1. Expected utility, risk aversion and stochastic dominance

Transcription

1 . Epected utility, risk aversion and stochastic dominance. Epected utility.. Description o risky alternatives.. Preerences over lotteries..3 The epected utility theorem. Monetary lotteries and risk aversion.. Monetary lotteries and the epected utility ramework.. Risk aversion..3 Measures o risk aversion..4 Risk aversion and portolio selection.3 Stochastic dominance.3. First degree stochastic dominance.3. Second degree stochastic dominance.3.3 Second degree stochastic monotonic dominance

2 The objective o this part is to eamine the choice o under uncertainty. We divide this chapter in 3 sections. The irst part begins by developing a ormal apparatus or modeling risk. We then apply this ramework to the study o preerences over risky alternatives. Finally, we eamine conditions o the preerences that guarantee the eistence o a utility unction that represents these preerences. In the second part, we ocus on the particular case in which the outcomes are monetary payos. Obviously, this case is very interesting in the area o inance. In this part we present the concept o risk aversion and its measures. In the last part, we are interested in the comparison o two risky assets in the case in which we have a limited knowledge o individuals preerences. This comparison leads us to the three concepts o stochastic dominance.

3 . Epected utility.. Description o risky alternatives Let us suppose that an agent aces a choice among a number o risky alternatives. Each risky alternative may result in one o a number o possible outcomes, but which outcome will occur is uncertain at the time that he must make his choice. Notation: X = The set o all possible outcomes. Eamples: X= A set o consumption bundles. X= A set o monetary payos. To simpliy, in this part we make the ollowing assumptions:. The number o possible outcomes o X is inite.. The probabilities o the dierent outcomes in a risky alternative are objectively known. The concept used to represent a risky alternative is the lottery. Deinition: A simple lottery L is a list L (,..., ; p,..., i i X, i =,, n n =, where, n p n ii p i 0 and p i =, where p i is interpreted as the probability o i= outcome i occurring. Generally to represent a lottery, we use a tree p p n n 3

4 Notice that in a simple lottery the outcomes are certain. A more general concept, a compound lottery, allows the outcomes o a lottery themselves to be simple lotteries. Deinition: Given K simple lotteries L k, the compound lottery ( L ;α,... α simple lottery L k with probability α k, k =,... K, and probabilities α 0 with α =, k K k =,..., L K K is the risky alternative that yields the k =,... K. k For any compound lottery, we can calculate its reduced lottery that is a simple lottery that generates the same distribution over the inal outcomes. Eercise: (Illustration o the derivation o a reduced lottery 4

5 .. Preerences over lotteries Having introduced a way to model risky alternatives, we now study the preerences over them corresponding to a ied agent. In this part we assume that or any risky alternative, only the reduced lottery over inal outcomes is o relevance to the agent. This means that given two dierent compound lotteries with the same reduced lottery, the decision maker is indierent between them. Let denote the set o all simple lotteries over the set o outcomes X. According to the previous assumption we assume that the agent s preerences ( are deined on. We make the ollowing assumptions A. The decision maker has a preerence relation on. This means that satisy the ollowing properties:. is releive ( L, L L. is complete ( L L, we either have L L or L L, 3. is transitive ( L L, L, such that L L and L L 3, then L L 3, 3 A. The Archimedian Aiom: L L, L, such that L L L 3, then there eists α, β ( 0, such that, 3 αl + ( α L3 L βl + ( β L3 Economic intuition: As L L, no matter how bad L 3 is, we can combine L and L 3 with α large enough (near to one such that αl + ( α L3 L. As L L 3, no matter how good L is, we can combine L and L 3 with β small enough (near to zero such that L βl + ( β L3. A.3 The Independence Aiom: 5

6 L L, L, and α (0, ], 3 L L i and only i αl + ( α L3 αl + ( α L3 Economic intuition: I we combine two lotteries, L and L, with a third lottery in the same way, the preerence ordering o the resulting lotteries does not depend on the third lottery...3 The epected utility theorem THE EXPECTED UTILITY THEOREM Suppose an agent whose preerences ( are deined on. Then, ( satisy (A.-(A.3 There eists u : X R such that L L n i= p u( i i m j= ( p j u( j L =,..., n; p,..., p, where L = (,,..., m; p,..., pm, n and u and v are two unctions that represent these preerences v=au+b, where a, b R, and a>0 Comments related to this theorem:. Consider an individual whose preerences satisy the previous hypothesis, then he has a utility unction that represents her preerences, i.e., There eists U : R such that L L U(L U(L, or all L,L.. This theorem also states that this utility unction U : R has a orm o a epected utility unction. For any L = (,..., ; p,...,, n p n 6

7 n U(L= pi u( i, where u : X R. i= Notice that any certain outcome has a utility level and the utility o a lottery is measured computing its epected utility level. 3. This utility unction is unique, ecept positive linear transormations. Notation: U : the von-neumann-morgenstern epected utility unction u: the Bernoulli utility unction 7

8 . Monetary lotteries and risk aversion In this section, we ocus on risky alternatives whose outcomes are amounts o money. In Economy, generally we consider money as a continuous variable. Until now we have stated the epected utility theorem assuming a inite number o outcomes. However, this theory can be etended to the case o an ininite domain. Net, we briely discuss this etension... Monetary lotteries and the epected utility ramework Notation: = amounts o money (continuous variable We can describe a monetary lottery by means o a cumulative distribution unction, that is, F : R [ 0, ] F = p( < Thereore, we will take the lottery space to be the set o all distribution unctions over nonnegative amounts o money ( or more general [ a, Epected utility theorem: Consider an agent whose preerences over satisy the assumptions o the theorem, then there eists a utility unction U that represents these preerences. Moreover, U has the orm o an epected utility unction, that is, u (. tq F U(F = u( df( = E(u(. In addition, u and v are two unctions that represent these preerences 8

9 v=au+b, where a, b R, and a>0 Remark: It is important to distinguish U and u. U( is deined on the space o simple lotteries and u( is deined on sure amounts o money. U : the von-neumann-morgenstern epected utility unction u: the Bernoulli utility unction Hypothesis: We will assume that u( is strictly increasing and dierentiable... Risk aversion We begin with a deinition o risk aversion very general, in the sense that it does not require the epected utility ormulation. Deinition: An individual is risk averse i or any monetary lottery F, the lottery that yields df ( with certainty is at least as good as the lottery F. An individual is risk neutral i or any monetary lottery F, the agent is indierent between the lottery that yields df ( with certainty and the monetary lottery F. An individual is strictly risk averse i or any monetary lottery F, the agent strictly preers the lottery that yields df ( with certainty than the lottery F. In this assumption we assume that the lottery F represents a risky alternative. Otherwise, the individual is indierent between these two lotteries. 9

10 Suppose that the decision maker has preerences that admit an epected utility unction representation. Let u(. be a Bernoulli utility unction corresponding to these preerences. Deinition: An individual is risk averse i and only i u df( u( df(, or all F ( Jensen s Inequality or equivalently, E( u( u( E( or all random variable. An individual is risk neutral i and only i u df( = u( df(, or all F, or equivalently, E ( u( = u( E( or all random variable. An individual is (strictly risk averse i and only i u df( < u( df(, or all F or equivalently, E ( u( < u( E( or all random variable. Proposition: Suppose a decision maker with a Bernoulli utility unction u ( u ( ehibits (strict risk aversion (. Then, u is (strictly concave Proo: We have to proo that and α [ 0,] u ( α ( α αu ( + ( α u (, +. 0

11 Consider the ollowing monetary lottery: α α Since Jensen s inequality holds or all the monetary lotteries, in particular or the previous one. E( u( u( E( Now, we develop both sides o this inequality ( + ( α u( E( u( = αu ( α + ( α u( E( = u Thereore, the previous inequality can be written as αu ( ( α u( u( α + ( α +. Q.E.D. Suppose that u( is concave, then u( u( E( + u'( E( ( E( u( u( E( + u'( E( ( E( (Taking epected values in both sides E( u( u( E( Q.E.D.

12 Net we introduce two concepts related to risk aversion. Consider a risk averse agent, with a Bernoulli utility unction u(. and an initial wealth W 0. Let z denote the outcome o a gamble. By risk aversion, we have that the individual preers E (z to z, that is E u( W + z u( W + E(. ( 0 0 z This inequality tells us that to avoid the risk the individual is willing to pay. The maimum amount o money that the individual is willing to pay is called the Pratt s risk premium or insurance risk premium. Deinition: Given a decision maker with a Bernoulli utility unction u( and a initial wealth W 0, the Pratt s risk premium or insurance risk premium o z is a certain amount, denoted by Π (z, such that E( u( W ( ( ( 0 + z = u W0 + E z Π z. The certain amount W + E( z Π( is called certainty equivalent o z, since 0 z it is the amount o money or which the individual is indierent between z and this certain amount. Eample: Let = W + 0 z. Suppose that it takes two values and equally likely.

