The Johnson Company purchased
The Johnson Company purchased a building for $200,000 at aninterest rate of 5% for 20 years. If they decided to pay $,2400additional a year toward the borrowed principle, while everythingelse remained constant, how many years from today can they pay offthe mortgage
Answer:
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 equalto 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 bem
Hence, present value of all the future payments should be equalto 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
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