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Lecture 04: Risk Preferences and Risk Preferences and Expected Fin 501: Asset Pricing Lecture 04: Risk Preferences and Expected Utility Theory • Prof. Markus K. Brunnermeier Slide 0404--11 Fin 501: Asset Pricing OiOverview: Risk P re ferences 1. StateState--byby--statestate dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Ax ioma tic f ound ati ons [DD3] 4. Risk aversion coefficients and portfolioportfolio choice [DD5,L4] 5. Prudence coefficient and precautionary savings [DD5] 6. MeanMean--variancevariance preferences [L4.6] Slide 0404--22 Fin 501: Asset Pricing Sta tee--byy--sstttate D omi nance - State-by-state dominance Ä incomppglete ranking -«riskier » Table 2.1 Asset Payoffs ($) t = 0 t = 1 Cost at t=0 Value at t=1 π1 = π2 = ½ s = 1 s = 2 investment 1 - 1000 1050 1200 itt2investment 2 - 1000 500 1600 investment 3 - 1000 1050 1600 - investment 3 state by state dominates 1. Slide 0404--33 Fin 501: Asset Pricing Sta tee--byy--sstttate Dom inance (td)(ctd.) Table 2.2 State Contingent ROR (r ) Sta te Con tingen t ROR ( r ) s = 1 s = 2 Er σ Investment 1 5% 20% 12.5% 7.5% Investment 2 -50% 60% 5% 55% Investment 3 5% 60% 32.5% 27.5% - Investment 1 mean-variance dominates 2 - BUT investment 3 does not m-v dominate 1! Slide 0404--44 Fin 501: Asset Pricing Sta tee--byy--sstttate Dom inance (td)(ctd.) Table 2.3 State Contingent Rates of Return State Contingent Rates of Return s = 1 s = 2 investment 4 3% 5% investment 5 3% 8% π1 = π2 = ½ E[r4] = 4%; σ4 = 1% E[r5] =55%;= 5.5%; σ5 =25%= 2.5% - What is the trade-off between risk and expected return? - I4hhihInvestment 4 has a higher Sharpe rat io (E[r]-rf)/σ thihan investment 5 for rf = 0. Slide 0404--55 Fin 501: Asset Pricing OiOverview: Risk P re ferences 1. StateState--byby--statestate dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Ax ioma tic f ound ati ons [DD3] c) Risk aversion coefficients [DD4,L4] 4. Risk aversion coefficients and portfolio choice [DD5, L4] 5. Prudence coefficient and precautionary savings [DD5] 6. Mean-variance preferences [L4.6] Slide 0404--66 Fin 501: Asset Pricing Stoc has tic D omi nance Still incomppglete 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 HkHomework: PidProvide an examp lhihble which can be rank kded according to FSD , but not according to state dominance. Slide 0404--77 Table 3-1 Sample Investment Alternatives Fin 501: Asset Pricing States of nature 1 2 3 Payoffs 10 100 2000 Proba Z1 .4 .6 0 Proba Z2 .4 .4 .2 EZ1 = 64, σ = 44 z1 EZ = 444, σ = 779 2 z 2 Pr obability F1 101.0 0.9 F2 0.8 070.7 0.6 0.5 F1 and F 2 0.4 0.3 0.2 010.1 Payoff 0 10 100 2000 Slide 0404--88 Fin 501: Asset Pricing Firs t O r der Stoc has tic D omi nance Definition 3. 1 : Let FA(x) and FB(x) , respectively, represent the cumulative distribution functions of two random variables ((pyff),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) HkHomework: PidProvide an examp lhihble which can be rank kded according to FSD , but not according to state dominance. Slide 0404--99 Fin 501: Asset Pricing Firs t O r der Stoc has tic D omi nance 1 0.9 0.8 FB 0.7 FA 0.6 0.5 0.4 0.3 0.2 0.1 0 01234567891011121314 X Slide 0404--1010 Fin 501: Asset Pricing Table 3-2 Two Independent Investments Investment 3 Investment 4 Payoff Prob. Payoff Prob. 4 0250.25 1 0330.33 5 0.50 6 0.33 12 0.25 8 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 012345678910111213 Figure 3-6 Second Order Stochastic Dominance Illustrated Slide 0404--1111 Fin 501: Asset Pricing SdOdSthtiDiSecond Order Stochastic Dominance ~ ~ Definition 3 .2: Let F A ( x ) , F B ( x ) , be two cumulative probability distribution for ~ random payoffs in [ a , b ] . We say that F A ( x ) second order stochastically dominates ~ (SSD) FB (x) if and only if for any x:x : x ∫ [ FB (t) - FA (t) ] dt ≥ 0 -∞ (with strict inequality for some meaningful interval of values of t). Slide 0404--1212 Fin 501: Asset Pricing Mean Preserving Spread xB = xA + z (()3.8) where z is independent of xA and has zero mean for normal distributions fxA ( ) fxB ( ) ~xPayoff, μ = = ∫ x fAB(x)dxx∫ f (x)dx Figure 3-7 Mean Preserving Spread Slide 0404--1313 Fin 501: Asset Pricing Mean Preserving Spread & SSD Theorem 3.4 : Let FA(•) and F B (•) be two distribution functions defined on the