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 ArrowPratt 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 statebystate dominance. These are of interest because they circumscribe the situations in which rankings among risky prospects are preferencefree, ie., can be defined independently of the specific tradeoffs (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 nondecreasing 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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