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Institute for Risk Management and Insurance

Part I: Decision Theory and Risk Management Motives

Lecture 1: Introduction and Expected Utility

Winter 2014/2015

Advanced Risk Management


• Your Instructors for Part I:

Prof. Dr. Andreas Richter

Email: [email protected]

Jun.-Prof. Dr. Richard Peter

Email: [email protected]

Office hour: By appointment

• Time & Location:

Thursday, 12-2pm, M 110

Monday, 4-8pm, A 125

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mailto:[email protected]

mailto:[email protected]


Course Outline – Part I (Richter/Peter)

Part I: Decision Theory and Risk Management Motives:

Introduction and Expected Utility

The Standard Portfolio Problem

Optimal Risk Sharing and Arrow-Lind Theorem

Risk Management Motives

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Course Outline – Parts II & III (Glaser, Elsas)

Part II: Market Risk:

Overview

VaR-Methods I

VaR-Methods II

Hedging

Part III: Credit Risk:

Overview/Introduction

Probability of Default/Rating

Asset-/Default-Correlation

Credit-Portfolio Models

Backup/Review Session

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Course outline - Part I

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• Course materials are available at:

http://www.inriver.bwl.uni-

muenchen.de/studium/winter_2014_2015/master/finance_risk/index.html

• How to navigate to the website:

www.inriver.bwl.uni-muenchen.de

Lehre Winter 2014/2015 Advanced Risk Management

• The password for protected files is:

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http://www.inriver.bwl.uni-muenchen.de/studium/winter_2014_2015/master/finance_risk/index.html



http://www.inriver.bwl.uni-muenchen.de/




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Master level classes at the INRIVER

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Master level classes at the INRIVER

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Risk as an interdisciplinary subject

Risk

Mathematics Stochastics and probability theory Mathematical statistics

Psychology Risk perception Cognitive processes

Decision theory Risk attitudes Decision-making under risk and uncer- tainty

Health care Risks to health (Multiplicative) risks associated with treatment

Economics Cost-benefit analysis under risk Behavior of econo- mic agents under risk

…

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Definition and classification of risk

Speculative Risk Describes a situation in which there is a possibility of loss but also a possibility of gain. Examples:

• Gambling

• Stock market investments

• Annual profit or loss of a company

Pure Risk Describes a situation in which there is only the possibility of a loss, i.e. the possible outcomes are either loss or no-loss. Examples:

• Personal risks: loss of income or assets

• Property risk: destruction, theft or damage of property

• Liability risk

• Risks arising from failure of others

Risk can be defined as the possibility of a (positive or negative) deviation from the expected outcome. (Ambivalent risk definition)

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Risk management

Risk management instruments

Risk management [in the traditional sense] is a scientific approach to dealing with pure risk by anticipating possible accidental losses and designing and implementing procedures that minimize the occurrence of loss or the financial impact of the losses that do occur. (Vaughan/Vaughan 2003)

Risk control:

• Risk avoidance

• Risk reduction

Risk financing:

• Risk retention (active or passive)

• Risk transfer (e.g. to an insurer)

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Risk management - Individual decision-making

• How can we evaluate risk? More specifically, how can we model decision-making in the

face of risk?

• A workhorse model for decision-making under risk is expected utility theory (EUT) which

has been applied to a multitude of problems.

We will analyze properties of EUT in this lecture.

Then, we will apply it to individual decision-making and exploit it to re-evaluate portfolio choice.

The next level is to analyze how two or more individuals deal with risk, i.e., risk sharing and

diversification.

Finally, we will analyze corporate risk management decisions.

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Expected utility theory – A basic model

Components

• Action space Set of all risky alternatives

• State space Set of all potential and relevant states

• Outcome space Set of all possible outcomes

• Outcome function maps every possible combination to an

outcome f(a,s)=z

),,( 1 maaA

},,{ 1 nssS

Z

ZSAf →:

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Decision matrix

In this setup, an action ai implies the associated outcome random variable (or “lottery”) zi

(often also written as {(zi1, p1), …, (zim, pm)}).

s1 s2 … sm

a1 z11 z12 z1. z1m

a2 z21 z22 z2. z2m

… z.1 z.2 z.. z.m

an zn1 zn2 zn. znm

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Expected utility axioms

i. Ordering Axiom:

