Question

The typical term of a fixed rate mortgage for a house is 30 years. The mortgage has a nominal annual interest rate of 4.75 percent compounded monthly, and payments are due at the end of each month.

(a). What is the monthly payment ($/month) if the borrowed principal is $100,000?

(b). Suppose that the mortgage has the same term but that instead of quoting an interest rate, you are simply quoted a monthly payment of $599.55 per month. Based on this monthly payment, what is the nominal annual interest rate (percent, two decimal places)? Note that you may need to iterate on this to find the solution.

(c). At an interest rate of 5.25 percent, the monthly payment is $552.20 per month. What is the amount of interest that will be paid over the life of the mortgage ($)?

(d). The interest rate is typically lower if the term is shorter. Suppose that you can get a 20-year loan at a nominal annual interest rate of 3.75 percent, compounded monthly. What would be the monthly payment ($/month)?

(e).Assume that the monthly payment for (d) is $600/month (this is not the correct answer to (d)). What would be the total amount of interest paid on this loan ($)?

Answer 1

a)

Number of payments=n=30*12=360

Principal loan amount=PV=$100000

Interest rate per month=i=4.75%/12==0.3958%=0.003958 per month

Monthly payment=PV/(P/A,0.003958,360)

Monthly payment=R=100000/191.700394=$521.65

b)

We are given R=$599.55

Let us see the value of PV factor at which monthly payment is 599.55.

(P/A,i,360)=PV/R=100000/599.55=166.7918

Let us calculate PV factor at i=5% (monthly 0.004167), 6% (monthly 0.005), 7% (monthly 0.00533)

We find that annual nominal rate of 6%, we PV factor quite close to desired PV factor i.e. 166.7918. Let us try PV factor at 6.01% i.e. 0.005008

Obtained value is quite away from 166.7918.

We can say that interest rate is 6.00% (correct up to two decimal places)

c)

Monthly payment amount=R=$552.20

Number of payments=n=360

Principal loan amount=PV=$100000

Total amount paid=552.20*360=$198792.00

Interest paid=198792-100000=$98792.00

d)

Number of payments=n=20*12=240

Principal loan amount=PV=$100000

Interest rate per month=i=3.75%/12=0.3125%=0.003125 per month

Monthly payment=PV/(P/A,0.003125,240)

Monthly payment=R=100000/168.6658=$592.89

e)

Monthly payment amount=R=$600.00

Number of payments=n=240

Principal loan amount=PV=$100000

Total amount paid=600*240=$144000

Interest paid=144000-100000=$44000.00