Question

A total debt of $ 1,000 due now, $4000 due 2 years from now, and $6000 due 5 years from now is to be repaid by 3 payments.

(1) The first payment is made now.

(2) The second payment, which is 80% of the first, is made at the end of 30 months from now.

(3) The third payment, which is 60% of the second, is made at the end of 4 years from now.

The annual interest rate is 4%, compounded semi-annually. Calculate the amount of each of the three payments. A timeline is required for full points.

Answer 1

The PV of total debt (discounted 4% semi-anually) = 1000 + 4000/(1.02^4) + 6000/(1.02^10) = $9617.47

Let the first payment be 'A', then the second payment = 0.8*A and the third payment = 0.6*0.8*A = 0.48*A

We discount the payments to PV

PV of total payments = A + (0.8*A)/(1.02^5) + (0.48*A)/(1.02^8)
This amount should be equal to the PV of total debt

A + (0.8*A)/(1.02^5) + (0.48*A)/(1.02^8) = $9617.47

A*[1+ (0.8)/(1.02^5) + (0.48)/(1.02^8)] =  $9617.47

A = $4506.23

Hence, the first payment = $4506.23

The second payment = 0.8*4506.23 = $3605

The third payment = 0.48*4506.23 = $2163

Year	Debt due	Payment
0	-1000	4506
0.5	0	0
1	0	0
1.5	0	0
2	-4000	0
2.5	0	3605
3	0	0
3.5	0	0
4	0	2163
4.5	0	0
5	-6000	0