Question

Assume that you are 10 years into a 30 year home loan at 5%. You owe $200,000 left on your home at this time. You can refinance your loan at 4% for 20 years; however the TOTAL closing costs will be around $3,000.

If you go for refinancing, how many more months would you need to live in the home in order to get the closing cost back? Ignore time value of money and tax credits of the interest payments (i.e. Simple Payback)_______________

2) also with time value of money with annual MARR 10% compounded monthly, solve.

Please help and show the solution for better understanding. I will rate if correct thanks a lot

Answer 1

1. If you go for refinancing, how many more months would you need to live in the home in order to get the closing cost back? Ignore time value of money and tax credits of the interest payments

We can use following Present Value of an Annuity formula to calculate the monthly mortgage payment with 5% interest rate on the loan

PV of loan = PMT * [1-(1+i) ^-n)]/i

Where,

Present value of loan (PV) = $200,000

Monthly loan payment PMT =?

Number of payments (remaining years of loan) n = 20 *12 = 240 months

Annual interest rate I =5%, therefore monthly interest rate i= 5%/12 =0.42% per month

Therefore

$200,000 = PMT * [1- (1+0.42%) ^-240]/0.42%

Or PMT = $1,319.91

Now PMT calculation at Annual interest rate I =4%, therefore monthly interest rate i= 4%/12 =0.33% per month

Therefore

$200,000 = PMT * [1- (1+0.33%) ^-240]/0.33%

Or PMT = $1,211.96

Therefore reduction in monthly payment

= $1,319.91 -$1,211.96

= $107.95

If you go for refinancing,

The number of more months would you need to live in the home in order to get the closing cost back

= refinancing cost / reduction in monthly payment due to refinancing

= $3,000 / $107.95           

= 27.79 months or 28 months (nearest whole number)

2. Also with time value of money with annual MARR 10% compounded monthly, solve.

We know that reduction in monthly payment due to refinancing is $107.95 and refinancing cost is $3,000

If discounting rate (MARR) is 10% per annum compounded monthly; the number of more months would you need to live in the home in order to get the closing cost back is the present value of monthly savings ($107.95) equal to refinancing cost ($3,000)

PV of monthly savings = PMT * [1-(1+i) ^-n)]/i

Where,

PV of monthly savings = $3,000

Monthly loan payment PMT =$107.95

Number of months n =?

Discounting rate (MARR) is 10% per annum, therefore monthly discount rate i= 10%/12 =0.833% per month

Therefore

$3,000 = $107.95* [1- (1+0.833%) ^-n]/0.833%

Or n = 31.74 months or 32 months (nearest whole number)

Therefore it will take 32 more months to recover refinancing cost if MARR is 10% per annum.