Fin 501: Asset PricingLecture Lecture 04: 04: Risk Preferences and Risk Preferences and Expected Expected Utility TheoryUtility Theory• Prof. Markus K. BrunnermeierSlide 04Slide 04--11Fin 501: Asset PricingOiOiRi k P fRi k P fOverview: Overview: Risk PreferencesRisk Preferences1.1. StateState--byby--state dominancestate dominance2.2. Stochastic dominanceStochastic dominance[DD4][DD4]3.3. vNMvNM expected utility theoryexpected utility theorya)a) IntuitionIntuition [L4][L4]b)b)A i ti f d tiA i ti f d tib)b)Axiomatic foundationsAxiomatic foundations [DD3][DD3]4.4. Risk aversion coefficients and Risk aversion coefficients and pportfolio choice ortfolio choice [DD5,L4][DD5,L4]55Prudence coefficient and precautionary savingsPrudence coefficient and precautionary savings[DD5][DD5]5.5.Prudence coefficient and precautionary savings Prudence coefficient and precautionary savings [DD5][DD5]6.6. MeanMean--variance preferencesvariance preferences [L4.6][L4.6]Slide 04Slide 04--22Fin 501: Asset PricingSt tSt tbbtt D itt D iStateState--byby--state Dominancestate Dominance- State-by-state dominance Ä incomplete rankingypg-«riskier »Table 2.1 Asset Payoffs ($) t = 0 t = 1 Cost at t=0 Value at t=1 ½π1 = π2 = ½ s = 1 s = 2 investment 1 itt2- 1000 10001050 5001200 1600investment 2 investment 3 -1000- 1000 500 1050 16001600 Slide 04Slide 04--33- investment 3 state by state dominates 1.Fin 501: Asset PricingSt tSt tbbtt D i (td)tt D i (td)StateState--byby--state Dominance (ctd.)state Dominance (ctd.)Table 2.2 State Contingent ROR (r ) St t C ti t ROR ( ) State Contingent ROR (r ) s = 1 s = 2 Er σ Investment 1 5% 20% 12.5% 7.5% Investment 2Investment 3 -50% 5% 60%60% 5% 32.5% 55% 27.5% - Investment 1 mean-variance dominates 2- BUT investment 3 does not m-v dominate 1!Slide 04Slide 04--44Fin 501: Asset PricingSt tSt tbbtt D i (td)tt D i (td)StateState--byby--state Dominance (ctd.)state Dominance (ctd.)Table 2.3 State Contingent Rates of Return State Contingent Rates of Return s = 1 s = 2investment 4 investment 5 3% 3% 5% 8% π1 = π2 = ½ E[r4] = 4%; σ4 = 1% E[r]=55%;σ=25%E[r5] = 5.5%; σ5= 2.5%- What is the trade-off between risk and expected return?I4hhihSh i(E[]f)/hi 5Slide 04Slide 04--55-Investment 4 has a higher Sharpe ratio(E[r]-rf)/σthan investment 5for rf= 0.Fin 501: Asset PricingOiOiRi k P fRi k P fOverview: Overview: Risk PreferencesRisk Preferences1.1. StateState--byby--state dominancestate dominance2.2. Stochastic dominanceStochastic dominance[DD4][DD4]3.3. vNMvNM expected utility theoryexpected utility theorya)a) IntuitionIntuition [L4][L4]b)b)A i ti f d tiA i ti f d tib)b)Axiomatic foundationsAxiomatic foundations [DD3][DD3]c)c) Risk aversion coefficientsRisk aversion coefficients [DD4,L4][DD4,L4]44Risk aversion coefficients andRisk aversion coefficients andpportfolio choiceortfolio choice[DD5 L4][DD5 L4]4.4.Risk aversion coefficients and Risk aversion coefficients and pportfolio choice ortfolio choice [DD5,L4][DD5,L4]5.5. Prudence coefficient and precautionary savings Prudence coefficient and precautionary savings [DD5][DD5]6.6.MeanMean--variance preferencesvariance preferences[L4.6][L4.6]Slide 04Slide 04--666.6.MeanMeanvariance preferencesvariance preferences[L4.6][L4.6]Fin 501: Asset PricingSt h ti D iSt h ti D iStochastic DominanceStochastic Dominance Stochastic dominance can be defined independently of the specific trade-offs (between py p(return, risk and other characteristics of probability distributions) represented by an agent's utility fi(“ikff”)function. (“risk-preference-free”) Less “demanding” than state-by-state dominanceSlide 04Slide 04--77Fin 501: Asset PricingSt h ti D iSt h ti D iStochastic DominanceStochastic Dominance Still incomplete orderingpg “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'scharacteristics of probability distributions) represented by an agent s utility function. (“risk-preference-free”) Next Section:  Complete preference ordering and utility representationsHkPid lhih b kdSlide 04Slide 04--8810:59 Lecture Risk AversionHomework:Provide an example which can be ranked according to FSD , but not according to state dominance.Fin 501: Asset Pricing Table 3-1 Sample Investment AlternativesStates of nature 1 2 3Payoffs 10 100 2000Proba Z1.4 .6 0Proba Z2.4 .4 .2EZ1= 64, 1zσ= 441zEZ2 = 444, 2zσ= 77910F1obabilityPr07 0.8 0.9 1.0F20.4 0.5 0.6 0.7F1 and F2010.4 0.2 0.3Slide 04Slide 04--990 10 100 20000.1PayoffFin 501: Asset PricingFi t O d St h ti D iFi t O d St h ti D iDefinition 3 1:LetF(x)andF(x)respectivelyFirst Order Stochastic DominanceFirst 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 (pyff), fgenerality 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)HkPid lhih b kdSlide 04Slide 04--1010Homework:Provide an example which can be ranked according to FSD , but not according to state dominance.Fin 501: Asset PricingFi t O d St h ti D iFi t O d St h ti D i1First Order Stochastic DominanceFirst Order Stochastic Dominance0.70.80.91FAFB0.40.50.600.10.20.3X001234567891011121314Slide 04Slide 04--1111Fin 501: Asset Pricing Table 3-2 Two Independent InvestmentsInvestment 3 Investment 4Payoff Prob. Payoff Prob.40251033140.2510.335 0.50 6 0.3312 0.25 8 0.330.70.80.910.30.40.50.6investment 400.10.2012345678910111213investment 3Slide 04Slide 04--1212Figure 3-6 Second Order Stochastic Dominance IllustratedFin 501: Asset PricingSdOdSthtiDiSdOdSthtiDiDefinition 3 2:Letbe two)x~(F)x~(FSecond Order Stochastic DominanceSecond Order Stochastic DominanceDefinition 3.2: Let , ,be two cumulative probability distribution for random payoffs inWe say that)x(FA)x(FB[]b,a)x~(FArandom payoffs in . We say that second order stochastically dominates(SSD)if and only if for anyx:[]b,a)x(FA)x~(FB(SSD)if and only if for anyx :)x(FB[]0dt(t)F(t)Fx≥∫(with strict inequality for some meaningful []0dt (t)F-(t)F AB-≥∫∞Slide 04Slide 04--1313interval of