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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

Eva is risk-averse. Currently she has \(\$ 50,000\) to invest. She faces the following choice: she can invest in the stock of a dot-com company, or she can invest in IBM stock. If she invests in the dot-com company, then with probability 0.5 she will lose \(\$ 30,000\), but with probability 0.5 she will gain \(\$ 50,000\). If she invests in IBM stock, then with probability 0.5 she will lose only \(\$ 10,000,\) but with probability 0.5 she will gain only $\$ 30,000$. Can you tell which investment she will prefer to make?

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?

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?

For each of the following situations, do the following: first describe whether it is a situation of moral hazard or of adverse selection. Then explain what inefficiency can arise from this situation and explain how the proposed solution reduces the inefficiency. a. When you buy a second-hand car, you do not know whether it is a lemon (low quality) or a plum (high quality), but the seller knows. A solution is for sellers to offer a warranty with the car that pays for repair costs. b. Some people are prone to see doctors unnecessarily for minor complaints like headaches, and health maintenance organizations do not know how urgently you need a doctor. A solution is for insurees to have to make a co-payment of a certain dollar amount (for example, \(\$ 10\) ) each time they visit a health care provider. All insurees are risk-averse. c. When airlines sell tickets, they do not know whether a buyer is a business traveler (who is willing to pay a lot for a seat) or a leisure traveler (who has a low willingness to pay). A solution for a profit-maximizing airline is to offer an expensive ticket that is very flexible (it allows date and route changes) and a cheap ticket that is very inflexible (it has to be booked in advance and cannot be changed). d. A company does not know whether workers on an assembly line work hard or whether they slack off. A solution is to pay the workers "piece rates," that is, pay them according to how much they have produced each day. All workers are risk-averse, but the company is not risk-neutral. e. When making a decision about hiring you, prospective employers do not know whether you are a productive or unproductive worker. A solution is for productive workers to provide potential employers with references from previous employers.

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?
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