13 The ollowing eample illustrates the use o the risk aversion concept. Eample: Demand or a risky asset Consider an individual that wants to invest an initial wealth W 0 in inancial assets. This individual has a Bernoulli utility unction, u, that holds u '> 0 and u '' < 0. Suppose that there eist two assets: Riskless asset with gross return (constant Risky asset with gross return (random variable The investor s problem consists in Ma A, B ( ( R B + RA E u where s. t. A + B = W 0 A: quantity invested in the risky asset B: quantity invested in the riskless asset. We study this problem assuming that A 0. It is important to point out that The act that we do not allow A<0 means that we are assuming that the investor cannot sell an asset that he does not own ( short-selling constraints. The act that A may be greater than W 0 means that the investor may borrow in the riskless asset. Since A + B = W 0, then B = W 0 A. Substituting this epression, the previous individual s choice problem can be reormulated as: Ma A ( ( R W + ( R R A E u 0. Notation: V ( A = E u( R W0 + ( R R A (. Property: V is a strictly concave unction. Derivating with respect to A, we have 3

14 and V V ''( A ( ' ( R W + ( R R A( R R '( A = E u 0 ( '' ( R W0 + ( R R A( R R < 0 = E u - + The epectation o a negative random variable is negative We distinguish 3 cases: V (A>0, A 0. In this case V is strictly increasing A inite solution does not eist. Eample: I R > R V (A>0 Economic Intuition: I R > R, then the investor will borrow in the riskless asset and will invest all in the risky asset. Since there is no restriction on the borrowing level, the investor will borrow an ininite quantity to obtain ininite proits. V (0 0 I V (0 0 V (A<0, A > 0 V is strictly decreasing in A A* = 0 Notice that V (0= E( u ( R W ( R R = u' ( R W ( E( R R E R R V (0 0 ( Intuition: ' 0 0. Thereore I the epected return o the risky asset is smaller than the return o the riskless asset, a risk averse agent will not invest in the risky asset. 3 From the previous two cases, we know that A * > 0 and A * inite implies that E ( R > R and R > R does not hold. Conclusion: E R > R A risk averse agent will invest in a risky asset ( 4

15 ..3 Measures o Risk Aversion Now we try to measure the etent o risk aversion. We begin with the most used measure o risk aversion which is the coeicient o absolute risk aversion (also called the Arrow-Pratt s coeicient The Coeicient o Absolute Risk Aversion Motivation o the coeicient o absolute risk aversion: Notice that risk aversion is equivalent to the concavity o u(, that is, u 0. Thereore, it seems logical to start considering one possible measure: u. However, this is not an adequate measure because is not invariant to positive linear transormations. To make it invariant, the simplest modiication is to normalize with u. Deinition: Given a Bernoulli utility unction u(, the coeicient o absolute risk aversion at is deined as R A u''( =. u'( Comments o this epression: This measure is invariant to linear transormations. The sign minus makes the epression be positive when u( is increasing and concave. Net, we use this measure in two comparative statics eercises. Comparison o risk attitudes across individuals with dierent utility unctions Comparison o risk attitudes or one individual at dierent levels o wealth. 5

16 Comparison across individuals Consider two individuals, individual and individual, with two Bernoulli utility unctions u and u, respectively, such that: u ' 0 and u ' 0, > > u '' 0 and u '' 0. < < Net, we want to prove that the coeicient o absolute risk aversion is an eective measure, that is, i individual is more risk averse than individual, then it holds RA RA,, and vice versa, where R ui '' =, i =,. u ' i A. i Notice that individual is more risk averse than individual G(., strictly increasing and concave such that u = G( u ( ( u is more concave than u. Thereore, we want to show ( Lemma : G(., strictly increasing and concave such that u = G( u ( ( RA RA, Suppose that u = G( u (. Then, Thereore, ( u = G'( u u' ( and ' ( u' G'( u u'' ( u '' = G''( u +. ( u' u''( G''( u( + G'( u u'' R A = = = u ' G'( u u' ( G' '( u u' G'( u u'' u'' = = RA (* G'( u u' G'( u u' u' 6

