Behavioral Finance 行为金融学

1. Behavioral Finance行为金融学 Expected Utility Theory and Experimental Evidences Li King King 李景景 SABS 2019 Spring

2. Objectives • Expected Utility Theory • Concept of Risk Attitude

3. Modern finance models are based on the models of economics, and neoclassical Economics is the dominant paradigm.

4. Neoclassical economics • People have rational preferences across possible outcomes or states of nature. • People maximize utility and firms maximize profits. • People make independent decisions based on all relevant information.

5. Preference • Suppose a person needs to choose between x and y. • The symbol means on choice is strictly preferred to another • means indifference • means weak preference

6. Rational Preferences • Two key assumptions about preferences: • Completeness (ordering) • A person can compare all possible choices and assess preference or indifference. • Transitivity • If y and z, then z

7. Utility Function • Utility theory is used to describe preferences. • Utility function, denoted as .), assign numbers to possible outcomes so that preferred choices receive higher numbers. • Can think utility as satisfaction received from a particular outcome. • Normally an outcome is characterized by a “bundle” of goods. • Example: Utility over goods: u(2 bread, 1 water) > u(1 bread, 2 water) • Example: Utility over money: u(w2) > u(w1) if w2 > w1

8. Properties of Utility Function • Ordering of outcomes by a utility function is important, the actual number assigned is immaterial. • The utility function is ordinal (i.e., order-preserving) but not cardinal (which would mean the exact utility value matters.)

9. Utility Maximization • To arrive at her optimal choice, an individual considers all possible bundles of goods that satisfy her budget constraint, and then choose the bundle that maximizes her utility.

10. Utility function (u(w) = ln(w))over wealth Consider a single good w example, with the utility function u(w)=ln(w). Note that the slop gets flatter as wealth increases.

11. Expected utility theory • Developed by John von Neumann and Oskar Morgenstern. • To describe rational behaviour when facing risk. • Distinction between risk & uncertainty: • Risk is when you know what the outcomes could be, and can assign probabilities • Uncertainty is when you can’tassign probabilities; or you can’tcome up with a list of possible outcomes

12. Wealth outcomes • Say there are a given number of states of the world: • A. rain or sun • B. cold or warm • Leading to 4 states: e.g., rain and cold • And individuals can assign probabilities to each of these states: • Probability of rain+cold is .1, etc. • Say income (or wealth) level can be assigned to each state of world. Think of an ice cream vendor: • Rain+cold: $100/day • Sun+warm: $500/day

13. Prospects • A prospect is defined as a series of wealth or income levels and associated probabilities. • Example: • $500 with probability .8 • $2,000 with probability .2 • P1(0.8, 500; 0.2, 2000) • When 2nd option is zero: • P2(0.8, 500; 0.2, 0) • Expected utility theory comes from a series of assumptions (axioms) on these prospects.

14. Expected utility • Say one has to choose between two prospects. • Based on assumptions such as ordering and transitivity (and others), it can be shown that when such choices over risky prospects are to be made, people should act as if they are maximizing expected utility: U(P) = pr A * u(wA) + (1-pr A) * u(wB) • Can generalize to more than two outcomes: U(P) = pr A * u(wA) + pr B * u(wB) + pr C * u(wC)

15. Why not use the expected value? St Petersburg’s paradox (1713) A fair coin will be tossed until a head appears; if the first head appears on the nth toss, then the payoff is 2n ducats. How much would you pay to play this game? The expected value of the gamble= ∑(2n)x(1/2)n=1+1+…=∞ Bernouilli (1738): utility depends on wealth and there is diminishing marginal utility, e.g., u(w)=log(w) ∑log(2n)x(1/2)n=log4

16. Expected utility example • u(w) = w.5 • Prospects: • P5(0.5, 1000; 0.5, 500) • P6(0.6, 1200; 0.4, 300) • For P5: U(P5) = .5 * 1000.5 + .5 * 500.5 = 26.99 • For P6: U(P6) = .6 * 1200.5 + .4 * 300.5 = 27.71 • So P6 P5.

17. Risk Attitudes • Risk Averse U(E(P))>U(P) • Risk Seeking U(E(P))<U(P) • Risk Neutral U(E(P))=U(P)

18. Risk aversion assumption • This comes from frequent observation that most people most of the time are not willing to accept a fair gamble: • Would you be willing to bet me $100 that you can predict a coin flip? • Most would say no. • And if one of you says yes, I will say no, since I am risk averse. • Risk aversion implies concavity.

