Question

1. Ann gets a fully amortizing 30-year fixed rate mortgage with quarterly payments for $1,000,000. The interest rate is 4%, compounded quarterly. She prepays the mortgage in 1 quarter (i.e. she makes the 1^st payment and immediately prepays the remaining balance). What is Ann’s APR?

   Notes: a quarter equals 3 months, one year consists of 4 quarters, APR is annual.

1. Modify question above: At the moment when Ann signs the mortgage, she must pay an origination fee that equals 2 points. Everything else stays the same, and she still prepays the mortgage in 1 quarter. What is Ann’s APR?

1. Modify question above: There is still an origination fee of 2 points, but now Ann decides to keep the mortgage for the whole term, i.e. she no longer plans to prepay it. What is Ann’s APR?

Answer 1

First Part:

Since interest rate for the loan at 4% is compounded quarterly, APR, in the absence of any closing costs, is 4% itself, when paid off in one quarter as follows:

Principal (P)= $1,000,000

Interest rate (R )= 4%

Period (N) = ¼ year

Interest for the first quarter= PRN

= 1,000,000*4%/4 = $10,000

Interest rate= (10000/1000000)*4*100 = 4%

Second part: First modification:

Interest for first quarter= $10,000 (as above)

Given, Origination fee= 2 points

Therefore, net loan received= 1,000,000*(1-2%) = $980,000

APR= (10,000/980,000)*4*100 = 4.081633%

Third part : Second modification:

APR, if not prepaid= 4.167966% as follows: