Suppose that you are evaluating two different alternatives. The inflated cost stream for alternative A is $8,000 for year 1, $9,000 for year 2, $12,000 for year 3, 12,000 for year 4, and $13,000 for year 5. The inflated cost stream for alternative B is $10,000 for year 1, $12,000 for year 2, $10,000 for year 3, $9,000 for year 4, and $9,000 for year 5. Assume that the cost of capital is 12%. Which alternative would you select? At which point in time will the selected alternative assume a point of preference over the other (i.e, break-even point)?

Question

Suppose that you are evaluating two different alternatives. The inflated cost stream for alternative A is $8,000 for year 1, $9,000 for year 2, $12,000 for year 3, 12,000 for year 4, and $13,000 for year 5. The inflated cost stream for alternative B is $10,000 for year 1, $12,000 for year 2, $10,000 for year 3, $9,000 for year 4, and $9,000 for year 5. Assume that the cost of capital is 12%. Which alternative would you select? At which point in time will the selected alternative assume a point of preference over the other (i.e, break-even point)?

Answer 1

To determine which alternative to select and at which point in time they will break even, we need to calculate the present value of the cost streams for each alternative and compare them.

To calculate the present value, we need to discount each cash flow using the cost of capital. We will use the formula:

PV = CF / (1 + r)^n

Where PV is the present value, CF is the cash flow, r is the discount rate (cost of capital), and n is the year.

For Alternative A:
PV_1 = 8,000 / (1 + 0.12)^1 = 7,142.86
PV_2 = 9,000 / (1 + 0.12)^2 = 7,698.41
PV_3 = 12,000 / (1 + 0.12)^3 = 8,057.46
PV_4 = 12,000 / (1 + 0.12)^4 = 7,184.19
PV_5 = 13,000 / (1 + 0.12)^5 = 8,389.47

For Alternative B:
PV_1 = 10,000 / (1 + 0.12)^1 = 8,928.57
PV_2 = 12,000 / (1 + 0.12)^2 = 9,547.01
PV_3 = 10,000 / (1 + 0.12)^3 = 7,691.47
PV_4 = 9,000 / (1 + 0.12)^4 = 5,788.23
PV_5 = 9,000 / (1 + 0.12)^5 = 5,178.10

Now, we calculate the net present value (NPV) for each alternative by summing up the present values:

NPV_A = PV_1 + PV_2 + PV_3 + PV_4 + PV_5 = 38,461.19
NPV_B = PV_1 + PV_2 + PV_3 + PV_4 + PV_5 = 36,133.38

Since the goal is to maximize the NPV, Alternative A has a higher NPV than Alternative B.

To find the break-even point, we need to calculate when the difference in NPVs crosses zero.

NPV_A - NPV_B = 38,461.19 - 36,133.38 = 2,327.81

Now, let's calculate the break-even point using interpolation. The break-even point lies between year 3 and 4, as that's where the difference in NPVs changes sign.

break-even point = year 3 + (NPV_A - 0) / (NPV_A - NPV_B) * (year 4 - year 3)
= 3 + (2,327.81 / (2,327.81 - (-5,527.32))) * (4 - 3)
= 3 + (2,327.81 / 7,855.13) * 1
≈ 3.296 years

Therefore, the break-even point is approximately 3.296 years.