Chapter 15

1. Chapter 15 Uncertainty and Information © 2004 Thomson Learning/South-Western

2. Probability • The probability of an event happening is, roughly speaking, the relative frequency with which an event occurs. • For example, the probability of a head coming up on a flip of a fair coin is ½. • That is, when a coin is flipped a large number of times, we can expect a head to come up in approximately one-half of the flips.

3. Expected Value • The expected value of a game with a number of uncertain outcomes is the size of the prize the player will win on average. • If, for example, on a single flip of a coin, Jones pays Smith $1 (X1 = +$1) if a tail comes up and Smith will pay Jones $1 (X2 = -$1) if a head comes up, the expected value of the game is

4. Expected Value • If the game was changes so that, from Smith’s point of view, X1 = $10, and X2 = -$1, the expected value for Smith would be • Because Smith would stand to win $4.50 on average, she might be willing to pay Jones up to this amount to play. • Fair games are games that cost their expected value.

5. APPLICATION 15.1: Blackjack Systems • Each player in Blackjack is dealt two cards (with the dealer playing last). • The dealer asks each player if he or she wishes another card. • The player getting a hand that totals closest to 21, without going over 21, is the winner. • If the receipt of a card puts a player over 21, that player automatically loses.

6. APPLICATION 15.1: Blackjack Systems • Played this way, the game offers a number of advantages to the dealer. • The dealer plays last, so other players can go over 21 (and lose) before the dealer plays. • Usually, the dealer also wins ties. • The dealer has a winning margin of 6 percent on average, with the player winning 47 percent of the hands and the dealer 53 percent.

7. APPLICATION 15.1: Betting Systems • To entice more people to play, casinos have eased the rules. • At many Las Vegas casinos, dealers must play under fixed rules and ties result in bets being returned to the players. • With these rules, the dealers advantage falls to as little as 0.1 percent. • If players use systems, such as card counting, they may win over 50 percent of the time.

8. APPLICATION 15.1: Casino’s Response • To deal with card counting, Las Vegas casinos have made several rule changes. • They use multiple decks to make counting more difficult. • They refuse admission to known system players. • This illustrates that small changes in expected values can have important implications.

9. Risk Aversion • When people are faced with a risky but fair situation, they will usually choose not to participate. • Risk aversion is the tendency for people to refuse to accept fair games. • A Swiss mathematician, Daniel Bernoulli, theorized that it is not strictly the monetary payoff of a game that matters to people, but the expected utility from the game’s prizes.

10. Diminishing Marginal Utility • Specifically, Bernoulli assumed that the utility associated with the payoffs in a risky situation increases less rapidly than the dollar value of these payoffs. • The extra (or marginal) utility that winning an extra dollar in prize money provides is assumed to decline as more dollars are won.

11. Diminishing Marginal Utility • Diminishing marginal utility is reflected in Figure 15.1, which shows the utility associated with possible prizes (or incomes) from $0 to $15,000. • The concave shape of the curve reflects assumed diminishing marginal utility. • The gain in utility due to an increase in income from $1000 to $2000 exceeds the gain from $14,000 to $15,000.

12. A graphical Analysis of Risk Aversion • In Figure 15.1, the person has three options. He or she may: • retain the current level of income ($10,000) without taking any risk; • take a fair bet with a 50-50 chance of winning or losing $2,000; • take a fair bet with a 50-50 chance of winning or losing $5,000.

13. FIGURE 15.1: Risk Aversion Utility U 0 5 8 10 12 15 Income (thousands of dollars)

14. A graphical Analysis of Risk Aversion • The current $10,000 provides utility of U3. • The utility of the $2,000 bet is the average of the utility of $12,000 (if he or she wins) and the utility of $8,000 (if he or she loses). • This average utility is U2 < U3. • The utility (U1 < U2) of the $5000 bet is the average of the utility of winning ($15,000) and losing ($5,000).

15. FIGURE 15.1: Risk Aversion Utility U U3 U2 U1 0 5 8 10 12 15 Income (thousands of dollars)

16. Willingness to Pay to Avoid Risk • With risk aversion and equal expected values ($10,000 for the three options in Figure 15.1), people will prefer risk-free incomes to risky incomes which offer less utility. • In Figure 16.1, a risk-free income of $9,500 provides the same utility as the $2,000 gamble. • The person would pay up to $500 to avoid the risk of the $2000 gamble.

