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

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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

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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

– h

Y

+ h

tangent lines

Asset Pricing

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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 c 2 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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4.2 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

_U

000

_{(Y )}

(Y )

≡ R

A

(Y )

(ii) relative risk aversion = −

YU

_U

000

_{(Y )}

(Y )

≡ R

R

(Y ).

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Absolute Risk Aversion and the Odds of a Bet

Relative Risk Aversion in Relation to the Odds of a Bet

Absolute

risk aversion = −

U

_U

000

_{(Y )}

(Y )

≡ R

A

(Y )

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Absolute Risk Aversion and the Odds of a Bet

Relative Risk Aversion in Relation to the Odds of a Bet

Relative

risk aversion = −

YU

_U

000

_{(Y )}

(Y )

≡ R

R

(Y ).

π(Y , θ) ∼

=

1

2 +

1

4 θR

R

(Y ).

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Asset Pricing

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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 )} 6= 1, then the inequality is strict.

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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 + e

Z ) = U(Y + CE (Y , e

Z ))

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=

U(Y + E ˜

Z − Π(Y , ˜

Z ))

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Asset Pricing

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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

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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

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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.

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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

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.

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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

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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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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]

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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 7 8 9 10 11 12 13 14 x F_A F B

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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( ).

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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

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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 7 8 9 10 11 12 13 A B C Investment 3 Investment 4 Asset Pricing

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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).

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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.

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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 ˜

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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 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)

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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