Advanced Risk Management

Transcription

1 Winter 2014/2015 Advanced Risk Management Part I: Decision Theory and Risk Management Motives Lecture 1: Introduction and Expected Utility

2 Your Instructors for Part I: Prof. Dr. Andreas Richter Jun.-Prof. Dr. Richard Peter Office hour: By appointment Time & Location: Thursday, 12-2pm, M 110 Monday, 4-8pm, A 125 1

3 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 2

4 Course Outline Parts II & III (Glaser, Elsas) Institute for Risk Management and Insurance Part II: Part III: Market Risk: Overview VaR-Methods I VaR-Methods II Hedging Credit Risk: Overview/Introduction Probability of Default/Rating Asset-/Default-Correlation Credit-Portfolio Models Backup/Review Session 3

5 Course outline - Part I 4

6 Course materials are available at: How to navigate to the website: Lehre Winter 2014/2015 Advanced Risk Management The password for protected files is: 5

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9 Master level classes at the INRIVER 8

10 Master level classes at the INRIVER 9

11 Risk as an interdisciplinary subject Economics Cost-benefit analysis under risk Behavior of economic agents under risk Risk Mathematics Stochastics and probability theory Mathematical statistics Health care Risks to health (Multiplicative) risks associated with treatment Decision theory Risk attitudes Decision-making under risk and uncertainty Psychology Risk perception Cognitive processes 10

12 Definition and classification of risk Risk can be defined as the possibility of a (positive or negative) deviation from the expected outcome. (Ambivalent risk definition) 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 noloss. Examples: Personal risks: loss of income or assets Property risk: destruction, theft or damage of property Liability risk Risks arising from failure of others 11

13 Risk management 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 management instruments Risk control: Risk avoidance Risk reduction Risk financing: Risk retention (active or passive) Risk transfer (e.g. to an insurer) 12

14 Risk management - Individual decision-making Institute for Risk Management and Insurance 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. 13

15 Expected utility theory A basic model Components A ( a 1,, a m ) Action space Set of all risky alternatives S { s 1,, s n } State space Set of all potential and relevant states Z Outcome space Set of all possible outcomes Outcome function f : A S Z maps every possible combination to an outcome f(a,s)=z 14

16 Decision matrix s 1 s 2 s m a 1 z 11 z 12 z 1. z 1m a 2 z 21 z 22 z 2. z 2m z.1 z.2 z.. z.m a n z n1 z n2 z n. z nm In this setup, an action a i implies the associated outcome random variable (or lottery ) z i (often also written as {(z i1, p 1 ),, (z im, p m )}). 15

17 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 z ~, z ~, z~ it holds that a) z~ z~ z~ ~ z ~ z~ z~ b) z~ z~ z~ z~ z~ z~ (Comparability, Completeness) (Transitivity) ii. Continuity Axiom: For any set of outcomes ~ z with ~ ~ ~ z, there is a probability p such that 1, ~ z2, ~ z 3 z1 z2 3 ~ z2 ~ ~ ~ { z1 p z3 }. 16

18 Expected utility axioms iii. Independence Axiom: Given two random variables ~z 1 and ~z 2 such that z~ ( ) ~. 1 z 2 Let ~z 3 be another random variable and let p be an arbitrary probability with Then, it holds that ~ ~ ~ ~ z p z ( ) z p z3 p (0,1). 17

19 Expected Utility Theorem Suppose that the decision-maker has preferences over all lotteries that are rationale 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 wellbeing 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. 18

20 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. 19

21 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. 20

22 Bernoulli utility functions and risk attitudes Institute for Risk Management and Insurance 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. E( ~ z ) ~ z. (Note: Risk aversion does not mean that a decision-maker avoids every risk!) In the expected utility context this translates to u( E( ~ z )) E( u( ~ z )). By Jensen s Inequality it follows that: u( E( z ~ )) E( u( z ~ )) ~ z, concave. for any non-trivial random variable if and only if u is strictly 21

23 The mathematical notion of concavity A function of one variable is concave if f ( tx (1 t) y) tf ( x) (1 t) f ( y) for all x and y and all t with 0 t 1. Graphically: u(z) Analytically: A function is concave if and only if f (x) 0 for all x. z Roughly speaking, a concave function grows more slowly than a linear function. 22

24 The mathematical notion of concavity A function of one variable is strictly concave if f ( tx (1 t) y) tf ( x) (1 t) f ( y) 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. 23

