Asset Pricing Chapter IV. Measuring Risk and Risk Aversion June 20, - PowerPoint PPT Presentation

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence 4.5 Assessing an Investor’s Level of Relative Risk Aversion 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Asset Pricing Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Utility function 4.5 Assessing an Investor’s Level of Relative Risk Aversion Indifference Curves 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Measuring Risk Aversion U ( Y ) tangent lines U ( Y + h ) U [0.5( Y + h ) + 0.5( Y – h )] 0.5 U ( Y + h ) + 0.5 U ( Y – h ) U ( Y – h ) Y – h Y Y + h Y Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Utility function 4.5 Assessing an Investor’s Level of Relative Risk Aversion Indifference Curves 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Indifference Curves State 2� Consumption I 1 I 2 c * 2 (c* 2 + c 2 ) /2 c 2 EU ( c) = k 2 EU ( c) = k 1 c * (c * 1 + c 1 )/ 2 c 1 State 1� 1 Consumption Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Absolute Risk Aversion and the Odds of a Bet 4.5 Assessing an Investor’s Level of Relative Risk Aversion Relative Risk Aversion in Relation to the Odds of a Bet 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Arrow-Pratt measures of risk aversion and their interpretations (i) absolute risk aversion = − U ′′ ( Y ) U ′ ( Y ) ≡ R A ( Y ) (ii) relative risk aversion = − YU ′′ ( Y ) U ′ ( Y ) ≡ R R ( Y ) . Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Absolute Risk Aversion and the Odds of a Bet 4.5 Assessing an Investor’s Level of Relative Risk Aversion Relative Risk Aversion in Relation to the Odds of a Bet 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Absolute risk aversion = − U ′′ ( Y ) U ′ ( Y ) ≡ R A ( Y ) π ( Y , h ) ∼ = 1 / 2 + ( 1 / 4 ) hR A ( Y ) , (1) Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Absolute Risk Aversion and the Odds of a Bet 4.5 Assessing an Investor’s Level of Relative Risk Aversion Relative Risk Aversion in Relation to the Odds of a Bet 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Relative risk aversion = − YU ′′ ( Y ) U ′ ( Y ) ≡ R R ( Y ) . = 1 2 + 1 π ( Y , θ ) ∼ 4 θ R R ( Y ) . (2) Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Jensen’s Inequality 4.5 Assessing an Investor’s Level of Relative Risk Aversion Certainty Equivalent 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts 4.4 Risk Premium and Certainty Equivalence Theorem ((4.1) Jensen’s Inequality) Let g ( ) be a concave function on the interval (a , b), and ˜ x be a random variable such that Prob { ˜ x ∈ ( a , b ) } = 1 . Suppose the expectations E (˜ x ) and Eg (˜ x ) exist; then E [ g (˜ x )] ≤ g [ E (˜ x )] . Furthermore, if g ( ) is strictly concave and Prob { ˜ x = E (˜ x ) } � = 1 , then the inequality is strict. Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Jensen’s Inequality 4.5 Assessing an Investor’s Level of Relative Risk Aversion Certainty Equivalent 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts EU ( Y + � U ( Y + CE ( Y , � Z ) = Z )) (3) U ( Y + E ˜ Z − Π( Y , ˜ (4) = Z )) Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence Jensen’s Inequality 4.5 Assessing an Investor’s Level of Relative Risk Aversion Certainty Equivalent 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Certainty Equivalent and Risk Premium: An illustration U(Y) U(Y 0 + Z 2 ) ~ U(Y 0 + E(Z)) ~ EU(Y 0 + Z) U(Y 0 + Z 1 ) ~ CE(Z) P ~ ~ Y 0 Y 0 + Z 1 CE(Y 0 + Z) Y 0 + E(Z) Y 0 + Z 2 Y Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence 4.5 Assessing an Investor’s Level of Relative Risk Aversion 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts 4.5 Assessing an Investor’s Level of Relative Risk Aversion ( Y + CE ) 1 − γ 1 2 ( Y + 50 , 000 ) 1 − γ 1 2 ( Y + 100 , 000 ) 1 − γ = + (5) 1 − γ 1 − γ 1 − γ Assuming zero initial wealth ( Y = 0), we obtain the following sample results (clearly, CE > 50,000): γ = 0 CE = 75,000 (risk neutrality) γ = 1 CE = 70,711 γ = 2 CE = 66,667 γ = 5 CE = 58,566 γ = 10 CE = 53,991 γ = 20 CE = 51,858 γ = 30 CE = 51,209 current wealth of Y = $100,000 and a degree of risk aversion of γ = 5, the equation results in a CE = $ 66,532. Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence First Order Stochastic Dominance 4.5 Assessing an Investor’s Level of Relative Risk Aversion Second Order Stochastic Dominance 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts 4.6 The Concept of Stochastic Dominance In this section we show that the postulates of Expected Utility lead to a definition of two alternative concepts of dominance which are weaker and this of wider application than the concept of state-by-state dominance. These are of interest because they circumscribe the situations in which rankings among risky prospects are preference-free, ie., 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. Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence First Order Stochastic Dominance 4.5 Assessing an Investor’s Level of Relative Risk Aversion Second Order Stochastic Dominance 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Table 4.1: Sample Investment Alternatives Payoffs 10 100 2000 Prob Z 1 .4 .6 0 Prob Z 2 .4 .4 .2 EZ 1 = 64, σ z 1 = 44 EZ 2 = 444, σ z 2 = 779 Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence First Order Stochastic Dominance 4.5 Assessing an Investor’s Level of Relative Risk Aversion Second Order Stochastic Dominance 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Probability F 1 1.0 0.9 F 2 0.8 0.7 0.6 0.5 F 1 and F 2 0.4 0.3 0.2 0.1 Payoff 0 10 100 2000 Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence First Order Stochastic Dominance 4.5 Assessing an Investor’s Level of Relative Risk Aversion Second Order Stochastic Dominance 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Definition 4.1: First Order Stochastic Dominance FSD Let F A (˜ x ) and F B (˜ 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 F A (˜ x ) first order stochastically dominates ( FSD ) F B (˜ x ) if and only if F A ( x ) ≤ F B ( x ) for all x ∈ [ a , b ] Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence First Order Stochastic Dominance 4.5 Assessing an Investor’s Level of Relative Risk Aversion Second Order Stochastic Dominance 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts First Order Stochastic Dominance: A More General Representation 1 0.9 0.8 0.7 0.6 F B 0.5 F A 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 Asset Pricing

4.1 Measuring Risk Aversion 4.2 Interpreting the Measures of Risk Aversion 4.4 Risk Premium and Certainty Equivalence First Order Stochastic Dominance 4.5 Assessing an Investor’s Level of Relative Risk Aversion Second Order Stochastic Dominance 4.6 The Concept of Stochastic Dominance 4.7 Mean Preserving Spreads 4.8 Key Concepts Theorem (4.2) Let F A (˜ x ) , F B (˜ x ) , be two cumulative probability distributions for random payoffs ˜ x ∈ [ a , b ] . Then F A (˜ x ) FSD F B (˜ x ) if and only if E A U (˜ x ) ≥ E B U (˜ x ) for all non-decreasing utility functions U ( ) . Asset Pricing