ZumBahlen Inc.is considering the following mutually exclusive projects: $0−$5,000−$5,00012003,00028003,00033,00080045,000200\begin{array}{lrr}\text { Year } & \begin{array}{r}\text { Project A } \\\text { Cash Flow }\end{array} & \begin{array}{r}\text { Project B } \\\text { Cash Flow }\end{array} \\\hline 0 & -\$ 5,000 & -\$ 5,000 \\1 & 200 & 3,000 \\2 & 800 & 3,000 \\3 & 3,000 & 800 \\4 & 5,000 & 200\end{array}$ At what cost of capital will the NPV of the two projects be the same? (That is,what is the "crossover" rate?)

A) 16.15%
B) 16.74%
C) 17.33%
D) 17.80%

Question

ZumBahlen Inc.is considering the following mutually exclusive projects:

0−$5,000−$5,00012003,00028003,00033,00080045,000200\begin{array}{lrr}\text { Year } & \begin{array}{r}\text { Project A } \\\text { Cash Flow }\end{array} & \begin{array}{r}\text { Project B } \\\text { Cash Flow }\end{array} \\\hline 0 & -\$ 5,000 & -\$ 5,000 \\1 & 200 & 3,000 \\2 & 800 & 3,000 \\3 & 3,000 & 800 \\4 & 5,000 & 200\end{array}

At what cost of capital will the NPV of the two projects be the same? (That is,what is the "crossover" rate?)

A) 16.15%
B) 16.74%
C) 17.33%
D) 17.80%

Nikki Eisen · Accepted Answer

Final Answer : A, Explanation :  The crossover rate is the cost of capital at which the NPV of two projects is the same. To find the crossover rate, we can use the following formula:Crossover rate = (PV of Project A - PV of Project B) / (Initial investment of Project A - Initial investment of Project B)In this case, the initial investment of both projects is $5,000. The PV of Project A can be calculated as follows:PV of Project A = -5,000 + 200/(1+r) + 800/(1+r)^2 + 3,000/(1+r)^3 + 5,000/(1+r)^4The PV of Project B can be calculated as follows:PV of Project B = -5,000 + 3,000/(1+r) + 3,000/(1+r)^2 + 800/(1+r)^3 + 200/(1+r)^4Setting the PV of Project A equal to the PV of Project B and solving for r, we get:-5,000 + 200/(1+r) + 800/(1+r)^2 + 3,000/(1+r)^3 + 5,000/(1+r)^4 = -5,000 + 3,000/(1+r) + 3,000/(1+r)^2 + 800/(1+r)^3 + 200/(1+r)^4200/(1+r) - 3,000/(1+r) + 800/(1+r)^2 - 3,000/(1+r)^2 + 5,000/(1+r)^4 - 200/(1+r)^4 = 0200/(1+r) - 2,200/(1+r) + 500/(1+r)^2 + 4,800/(1+r)^4 = 0(200 - 2,200r + 500r^2 + 4,800r^4)/(1+r)^4 = 0200 - 2,200r + 500r^2 + 4,800r^4 = 0Using a financial calculator, we can solve for r and get the crossover rate of 16.15%.

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