1. Part 3 Uncertainty and Strategy © 2006 Thomson Learning/South-Western

2. Chapter 5 Uncertainty © 2006 Thomson Learning/South-Western

3. 3 Probability Probability of an event happening; relative frequency with which an event occurs Probability of “heads” coming up on flip of fair coin is ½. That is, when coin is flipped many times, we can expect “heads” to come up in approximately one-half of flips.

4. 4 Expected Value Expected value of game with a number of uncertain outcomes: size of prize that player will win on average. On a single flip of a coin, Jones pays Smith $1 (X 1 = +$1) if a tail comes up and Smith will pay Jones $1 (X 2 = -$1) if a head comes up, the expected value of game for both players is

5. 5 Expected Value If game changes so that, from Smith’s point of view, X 1 = $10, and X 2 = -$1, 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 precisely their expected value.

6. 6 Risk Aversion When people face risky but fair situations, they will usually choose not to participate. Risk aversion is tendency for people to refuse to accept fair games. Swiss mathematician Daniel Bernoulli theorized that not only the monetary payoff of a game matters to people; expected utility from the game’s prizes also affects people’s willingness to play.

7. 7 Diminishing Marginal Utility Bernoulli assumed that utility associated with payoffs in risky situation increases less rapidly than dollar value of payoffs. Extra (marginal) utility obtained from winning an extra dollar in prize money is assumed to decline as additional dollars are won.

8. 8 Diminishing Marginal Utility Fig. 5.1 reflects diminishing marginal utility; shows utility associated with possible prizes (or incomes) from $0 to $50,000. Concave shape of curve reflects assumed diminishing marginal utility. Gain in utility due to increased income from $1000 to $2000 exceeds gain from $49,000 to $50,000.

9. 9 Graphical Analysis of Risk Aversion Figure 5-1 shows person with three options. Contender may: retain current income level ($35,000) without taking any risk; take fair bet with 50-50 chance of winning or losing $5,000; take fair bet with a 50-50 chance of winning or losing $15,000.

10. 10 U Income (thousands of dollars) 03540503020 Utility FIGURE 5-1: Risk Aversion 33

11. 11 A graphical Analysis of Risk Aversion Current $35,000 provides utility of U 3. Utility of $5,000 bet is average of utility of $40,000 (if player wins) and utility of $30,000 (if player loses). Average utility is U 2 < U 3. The utility (U 1 < U 2 ) of the $15,000 bet is average of utility of winning ($50,000) and losing ($20,000).

12. 12 U U3U3 U2U2 U1U1 Income (thousands of dollars) 03540503020 Utility FIGURE 5-1: Risk Aversion 33

13. 13 Willingness to Pay to Avoid Risk With risk aversion and equal expected values ($35,000 for three options in Figure 5-1), contender will prefer risk- free incomes to risky incomes that offer less utility. Figure 5-1,risk-free income of $33,000 provides same utility as $5,000 gamble. Player would pay up to $2,000 to avoid taking risk.

14. 14 Methods of Reducing Risk: Insurance Figure 5-2 shows motive for buying insurance. Assume that during next year person with $35,000 current income faces 50 percent chance of incurring $15,000 in unexpected medical bills. Without insurance, person’s utility would be U 1 --utility of average of $20,000 and $35,000.

15. 15 U U1U1 Income (thousands of dollars) 025 35 20 Utility FIGURE 5-2: Insurance Reduces Risk

16. 16 Fair Insurance Fair insurance: premium equals expected value of loss. Figure 5-2, fair insurance would cost $7,500--expected value of what insurance companies would have to pay each year in health claims. Would guarantee income of $27,500--yield utility of U 2

17. 17 U U2U2 U1U1 Income (thousands of dollars) 027.5253520 Utility FIGURE 15.2: Insurance Reduces Risk

18. 18 Unfair Insurance Since insurance companies have costs beyond paying benefits, they can not sell insurance at actuarially fair premiums. Figure 5-2: client would be willing to pay up to $10,000 for health insurance since $25,000 of risk-free income yields as much utility (U 1 ) as going without any insurance. $12,000 premium would reduce utility to U 0.

19. 19 U U2U2 U1U1 U0U0 Income (thousands of dollars) 027.525233520 Utility FIGURE 5-2: Insurance Reduces Risk

20. 20 Uninsurable Risks Some risks so unique or difficult to evaluate that insurers unable to set premium rates--risks become uninsurable. If events so infrequent or unpredictable (such as wars, “Acts of God” etc.) insurers have no basis for establishing premiums.

21. 21 Methods of Reducing Risk: Diversification Diversification: economic version of “Don’t put all your eggs in one basket.” Diversification spreads risk among several options rather than choosing only one. Suitably spreading risk may increase utility above that obtain by a single transaction.

22. 22 Diversification Figure 5-3 shows income utility for individual with current income of $35,000 who must invest $15,000 in risky assets. Assume only two such assets: shares of company A or company B stock Each company’s stock costs $1, but value will increase to $2 if company does well during next year.

