Question

A couple is looking to purchase their first home. The house is for sale at $125,000.
They can obtain an 80% LTV with a 7% interest rate and monthly payments. The loan is to be fully amortized over 30 years.
Alternatively, they could obtain a 90% LTV at a 7.5% interest rate amortized over the same term.
What is the incremental cost of borrowing the additional funds? Assuming the couple took the 80% loan, after fifteen years, the couple has the opportunity to refinance their mortgage to a new rate of 4% with closing costs for the new loan being $2,000. What is the incremental cost of refinancing and how long will it take for the couple to recover their costs?

Answer 1

Value of house= 125000

Loan amt.(option 1) =125000*80%= 100000

Monthly pmts. On the loan at 7% /12=0.5833% p.m. for 360 months =

Mthly .pmt.=PV of Loan/Annuity factor for 0.5833% , 360 months

ie. 100000/((1-1.005833^-360)/0.005833)=

665.28

Loan amt.(option 2) =125000*90%= 112500

Monthly pmts. On the loan at 7.5% /12=0.625% p.m. for 360 months =

Mthly .pmt.=PV of Loan/Annuity factor for 0.625% , 360 months

112500/((1-1.00625^-360)/0.00625)=

786.62

so, the incremental cost of borrowing the additional funds

(786.62*360)-(665.28*360)=

43682.4

Loan balance after 15 yrs. Ie. At end of 15*12=180 months=

FV of original loan-FV of mothly annuities at end of 180 mths.

ie.(100000*(1+0.005833)^180)-(665.28*(1.005833^180-1)/0.005833)=

74016.33

so, the loan balance at end of 15 yrs.(180 months)= 74016.33

now, the new loan amt, with closing costs will be 74016.33+2000= 76016.33

the monthly pmt.on this new loan at the new interest rate of 4%/12 =0.3333% will be

76016.33/((1-1.003333^-180)/0.003333)=

562.27

ie. Monthly savings of $ 665.28-562.27=

103.01

in annuity payments

so, the

incremental cost of refinancing=Total mthly. Annuities on the old loan-Total monthly annuities on the new refinanced loan

ie.(665.28*180)-(562.27*180)=

18541.80

Total savings in interest costs

The closing costs of $ 2000 will be recovered in----

can be found by solving the following equation for n , no.of mthly. pmts.

2000=103.01*(1-1.003333^-n)/0.003333

n= 20 months

ie. 1.67 yrs.