%)^12-1=

4.53%

ANSWER: The Effective Borrowing Cost for the loan if we prepay at the end of the third year=   4.53% (APR)