Question

10. (6 marks) A person amortizes a loan of $170,000 for a new home by obtaining a 20-year mortgage at the rate of 7.5% compounded monthly. Find (a) the monthly payment, (b) the total interest charges, and (c) the principal remaining after 5 years.

Answer 1

(a)

Loan Amount P = $170000

Interest Rate = r = 7.5% or 0.075/12 monthly

Number of payment periods = N = 25*12 = 240 months

Let monthly payments made be X

Hence, the sum of present value of monthly 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] = (0.07/12)*170000*(1+0.07/12)²⁴⁰/[(1+0.07/12)²⁴⁰-1] = $1318.00

(b) Total Interest Charges = Total Amount paid in 240 years - Loan Amount

= 1318.00*240 - 170000

= $146320

(c)

Let the balance principal after 5 years (p = 60 months) be Z

The Present Value of monthly 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

=> [(0.075/12)(170000)(1+0.075/12)²⁴⁰/[(1+0.075/12)²⁴⁰-1]]*[1- (1+0.075/12)^-60]/(0.075/12) + Z/(1+0.075/12)⁶⁰ = 170000

=> 68345.74 + Z/(1+0.075/12)⁶⁰ = 170000

=> Z = (170000 - 68345.74)(1+0.075/12)⁶⁰ = $147733.57