Question

You are going to buy a refrigerator. Brand A will cost $700 and cost $80 per year to perate. Brand B will cost $800 and cost $55 per year to operate. You want to earn 12% per year on your money. Neither will have any salvage value. How long will you need to keep your refrigerator to make brand B attractive?

Answer 1

Let both brands become equally attractive in year N. Then by equivalence,

PW of Brand A = PW of Brand B

700 + 80 x P/A(12%, N) = 800 + 55 x P/A(12%, N)

(80 - 55) x P/A(12%, N) = 100

25 x P/A(12%, N) = 100

P/A(12%, N) = 4

[1 - (1.12)^-N] / 0.12 = 4

[1 - (1.12)^-N] = 0.48

(1.12)^-N = 0.52

(1.12)^N = 1.9231

Taking natural logarithm on each side,

N x ln 1.12 = ln 1.9231

N x 0.1133 = 0.6539

N = 5.77 years

In year 6,

PW of brand A = 700 + 80 x P/A(12%, 6) = 700 + 80 x 4.1114 = 700 + 329 = 1,029

PW of brand B = 800 + 55 x P/A(12%, 6) = 800 + 55 x 4.1114 = 800 + 226 = 1,026 < PW of brand A

Therefore I need to keep the refrigerator for 6 years to make brand B attractive.

Note: Since we are computing PW of costs, the brand with lower PW will be more attractive.