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Chapter 20: Problem 1

For each of the following situations, calculate the expected value. a. Tanisha owns one share of IBM stock, which is currently trading at $\$ 80 .\( There is a \)50 \%\( chance that the share price will rise to \)\$ 100$ and a \(50 \%\) chance that it will fall to \(\$ 70\). What is the expected value of the future share price? b. Sharon buys a ticket in a small lottery. There is a probability of 0.7 that she will win nothing, of 0.2 that she will win \(\$ 10,\) and of 0.1 that she will win \(\$ 50 .\) What is the expected value of Sharon's winnings? c. Aaron is a farmer whose rice crop depends on the weather. If the weather is favorable, he will make a profit of \(\$ 100\). If the weather is unfavorable, he will make a profit of \(-\$ 20\) (that is, he will lose money). The weather forecast reports that the probability of weather being favorable is 0.9 and the probability of weather being unfavorable is \(0.1 .\) What is the expected value of Aaron's profit?

Short Answer

Expert verified
Answer: The expected values are as follows: a. IBM share price: $85 b. Sharon's lottery winnings: $7 c. Aaron's profit: $88

Step by step solution

01

a. Expected value of IBM share price

To calculate the expected value of the future IBM share price, we multiply the value of each outcome (share price) by its probability, and then sum the results. Hence, the expected value is: (Expected value) = (share price when rises * probability of rise) + (share price when falls * probability of fall) = \(( 100 * 0.50) + (70 * 0.50)\) = \(50 + 35\) = \(85\) Therefore, the expected value of the future IBM share price is \( \$ 85\).
02

b. Expected value of Sharon's winnings in the lottery

To calculate the expected value of Sharon's winnings in the lottery, we multiply the value of each outcome (winnings) by its probability, and then sum the results. Hence, the expected value is: (Expected value) = (winnings when nothing * probability of nothing) + (winnings when 10 dollars * probability of 10 dollars) + (winnings when 50 dollars * probability of 50 dollars) = \(( 0 * 0.7) + (10 * 0.2) + (50 * 0.1)\) = \(0 + 2 +5\) = \(7\) Therefore, the expected value of Sharon's winnings in the lottery is \( \$ 7.\)
03

c. Expected value of Aaron's profit

To calculate the expected value of Aaron's profit, we multiply the value of each outcome (profit) by its probability, and then sum the results. Hence, the expected value is: (Expected value) = (profit when weather is favorable * probability of favorable weather) + (profit when weather is unfavorable * probability of unfavorable weather) = \((100 * 0.9) + (-20 * 0.1)\) = \(90 - 2\) = \(88\) Therefore, the expected value of Aaron's profit is \( \$ 88.\).

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Most popular questions from this chapter

You have \(\$ 1,000\) that you can invest. If you buy General Motors stock, then, in one year's time: with a probability of 0.4 you will get \(\$ 1,600\); with a probability of 0.4 you will get \(\$ 1,100\); and with a probability of 0.2 you will get \(\$ 800\). If you put the money into the bank, in one year's time you will get \(\$ 1,100\) for certain. a. What is the expected value of your earnings from investing in General Motors stock? b. Suppose you prefer putting your money into the bank to investing it in General Motors stock. What does that tell us about your attitude to risk?

Suppose you have \(\$ 1,000\) that you can invest in Ted and Larry's Ice Cream Parlor and/or Ethel's House of Cocoa. The price of a share of stock in either company is \(\$ 100\). The fortunes of each company are closely linked to the weather. When it is warm, the value of Ted and Larry's stock rises to \(\$ 150\) but the value of Ethel's stock falls to \$60. When it is cold, the value of Ethel's stock rises to \(\$ 150\) but the value of Ted and Larry's stock falls to \(\$ 60\). There is an equal chance of the weather being warm or cold. a. If you invest all your money in Ted and Larry's, what is your expected stock value? What if you invest all your money in Ethel's? b. Suppose you diversify and invest half of your \(\$ 1,000\) in each company. How much will your total stock be worth if the weather is warm? What if it is cold? c. Suppose you are risk-averse. Would you prefer to put all your money in Ted and Larry's, as in part a? Or would you prefer to diversify, as in part b? Explain your reasoning.

From 1990 to 2013,1 in approximately every 277 cars produced in the United States was stolen. Beth owns a car worth \(\$ 20,000\) and is considering purchasing an insurance policy to protect herself from car theft. For the following questions, assume that the chance of car theft is the same in all regions and across all car models. a. What should the premium for a fair insurance policy have been in 2013 for a policy that replaces Beth's car if it is stolen? b. Suppose an insurance company charges \(0.6 \%\) of the car's value for a policy that pays for replacing a stolen car. How much will the policy cost Beth? c. Will Beth purchase the insurance in part b if she is risk-neutral? d. Discuss a possible moral hazard problem facing Beth's insurance company if she purchases the insurance.

You own a company that produces chairs, and you are thinking about hiring one more employee. Each chair produced gives you revenue of \(\$ 10\). There are two potential employees, Fred Ast and Sylvia Low. Fred is a fast worker who produces ten chairs per day, creating revenue for you of \(\$ 100\). Fred knows that he is fast and so will work for you only if you pay him more than \(\$ 80\) per day. Sylvia is a slow worker who produces only five chairs per day, creating revenue for you of \(\$ 50 .\) Sylvia knows that she is slow and so will work for you if you pay her more than \$ 40 per day. Although Sylvia knows she is slow and Fred knows he is fast, you do not know who is fast and who is slow. So this is a situation of adverse selection. a. Since you do not know which type of worker you will get, you think about what the expected value of your revenue will be if you hire one of the two. What is that expected value? b. Suppose you offered to pay a daily wage equal to the expected revenue you calculated in part a. Whom would you be able to hire: Fred, or Sylvia, or both, or neither? c. If you know whether a worker is fast or slow, which one would you prefer to hire and why? Can you devise a compensation scheme to guarantee that you employ only the type of worker you prefer?

You have \(\$ 1,000\) that you can invest. If you buy Ford stock, you face the following returns and probabilities from holding the stock for one year: with a probability of 0.2 you will get \(\$ 1,500\); with a probability of 0.4 you will get \(\$ 1,100\); and with a probability of 0.4 you will get \(\$ 900 .\) If you put the money into the bank, in one year's time you will get \(\$ 1,100\) for certain. a. What is the expected value of your earnings from investing in Ford stock? b. Suppose you are risk-averse. Can we say for sure whether you will invest in Ford stock or put your money into the bank?
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