23. 23 Diversification If either company does poorly, its stock will be worthless. Each company has a 50-50 chance of doing well. If A and B are unrelated to one another, holding both stocks will reduce investor’s risks.

24. 24 Diversification Investing in 15,000 shares of company A yields a 50 percent chance of having $50,000 and a 50 percent chance of having $20,000. Yields a utility level of U 1. If person invests in 7,500 shares of each company, faces four possible outcomes shown in Table 5-1.

25. 25 U U1U1 Income (thousands of dollars) 0355020 Utility FIGURE 5-3: Diversification Reduces Risk

26. 26 TABLE 5-1: Possible Outcomes from Investing in Two Companies

27. 27 Diversification Each of four outcomes is equally likely; with half of cases, investor ends up with original $35,000. Diversification strategy, while it still has an expected value of $35,000, has less risk. Figure 5-3, 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 U 2 > U 1.

28. 28 U D E C U2U2 U1U1 Income (thousands of dollars) 0355020 Utility FIGURE 5-3: Diversification Reduces Risk

29. 29 Attributes of Options Specification of underlying transaction Definition of period during which option may be exercised Price of option

30. 30 Value of Underlying Transaction Affects Option Value Because transaction that underlies an option will occur in the future, underlying transaction’s value subject to many uncertainties. The value of underlying transaction in option has two general dimensions Expected value of transaction Variability of value of transaction.

31. 31 Duration of Option Affects Its Value The longer an option lasts, the more valuable it is. Intuitively, more time you have to take advantage of flexibility an option offers, more likely it is that you will want to do so.

32. 32 Investors’ Market Options Figure 5-4 shows simplified illustration of market options open to a would-be investor in financial assets. Points on figure represent options available. For example, point A represents a risk-free asset such as money in a checking account. Asset B represents relatively risky stock. All other points on Figure 5-4 represent risks and returns associated with assets that investors might buy.

33. 33 Annual return Risk A C Market Line B Figure 5-4: Market Options for Investors

34. 34 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 risk-free asset (A), they can choose any point along the line AC. Market line: shows possible combinations of annual returns and risk that investors can achieve by taking advantage of what market the offers.

35. 35 Choices by Individual Investors The market line in Figure 5-4 provides a constraint on the options that financial markets provide to individual investors. These investors then choose among the available options on the basis of their own attitudes toward risk. This process is illustrated in Figure 5-5.

36. 36 Choices by Individual Investors The figure shows a typical indifference curve for three different types of investors. The three investors illustrated in Figure 5-5 have different attitudes toward risk. Investor I has a very low tolerance for risk. He will opt for mix of investments that include a lot of the risk-free option (point L).

37. 37 Choices by Individual Investors Investor II has a moderate toleration for risk. She will opt for a combination of assets that are reasonably representative of the overall market (M). Finally, investor III is a real speculator. She will accept a risky combination of assets (N) – more risky than the overall market.

38. 38 Annual return Risk A L Market Line Figure 5-5: Choices by Individual Investors M N UIUI U II U III

39. 39 The Economics of Information A Utility-Maximizing Model The basic model is shown in Figure 5-6 where an individual is assumed to face two possible outcomes (sometimes called states of the world), but he or she does not know what outcome will occur. The person’s consumption in the two states is denoted C 1 and C 2, and possible values are recorded in the axes.

40. 40 U1U1 U2U2 D B A E Certainty line C2C2 E C2C2 A C1C1 CE1CE1 CA1CA1 C2C2 FIGURE 5-6: Utility Maximization under Uncertainty

41. 41 A Utility-Maximizing Model At point A, the individual has considerably more consumption in state 1 than in state 2. This person might be willing to give up some consumption in state 1 to consume more in state 2. This might be accomplished by paying an insurance premium in state 1 in order to increase consumption in state 2 (when things go wrong).

42. 42 A Utility-Maximizing Model For example, if the terms at which insurance can be bought are reflected in the slope of the line AE, this person could increase utility from U 1 to U 2 by purchasing complete insurance and moving to point E. Buying complete insurance has allowed this person to obtain C E 1 (which equals C E 2 ) with certainty.

43. 43 U1U1 U2U2 D B A E Certainty line C2C2 E C2C2 A C1C1 C1C1 E C1C1 A C2C2 FIGURE 5-6: Utility Maximization under Uncertainty

44. 44 Balancing the Gains and Costs of Information Another way to improve his or her situation would be to gather additional information. Consumption will decrease in the good outcome but increase in the bad outcome. The issue is whether acquiring information will raise utility above U 1. Point B, for example, represents a utility-improving investment, but point D is a poor investment in information.

45. 45 Information Differences among Economic Actors The level of information that an individual acquires will depend on how much the information costs. There are reasons to believe that information costs may differ significantly among individuals. Sellers or large-scale repeat buyers of a good may have greater access to information than first-time buyers.