Question

An engineer borrowed $3000 from the bank, payable in six equal end-of-year payments at 8%. The bank agreed to reduce the interest on the loan if interest rates declined in the United States before the loan was fully repaid. At the end of 3 years, at the time of the third payment, the bank agreed to reduce the interest rate from 8% to 7% on the remaining debt. What was the amount of the equal annual end-of-year
payments for each of the first 3 years? What was the amount of the equal annual end-of-year payments for each of the last 3 years?

Answer 1

Amount of the equal annual end-of-year payments for each of the first 3 years:

We are given the following information

Annual payment	PMT	To be calculated
rate of interest	r	8.00%
number of years	n	6
Loan Amount	PV	$            3,000.00

We need to solve the following equation to arrive at the required PMT

So the PMT is $648.95 for the first 3 years

Amount of the equal annual end-of-year payments for each of the last 3 years:

Step1) Create amortization schedule to find out the amount of loan outstanding at the end of year 3 when the interest rate fell

Year	Opening Balance	Interest	Principal repayment	Closing Balance
1	$              3,000.00	$             240.00	$                         408.95	$           2,591.05
2	$              2,591.05	$             207.28	$                         441.66	$           2,149.39
3	$              2,149.39	$             171.95	$                         476.99	$           1,672.40
4	$              1,672.40	$             133.79	$                         515.15	$           1,157.24
5	$              1,157.24	$               92.58	$                         556.37	$               600.88
6	$                  600.88	$               48.07	$                         600.88	$                    0.00
		$             893.68	$                     3,000.00

Opening balance = previous year's closing balance
Closing balance = Opening balance-Principal repayment
PMT is calculated as per the above formula
Interest = 0.08 x opening balance
Principal repayment = PMT - Interest
So after year 3, the closing balance or the outstanding loan is $1,672.40

Step 2) Calculate the annual payment for the next three years

We are given the following information

Annual payment	PMT	To be calculated
rate of interest	r	7.00%
number of years	n	3
Present value	PV	$            1,672.40

We need to solve the following equation to arrive at the required PMT

So the PMT is $637.27 for the last 3 years