Project A has cash flows of −$50,000, $49,400, $27,200, and $24,500 for Years 0 to 3, respectively. Project B has an initial cost of $50,000 and an annual cash inflow of $18,500 for four years. These are mutually exclusive projects. What is the crossover rate?

Crossover rate is the rate at which the NPV of two projects are equal is called crossover rate. The NPV of any project is computed by the following formula:

NPV = CF1/(1 + r)^1 + CF2/(1 + r)^2 + CF3/(1 + r)^3 − Initial Investment

Now for Project A and B, we have following equations:

NPV1 = CF1/(1 + r)^1 + CF2/(1 + r)^2 + CF3/(1 + r)^3 − A

NPV2 = CF1/(1 + r)^1 + CF2/(1 + r)^2 + CF3/(1 + r)^3 + CF4/(1 + r)^4 − B

By putting values we have:

NPV1 = 49400/(1 + r)^1 + 27200/(1 + r)^2 + 24500/(1 + r)^3 − 50,000

And

NPV2 = 18500/(1 + r)^1 + 18500/(1 + r)^2 + 18500/(1 + r)^3 + 18500/(1 + r)^4 − 50,000

Now at crossover rate (a point of intersection of equation of project A and project B), the NPV of project A equals the NPV of project B. This means

NPV equation of project A = NPV equation of project B

49400/(1 + r)^1 + 27200/(1 + r)^2 + 24500/(1 + r)^3 − 50,000 = 18500/(1 + r)^1 + 18500/(1 + r)^2 + 18500/(1 + r)^3 + 18500/(1 + r)^4 − 50,000

Adding +50,000 on both sides of equal sign will cancel out the -50,000 on both sides. Now the equation is as under

49400/(1 + r)^1 + 27200/(1 + r)^2 + 24500/(1 + r)^3 = 18500/(1 + r)^1 + 18500/(1 + r)^2 + 18500/(1 + r)^3 + 18500/(1 + r)^4

Moving left hand side to right hand side will change the signs

0 = 18500/(1 + r)^1 + 18500/(1 + r)^2 + 18500/(1 + r)^3 + 18500/(1 + r)^4 - 49400/(1 + r)^1 - 27200/(1 + r)^2 - 24500/(1 + r)^3

Now rearranging according to the denominator,

0 = [18500/(1 + r)^1 - 49400/(1 + r)^1 ] + [18500/(1 + r)^2 - 27200/(1 + r)^2] + [18500/(1 + r)^3 - 24500/(1 + r)^3] + 18500/(1 + r)^4

Now taking denominators common from brackets we have:

0 = [18500 - 49400]/(1 + r)^1 + [18500 - 27200]/(1 + r)^2 + [18500 - 24500]/(1 + r)^3 + 18500/(1 + r)^4

Adding amounts in the brackets we have:

0 = -30,900/(1 + r)^1 - 8,700/(1 + r)^2 - 6,000/(1 + r)^3 + 18500/(1 + r)^4

Multiplying by Minus 1 on both sides we have:

0 = 30,900/(1 + r)^1 + 8,700/(1 + r)^2 + 6,000/(1 + r)^3 - 18500/(1 + r)^4

Now the equation we have can be used for finding "R" by using IRR method.

Using Internal Rate of Return method:

Year Cash flow Lower rate@30% NPV at lower rate

1 30,900 0.769 23769

2 8,700 0.592 5148

3 6,000 0.455 2731

4 (18,500) 0.350 (6477)

Total 25171

Now using higher rate we will find NPV at higher rate:

Year Cash flow Higher rate@32% NPV at higher rate

1 30,900 0.758 23409

2 8,700 0.574 4993

3 6,000 0.435 2609

4 (18,500) 0.330 (6094)

Total 24917

IRR equation= Lower rate + [NPV at lower rate/(NPV at lower rate - NPV at higher rate)] * (Higher rate -lower rate)

Now putting values in the IRR equation we can find the crossover rate:

Crossover rate= 30% + [25171/(25171-24917)]*(32-30)%

Crossover rate = 30% + 1.98%= Almost 32%