19. Expected utility of a prospect • Consider prospect P7: • P7(0.4, 50,000; 0.6, 1,000,000) • Use expected utility formula: U(P7) = 0.40u(50,000) + 0.60u(1,000,000) • Using logarithmic utility function, we have: U(P7) = 0.40(1.6094) + 0.60(4.6052) = 3.4069 • Graph also shows utility of exp. value of prospect: E(w)=0.40(50,000) + 0.60(1,000,000)=$620,000 u(E(w)) = ln(62) = 4.1271 • If Risk Averse: U(E(P))>U(P)

20. Expected utility on graph for a Risk-Averse Individual

21. Certainty equivalents • Certainty equivalent is defined as that wealth level which leads decision-maker to be indifferent between a particular prospect and a certain wealth level. • We need to solve for w below: U(P7) = 3.4069 = u(w) • Solution is w = 30.17

22. Problems with expected utility theory • A number of violations of expected utility have been discovered. • Most famous is Allais paradox. • Alternative theories have been developed which seek to account for these violations. • Best-known is prospect theory of Daniel Kahneman and Amos Tversky.

23. Allais Paradox Nobel Prize Winner in 1988

24. A thought experiment Q1. Q2.

25. Typical choice pattern • Choose A in Q1, and B in Q2. • However such behavior cannot be explained by the expected utility theory, because: • Choosing A in Q1 implies • U(1M) > 0.01 U(0)+0.89U(1M) + 0.1U(5M) • 0.11U(1m)>0.01U(0)+0.1U(5M) • Choosing B in Q2 implies • 0.9U(0)+0.1U(5M)>0.89U(0)+0.11U(1M) • 0.01U(0)+0.1U(5M)>0.11U(1M) • Contradiction

26. Allais Paradox Nobel Prize Winner in 1988

27. Probability weighting • The result of the Allais Paradox experiment suggest that people may weight lotteries differently as assumed by expected theory. • For example, people prefer sure outcome (preference for certainty) • People may also prefer lotteries with “super high” payoff even the probability of winning is very low (long-shot bias)

28. Application: Implications for Lotto design Source: Cook and Clotfelter (1993, AER)

29. Evidence of long-shot bias Source: Snowberg and Wolfers (2010, JPE) Explaining the Favorite-Longshot Bias: Is it Risk-Love or Misperceptions

31. Implications for business strategies • Investment opportunities • Contract design

32. An Example: An Experiment on Illusion of Control

33. What is illusion of control? • Langer (1975) finds that individuals have a higher valuation of lottery tickets if they can choose their own numbers than when they are assigned random numbers. The author attributes the difference in valuation to illusion of control, which refers to the belief that the probability of winning is higher when one can choose the numbers oneself. • This observation represents a challenge for the validity of expected utility theory (von Neumann and Morgenstern 1944), which predicts that an individual should be indifferent between these methods because the probabilities of winning are objectively the same.

34. The paper aims to discriminate against the following theories on explaining preference towards control • Expected utility theory (von Neumann and Morgenstern 1944) • an individual should be indifferent between these methods because the probabilities of winning are objectively the same. • Source preference theory • Tversky and Wakker 1995; Chew and Sagi 2008; Abdellaoui et al. 2011 • An individual is said to exhibit source preference if she prefers one source of uncertainty (e.g., choosing numbers herself) to another (e.g., numbers randomly generated by the computer), even when she believes the probability of winning is equally likely. • Subjective expected utility theory (Savage 1954) • Illusion of control

35. Related literature • There is no existing study which shows that some economic agents prefer not to control the process, or have preference for randomization, instead of the more conventional preference for control. • Charness and Gneezy (2010) investigate whether individuals exhibit illusion of control and if such illusion influences the level of investment in a risky gamble.

36. Charness and Gneezy (2010) • The authors find that subjects do exhibit illusion of control (prefer to roll the die themselves), while such illusion does not influence the level of investments. However, they assume that the fact that subjects prefer to roll the die is due to illusion of control. • Since they do not elicit subjects’ probability belief, it remains unclear if individuals prefer to control because of illusion of control or because of source preference.

38. Experiment 1: control vs. no control • Subjects are endowed with 10,000 points (1,000 points=0.5 euro) each, are asked how much they will allocate to a risky gamble, and are required to choose between two different methods of control when picking three numbers to bet on. • The outcome of the gamble depends on which ball is drawn from an urn that contains 10 balls numbered from 1 to 10. Participants win 2.5 times of the amount bet if the ball drawn is one of the three numbers chosen.

40. Experiment 1a: Free to choose

41. Experiment 1b: Pay to gain control

42. Preference for randomization