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Chapter 5: Q4. (page 206)

Suppose an investor is concerned about a business choice in which there are three prospects—the probability and returns are given below:

PROBABILITY
RETURN
4\(100
3\)30
3-$30

What is the expected value of the uncertain investment? What is the variance.

Short Answer

Expert verified

The expected value and the variance of the uncertain investment according to the given probability and return will be $40 and $2940, respectively.

Step by step solution

01

Expected value of the uncertain investment 

The expected value of the uncertain investment can be found out using the given details in the table. In the table, the probability of each return is given. With the probability and its return at each point, the expected value of the uncertain investment will be:

EV=.4100+.330+.3-30=40+9-9=$40

The expected value of the uncertain investment is $40.

02

 Variance of the uncertain investment

The variance of the investment can be obtained from the sum of squared deviations from the mean weighted by their probabilities. The mean is the expected value that is 40.

The variance of the investment will be:

Variance=.4100-402+.330-402+.3-30-402=1440+30+1470=2940

The variance of the uncertain investment is 2940.

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Most popular questions from this chapter

Richard is deciding whether to buy a state lottery ticket. Each ticket costs \(1, and the probability of winning payoffs is given as follows:

PROBABILITY
RETURN
.5\)0.00
.25\(1.00
.2\)2.00
.05$7.50

a. What is the expected value of Richard's payoff if he buys a lottery ticket? What is the variance?

b. Richard's nickname is "No-Risk Rick" because he is an extremely risk-averse individual. Would he buy the ticket?

c. Richard has been given 1000 lottery tickets. Discuss how you would determine the smallest amount for which he would be willing to sell all 1000 tickets.

d. In the long run, given the price of the lottery tickets and the probability/return table, what do you think the state would do about the lottery?

Consider a lottery with three possible outcomes:

• \(125 will be received with probability .2

• \)100 will be received with probability .3

• $50 will be received with probability .5

a. What is the expected value of the lottery?

b. What is the variance of the outcomes?

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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?

Suppose you have invested in a new computer company whose profitability depends on two factors: (1) whether the U.S. Congress passes a tariff raising the cost of Japanese computers and (2) whether the U.S. economy grows slowly or quickly. What are the four mutually exclusive states of the world that you should be concerned about?

As the owner of a family farm whose wealth is \(250,000, you must choose between sitting this season out and investing last year’s earnings (\)200,000) in a safe money market fund paying 5.0 percent or planting summer corn. Planting costs \(200,000, with a six-month time to harvest. If there is rain, planting summer corn will yield \)500,000 in revenues at harvest. If there is a drought, planting will yield \(50,000 in revenues. As a third choice, you can purchase AgriCorp drought-resistant summer corn at a cost of \)250,000 that will yield \(500,000 in revenues at harvest if there is rain, and \)350,000 in revenues if there is a drought. You are risk-averse, and your preference for family wealth (W) is specified by the relationship U(W) = √W. The probability of summer drought is 0.30, while the probability of summer rain is 0.70. Which of the three options should you choose? Explain.
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