Eco 525 Financial Economics I Lecture 02 Risk Preferences and Savings Portfolio Choice Prof Markus K Brunnermeier 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 1 Eco 525 Financial Economics I State by state Dominance State by state dominance incomplete ranking riskier Table 2 1 Asset Payoffs t 0 Cost at t 0 investment 1 investment 2 investment 3 1000 1000 1000 t 1 Value at t 1 1 2 s 1 s 2 1050 1200 500 1600 1050 1600 investment 3 state by state dominates 1 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 2 Eco 525 Financial Economics I State by state Dominance ctd Table 2 2 State Contingent ROR r Investment 1 Investment 2 Investment 3 State Contingent ROR r s 1 s 2 Er 5 20 12 5 7 5 50 60 5 55 5 60 32 5 27 5 Investment 1 mean variance dominates 2 BUT investment 3 does not m v dominate 1 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 3 Eco 525 Financial Economics I State by state Dominance ctd Table 2 3 State Contingent Rates of Return investment 4 investment 5 State Contingent Rates of Return s 1 s 2 3 5 3 8 1 2 E r4 4 4 1 E r5 5 5 5 2 5 What is the trade off between risk and expected return Investment 4 has a higher Sharpe ratio E r rf than investment 5 for rf 0 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 4 Eco 525 Financial Economics I Stochastic Dominance Stochastic dominance can be defined 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 Less demanding than state by state dominance 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 5 Eco 525 Financial Economics I Stochastic Dominance 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 Homework Provide an example which can be ranked according to FSD but not according to state dominance 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 6 Eco 525 Financial Economics I Table 3 1 Sample Investment Alternatives States of nature Payoffs Proba Z 1 Proba Z 2 1 2 10 100 4 6 4 4 EZ 1 64 z 1 44 3 2000 0 2 EZ 2 444 z 2 779 Pr obability F1 1 0 0 9 F2 0 8 0 7 0 6 0 5 F 1 and F 2 0 4 0 3 0 2 0 1 Payoff 0 21 58 Lecture 02 10 100 Risk Preferences Portfolio Choice 2000 Slide 2 7 Eco 525 Financial Economics I First Order Stochastic Dominance Definition 3 1 Let FA x and FB x respectively represent the cumulative distribution functions of two random variables cash payoffs that without loss of generality assume values in the interval a b We say that FA x first order stochastically dominates FSD FB x if and only if for all x a b FA x FB x Homework Provide an example which can be ranked according to FSD but not according to state dominance 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 8 Eco 525 Financial Economics I 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 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 9 Eco 525 Financial Economics I Table 3 2 Two Independent Investments Investment 3 Payoff 4 5 12 Investment 4 Prob 0 25 0 50 0 25 Payoff 1 6 8 Prob 0 33 0 33 0 33 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 Figure 3 6 Second Order Stochastic Dominance Illustrated 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 10 Eco 525 Financial Economics I Second Order Stochastic Dominance Definition 3 2 Let FA x FB x be two cumulative probability distribution for random payoffs in a b We say that FA x second order stochastically dominates SSD FB x if and only if for any x x FB t FA t dt 0 with strict inequality for some meaningful interval of values of t 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 11 Eco 525 Financial Economics I Mean Preserving Spread xB xA z 3 8 where z is independent of xA and has zero mean for normal distributions fA x f B x x fA x dx x f B x dx x Payoff Figure 3 7 Mean Preserving Spread 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 12 Eco 525 Financial Economics I Mean Preserving Spread SSD Theorem 3 4 Let FA and FB be two distribution functions defined on the same state space with identical means Then the follow statements are equivalent x SSD FB x FA x x is a mean preserving spread of FA FB in the sense of Equation 3 8 above 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 13 Eco 525 Financial Economics I Expected Utility Stochastic Dominance Theorem 3 2 Let FA x FB x be two cumulative x a b probability distribution for random payoffs Then FA x FSD FB x if and only if for all non decreasing utility functions U E A U x E B U x 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 14 Eco 525 Financial Economics I Expected Utility Stochastic Dominance Theorem 3 3 Let FA x FB x be two cumulative probability distribution x defined on a b for random payoffs Then FA x SSD FB x if and only if E A U x E B U x for all non decreasing and concave U 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 15 Eco 525 Financial Economics I Arrow Pratt measures of risk aversion and their interpretations absolute risk aversion U Y R A Y U Y relative risk aversion Y U Y R R Y U Y risk tolerance 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 16 Eco 525 Financial Economics I Absolute risk aversion coefficient Y h 1 Y h Y 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 17 Eco 525 Financial Economics I Relative risk aversion coefficient Y 1 1 Y 1 Y Homework Derive this result 21 58 Lecture 02 Risk Preferences Portfolio Choice Slide 2 18 Eco 525 Financial Economics I CARA and CRRA utility functions Constant Absolute RA utility function Constant Relative RA utility function …