Question

I am stuck on how to approach this question - if someone could provide me some guidance, that would be very appreciated!!

One year ago, after graduating from university, you bought a nice house. You plan to stay in this house for at most 10 more years from now. Then you will move into an even better one and sell the current one when your children grow older. When buying the house, you took a 10- year-ARM mortgage from a bank. The APR of the mortgage is 4%. The rate is expected to rise to 4.75% after the 10-year (from the origination of the mortgage) period. The remaining balance is $1 million. Noticing that the mortgage rates decrease recently, you are considering whether to refinance your mortgage or not. To make a more informed decision, you contacted a few mortgage brokers. From these brokers, the best options you find are the following.

Option 1: 30 years fixed-rate mortgage with an APR of 4%. The closing fee is $4000.

Option 2: 10 year ARM with and APR of 3.875%. The closing fee is $1000.

Option 3: 10 year ARM with and APR of 3.75%. The closing fee is $8000.

Besides looking into these options, you are also aware that all the interest rates have a 50% chance to increase by 0.5%, and a 50% chance to drop by 0.25% in half a year (when you can refinance again). Are you going to refinance your mortgage now? If yes, which option would you choose and what is the benefit (in $ amount)? Please explain the reason behind your choice.

Answer 1

Taken loan at adjustable rate mortgage(ARM) of 4% annual .

for remaining balance $ 1 Million . we will refinance only if we have any benefit. currently we have taken at adjustable rate mortgage , we will have loss if rate increase in future.

if we wont refinance , in next half year increase in rate =[ 0.5*0.005*(1/2)] -  [ 0.5*0.0025*(1/2)] = 0.00125 - 0.000625 = 0.000625 in half yearly or 0.125% yearly.

loss if we wont refinance in half year = $1000000* 0.000625 = $625

Following option under we can refinance

Option 1: 30 years fixed-rate mortgage with an APR of 4%. The closing fee is $4000

here we will have fixed rate 4% so we will have above benefit $625 in half year but we have to pay closing fee $4000

so over  all loss = $4000- $625 = $3375 in next half year.

Option 2: 10 year ARM with and APR of 3.875%. The closing fee is $1000

here we have option of lower adjustable rate

benefit from current rate = 0.04 - 0.03875 = 0.00125= 0.00125*0.5= 0.000625 in six months

profit = $1000000 * 0.000625 = $625 - $1000 fees = -$375

Option 3: 10 year ARM with and APR of 3.75%. The closing fee is $8000.

here we have option of lower adjustable rate

benefit from current rate = 0.04 - 0.0375 = 0.0025= 0.0025*0.5= 0.00125 in six months

profit = $1000000 * 0.00125 = $1250 - $8000 fees = -$6750

we can see that we will have loss in all option but option 2 give lower loss

hence we will refinance , we will select option 2 for refinance.