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 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 Utility function Indifference Curves

Measuring Risk Aversion

U(Y + h) U(Y) U[0.5(Y + h) + 0.5(Y – h)] 0.5U(Y + h) + 0.5U(Y – h) U(Y – h) Y Y – h Y + h Y tangent lines

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 Utility function Indifference Curves

Indifference Curves

c*

1

c1 c2 c*

2

State 2 Consumption State 1 Consumption (c*2 + c2)/2 EU(c) = k2 EU(c) = k1 (c*

1 + c1)/ 2

I1 I2 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 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) ≡ RA(Y)

(ii) relative risk aversion = −YU′′(Y)

U′(Y) ≡ RR(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 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) ≡ RA(Y)

π(Y, h) ∼ = 1/2 + (1/4)hRA(Y), (1)

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 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) ≡ RR(Y).

π(Y, θ) ∼ = 1 2 + 1 4θRR(Y). (2)

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 Jensen’s Inequality Certainty Equivalent

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 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 Jensen’s Inequality Certainty Equivalent

EU(Y + Z) = U(Y + CE(Y, Z)) (3) = U(Y + E ˜ Z − Π(Y, ˜ Z)) (4)

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 Jensen’s Inequality Certainty Equivalent

Certainty Equivalent and Risk Premium: An illustration

Y0 Y0 + Z1 Y0 + Z2 U(Y0 + Z2) U(Y0 + Z1) U(Y0 + E(Z)) ~ EU(Y0 + Z) ~ CE(Y0 + Z) ~ Y0 + E(Z) ~ Y U(Y) CE(Z) ~ P 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 − γ =

1 2(Y + 50, 000)1−γ

1 − γ +

1 2(Y + 100, 000)1−γ

1 − γ (5)

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 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 First Order Stochastic Dominance Second Order Stochastic Dominance

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

• f 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 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 First Order Stochastic Dominance Second Order Stochastic Dominance

Table 4.1: Sample Investment Alternatives Payoffs 10 100 2000 Prob Z1 .4 .6 Prob Z2 .4 .4 .2 EZ1 = 64, σz1= 44 EZ2 = 444, σz2= 779

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 First Order Stochastic Dominance Second Order Stochastic Dominance 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 1.0 0.9 10 100 2000 Payoff Probability F1 and F2 F2 F1 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 First Order Stochastic Dominance Second Order Stochastic Dominance

Definition 4.1: First Order Stochastic Dominance FSD Let FA(˜ x) and FB(˜ 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 FA(˜ x) first order stochastically dominates (FSD) FB(˜ x) if and only if FA(x) ≤ FB(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 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 First Order Stochastic Dominance Second Order Stochastic Dominance

First Order Stochastic Dominance: A More General Representation

0.3 0.5 0.6 0.8 0.1 0.2 0.4 1 0.9 0.7 1 2 3 4 5 6 8 10 12 7 9 11 13 14 x FA FB 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 First Order Stochastic Dominance Second Order Stochastic Dominance

Theorem (4.2) Let FA(˜ x), FB(˜ x), be two cumulative probability distributions for random payoffs ˜ x ∈ [a, b]. Then FA(˜ x) FSD FB(˜ x) if and only if EAU (˜ x) ≥ EBU (˜ x) for all non-decreasing utility functions U( ).

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 First Order Stochastic Dominance Second Order Stochastic Dominance

Table 4.2: Two Independent Investments Investment 3 Investment 4 Payoff Prob. Payoff Prob. 4 0.25 1 0.33 5 0.50 6 0.33 9 0.25 8 0.33

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 First Order Stochastic Dominance Second Order Stochastic Dominance

Second Order Stochastic Dominance Illustrated

0.3 0.5 0.6 0.8 0.1 0.2 0.4 1 0.9 0.7 1 2 3 4 5 6 10 13 8 12 7 11 9 A B C Investment 3 Investment 4 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 First Order Stochastic Dominance Second Order Stochastic Dominance

Definition 4.2: Second Order Stochastic Dominance Let FA(˜ x), FB(˜ x), be two cumulative probability distributions for random payoffs in [a, b]. We say that FA(˜ x) second order stochastically dominates (SSD) FB(˜ x) if and only if for any x :

x

∫

−∞

[ FB(t) − FA(t)] dt ≥ 0. (with strict inequality for some meaningful interval

• f values of t).

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 First Order Stochastic Dominance Second Order Stochastic Dominance

Theorem (4.3) Let FA(˜ x), FB(˜ x), be two cumulative probability distributions for random payoffs ˜ x defined on [a, b]. Then, FA(˜ x) SSD FB(˜ x) if and only if EAU (˜ x) ≥ EBU (˜ x) for all nondecreasing and concave U.

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

4.7 More or less risky ∼ = mean preserving spread

EA(x) = xf A(x)dx = xf B(x)dx = EB(x) f A(x) f B(x) x, Payoff ˜ 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

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

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

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

Asset Pricing