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PowerPoint PresentationState-by-state DominanceState-by-state Dominance (ctd.)Slide 4Stochastic DominanceSlide 6Slide 7First Order Stochastic DominanceSlide 9Slide 10Second Order Stochastic DominanceMean Preserving SpreadMean Preserving Spread & SSDRepresentation of PreferencesPreferences over Prob. DistributionsSlide 16Expected Utility TheorySlide 18vNM Expected Utility TheoryIndependence AxiomAllais Paradox – Violation of Independence AxiomSlide 22Slide 23vNM EU TheoremJensen’s InequalitySlide 26Expected Utility & Stochastic DominanceSlide 28Digression: Subjective EU TheoryDigression: Ellsberg ParadoxDigression: Prospect TheoryMeasuring Risk aversionSlide 33Certainty Equivalent and Risk PremiumSlide 35Arrow-Pratt measures of risk aversion and their interpretationsAbsolute risk aversion coefficientRelative risk aversion coefficientCARA and CRRA-utility functionsInvestor ’s Level of Relative Risk AversionRisk aversion and Portfolio AllocationSlide 42Portfolio as wealth changesSlide 44Log utility & Portfolio AllocationPortfolio of risky assets as wealth changesLRT/HARA-utility functionsRisk Aversion and Savings BehaviorPre-cautionary SavingSlide 50Prudence & Pre-cautionary SavingExtra material follows!Joint saving-portfolio problemSlide 54Multi-period SettingMulti-period Portfolio ChoiceDigression: Preference for the timing of uncertainty resolution06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset PricingLecture 03: Risk Preferences and Lecture 03: Risk Preferences and Expected Utility Expected Utility TheoryTheory•Prof. Markus K. Brunnermeier06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset PricingState-by-state DominanceState-by-state Dominance- State-by-state dominance  incomplete ranking- « riskier » Table 2.1 Asset Payoffs ($) t = 0 t = 1 Cost at t=0 Value at t=1 1 = 2 = ½ s = 1 s = 2 investment 1 investment 2 investment 3 - 1000 - 1000 - 1000 1050 500 1050 1200 1600 1600 - investment 3 state by state dominates 1.06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset PricingTable 2.2 State Contingent ROR (r ) State Contingent ROR (r ) s = 1 s = 2 Er Investment 1 Investment 2 Investment 3 5% -50% 5% 20% 60% 60% 12.5% 5% 32.5% 7.5% 55% 27.5% - Investment 1 mean-variance dominates 2- BUT investment 3 does not m-v dominate 1!State-by-state Dominance (ctd.)State-by-state Dominance (ctd.)06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset PricingTable 2.3 State Contingent Rates of Return State Contingent Rates of Return s = 1 s = 2 investment 4 investment 5 3% 3% 5% 8% 1 = 2 = ½ 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 than investment 5State-by-state Dominance (ctd.)State-by-state Dominance (ctd.)06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset Pricing•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 dominanceStochastic DominanceStochastic Dominance06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset PricingStochastic DominanceStochastic 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 representationsHomework: Provide an example which can be ranked according to FSD , but not according to state dominance.06:32 Lecture 03Risk AversionRisk AversionFin 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, 1z= 44EZ2 = 444, 2z= 7790 1 0 1 0 0 2 0 0 00 . 10 . 4 0 . 2 0 . 3 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0F1 a n d F2F1F2PayoffobabilityPr06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset Pricing•Definition 3.1 : Let and , 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 . b,a)~(xFA)~(xFB)~(xFB)~(xFAFirst Order Stochastic DominanceFirst Order Stochastic DominanceHomework: Provide an example which can be ranked according to FSD , but not according to state dominance.06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset PricingX00.10.20.30.40.50.60.70.80.910 1 2 3 4 5 6 7 8 9 10 11 12 13 14FAFBFirst Order Stochastic DominanceFirst Order Stochastic Dominance06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset Pricing00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 910 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3i n v e s t m e n t 3i n v e s t m e n t 4 Table 3-2 Two Independent InvestmentsInvestment 3 Investment 4Payoff Prob. Payoff Prob.4 0.25 1 0.335 0.50 6 0.3312 0.25 8 0.33Figure 3-6 Second Order Stochastic Dominance Illustrated06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset Pricing•Definition 3.2: Let , , be two cumulative probability distribution for random payoffs in . We say that second order stochastically dominates (SSD) if and only if for any x : (with strict inequality for some meaningful interval of values of t). )x~(FA)x~(FB b,a)x~(FA)x~(FB 0 dt (t)F - (t)F ABx-Second Order Stochastic DominanceSecond Order Stochastic Dominance06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset PricingfxAfxB~,xPayoffxfxdxxfxdxABFigure 3-7 Mean Preserving SpreadMean Preserving SpreadMean Preserving Spreadfor normal distributionsxB = xA + z where z is independent of xA06:32 Lecture 03Risk AversionRisk AversionFin 501: Asset Pricing•Theorem 3.4 : Let (•) and (•) be two distribution functions defined on the same


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Princeton FIN 501 - Lecture 3

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