Question

A young couple took a 10-year Canadian mortgage with the Toronto Dominion Bank for

$120,000 with equal monthly payments (rounded to the nearest cent). The stated annual rate

on the mortgage is 8%, compounded semiannually. Assume that the mortgage rate remains at

8% for the remaining time of the mortgage. After they have paid one fifth of their mortgage

payments, the amount still remaining to be paid on the mortgage is

Answer 1

Loan Amount P = $120000

Semiannual Interest Rate = r = 0.08/2 = 0.04

Number of payment periods = N = 10*2 = 20 semiannual payments

Let semiannual payments made be X

Hence, the sum of present value of semiannual payments must be equal to the value of the loan amount

=> X/(1+r) + X/(1+r)² +....+ X/(1+r)^N = P

=> X[1- (1+r)^-N]/r = P

=> X = rP(1+r)^N/[(1+r)^N-1]

We need to find the balance principal after p = 20/5 = 4 payments are made

Let the balance principal after p payments be Z

The Present Value of semiannual payments and balance principal should be equal to the loan amount

=> X/(1+r) + X/(1+r)² + ..... X/(1+r)^p + Z/(1+r)^p = P

=> X[1- (1+r)^-p]/r + Z/(1+r)^p = P

substituting X = rP(1+r)^N/[(1+r)^N-1] in the above equation

=> rP(1+r)^N/[(1+r)^N-1][1- (1+r)^-p]/r + Z/(1+r)^p = P

=> [(0.04)(120000)(1+0.04)²⁰/[(1+0.04)²⁰-1]]*[1- (1+0.04)^-4]/(0.04) + Z/(1+0.0)⁴ = 120000

=> 32051.29 + Z/(1+0.04)⁴ = 120000

=> Z = (120000 - 32051.29)(1+0.04)⁴ = $102,887.55

Hence, amount remaining = $102,887.55