Question

You are an insurance agent who must write a policy for a new client named Sam. His company, Society for Creative Alternatives to Mayonnaise (SCAM), is working on a low-fat, low-cholesterol mayonnaise substitute for the sandwich-condiment industry. The sandwich industry will pay top dollar to the first inventor to patent such a mayonnaise substitute. Sam’s SCAM seems like a very risky proposition to you. You have calculated his possible returns table as follows:

Probability	Return	Outcome
.999	-\(1,000,000	(he fails)
.001	\)1,000,000,000	(he succeeds and sell his formula)

a. What is the expected return of Sam’s project? What is the variance?

b. What is the most that Sam is willing to pay for insurance? Assume Sam is risk-neutral.

c. Suppose you found out that the Japanese are on the verge of introducing their own mayonnaise substitute next month. Sam does not know this and has just turned down your final offer of $1000 for the insurance. Assume that Sam tells you SCAM is only six months away from perfecting its mayonnaise substitute and that you know what you know about the Japanese. Would you raise or lower your policy premium on any subsequent proposal to Sam? Based on his information, would Sam accept?

Short answer

a. The expected return on Sam’s project is $1000. The variance of the project will be 1,000, 998, 999, 000, 000.

b. Since Sam is risk-neutral, he won’t pay for insurance.

c. The expected outcome of Sam in the new scenario will be $499,500, and Sam won’t accept the new proposal.

Step-by-step solution

Expected return and variance of Sam

The expected return of the Sam can be calculated using the data given in the table

With probability .999, the return will be -$1,000,000.

With probability .001, the return will be $1,000,000,000.

The expected return on Sam’s project will be:

$$\begin{gathered} ER=\underset{}{\overset{}{\sum P}}\left(x\right)\times X \\ ER=\left(0.999\right)\left(-1000000\right)+\left(.001\right)\left(1,000,000,000\right) \\ =999000+1000000 \\ =1000 \end{gathered}$$

The expected return on the project is $1000.

The variance of the project will be:

$$\begin{gathered} Variance=\sum _{}^{}P\left(X\right){\left(X-ER\right)}^2 \\ Variance=\left(0.999\right){\left(-1000000-1000\right)}^2+\left(.001\right){\left(1000000000-1000\right)}^2 \\ =1,000,998,999,000,000 \end{gathered}$$

The variance of the project will be 1,000,998,999,000,000. (value is an approximate one)

 Amount Sam will be paying for insurance

The insurance company guarantees a minimum amount of $1000 without considering the outcome of the project. Since Sam is risk-neutral, the amount he is expecting will also be the same as the expected return, which is $1000. Sam will anyway get a minimum amount of expected return whether he pays an amount or not. So he won’t pay any amount as insurance premium as he has a guaranteed outcome of $1000.

Sam will refuse the offer

The payoff of Sam will be low after the entry of the new firm. Assuming the probability to be reduced to half, the new probability and return will be:

With the probability of 0.4995, the return will be -$1,000,000.

With the probability 0.005, the return will be $1,000,000,000.

The expected return of the project in this situation will be:

$$\begin{gathered} ER=\left(.4995\right)\left(-1,000,000\right)+\left(0.0005\right)\left(1,000,000,000\right) \\ =499500+500000 \\ =500 \end{gathered}$$

The expected return has been reduced to $500 due to the entry of a Japanese firm.

Since Sam is unaware of the entry of a new firm, he continues to refuse the offers without knowing that his expected return has been reduced.