Question

8. (6pts) A loan is being repaid by payments of $2000 at the end of each half-year at nominal rate of 6% convertible semiannually, and a smaller final payment is made after the last regular payment. If the loan balance at the end of the third year is $15,000, find

(1) the amount of the loan. (Answer: $23,396.65)

(2) the loan balance right after the last regular payment. (Answer: $1,216.88)

(3) the amount of principal and the amount of interest paid in the payment paid at the end of third year. (Answers: $1504.85, $495.15)

Financial Math three-part Question: Chapter on Annuities and Outstanding Loan Balances

Answer 1

1
Calculation of Loan amount if the balance at end of third year is $15,000
Loan amount = Present value of annuity + Present value of loan amount at end of third year
Loan amount = 2000*(1-(1.03^-6)/0.03) + 15000*(1/1.03^6)
Loan amount = 2000*(1-(1.03^-6)/0.03) + 15000*(1/1.03^6)
Loan amount = 2000*5.41719 + 15000*0.83748
Loan amount	$23,396.65
The loan amount is $23,396.65
2
We would prepare amortization table for the loan to calculate the balance after last regular payment
Period	Interest @ 3%	Principal	Installment	Loan balance
	(Previous period loan balance*3%)	(Installment - Interest)		(Previous period loan balance - Principal)
0				$23,396.65
1	$701.90	$1,298.10	$2,000	$22,098.55
2	$662.96	$1,337.04	$2,000	$20,761.50
3	$622.85	$1,377.15	$2,000	$19,384.35
4	$581.53	$1,418.47	$2,000	$17,965.88
5	$538.98	$1,461.02	$2,000	$16,504.85
6	$495.15	$1,504.85	$2,000	$15,000.00
7	$450.00	$1,550.00	$2,000	$13,450.00
8	$403.50	$1,596.50	$2,000	$11,853.50
9	$355.61	$1,644.40	$2,000	$10,209.11
10	$306.27	$1,693.73	$2,000	$8,515.38
11	$255.46	$1,744.54	$2,000	$6,770.84
12	$203.13	$1,796.87	$2,000	$4,973.96
13	$149.22	$1,850.78	$2,000	$3,123.18
14	$93.70	$1,906.30	$2,000	$1,216.88
Therefore the balance of the loan after last regular payment is $1,216.88
3
From the amortization above, we can see that for the 6th semi annual payment which is third year payment
The interest payment is $495.15 and principal payment is $1,504.85