The decision maker can order all possible actions, i.e. a complete weak preference relation

exists over A. For any three random variables it holds that

ii. Continuity Axiom:

For any set of outcomes with , there is a probability such that

~

321~,~,~ zzz

(Comparability, Completeness)

(Transitivity)

212121~~~~~~~a) zzzzzz

313221~~~~~~b) zzzzzz ⇒

}.~~{ 31 zpz

p321

~~~ zzz 321~,~,~ zzz

2~z

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Expected utility axioms

iii. Independence Axiom:

Given two random variables and such that

Let be another random variable and let p be an arbitrary probability with

Then, it holds that

.~)(~21 z z

).1,0(p .~~)(~~

3231 z pzz p z

1~z 2

~z

3~z

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Expected Utility Theorem

Suppose that the decision-maker has preferences over all lotteries that are ra-

tionale and satisfy continuity and independence. Then, this preference relation

can be represented by a preference functional that is linear in probabilities.

This is, there is a utility function over outcomes which measures the well-

being of the consumer and we can determine the consumer’s satisfaction by

evaluating the expected utility of a particular lottery.

In other words: The decision-maker chooses the action

that maximizes expected utility.

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Expected Utility Theorem

• The expected utility theorem (among other things) provides the existence of the utility

function.

• The utility function is also called Bernoulli utility function.

• Obtaining a Bernoulli utility function can be a challenging task in a real life situation.

• It is unique up to a positive affine transformation

• Why?

• In other words, utility in EUT is ordinal.

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Daniel Bernoulli

• February 8, 1700 (Groningen) - March 17, 1782 (Basel)

• Swiss mathematician and physicist

• He worked on…

… the mechanics of fluids. (Strömungsmechanik)

… differential equations.

… the St. Petersburg Paradox.

• He proposed to use a utility function to evaluate risky

gambles. This resolves the original St. Petersburg

Paradox.

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Bernoulli utility functions and risk attitudes

• A risk-averse decision-maker prefers a certain payment to a (non-trivial) lottery with an

expected value equal to the certain payment, i.e.

(Note: Risk aversion does not mean that a decision-maker avoids every risk!)

• In the expected utility context this translates to

.~)~( zzE

)).~(())~(( zuEzEu

By Jensen’s Inequality it follows that:

for any non-trivial random variable if and only if u is strictly

concave.

))~(())~(( zuEzEu ,~z

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The mathematical notion of concavity

• A function of one variable is concave if

for all x and y and all t with 0 t 1.

• Graphically:

• Analytically: A function is concave if and only if f’’(x) 0 for all x.

• Roughly speaking, a concave function grows “more slowly” than a linear function.

)()1()())1(( yftxtfyttxf

u(z)

z

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The mathematical notion of concavity

• A function of one variable is strictly concave if

for all x and y and all t with 0 t 1.

• Analytically: A function is strictly concave if f’’(x) < 0 for all x.

• We can define convexity analogously.

)()1()())1(( yftxtfyttxf

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• A risk-averse decision-maker prefers a certain payment to a (non-trivial) lottery with an

expected value equal to the certain payment, i.e.

• In the expected utility context this translates to

• This holds for all non-trivial risks if and only if u is strictly concave.

• A decision-maker is risk-loving if and only

• This holds for all non-trivial risks if and only if u is strictly convex.

• A decision-maker is risk-neutral if and only if

• This holds for all non-trivial risks if and only if u is linear.

.~)~( zzE

)).~(())~(( zuEzEu

)).~(())~(( zuEzEu

)).~(())~(( zuEzEu

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• Linear utility functions imply risk-neutrality

for instance

• (Strictly) convex utility functions imply a risk-loving attitude

for instance

• (Strictly) concave utility functions imply risk-aversion

for instance

zzu 5.210)(1

0z ,)( 2

2 zzu

0z ,)(3 zzu

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u(z)

z

u2(z)

u3(z)

u1(z)

strictly convex

linear

strictly concave

0u

0u

0u

0u

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Measures of risk aversion

• We can measure the intensity of a decision-maker‘s risk aversion.

• Both measures are local measures of risk aversion.

Consider a decision-maker with utility function u(z).

The Arrow-Pratt measure of absolute risk aversion is defined as

