Question

Warren purchased a $100,000 house 5 years ago. The mortgage was to be repaid with monthly payments in arrears over 15 years. The nominal rate of interest is 8% compounded monthly. In order to shorten the term of the mortgage by 5 years so as to reduce interest, Warren negotiates with the lender so that the remaining outstanding loan balance is to be repaid in 5 years by larger monthly level payments. How much interest can Warren save?

Answer 1

To calculate the interest saved, we compare the total cost of the original mortgage with the total cost of the new mortgage. The monthly payment of the original mortgage is $945.83. The remaining outstanding loan balance after 5 years is $64,087.72. The new monthly payment of the new mortgage is $1,271.83. Therefore, Warren can save $94,138.80 in interest.

To calculate the amount of interest that Warren can save, we need to compare the total cost of the original mortgage with the total cost of the new mortgage after renegotiation.

First, let's calculate the total cost of the original mortgage. The monthly payment can be calculated using the formula for the present value of an annuity:

`PMT = PV * r / (1 - (1+r)^((-n)))`

Where PMT is the monthly payment, PV is the loan amount, r is the monthly interest rate, and n is the number of months.

Using the given information, we can calculate the monthly payment of the original mortgage:

PV = $100,000, r = 8% / 12 = 0.08 / 12 = 0.00667, n = 15 * 12 = 180

PMT = $100,000 * 0.00667 /
`(1 - (1 + 0.00667)^((-180)))`

= $945.83

Now, let's calculate the remaining outstanding loan balance after 5 years:

The number of remaining months is 15 * 12 - 5 * 12 = 120 months.

The remaining balance can be calculated using the formula for the future value of an ordinary annuity:

`FV = PMT * ((1 + r)^(n - 1)) / r`

Using the given information, we can calculate the remaining outstanding loan balance:

FV = $945.83 *
`((1 + 0.00667)^(120 - 1))` / 0.00667

= $64,087.72

Finally, let's calculate the monthly payment of the new mortgage with a term of 5 years:

The number of months is 5 * 12 = 60 months.

The new monthly payment can be calculated using the formula for the present value of an annuity:

PMT_new = FV * r /
`(1 - (1 + r)^((-n)))`

Using the given information, we can calculate the new monthly payment:

PMT_new = $64,087.72 * 0.00667 / (1 -
`(1 + 0.00667)^((-60)))`

= $1,271.83

Now we can calculate the interest saved by subtracting the total cost of the new mortgage from the total cost of the original mortgage:

Total cost of the original mortgage = $945.83 * 180 = $170,448.60

Total cost of the new mortgage = $1,271.83 * 60 = $76,309.80

Interest saved = $170,448.60 - $76,309.80 = $94,138.80