An investor estimates that investing €5 million in a particular venture capital project can return
$40 million at the end of five years if it succeeds; however, she realizes that the project may fail at any time between now and the end of the fifth year. The investor is considering an equity investment in the project and her cost of equity for a project with this level of risk is 15%. In the table below are the investor's estimates of certain probabilities of failure for the project. First, 0.30 is the probability of failure in year one. The probability that project fails in the second year, given that it has survived through year one, is 0.25; and so forth:
$$\begin{array} { l c c c c c }
\text { Year } & 1 & 2 & 3 & 4 & 5 \
\hline \text { Failure Prabability } & 0.30 & 0.25 & 0.20 & 0.20 & 0.20 \
\hline
\end{array}$$ a. Determine the expected net present value of the project.
b. Recommend whether the project should be undertaken.

Question

An investor estimates that investing €5 million in a particular venture capital project can return
$40 million at the end of five years if it succeeds; however, she realizes that the project may fail at any time between now and the end of the fifth year. The investor is considering an equity investment in the project and her cost of equity for a project with this level of risk is 15%. In the table below are the investor's estimates of certain probabilities of failure for the project. First, 0.30 is the probability of failure in year one. The probability that project fails in the second year, given that it has survived through year one, is 0.25; and so forth:
 $$\begin{array} { l c c c c c } 
\text { Year } & 1 & 2 & 3 & 4 & 5 \
\hline \text { Failure Prabability } & 0.30 & 0.25 & 0.20 & 0.20 & 0.20 \
\hline
\end{array}$$ a. Determine the expected net present value of the project.
b. Recommend whether the project should be undertaken.

Quizplus · Accepted Answer

The Answer is a. The probability that the project survives to the end of the first year is (1 - 0.30) = 0.70. The probability that it survives to the end of second year is the product of the probability it survives the first year times the probability it survives the second year, or (1 -0.30) (1 - 0.25). Using this pattern, the probability that the firm survives to the end of the fifth year is (1- 0.30) (1 - 0.25)
(1 -0.20)³= (0.70) (0.75) (0.80)³ = 0.2688 or 26.88%.
There are two ways to compute the probability-weighted net present value.
A first approach is to weight each cash flow by its probability of occurrence. The initial investment of 1 million is certain (probability of 100%), the cash flow in five years has a probability of 26.88%. Hence the NPV is equal to:

ext { NPV } = € 5 ext { million } + \frac { € 40 ext { million } imes 26.88 \% } { 1.15 ^ { 5 } } = € 345,644

A second approach is to compute the NPV assuming first failure and then no failure, and to weight these NPVs by the probability of occurrence.
- The net present value of the project given that it survives to the end of the fifth year and thus earns €40 million equals €14.8871 million = -€5 million + €40 million/1.15⁵.
- The net present value of project given that it fails is -€5 million.
Thus the project's expected NPV is a probability-weighted average of these two amounts, or (0.2688)(€14.8871 million) + (0.7312)( -€5 million) = €345,644.
Both approaches yield the same NPV.
b. Based on its positive net present value, the recommendation is to undertake the investment.

An Investor Estimates That Investing €5 Million in a Particular