Question

The Johnson Company purchased a building for $200,000 at an interest rate of 5% for 20 years. If they decided to pay $,2400 additional a year toward the borrowed principle, while everything else remained constant, how many years from today can they pay off the mortgage

Answer 1

Loan Amount P = $200000

Interest Rate = 5%

Number of payment periods = n = 20 years

Let the required annual payments made be X

Hence, the sum of present value of annual 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]

Hence, Annual Payments =  rP(1+r)^N/[(1+r)^N-1]

= 200000*( 0.05)*(1+ 0.05)²⁰/((1+ 0.05)²⁰-1) = $16048.52

However, actual payment made = A = 16048.52 + 2000 = 18048.52

Let the number of years in which mortgage will be paid off be m

Hence, present value of all the future payments should be equal to the loan amount

Using the formula above,

A[1- (1+r)^-m]/r = P

=> 18048.52[1- (1+0.05)^-m]/0.05 = 200000

=> [1- (1+0.05)^-m] = 200000*0.05/18048.52

=> (1+0.05)^-m = 1 - 200000*0.05/18048.52

=> (1+0.05)^-m = 0.4459

Take log on both sides

=> m = - ln (0.4459) / ln (1+0.05) = 16.55

Hence, the loan will be paid off in 16.55 years