The Arrow-Pratt measure of relative risk aversion is defined as

.)('

)('')(

zu

zuzrA

.)('

)('')(

zu

zuzzrR

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Measures of risk aversion

• We can measure the intensity of a decision-maker‘s risk aversion.

• Both measures are local measures of risk aversion.

Why?

Consider a decision-maker with utility function u(z).

The Arrow-Pratt measure of absolute risk aversion is defined as

The Arrow-Pratt measure of relative risk aversion is defined as

.)('

)('')(

zu

zuzrA

.)('

)('')(

zu

zuzzrR

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Your own degree of risk aversion

• Consider that you either gain or lose 10% of your wealth with equal probability (1/2).

• What is the share of your wealth, , you are willing to pay to escape this risk?

• Take your time to think about the problem and take down your answer.

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• Under the assumption of constant relative risk aversion with parameter we have that

• We can solve for and obtain

.1

)1(

1

1.15.0

1

9.05.0

111

.1.15.09.05.01)(1 11

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• This is depicted in the following figure:

RRA

= 0.5 0.003

= 1 0.005

= 4 0.02

= 10 0.044

= 40 0.084

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State-by-state dominance

Example:

Lottery A dominates lottery B state-by-state if A yields a better outcome than B

in every possible state of nature.

s1 s2 s3

A 10 4 7

B 7 1 6

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First-order stochastic dominance

• Lottery A first-order stochastically dominates B, if for any outcome z the likelihood of

receiving an outcome equal to or better than z is greater for A than for B.

Cumulative distribution function FA(·) first-order stochastically dominates

cumulative distribution function FB(·) ( ) if and only if

for all z with for some z. )z(F)z(F BA

)z(F)z(F BA )(F)(F BFSDA

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)(F)(F BFSDA

z

)z(F

maxz

1

BF

AF

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Example

EUR

50

40

30

20

10

Likelihood

(Lottery B)

0

0.5

0

0

0.5

Likelihood

(Lottery A)

0.5

0

0.25

0.25

0

EUR 10 20 30 40 50

F(z)

1

0,75

0,5

0,25

0

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• If distribution A first-order stochastically dominates B, any expected utility maximizing

individual with positive marginal utility will prefer A to B.

First-order stochastic dominance theorem:

and .0 uEuEu BA )()( BFSDA FF

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• If distribution A first-order stochastically dominates B, any expected utility maximizing

individual with positive marginal utility will prefer A to B.

• Proof?

First-order stochastic dominance theorem:

and .0 uEuEu BA )()( BFSDA FF

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Second-order stochastic dominance

• As a prerequisite define the area under the cumulative distribution function F up to z:

• First-order stochastic dominance implies second-order stochastic dominance.

z

dxxFzT )()(

Cumulative distribution function FA(·) second-order stochastically dominates


for all z with for some z.

)()( BSSDA FF

)()( zTzT BA

)()( zTzT BA

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• As a prerequisite define the area under the cumulative distribution function F up to z:

• First-order stochastic dominance implies second-order stochastic dominance.

Why?

z

dxxFzT )()(

Cumulative distribution function FA(·) second-order stochastically dominates


for all z with for some z.

)()( BSSDA FF

)()( zTzT BA

)()( zTzT BA

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EUR

40

30

20

10

Likelihood

(Lottery A)

0

0.5

0.5

0

Likelihood

(Lottery B)

0.25

0.25

0.25

0.25

F(z)

1

0,75

0,5

0,25

0

20 30 40 EUR 10 30 40 EUR

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• If FA second-order stochastically dominates FB, then every expected utility maximizer with

positive and decreasing marginal utility prefers FA to FB.

Second-order stochastic dominance theorem:

and .0'',0 uEuEuu BA )()( BSSDA FF

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Application: The merits of diversification

• Assume that there are n assets that are assumed to be i.i.d.

• A feasible strategy is characterized by a vector A = (1, …, n) where i is one’s share in

the ith asset, and

• This yields a net payoff of

• The perfect diversification strategy is given by

.~,,~,~21 nxxx

n

ii

1.1

n

iii xy

1.~~

The distribution of final wealth generated by the perfect diversification strategy

second-order stochastically dominates any other feasible strategy

.1

,,1

nnA

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