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Prospect Theory

Prospect Theory. Non-Expected Utility Theories. Prospect theory. Almost all models in finance assumes that investors evaluate risk according to The expected utility. However, there are many empirical and experimental Evidences showing people in fact systematically violate the expected utility

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Prospect Theory

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  1. Prospect Theory Non-Expected Utility Theories

  2. Prospect theory Almost all models in finance assumes that investors evaluate risk according to The expected utility. However, there are many empirical and experimental Evidences showing people in fact systematically violate the expected utility Theory. In our previous lecture, we have discussed these experimental Evidences. In light of the violations, some non-expected utility theories are proposed to explain these patterns. Among them, prospect theory is the most famous one.

  3. Prospect theory (Kahneman and Tversky1979) • Consider the gamble (x,p; y, 1-p). Under the expected utility, the decision maker assigns a value using: pu(x)+(1-p)u(y) • Under prospect theory, the decision maker assigns a value using: p(p)v(x)+p(1-p)v(y) where p is the probability weighting function called decision weight v is the value function. Kahneman, Daniel, and Amos Tversky. "Prospect Theory: An Analysis of Decision Under Risk". Econometrica. XLVII (1979): 263-291.

  4. Prospect Theory: Value Function Consider the following problems: Problem 11: You have been given 1000. You are now asked to choose between A: (1,000, 0.5; 0, 0.5) and B: (500) Problem 12:You have been given 2000. You are now asked to choose between A: (-1,000, 0.5; 0, 0.5) and B: (-500) It is observed that most people would choose B in problem 11 and A in problem 12. Thus, It suggests that subjects did not integrate the bonus with the lotteries. However, under expected utility, it is predicted they should make the same choices . For example, the same utility is assigned to a wealth of $100000, regardless of whether it was reached from a prior wealth of $95000 or $105000. This suggests that an alternative theory should be reference point dependent and concerns with the change relative to the reference point rather than the absolute value of the outcome.

  5. Prospect theory: Value Function The value function is: Define on deviations from the reference point Generally concave for gains and commonly convex for losses Steeper for losses than for gains

  6. Prospect theory: Value Function Consider the following problems. Problem 13 A.(6000, .25; 0, 0.75) or B. (4000, 0.25; 2000, 0.25; 0, 0.5) Problem 14 C. (-6000, 0.25; 0, 0.75) or D. (-4000, 0.25; -2000, 0.25; 0, 0.5) It is found that most people would choose B in problem 13, and C in problem 14. Applying prospect theory, this implies: p(0.25)v(6000) < p(0.25)[v(4000)+v(2000)] Hence, v(6000) < v(4000)+v(2000). Thus, this is inline with the hypothesis that the value function is concave under gain. p(0.25)v(-6000) > p(0.25)[v(-4000)+v(-2000)] Hence, v(-6000) > v(-4000)+v(-2000). Thus, this is inline with the hypothesis that the value function is convex under loss.

  7. Another Example Consider the following problems. Decision i. A.(240) or B. (1000, 0.25; 0, 0.75) Decision ii. C. (-750) or D. (-1000, 0.75; 0. 0, 0.25) Most people chose A in i (i.e., being risk averse), and D in ii (i.e., being risk loving). Such behavior is inconsistent with expected utility theory. Prospect theory, however, can account for this behavior.

  8. Prospect theory: Value Function Another feature of the value function is that losses loom larger than gains. Note that most people will find the gamble (x, 0.5; -x, 0.5) unattractive. Moreover, the aversiveness of symmetric fair bets generally increases with the size of the Stake. That is, if x>y>=0, then (y, 0.5; -y, 0.5) is preferred to (x, 0.5; -x, 0.5) . Applying the prospect theory, we have v(y) +v(-y) > v(x) +v(-x) and v(-y) – v(-x) > v(x) - v(y) Setting y=0 yields v(x) < -v(-x), and letting y approach x yields v’(x) < v’(-x), provided v’ exists. Thus, the value function for losses is steeper than the value function for gains.

  9. Prospect theory: Probability Weighting Function • is an increasing function of p, with p (0)=0 and p (1)=1. • Properties of weighting function for small probabilities. • Consider the following choice problem • Problem 14. (5000, 0.001) or (5) • Problem 14’. (-5000, 0.001) or (-5) • It is observed that most people will choose the lottery in problem 14. On the other hand, • most people will choose to pay $5 in problem 14’. • Applying the prospect theory, choosing lottery in problem 1 implies • (0.001)v(5000)>v(5) • (0.001)>v(5)/v(5000)>0.001, assuming the value function for gain is concave.

  10. Choosing -% in problem 2 implies v(-5)> p(0.001)v(-5000) p(0.001)> v(-5)>/v(-5000)>0.001, assuming the value function is convex under losses. Thus, very low probabilities are generally over-weighted, i.e., p(p)>p for low probabilities.

  11. Another example Choose between: Decision (i) A. (6000, 0.45; 0, 0.55) and B. (3000, 0.9; 0, 0.1) Decision (ii) A. (6000, 0.001; 0, 0.999) and B. (3000, 0.002; 0, 0.998) It is found that 86% of subjects choose B (risk averse) in decision i and 73% choose A (risk taking) in decision ii. This implies overweighting is greater for 0.001 than for 0.002, suggesting overweighting is greatest at the lowest probabilities.

  12. Certainty Effect Choose between: Decision (i) A. (4000, 0.80; 0, 0.2) and B. (3000) Decision (ii) A. (4000, 0.20; 0, 0.8) and B. (3000, 0.25; 0, 0.75) It is found that 80% of subjects choose B in decision i and 65% choose A in decision ii. Decision ii is the same as decision i except that decision that probabilities are multiplied by 0.25. I appears that lowering probability from 100% to 25% has a larger effect than lowering probability from 80% to 20%. This is an example of the common ratio effect.

  13. KT aruge that people value what is certain relative to what is merely probability. • That is, people overweight certain outcomes. • This phenomenon is called the certainty effect.

  14. Prospect theory: Probability Weighting Function

  15. Framing

  16. Imagine that the United States is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the diseases have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows: Program A: 200 people will be saved. Program B: 1/3 probability that 600 people will be saved, 2/3 probability that no people will be saved. Which program will you choose?

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