Question

Anne and Frank Smith each borrow $12,000 from their father. Anne and Mr. Smith have agreed that she will repay her loan in full by paying $6,000 in two years and $8,000 in four years. Frank prefers to make one lump payment of $15,000 to fully repay his loan. When should he make that payment so that he and his sister will each have the same effective interest rate?

Answer 1

Answer:

4.52 years

Explanation:

the formula that we need to calculate by hand is the following:

PV = future value / (1 + r)ⁿ

Anne

12,000 = 6,000/(1 + r)² + 8,000/(1 + r)⁴

Frank

12,000 = 15,000/(1 + r)ⁿ

we need to equal both:

6,000/(1 + r)² + 8,000/(1 + r)⁴ = 15,000/(1 + r)ⁿ

the math is really complex if you do it by hand, but instead we can use an excel spreadsheet:

Anne

NCF year 0 = -12,000

NCF year 1 = 0

NCF year 2 = 6,000

NCF year 3 = 0

NCF year 4 = 8,000

we must determine the IRR of the cash flows = 5.07%

now we can determine Frank's payment:

12,000 = 15,000/(1 + 5.07%)ⁿ

1.0507ⁿ = 15,000/12,000 = 1.25

n = log 1.25 / log 1.0507 = 0.096910013 / 0.021478732 = 4.5119 ≈ 4.52 years