same state space with identical means. Then the follow statements are equivalent : ~ ~ FA (x) SSD FB (x) ~ ~ FB (x) is a mean p rese rving sp read of FA (x) in the sense of Equation (3.8) above. Slide 0404--1414 Fin 501: Asset Pricing OiOverview: Risk P re ferences 1. StateState--byby--statestate dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Ax ioma tic f ound ati ons [DD3] 4. Risk aversion coefficients and portfolioportfolio choice [DD4,5,L4] 5. Prudence coefficient and precautionary savings [DD5] 6. MeanMean--variancevariance preferences [L4.6] Slide 0404--1515 Fin 501: Asset Pricing A Hypo the tical G ambl e Suppose someone offers you this gamble: "I have a fair coin here. I'll flip it, and if it's tail I pay you $1 and the gamble is over. If it's head, I'll flip again. If it's tail 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'stail s tail." Sounds like a good deal. After all, you can't loose. So here ' s the question: How much are you willing to pay to take this gamble? Slide 0404--1616 Fin 501: Asset Pricing Pl1Proposal 1: EtdVlExpected Value 1 1 0 With probability 1/2 you get $1. (2 ) times 2 1 2 1 With probability 1/4 you get $2 . (2 ) times 2 With probability 1/8 you get $4. 1 3 2 (2 ) times 2 etc. Ê The expected payoff is the sum of these payoffs, weighted with their probabilities, t ∞ so ∞ ⎛ 1 ⎞ 1 ⋅ 2t −1 = ∑ ⎜ ⎟ ∑ 2 = ∞ t =1 ⎝ 2 ⎠ t =1 { 123 probability payoff Slide 0404--1717 Fin 501: Asset Pricing An Infinitely Valuable Gamble? You should pay probability 0.5 everything you own and more to purchase the right 0.4 to take this gamble! 0.3 Yet, in practice, no one is 0.2 prepared to pay such a 0.1 high price. Why? 0 0204060$ EthhthtdEven though the expected With 93% probability payoff is infinite, the we get $8 or less, with distribution of payoffs is 99% probbilibability we get not attractive… $64 or less. Slide 0404--1818 Fin 501: Asset Pricing What Should We Do? How can we decide in a rational fashion about such gambles (or investments)? 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. ] t ∞ ⎛1⎞ expected utility ∑ ⎜ ⎟ ⋅ ln(2t −1) = ln(2) = < ∞ ⎝2⎠ of gamble t =1 { 142 43 probability utility of payoff Ê Bernoulli would have paid at most eln(2) = $2 to participate in this gamble. Slide 0404--1919 Fin 501: Asset Pricing Risk -AiAversion an dCitd Concavity u(x) The shape of the von Neumann Morgenstern (NM) utility function contains a lot of information. u(x1) Consider a fifty-fifty lottery with final wealth of x0 or x1 E{u(x)} u(x0) x0 E[x] x1 x Slide 0404--2020 Fin 501: Asset Pricing Risk --aversaversidion and concavit y u(x) Risk-aversion means that the certainty equivalent is smaller than the expected prize . Ê We conclude that a u(x1) risk-averse vNM u(E[x]) utility function must be concave. E[u(x)] u(x0) x0 E[x] x1 x u-1(E[u(x)]) Slide 0404--2121 Fin 501: Asset Pricing Jensen ’s Inequa lity Theorem 3 .1 (Jensen ’s Inequality): Let g( ) be a concave function on the interval [a,b], and be a random variable such that Prob (x ∈[a,b]) =1 Suppose the expectations E(x) and E[g(x)] exist; then E[g(~x)]≤ g[E(~x)] Furthermore, if g(•) is strictly concave, then the inequality is strict. Slide 0404--2222 Fin 501: Asset Pricing RttifPfRepresentation 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. (c0,c1,…,cS)  (c’0,c’1,…,c’S) ⇔ U(c0,c1,…,cS) > U(c’0,c’1,…,c’S) (preference ordering over lotteries – (S+1)-dimensional spp)ace) Slide 0404--2323 Fin 501: Asset Pricing PfPreferences over PbDitibtiProb. Distributions Consider c0 fixed, c1 is a random variable Preference ordering over probability distributions Let P be a se t o f pro ba bility di st rib uti ons with a fi nit e support over a set X,  a (strict) preference ordering over P, and Define % by p % q if q ¨ p Slide 0404--2424 Fin 501: Asset Pricing S states of the world Set of all possible lotteries Space with S dimensions Can we s implify lif th e utility tilit represent ati on of preferences over lotteries? Space w ith one dimens ion – income We need to assume further axioms Slide 0404--2525 Fin 501: Asset Pricing Expec ted Utility Theory A binary relation that satisfies the following three axioms if and only if there exists a function u(•) such that p  q ⇔ ∑ u(c) p(c) > ∑ u(c) q(c) i.e.
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