Question

What will you still owe after the 13th year of a 15-year, $450,000 mortgage at 4.35% APR if you only make the minimum monthly payments?

A. $77,826

B. $114,827

C. $791,338

D. $78,201

Please work out the problem!

Answer 1

Loan Amount P = $450000

Interest Rate = r = 4.35% or 0.0435/12 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 13 years 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

=> [(0.0435/12)(450000)(1+0.0435/12)¹⁸⁰/[(1+0.0435/12)¹⁸⁰-1]]*[1- (1+0.0435/12)^-156]/(0.0435/12) + Z/(1+0.0435/12)¹⁵⁶ = 450000

=> 405530.38 + Z/(1+0.0435/12)¹⁵⁶ = 450000

=> Z = (450000 - 405530.38)(1+0.0435/12)¹⁵⁶ = $78201