FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (1) Markus K. Brunnermeier LECTURE 4: RISK PREFERENCES & EXPECTED UTILITY THEORY FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (2) specify observe/specify Preferences & existing Technology Asset Prices • evolution of states NAC/LOOP NAC/LOOP • risk preferences • aggregation State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing LOOP derive derive Asset Prices Price for (new) asset Only works as long as market completeness doesn’t change FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (3) Overview: Risk Preferences 1. State-by-state dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Axiomatic foundations [DD3] 4. Risk aversion coefficients and portfolio choice [DD5,L4] 5. Uncertainty/ambiguity aversion 6. Prudence coefficient and precautionary savings [DD5] 7. Mean-variance preferences [L4.6] FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (4) State-by-state Dominance - State-by-state dominance is an incomplete ranking 풕 = ퟎ 풕 = ퟏ Cost Payoff 휋1 = 휋2 = 1/2 푠 = 1 푠 = 2 Investment 1 -1000 1050 1200 Investment 2 -1000 500 1600 Investment 3 -1000 1050 1600 - Investment 3 state-by-state dominates investment 1 - Recall: 푦, 푥 ∈ ℝ푆 . 푦 ≥ 푥 ⇔ 푦푠 ≥ 푥푠 for each 푠 = 1, … , 푆 . 푦 > 푥 ⇔ 푦 ≥ 푥, 푦 ≠ 푥 . 푦 ≫ 푥 ⇔ 푦푠 > 푥푠 for each 푠 = 1, … , 푆 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (5) State-by-state Dominance (ctd.) 풕 = ퟎ 풕 = ퟏ Cost Return 휋1 = 휋2 = 1/2 E[Return] 휎 푠 = 1 푠 = 2 Investment 1 -1000 + 5% + 20% 12.5% 7.5% Investment 2 -1000 - 50% + 60% 5% 55% Investment 3 -1000 + 5% + 60% 32.5% 27.5% • Investment 1 mean-variance dominates 2 • But, investment 3 does not mean-variance dominate 1 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (6) State-by-state Dominance (ctd.) 풕 = ퟎ 풕 = ퟏ Cost Return 휋1 = 휋2 = 1/2 E[Return] 휎 푠 = 1 푠 = 2 Investment 4 -1000 + 3% + 5% 4.0% 1.0% Investment 5 -1000 + 3% + 8% 5.5% 2.5% • What is the trade-off between risk and expected return? 퐸 푟 −푟 • Investment 4 has a higher Sharpe ratio 푓 than investment 5 for 푟 = 0 푓 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (7) Overview: Risk Preferences 1. State-by-state dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Axiomatic foundations [DD3] 4. Risk aversion coefficients and portfolio choice [DD5,L4] 5. Prudence coefficient and precautionary savings [DD5] 6. Mean-variance preferences [L4.6] FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (8) Stochastic Dominance • No state-space – probabilities are not assigned specific states . Only applicable for final payoff gamble • Not for stocks/lotteries that form a portfolio (whose payoff is final) . Random variables before introduction of (Ω, ℱ, 푃) • Still incomplete ordering . “More complete” than state-by-state ordering . State-by-state dominance ⇒ stochastic dominance . Risk preference not needed for ranking! • independently of the specific trade-offs (between return, risk and other characteristics of probability distributions) represented by an agent's utility function. (“risk-preference-free”) • Next Section: . Complete preference ordering and utility representations FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (9) From payoffs per state to probability state 풔ퟏ 풔ퟐ 풔ퟑ 풔ퟒ 풔ퟓ probability 휋1 휋2 휋3 휋4 휋5 Payoff x 10 10 20 20 20 Payoff y 10 20 20 20 30 Expressed in “probability lotteries” – only useful for final payoffs (since some cross correlation information ins lost) payoff 10 20 30 Prob x 푝10 = 휋1 + 휋2 푝20 = 휋3 + 휋4 + 휋5 푝30 = 0 Prob y 푞10 = 휋1 푞20 = 휋2 + 휋3 + 휋4 푞30 = 휋5 푆 Preference 푥 ≻ 푦 ∈ ℝ expressed in probabilities 푝푥 ≻ 푞푦 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (10) Expected Standard Payoff Deviation Payoffs 10 100 2000 Probability 1 .4 .6 0 64 44 Probability 2 .4 .4 .2 444 779 Pr obability F1 1.0 0.9 F2 0.8 0.7 0.6 0.5 F1 and F 2 0.4 0.3 Note: Payoff of $10 is 0.2 equally likely, but can be 0.1 in different states of the world Payoff 0 10 100 2000 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (11) First Order Stochastic Dominance • Definition: Let 퐹퐴 푥 , 퐹퐵 푥 , respectively, represent the cumulative distribution functions of two random variables (cash payoffs) that, without loss of generality assume values in the interval 푎, 푏 . We say that 퐹퐴 푥 first order stochastically dominates (FSD) 퐹퐵 푥 if and only if for all 푥 ∈ 푎, 푏 퐹퐴 푥 ≤ 퐹퐵 푥 Homework: Provide an example which can be ranked according to FSD, but not according to state dominance. