Question

In 16 months Rebecca is to pay back $10,000 to a loan he took out from Eric. He’s able to renegotiate to paying Jabba $2000 in 3 months, two equal payments of $X in 6 and 11 months, and $3500 in 16 months. What is X if the interest rates are j12= 7% for 5 months and j12= 9% thereafter?

Answer 1

Loan of $10000 has to be paid in 16 months

We are assuming, payment is done at the end of the month

It has been negotiated in 4 installments. The Future value of these installments should add upto $10000

On a time scale it should look like this

Time (T)	Deposits	Explanation	FV of Cash Flow
1
2
3	2,000	Find Future value of 2000 for 13 months	2196.73
4
5
6	X	Find Future value of X for 10 months at 9% pa	=(1+9%/12)^10 = 1.0776 x
7
8
9
10
11	X	Find Future value of X for 5 months at 9% pa	=(1+9%/12)^5 = 1.0381x
12
13
14
15
16	3,500	Find Future value of 3500 for 0 months at 9% pa	3500

e start with 2000 deposit at the end of 3 months. For 2 months interest on this would be at 7% pa and from then at 9% pa for 11 months

FV of $2000 at end of 16 months = 2000 X (1+ (7%/12))^2 X (1+9%/12)^11 = 2196.73

FV of x deposited at end of month 6 = (1+ 9%/12)^10 x = 1.0766x

FV of x deposited at end of month 11 = (1+ 9%/12)^5 x = 1.0381x

FV of $3500 deposited at end of month 16 = 3500

All these FV should add upto 10000

2196.73 + 1.0766x + 1.0381x + 3500 = 10000

2.1156x = 4303.27

x = 2034.06

$2304.06 should be deposited at the end of month 6 & 11