Question

Please show the steps to calculate

X 5.3.17 A borrower had a loan of $10,000.00 at 6% compounded annually, with 9 annual payments. Suppose the borrower paid off the loan after 5 years. Calculate the amount needed to pay off the loan. The amount needed to pay off this loan after 5 years is $. (Round to the nearest cent as needed.)

Answer 1

Loan Amount P = $10000

Interest Rate = r = 6%

Number of payment periods = N = 9 years

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 p = 5 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

=> [(0.06)(10000)(1+0.06)⁹/[(1+0.06)⁹-1]]*[1- (1+0.06)^-5]/(0.06) + Z/(0.06)⁵ = 10000

=> 6193.11 + Z/(1+0.06)⁵ = 10000

=> Z = (10000 - 6193.11)(1+0.06)⁵ = $5094.47