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (12) First Order Stochastic Dominance 1 0.9 0.8 FB 0.7 FA 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 X FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (13) Payoff 1 4 5 6 8 12 Probability 3 0 .25 0.50 0 0 .25 Probability 4 .33 0 0 .33 .33 0 1 0.9 0.8 0.7 0.6 investment 4 0.5 0.4 0.3 0.2 investment 3 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 CDFs of investment 3 and 4 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (14) Second Order Stochastic Dominance • Definition: Let 퐹퐴 푥 , 퐹퐵 푥 be two cumulative probability distribution for random payoffs in 푎, 푏 . We say that 퐹퐴 푥 second order stochastically dominates (SSD) 퐹퐵 푥 if and only if for any 푥 ∈ 푎, 푏 푥 퐹퐵 푡 − 퐹퐴 푡 푑푡 ≥ 0 −∞ (with strict inequality for some meaningful interval of values of t). FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (15) Mean Preserving Spread 푥퐵 = 푥퐴 + 푧 (where 푧 is independent and has zero mean) Mean Preserving Spread: f x (for normal distributions) A fB x ~x, Payoff x fA xdx x fBxdx FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (16) Mean Preserving Spread & SSD • Theorem: Let 퐹퐴 푥 and 퐹퐵 푥 be two distribution functions defined on the same state space with identical means. Then the following statements are equivalent : . 퐹퐴 푥 SSD 퐹퐵 푥 . 퐹퐵 푥 is a mean-preserving spread of 퐹퐴 푥 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (17) Overview: Risk Preferences 1. State-by-state dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Axiomatic foundations [DD3] 4. Risk aversion coefficients and portfolio choice [DD5,L4] 5. Prudence coefficient and precautionary savings [DD5] 6. Mean-variance preferences [L4.6] FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (18) A Hypothetical Gamble • Suppose someone offers you this gamble: . "I have a fair coin here. I'll flip it, and if it's tails I pay you $1 and the gamble is over. If it's heads, I'll flip again. If it's tails then, I pay you $2, if not I'll flip again. With every round, I double the amount I will pay to you if it turns up tails." • Sounds like a good deal. After all, you can't lose. So here's the question: . How much are you willing to pay to take this gamble? FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (19) Proposal 1: Expected Value 1 1 1 • With probability you get $1, × 20 2 2 1 1 2 • With probability you get $2, × 21 4 2 1 1 3 • With probability you get $4, × 22 8 2 • … . The expected payoff is given by the sum of all these terms, i.e. ∞ 푡 ∞ 1 1 × 2푡−1 = = ∞ 2 2 푡=1 푡=1 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (20) St. Petersburg Paradox • You should pay everything you own and more to purchase the 0.5 right to take this gamble! • Yet, in practice, no one is 0.4 prepared to pay such a high price. Why? 0.3 • Even though the expected 0.2 payoff is infinite, the distribution of payoffs is not 0.1 attractive… 0 . with 93% probability 0 20 40 60 we get $8 or less; . with 99% probability we get $64 or less FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (21) Proposal 2 • Bernoulli suggests that large gains should be weighted less. He suggests to use the natural logarithm. [Cremer - another great mathematician of the time - suggests the square root.] ∞ 푡 1 × ln 2푡−1 = ln 2 < ∞ 2 푡=1 According to this Bernoulli would pay at most 푒ln 2 = 2 to participate in this gamble FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (22) Representation of Preferences A preference ordering is (i) complete, (ii) transitive, (iii) continuous and [(iv) relatively stable] can be represented by a utility function, i.e. ′ ′ ′ 푐0, 푐1, … , 푐푆 ≻ 푐0, 푐1, … , 푐푆 ′ ′ ′ ⇔ 푈 푐0, 푐1, … , 푐푆 > 푈 푐0, 푐1, … , 푐푆 (preference ordering over lotteries – (푆 + 1)-dimensional space) FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (23) Indifference curves in ℝ2 (for 푆 = 2) 푐2 흅 푧 45° 푐1 푧 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (24) Preferences over Prob. Distributions • Consider 푐0 fixed, 푐1 is a random variable • Preference ordering over probability distributions • Let . 푃 be a set of probability distributions with a finite support over a set 푋, . ≽ preference ordering over 푃 (that is, a subset of 푃 × 푃) FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (25) Prob.