Question

Projects A and B are mutually exclusive. Project A has cash flows of −$10,000, $5,100, $3,400, and $4,500 for Years 0 to 3, respectively. Project B has cash flows of −$10,000, $4,500, $3,400, and $5,100 for Years 0 to 3, respectively. What is the crossover rate for these two projects?Projects A and B are mutually exclusive. Project A has cash flows of −$10,000, $5,100, $3,400, and $4,500 for Years 0 to 3, respectively. Project B has cash flows of −$10,000, $4,500, $3,400, and $5,100 for Years 0 to 3, respectively. What is the crossover rate for these two projects?

Answer 1

The crossover rate for these two projects is the rate of return at which the net present value (NPV) of the two projects becomes equal. To find the crossover rate, we need to calculate the NPV of both projects at different rates of return and find the rate at which the NPV of both projects is the same. The crossover rate for these two projects is approximately 6.528%.

Step-by-step explanation:

The crossover rate for these two projects is the rate of return at which the net present value (NPV) of the two projects becomes equal. To find the crossover rate, we need to calculate the NPV of both projects at different rates of return and find the rate at which the NPV of both projects is the same. Let's calculate the NPVs for both projects at various rates of return:

At 6% rate of return:

Project A: -$10,000 + $5,100/(1+0.06) + $3,400/(1+0.06)² + $4,500/(1+0.06)³ = -$10,000 + $4,811 + $3,213 + $3,929 = $1,953

Project B: -$10,000 + $4,500/(1+0.06) + $3,400/(1+0.06)² + $5,100/(1+0.06)³ = -$10,000 + $4,245 + $3,180 + $3,798 = $1,223

At 9% rate of return:

Project A: -$10,000 + $5,100/(1+0.09) + $3,400/(1+0.09)² + $4,500/(1+0.09)³ = -$10,000 + $4,678 + $2,825 + $3,366 = $475

Project B: -$10,000 + $4,500/(1+0.09) + $3,400/(1+0.09)² + $5,100/(1+0.09)³ = -$10,000 + $4,128 + $3,139 + $3,694 = $961

Based on the NPVs calculated at different rates of return, we can see that the crossover rate is somewhere between 6% and 9%. To find the exact crossover rate, we can use interpolation or trial and error to find the rate at which the NPVs of both projects are equal. For example, let's use interpolation:

Interpolation:

1. Calculate the difference between the NPVs of each project at the two rates:
2. Project A: $1,953 - $475 = $1,478
3. Project B: $1,223 - $961 = $262
4. Calculate the difference ratio:
5. Project A: $1,478 / ($1,478 + $262) = 0.85
6. Project B: $262 / ($1,478 + $262) = 0.15
7. Calculate the crossover rate:
8. Crossover rate = 6% + (9% - 6%) * (0.15 / 0.85) = 6% + 3% * (0.15 / 0.85) = 6% + 3% * 0.176 = 6% + 0.528% = 6.528%

Therefore, the crossover rate for these two projects is approximately 6.528%.

Answer 2

Is that the subject math?