Question

Suppose that you are borrowing $1,000,000 to buy a beach house. The lender offers you the following options:

Option 1: A 15-year fixed conventional mortgage at a nominal rate of 5.0% and zero discount points. The mortgage is payable monthly.

Option 2: A 15-year fixed conventional mortgage at a rate of 4.75% and 2 discount points. The mortgage is payable monthly.

Questions (answer all three parts and document any calculations):

A) What is the monthly payment required for Option 1? What is the loan's effective annual rate (EAR)?

B) What is the monthly payment required for Option 2?  What is the loan's effective annual rate (EAR), assuming you hold the note for the full 15-year term?

C) Estimate how long (i.e. # months) you would need to stay in the discount points mortgage (Option 2) for it to be less costly than the no-points mortgage (Option 1) to the borrower

Answer 1

A: amount borrowed = 1000000

P: monthly emi

Option 1:

Interest rate = 5% pa = 5%/12 monthly

time = 15 years = 15*12 or 180 months

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

1000000 = (P/(5%/12))*(1-1/(1+5%/12)^(15*12))

P = $7907.9 (monthly payment)

(1+x) = (1+5%/12)^12

x = 5.12% (EAR)

Option 2:

1000000 = (P/(4.75%/12))*(1-1/(1+4.75%/12)^(15*12))

P = $7778.3 (monthly payment)

Discount points = 2%*1000000 = 20000

Net loan amount = 1000000-20000 = 980000

Let r be the monthly interest rate

980000 = (7778.3/r)*(1-1/(1+r)^180)

Solving the above equation, we get r = 0.42%

1+x = (1+0.42%)^12

x = 5.16% (EAR)

(1+r)^12 = (1+5.16%)

r = 0.42% (monthly rate)

C)

Let t be the total months required in option 2

(1/(5%/12))*(1-1/(1+5%/12)^(15*12)) = (1/(0.42%))*(1-1/(1+0.42%)^(t))

t = 181 months

1 month extra