Question

Sarah secured a bank loan of $190,000 for the purchase of a house. The mortgage is to be amortized through monthly payments for a term of 15 years, with an interest rate of 3%/year compounded monthly on the unpaid balance. She plans to sell her house in 10 years. How much will Sarah still owe on her house at that time? (Round your answer to the nearest cent.)

Answer 1

Loan Amount P = $190000

Interest Rate = r = 3% or 0.03/12 monthly

Number of payment periods = N = 15*12 = 180 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]

Let the balance principal after 10 years be Z (p = 10*12 = 120)

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

=> [(0.03/12)(190000)(1+0.03/12)¹⁸⁰/[(1+0.03/12)¹⁸⁰-1]]*[1- (1+0.03/12)^-120]/(0.03/12) + Z/(1+0.03/12)¹²⁰ = 190000

=> 135883.90 + Z/(1+0.03/12)¹²⁰ = 190000

=> Z = (190000 - 135883.90)(1+0.03/12)¹²⁰ = $73,021.75

Hence, amount still owed = $73,021.75