asset pricing - chapter iv. measuring risk and risk aversion · pdf file ·...

4.1 Measuring Risk Aversion4.2 Interpreting the Measures of Risk Aversion

4.4 Risk Premium and Certainty Equivalence4.5 Assessing an Investor’s Level of Relative Risk Aversion

4.6 The Concept of Stochastic Dominance4.7 Mean Preserving Spreads

4.8 Key Concepts

Asset PricingChapter IV. Measuring Risk and Risk Aversion

June 20, 2006

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4.8 Key Concepts

Utility functionIndifference 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)

YY – h Y + hY

tangent lines

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4.8 Key Concepts

Utility functionIndifference 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

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4.8 Key Concepts

Absolute Risk Aversion and the Odds of a BetRelative Risk Aversion in Relation to the Odds of a Bet

Arrow-Pratt measures of risk aversion and theirinterpretations

(i) absolute risk aversion = −U′′(Y )U′(Y ) ≡ RA(Y )

(ii) relative risk aversion = −YU′′(Y )U′(Y ) ≡ RR(Y ).

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4.8 Key Concepts


Absoluterisk aversion = −U′′(Y )

U′(Y ) ≡ RA(Y )

π(Y , h) ∼= 1/2 + (1/4)hRA(Y ), (1)

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4.8 Key Concepts


Relativerisk aversion = −YU′′(Y )

U′(Y ) ≡ RR(Y ).

π(Y , θ) ∼=12

+14θRR(Y ). (2)

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4.8 Key Concepts

Jensen’s InequalityCertainty 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 x̃ be arandom variable such that Prob {x̃ ∈ (a, b)} = 1. Suppose theexpectations E(x̃) and Eg(x̃) exist; then

E [g(x̃)] ≤ g [E(x̃)] .

Furthermore, if g( ) is strictly concave and Prob{x̃ = E(x̃)} 6= 1, then the inequality is strict.

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4.8 Key Concepts


EU(Y + Z̃ ) = U(Y + CE(Y , Z̃ )) (3)= U(Y + EZ̃ − Π(Y , Z̃ )) (4)

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4.8 Key Concepts


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

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4.8 Key Concepts

4.5 Assessing an Investor’s Level of Relative Risk Aversion

(Y + CE)1−γ

1− γ=

12(Y + 50, 000)1−γ

1− γ+

12(Y + 100, 000)1−γ

1− γ(5)

Assuming zero initial wealth (Y = 0), we obtain the following sampleresults (clearly, CE > 50,000):γ = 0 CE = 75,000 (risk neutrality)γ = 1 CE = 70,711γ = 2 CE = 66,667γ = 5 CE = 58,566γ = 10 CE = 53,991γ = 20 CE = 51,858γ = 30 CE = 51,209current wealth of Y = $100,000 and a degree of risk aversion of γ = 5,the equation results in a CE= $ 66,532.

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4.8 Key Concepts

First Order Stochastic DominanceSecond Order Stochastic Dominance

4.6 The Concept of Stochastic Dominance

In this section we show that the postulates of ExpectedUtility lead to a definition of two alternative concepts ofdominance which are weaker and this of wider applicationthan the concept of state-by-state dominance. These areof interest because they circumscribe the situations inwhich rankings among risky prospects are preference-free,ie., can be defined independently of the specific trade-offs(between return, risk and other characteristics ofprobability distributions) represented by an agent’s utilityfunction.

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4.8 Key Concepts


Table 4.1: Sample Investment AlternativesPayoffs 10 100 2000Prob Z1 .4 .6 0Prob Z2 .4 .4 .2

EZ1 = 64, σz1= 44EZ2 = 444, σz2= 779

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4.8 Key Concepts


0.1

0.2

0.3

0.4

0.5

0.6

0.8

0.7

1.0

0.9

0 10 100 2000

Payoff

Probability

F1 and F2

F2

F1

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4.8 Key Concepts


Definition 4.1: First Order Stochastic Dominance FSD LetFA(x̃) and FB(x̃), respectively, represent thecumulative distribution functions of two randomvariables (cash payoffs) that, without loss ofgenerality assume values in the interval [a, b]. Wesay that FA(x̃) first order stochasticallydominates (FSD) FB(x̃) if and only ifFA(x) ≤ FB(x) for all x ∈ [a, b]

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4.8 Key Concepts


First Order Stochastic Dominance: A More General Representation

0

0.3

0.5

0.6

0.8

0.1

0.2

0.4

1

0.9

0.7

0 1 2 3 4 5 6 8 10 127 9 11 13 14

x

FA

FB

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4.8 Key Concepts


Theorem (4.2)

Let FA(x̃), FB(x̃), be two cumulative probability distributions forrandom payoffs x̃ ∈ [a, b]. Then FA(x̃) FSD FB(x̃) if and only ifEAU (x̃) ≥ EBU (x̃) for all non-decreasing utility functions U( ).

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4.8 Key Concepts


Table 4.2: Two Independent Investments

Investment 3 Investment 4Payoff Prob. Payoff Prob.4 0.25 1 0.335 0.50 6 0.339 0.25 8 0.33

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4.8 Key Concepts


Second Order Stochastic Dominance Illustrated

0

0.3

0.5

0.6

0.8

0.1

0.2

0.4

1

0.9

0.7

0 1 2 3 4 5 6 10 138 127 119

A

B

C

Investment 3

Investment 4

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4.8 Key Concepts


Definition 4.2: Second Order Stochastic Dominance LetFA(x̃), FB(x̃), be two cumulative probabilitydistributions for random payoffs in [a, b]. We saythat FA(x̃) second order stochasticallydominates (SSD) FB(x̃) if and only if for any x :

x∫−∞

[ FB(t)− FA(t)] dt ≥ 0.

(with strict inequality for some meaningful intervalof values of t).

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4.8 Key Concepts


Theorem (4.3)

Let FA(x̃), FB(x̃), be two cumulative probability distributions forrandom payoffs x̃ defined on [a, b]. Then, FA(x̃) SSD FB(x̃) ifand only if EAU (x̃) ≥ EBU (x̃) for all nondecreasing andconcave U.

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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˜

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4.8 Key Concepts

Theorem (4.4)

Let FA( ) and FB( ) be two distribution functions defined on thesame state space with identical means. If this is true, thefollowing statements are equivalent:(i) FA(x̃) SSD FB(x̃)(ii) FB(x̃) is a mean preserving spread of FA(x̃) in the sense ofEquation

x̃B = x̃A + z̃ (6)

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4.8 Key Concepts

Key Concepts

Absolute and relative measures of risk aversionCertainty equivalence and risk premiumStochastic dominance and the reason for searching for thebroadest concept of dominance

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asset pricing - chapter iv. measuring risk and risk aversion · pdf file ·...

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