Question

the amount is 1125.5, but not entirely sure how to get there. No excel please

A $10,000 loan is being paid off by annual payments of $2,000 plus a smaller final payment. If the annual effective rate of interest is 15%, and the first payment is made one year after the time of the loan, find the amount of interest contained in the fifth payment. (A) X < $800 (B) $800 < X < $900 (C) $900 < X < $1000 (D) $1000 < X < $1100 (E) $1100 < x

Answer 1

Annual Payments = $ 2000, Interest Rate = 15% and Loan = $ 10000

Let the required final payment be $ P

Further, the number of payments is at least 5 as 5 x 2000 = $ 10000 in case the discount rate is zero.

Therefore, 2000 x (1/0.15) x [1-{1/(1.15)^(5)}] = $ 6704.31

Remaining Payment = $ 10000 - 6704.31 = $ 3295.69

PV of Another $ 2000 coming in at the end of Year 6 = 2000 / (1.15)^(6) = $ 864.655

Remaining Payment = 3295.69 - 864.655 = $ 2431.03

PV of Another $ 2000 coming in at the end of Year 7 = 2000 / (1.15)^(7) = $ 751.874

Remaining Payment = 2431.03 - 751.874 = $ 1679.16

PV of Another Payment coming in at the end of Year 8 = 2000 / (1.15)^(8) = $ 653.804

Remaining Payment = 1679.16 - 653.804 = $ 1025.36

PV of Another Payment coming in at the end of Year 9 = 2000 / (1.15)^(9) = $ 568.525

Remaining Payment = 1025.36 - 568.525 = $ 456.832

PV of Another $ 2000 at the end of Year 10 is higher than the outstanding payment, thereby implying that the final payment P is made at the end of Year 10

Therefore, P = 456.832 x (1.15)^(10) = $ 1848.14

Loan Outstanding at the end of Year 4 = 2000 x (1/0.15) x [1-{1/(1.15)^(5)}] + 1848.14 / (1.15)^(6) = $ 7503.31

Interest Accrued on Outstanding Loan during year 5 = 7503.31 x 0.15 = $ 1125.5

Interest Portion of 5th Payment = Interest Accrued on Outstanding Loan during Year 5 = $ 1125.5

Hence, the correct option is (E)