Question

MORTGAGE AMORTIZATION:

Suppose you are considering buying a house with a market price of $350,000. You plan on making a down payment of 20% and financing the remainder using a fully amortizing, 30-year, monthly payment mortgage with a fixed interest rate of 4.50%. Assuming your first payment is due exactly one month from today...

• What is your required monthly payment?

• During the first five years (i.e., 60 months), what is the percentage of your total payments which go toward the repayment of interest charges?

• During years 3 – 7 (hint: that's a 5-year interval), what is the total amount of principal retired?

• Your friend suggests you might be able to afford higher monthly payments and might want to consider a 15-year loan. Assuming interest rates on 15- year loans are only 4.00%, how much higher would your monthly payment be if you chose the 15-year loan instead?

• Finally, suppose you chose the 30-year option, have made payments as scheduled for the past 7 years (84 months), and now interest rates have declined to 3.00%. If you refinance the outstanding balance using a new 30- year, monthly payment loan, what will your new (and lower) payment be?

Answer 1

1. Required monthly payment on the mortgage

Mortgage amount= 350000*(1-20%)= 280000

Monthly pmt.=Present value of mortgage/PV Factor at 4.5%/12 p.m.for 30 yr.*12=360 months

ie. 280000/((1-(1+(4.5%/12))^-360)/(4.5%/12))=

1418.72

2.During the first five years (i.e., 60 months), %age of total payments which go toward the repayment of interest charges

First we will find the principal balance at end of 5 yrs., ie. 60 months

Principal balance=(FV of the single sum of original loan at end of 60 mths.)-(FV of the monthly annuity for 60 months)---both at 4.5%/12 per month

ie.(280000*(1+(4.5%/12))^60)-(1418.72*((1+(4.5%/12))^60-1)/(4.5%/12))=

255242.09

So, pmt. Towards interest charges=Total annuities paid so far-Amt. paid tow. Principal so far , which is --(Original loan amt.-Principal bal.at end of 60 mths.)

(1418.72*60)-(280000-255242.09)=

60365.29

(Answer)

3..During years 3 – 7 (hint: that's a 5-year interval), what is the total amount of principal retired?

ie. As above we need to find principal balances as at end of 24 months   & 84 months

& find the difference

so,using the same formula, as in 2 above,

Principal balance at ane of 24 mths.=(FV of the single sum of original loan at end of 60 mths.)-(FV of the monthly annuity for 60 months)---both at 4.5%/12 per month

ie.(280000*(1+(4.5%/12))^24)-(1418.72*((1+(4.5%/12))^24-1)/(4.5%/12))=

270758.39

Bal. at end of 84 mths.

ie.(280000*(1+(4.5%/12))^84)-(1418.72*((1+(4.5%/12))^84-1)/(4.5%/12))=

243673.48

Now, the total amount of principal retired between yrs. 3 & 7 is

270758.39-243673.48=

27084.91

(Answer)

4.Assuming interest rates on 15- year loans are only 4.00%, how much higher would your monthly payment be if you chose the 15-year loan instead?

Monthly pmt.=Present value of mortgage/PV Factor at 4.0%/12 p.m.for 15 yr.*12=180 months

ie. 280000/((1-(1+(4.0%/12))^-180)/(4.0%/12))=

2071.13

Mthly. Pmt. Will be Higher by 2071.13-1418.72=

652.41

(Answer)

5…. Loan Principal balance at end of 7 yrs.

As found out in 3. above, Bal. at end of 84 mths.

ie.(280000*(1+(4.5%/12))^84)-(1418.72*((1+(4.5%/12))^84-1)/(4.5%/12))=

243673.48

In case of refinancing the above outstanding balance using a new 30- year, 3% p.a. loan,monthly payment will be

Monthly pmt.=Present value of mortgage/PV Factor at 3.0%/12 p.m.for 30 yr.*12=360 months

ie. 243673.48/((1-(1+(3.0%/12))^-360)/(3.0%/12))=

1027.34

Monthly pmt. Will be Lower by

1418.72-1027.34=

391.38

(Answer)