Asset Pricing. Chapter IV. Measuring Risk and Risk Aversion. June 20, 2006

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1 Chapter IV. Measuring Risk and Risk Aversion June 20, 2006

2 Measuring Risk Aversion Utility function Indifference Curves U(Y) tangent lines U(Y + h) U[0.5(Y + h) + 0.5(Y h)] 0.5U(Y + h) + 0.5U(Y h) U(Y h) Y h Y Y + h Y

3 Indifference Curves State 2 Consumption Utility function Indifference Curves I 1 I 2 c * 2 (c* 2 + c 2 )/2 c 2 EU(c) = k 2 EU(c) = k 1 c * 1 (c * 1 + c 1 )/ 2 c 1 State 1 Consumption

4 Absolute Risk Aversion and the Odds of a Bet Relative Risk Aversion in Relation to the Odds of a Bet 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 ).

5 Absolute Risk Aversion and the Odds of a Bet Relative Risk Aversion in Relation to the Odds of a Bet Absolute risk aversion = U (Y ) U (Y ) R A(Y ) π(y, h) = 1/2 + (1/4)hR A (Y ), (1)

6 Absolute Risk Aversion and the Odds of a Bet Relative Risk Aversion in Relation to the Odds of a Bet Relative risk aversion = YU (Y ) U (Y ) R R (Y ). π(y, θ) = θr R(Y ). (2)

7 Jensen s Inequality Certainty Equivalent 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.

8 Jensen s Inequality Certainty Equivalent EU(Y + Z ) = U(Y + CE(Y, Z )) (3) = U(Y + E Z Π(Y, Z )) (4)

9 Jensen s Inequality Certainty Equivalent 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

10 (Y + CE) 1 γ 1 γ = 1 2 (Y + 50, 000)1 γ + 1 γ 1 2 (Y + 100, 000)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. (5)

11 First Order Stochastic Dominance Second Order 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.

12 First Order Stochastic Dominance Second Order Stochastic Dominance Table 4.1: Sample Investment Alternatives Payoffs Prob Z Prob Z EZ 1 = 64, σ z1 = 44 EZ 2 = 444, σ z2 = 779

13 First Order Stochastic Dominance Second Order Stochastic Dominance Probability 1.0 F F F 1 and F Payoff

14 First Order Stochastic Dominance Second Order Stochastic Dominance 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]

15 First Order Stochastic Dominance Second Order Stochastic Dominance First Order Stochastic Dominance: A More General Representation F B F A x

16 First Order Stochastic Dominance Second Order Stochastic Dominance 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( ).

17 First Order Stochastic Dominance Second Order Stochastic Dominance Table 4.2: Two Independent Investments Investment 3 Investment 4 Payoff Prob. Payoff Prob

18 First Order Stochastic Dominance Second Order Stochastic Dominance Second Order Stochastic Dominance Illustrated C Investment 4 B A Investment

19 First Order Stochastic Dominance Second Order Stochastic Dominance Definition 4.2: Second Order Stochastic Dominance Let F A ( x), F B ( x), be two cumulative probability distributions for random payoffs in [a, b]. We say that F A ( x) second order stochastically dominates (SSD) F B ( x) if and only if for any x : x [ F B (t) F A (t)] dt 0. (with strict inequality for some meaningful interval of values of t).

20 First Order Stochastic Dominance Second Order Stochastic Dominance Theorem (4.3) Let F A ( x), F B ( x), be two cumulative probability distributions for random payoffs x defined on [a, b]. Then, F A ( x) SSD F B ( x) if and only if E A U ( x) E B U ( x) for all nondecreasing and concave U.

21 4.7 More or less risky = mean preserving spread f A (x) f B (x) E A (x) = xf A (x)dx = xf B (x)dx = E B (x) x, Payoff

22 Theorem (4.4) Let F A ( ) and F B ( ) be two distribution functions defined on the same state space with identical means. If this is true, the following statements are equivalent: (i) F A ( x) SSD F B ( x) (ii) F B ( x) is a mean preserving spread of F A ( x) in the sense of Equation x B = x A + z (6)

23 Key Concepts Absolute and relative measures of risk aversion Certainty equivalence and risk premium Stochastic dominance and the reason for searching for the broadest concept of dominance

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