17. FIGURE 15.1: Risk Aversion Utility U U3 U2 U1 0 5 8 9.5 10 12 15 Income (thousands of dollars)

18. APPLICATION 15.2: Do HMO Patients Need a Bill of Rights? • Much of the controversy over HMOs can be clarified by considering how the issue of moral hazard operates in health insurance markets. • Figure 1 illustrates the moral hazard dilemma as it applies to medical care. • In the absence of insurance, people would by Q* units of medical care at a price of P*.

19. FIGURE 1: Moral Hazard in the Medical Care Market P . A C P* S . B P*/3 Demand Q 0 Q* Q’

20. APPLICATION 15.2: Do HMO Patients Need a Bill of Rights? • Because most medical care purchases are insured, consumers do not pay P* for their care. • Instead they may pay only, say, one-third of the market price for the care that they receive and they therefore increase the quantity demanded to Q’. • Because the value of this extra care to consumers is less than the marginal cost of the care (assumed to be P*), there will be a deadweight loss from the insurance given by the area ABC.

21. APPLICATION 15.2: Do HMO Patients Need a Bill of Rights? • Because HMOs operate under predetermined budget constraints, they have an incentive to be sure that the care they provide meets some sort of cost-benefit standard. • They may therefore be reluctant to provide care up to the amount that insured consumers want to buy (Q’).

22. APPLICATION 15.2: Do HMO Patients Need a Bill of Rights? • These firms will tend to push the quantity of care toward Q* -- a level of care for which marginal benefit is equal to marginal cost. • Much of the consumer dissatisfaction with HMOs may reflect the fact that established spending patterns were already exaggerated by the prevalence of insurance in medical care markets.

23. Methods of Reducing Risk: Insurance • Figure 15.2 shows the motive for buying insurance. • Assume that during the next year this person, with a $10,000 current income, faces a 50 percent change of having $4,000 in unexpected medical bills. • Without insurance, this person’s utility would be U1, the utility of the average of $6000 and $10,000.

24. FIGURE 15.2: Insurance Reduces Risk Utility U U1 Income (thousands of dollars) 0 6 7.5 10

25. Fair Insurance • Fair insurance is insurance for which the premium is equal to the expected value of the loss. • In Figure 15.2, fair insurance would cost $2,000 which is the expected value of what insurance companies would have to pay each year in health claims. • This would guarantee income of $8,000 which would yield utility of U2

26. FIGURE 15.2: Insurance Reduces Risk Utility U U2 U1 Income (thousands of dollars) 0 6 7.5 8 10

27. Unfair Insurance • Since insurance companies have costs beyond paying benefits, they can not sell insurance at actuarially fair premiums. • In Figure 15.2 the person would be willing to pay up to $2,500 for health insurance since $7,500 of risk-free income yields as much utility (U1) as without any insurance. • A premium of $3,500, however, would reduce utility to U0.

28. FIGURE 15.2: Insurance Reduces Risk Utility U U2 U1 U0 Income (thousands of dollars) 0 6 6.5 7.5 8 10

29. Uninsurable Risks • Some risks are so unique or difficult to evaluate that insurers are unable to set premium rates so that risks become uninsurable. • If events are so infrequent or unpredictable, such as wars, etc. insurers have no basis for establishing premiums.

30. Uninsurable Risks • When buyers and sellers have different information, market outcomes may exhibit adverse selection--the quality of goods or services traded will be biased toward market participants with better information. • Those who expect large losses will buy insurance so the insurer will be paying out more in losses than expected.

31. Uninsurable Risks • Moral hazard is the effect that having insurance has on the behavior of the insured. • Having insurance may cause people to be more likely to incur losses. • For example, if people have insurance on the cash they carry they are more likely to lose it through carelessness.

32. Methods of Reducing Risk: Diversification • Diversification is the economic principle underlying the adage, “Don’t put all your eggs in one basket.” • Diversification is the spreading of risk among several options rather than choosing only one. • Suitably spreading risk around may increase utility above that obtain by a single action.

33. Diversification • Figure 15.3 shows the utility of income for an individual with a current income of $10,000 who must invest $4,000 in risky assets. • Assume only two such assets, shares of stock in company A or company B. • The stock costs $1, but will increase to $2 if the company does well during the next year.

