Question

Graham purchased a business by agreeing to make three payments of ​$18 comma 000.00 each in 3 ​months, 9 ​months, and 10 months. Because of cash flow​ difficulties, he renegotiated the payment schedule so that he would pay ​$33 comma 800.00 in 4 ​months, ​$11 comma 000.00 in 15 ​months, and a third payment of ​$11 comma 000.00. In how many years should he make the third payment if interest is 9​% compounded monthly​?

State your answer in years and months​ (from 0 to 11​ months).

Answer 1

1. Formula for calculating present value

PV = FV ÷ ( 1+r)ⁿ

Here, PV = present value

FV = future value

r = rate per period

n = number of compounding periods

For calculating Initial PV₁ , we have

FV₁ = $18000

r = 9 ÷ ( 12×100) = 0.0075

n = (3 ÷ 12 ) × 12 = 3

Putting values in the formula we get

PV₁ = 18000 ÷ (1+0.0075)³ = $17601

For PV₂ we have

n = 9

PV₂ = 18000 ÷ (1+0.0075)⁹ = $16829.34

For PV₃ we have

n = 10

PV₃ = 18000 ÷ (1+0.0075)¹⁰ = $16704.06

Total present value of payments before renegotiation = 17601 + 16829.34 + 16704.06 = $51134.40

PV₁ after renegotiation

FV₁ = $33000

n = 4

PV₁ = 33000 ÷ (1+0.0075)⁴ = $32028.29

PV₂ after renegotiation

FV₂ = $11000

n = 15

PV₂ = 11000 ÷ (1+0.0075)¹⁵ = $9833.70

PV₃ after renegotiation

FV₃ = $11000

PV₃ = 11000 ÷ (1+0.0075)ⁿ

Present values of payments before negotiation and after negotiation should be equal

51134.40 = 32028.29 + 9833.70 + { 11000 ÷ (1+0.0075)ⁿ }

9272.41 = 11000 ÷ (1.0075)ⁿ

(1.0075)ⁿ = 11000 ÷ 9272.41

Taking ln of both sides

n × ln(1.0075) = ln(11000 ÷ 9272.41)

n = 22.87 = 23 months = 1 year 11 months

Answer 2

PV1 should have been 33800 not 33000 fyi