25 Bernoulli utility functions and risk attitudes Institute for Risk Management and Insurance 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. E( ~ z ) ~ z. In the expected utility context this translates to u( E( ~ z )) E( u( ~ z )). This holds for all non-trivial risks if and only if u is strictly concave. A decision-maker is risk-loving if and only u( E( ~ z )) E( u( ~ z )). 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 u( E( ~ z )) E( u( ~ z )). This holds for all non-trivial risks if and only if u is linear. 24

26 Bernoulli utility functions and risk attitudes Institute for Risk Management and Insurance Linear utility functions imply risk-neutrality for instance u1( z) z (Strictly) convex utility functions imply a risk-loving attitude for instance 2 u2( z) z, z 0 (Strictly) concave utility functions imply risk-aversion for instance u3( z) z, z 0 25

27 Bernoulli utility functions and risk attitudes u 0 strictly convex u 0 linear u 0 strictly concave u 0 u(z) u 2 (z) u 1 (z) u 3 (z) z 26

28 Measures of risk aversion We can measure the intensity of a decision-maker s 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 r R u''( z) r A ( z). u'( z) u''( z) ( z) z. u'( z) Both measures are local measures of risk aversion. 27

29 Measures of risk aversion We can measure the intensity of a decision-maker s 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 r R u''( z) r A ( z). u'( z) u''( z) ( z) z. u'( z) Both measures are local measures of risk aversion. Why? 28

30 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. 29

31 Your own degree of risk aversion Under the assumption of constant relative risk aversion with parameter we have that We can solve for and obtain ( ) (1 )

32 Your own degree of risk aversion This is depicted in the following figure: RRA = = = = =

33 State-by-state dominance Lottery A dominates lottery B state-by-state if A yields a better outcome than B in every possible state of nature. Example: s 1 s 2 s 3 A B

34 First-order stochastic dominance Cumulative distribution function F A ( ) first-order stochastically dominates cumulative distribution function F B ( ) ( F for all z with F ( z ) F ( z ) for some z. A B A ( ) FSD F ( ) B ) if and only if F A ( z ) F B ( z ) 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. 33

35 First-order stochastic dominance F A ( ) FSD F B ( ) F(z) 1 F B F A z z max 34

36 First-order stochastic dominance Example EUR Likelihood (Lottery B) Likelihood (Lottery A) F(z) , , , EUR 35

37 First-order stochastic dominance First-order stochastic dominance theorem: F A ( ) F ( ) FSD B and u 0 E u E u. A B If distribution A first-order stochastically dominates B, any expected utility maximizing individual with positive marginal utility will prefer A to B. 36

38 First-order stochastic dominance First-order stochastic dominance theorem: F A ( ) F ( ) FSD B and u 0 E u E u. A B If distribution A first-order stochastically dominates B, any expected utility maximizing individual with positive marginal utility will prefer A to B. Proof? 37

39 Second-order stochastic dominance As a prerequisite define the area under the cumulative distribution function F up to z: T ( z) F( x) dx z Cumulative distribution function F A ( ) second-order stochastically dominates cumulative distribution function F B ( ) ( F for all z with T ( z) T ( z) for some z. A B A ( ) F ( ) SSD B ) if and only if T A ( z) T ( z) B First-order stochastic dominance implies second-order stochastic dominance. 38

40 Second-order stochastic dominance As a prerequisite define the area under the cumulative distribution function F up to z: T ( z) F( x) dx z Cumulative distribution function F A ( ) second-order stochastically dominates cumulative distribution function F B ( ) ( F for all z with T ( z) T ( z) for some z. A B A ( ) F ( ) SSD B ) if and only if T A ( z) T ( z) B First-order stochastic dominance implies second-order stochastic dominance. Why? 39

41 Second-order stochastic dominance EUR Likelihood (Lottery A) Likelihood (Lottery B) F(z) , , , EUR 40

42 Second-order stochastic dominance Second-order stochastic dominance theorem: F A ( ) F ( ) SSD B and u 0, u'' 0 E u E u. A B If F A second-order stochastically dominates F B, then every expected utility maximizer with positive and decreasing marginal utility prefers F A to F B. 41

43 Application: The merits of diversification Assume that there are n assets that are assumed to be i.i.d. ~ x 1 2 n, ~ x,, ~ x. 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 n i 1 1. The perfect diversification strategy is given by i ~ y n ~. i 1 ix i 1 1 A,,. n n The distribution of final wealth generated by the perfect diversification strategy second-order stochastically dominates any other feasible strategy 42