34. Diversification • If the company does poorly, the stock will be worthless. • Each company has a 50-50 chance of doing well. • If the two companies’ prospects are unrelated to one another, holding both stocks will reduce the person’s risks.

35. Diversification • Investing in 4,000 shares of company A yields a 50 percent chance of having $14,000 and a 50 percent chance of having $6,000. • This yields a utility level of U1. • If the person invests in 2,000 shares of each company, there are four possible outcomes which are shown in Table 15.1.

36. FIGURE 15.3: Diversification Reduces Risk Utility U U1 Income (thousands of dollars) 0 6 10 14

37. TABLE 15.1: Possible Outcomes from Investing in Two Companies

38. Diversification • Each of these four outcomes is equally likely, with half the cases where the person ends up with his or her original $10,000. • The diversification strategy, while it still has an expected value of $10,000, has less risk. • In Figure 15.3, point C represent when B does poorly and D represent when B does well. • Point E, (the average of C and D) results from diversification and yields utility U2 > U1.

39. FIGURE 15.3: Diversification Reduces Risk Utility D U U2 U1 E C Income (thousands of dollars) 0 6 10 14

40. APPLICATION 15.3: Mutual Funds • Mutual funds pool money from many investors to buy shares in several different companies. • For this service, investors pay annual management fees of 1 to 1.5 percent of the value of the money they have invested.

41. APPLICATION 15.3: Diversification and Riskiness of Funds • A single investor trying to buy shares in, say, 100 companies would find that most of his or her funds would be used for brokerage fees. • Because mutual funds deal in large volumes, brokerage commissions are lower. • This allows the investor to reduce risk through diversification at lower cost.

42. APPLICATION 15.3: Diversification and Riskiness of Funds • Fund managers offer products with varying amounts of risk. • Money market and short-term bond funds offer little risk, balanced funds (stock and bonds) offer more risk, and growth funds generally offer the most risk. • One study found a 10 percent increase in riskiness increases average yield by 1 percentage point.

43. APPLICATION 15.3: Portfolio Management • Mutual fund managers can reduce risk further by their choice of stocks. • If their investments tend to move in opposite directions, the value of diversification increases. • Examples include holding mining stock, which tends to move opposite of the market, or holding stock from many different countries.

44. Pricing of Risk in Financial Assets • Because people are willing to pay something to avoid risks, it seems that one should be able to study the process directly. • We could treat “risk” like any other commodity and study the factors that influence its demand and supply. • With financial assets, the risks people face are purely monetary and relatively easy to measure.

45. Investors’ Market Options • Figure 15.4 shows a simplified illustration of the market options open to a would-be investor in financial assets. • The points on the figure represent the options available. • For example, point A represents a risk-free asset such as money in a checking account. • Asset B represents a relatively risky stock. • All other points in Figure 15.4 represent the risks and returns associated with assets that an investor might buy.

46. Figure 15.4: Market Options for Investors Market Line Annual return C B A Risk

47. Investors’ Market Options • Investors like high annual returns but dislike risk, so they will choose to hold combinations of these available assets that lie on their “northwest” periphery. • By mixing various risky assets with the risk-free asset (A), they can choose any point along the line AC. • This line is labeled the market line because it shows the possible combinations of annual returns and risk that an investor can achieve by taking advantage of what market the offers.

48. APPLICATION 15.4: The Equity Premium Puzzle—Historical Rates of Return • The risk associated with various assets is sometimes measured by the “standard deviation” of their annual returns. • This measure shows the range in which roughly two-thirds of the returns fall. • As shown in Table 1, in two-thirds of the years common stocks returned more than -8 percent and less than +32.4 percent.

49. APPLICATION 15.4: The Equity Premium Puzzle--The Excess Return on Common Stocks • The quantitative nature of the extra returns to common stock holding are inconsistent with many other studies of risk. • Other studies suggest that individuals would accept the extra risk associated with stocks for an extra return of between 1 and 2 percent per year--significantly less than the 7 percent actually provided.

50. APPLICATION 15.4: The Equity Premium Puzzle--The Excess Return on Common Stocks • One possible explanation is that the figures in Table 1 underestimate the risk of stocks. • If returns on stock are highly correlated with the business cycle, this would cause a double risk in economic downturns • A fall in income • A fall in returns from investment • However, the actual correlation does not appear high enough for this extra risk to be very large.