A person must decide whether or not to proceed with a particular investment project. If the project succeeds, She will gain $15 million. If the project fails, she will lose $3 million. She estimates there is a 20% chance that the project will succeed and an 80% chance it will fail.
There is a consultant that could tell her with certainty if the project succeed or fail, but only for a fee. What is the most that she should be willing to pay the consultant for the information? Explain. Assume that she correctly estimated the probabilities of the project’s likely success and failure.
I AM LOST on this one - can anyone please help?
First, lets assume the person is RISK NEUTRAL -- where the change in utility from an expected dollar loss is equal to the change in utility from an expected dollar gain.
That said, this is simply a comparison of the expected return on the person's investment. Without the consultant, there is a 20% chance she gets 15M and an 80% chance she loses 3M.
E(return) = .2*15 - .8*3 = 0.6
If she hires the consultant, she will only make the investment if its a win. However, taking into account of the fee, her expected return must greater than what she could do without the consultant. So:
E(return) = .2*(15 - F) - .8*F - 0.6
I get the Fee=2.4
To determine the maximum amount she should be willing to pay the consultant for the information on whether the project will succeed or fail, we need to analyze the expected value of the investment project.
The expected value is calculated by multiplying each outcome by its respective probability and summing them up. In this case, the potential gain is $15 million if the project succeeds, and the potential loss is $3 million if the project fails. The probabilities are 20% for success and 80% for failure.
Expected Value = (Probability of Success * Gain) + (Probability of Failure * Loss)
Expected Value = (0.20 * $15 million) + (0.80 * -$3 million)
Expected Value = $3 million - $2.4 million
Expected Value = $0.6 million
So, the expected value of this investment project is $0.6 million. This means, on average, she can expect to gain $0.6 million from this project.
Now let's consider the consultant's fee. She should be willing to pay the consultant up to the point where the expected value of the investment project is equal to the fee, as paying more than that would reduce her expected gains.
In this scenario, the maximum fee she should pay the consultant is $0.6 million. If the consultant charges more than that, it would result in reducing her expected gains. Therefore, she should not be willing to pay the consultant more than $0.6 million for the information on whether the project will succeed or fail.
It is important to note that this analysis assumes perfect accuracy in estimating the probabilities and the consultant's information being completely reliable.