17 Now we assume that RA RA,. Notice that since u and u are strictly increasing, it is true that eits G(. such that u = G( u (. (Notice thatg = u u and it is dierentiable. ( Thereore, u = G'( u u' (, which implies that G >0. ' Moreover, using (*, we know that G' '( u u' R A = + RA. G'( u u' ( Using the act that RA RA, the previous equality tells us that G''( u G'( u u' ( u' are strictly increasing unctions. 0, which implies that G ''( u 0 because u and G Note: The relation more-risk averse-than relation is a partial ordering o Bernoulli utility unctions, since it is not complete. Typically, given two Bernoulli utility unctions RA RA at some, but the contrary inequality holds or other levels. Lemma : Consider two individuals with strictly increasing and strictly concave Bernoulli utility unctions u and u, and with identical initial wealth W 0. Then RA RA Π ( ( z Π z z Proo: By Lemma, it suices to show G(. strictly increasing and concave such that u = G( u ( ( Π z Π ( z ( z Using the deinition o the Pratt s risk premium or the individual, we have u W + E( z Π ( z = E( u ( W + z = E( G( u ( W + z G( E( u ( W + = ( z G ( u W + E( z Π ( z = u ( W + E( z Π ( ( 0 0 z 7

18 Thereore, u W + E( z Π ( u W + E( z Π (. ( 0 z ( 0 z Using the act that u is strictly increasing, this inequality implies that W + E z Π ( W + E z Π (, 0 ( z or equivalently, Π z Π (. ( z 0 ( z Notice that since u and u are strictly increasing, it is true that eits G(. such that u = G( u ( (Notice that ( G = u u and it is dierentiable. Thereore, u = G'( u u' (, which implies that G >0. ' To show the concavity o G, we will prove Jensen s inequality, that is, E( G( G( E(,. Fi. Then there eists z such that = u ( W +. Thereore, 0 z ( G( u ( W + z = E( u ( W + = E G( = E z ( 0 0 u + ( ( W0 E z Π z ( ( W + E( z Π ( z = G( E( u ( W + z G( E( u W + E( z Π ( z = G u ( = ( Comparison across wealth levels Typically, richer people are more willing to accept risk than poorer people. Although this might be due to dierences in utility unctions across people, it is more likely that this is due to dierences in the wealth levels. Then, the way to ormalize this risk attitude is to assume that R A ( is a decreasing unction o. Lemma: R A ( is a decreasing unction o u' '' > 0 8

19 The Coeicient o Relative Risk Aversion To understand the concept o relative risk aversion, it is important to point out that the concept o absolute risk aversion is used to compare attitudes toward risky alternatives whose outcomes are absolute gains or absolute losses. But sometimes we consider risky alternatives whose outcomes are percentage gains or losses o current wealth. In this case, we measure the risk aversion by means o the coeicient o relative risk aversion. Deinition: Given a Bernoulli utility unction u(, the coeicient o relative risk aversion at is deined as R u''( = = RA (. u'( R Lemma(Relationship between the two coeicients o risk aversion Consider an individual with a strictly increasing and strictly concave Bernoulli utility unction u. Then drr dra 0 < 0. d d drr dra Proo: Directly ollows rom = + RA. d d 9

20 ..4 Risk aversion and portolio selection Consider again the portolio selection problem or an agent with a strictly increasing and strictly concave Bernoulli utility unction. Moreover we assume E R > R. Then that ( Ma A ( ( R W + ( R R A E u 0. F.O.C: E( u' ( R ( ( W0 + R R A R R = 0 S.O.C: E u'' ( R W ( R R A( R R ( 0 + < 0 Net, we perorm some comparative statics eercises. Comparison o the investment in the risky asset o two agents who dier in their risk attitude. Comparison o the investment in the risky asset o an agent with two distinct initial wealth levels. Consider two individuals with strictly increasing and strictly concave Bernoulli utility unctions u and u, and with identical initial wealth W 0. Proposition : Let Ai be the investment in risky asset o agent i, i=,. Then, RA RA A A (I individual is more risk averse than individual, then the quantity invested in the risky asset o agent is smaller than the one o agent 0

21 Now, we are interested in studying how vary the investment in the risky asset o an agent when her initial wealth varies. Proposition : dr A d dr A d dr A d < 0 = 0 > 0 da dw 0 da dw 0 da dw 0 > 0 = 0 < 0 Suppose that an individual has a Bernoulli utility unction that ehibits decreasing absolute risk aversion then i the agent becomes richer then he will invest more in the risky asset.

22 Proposition 3: Let Ai a = be the proportion o the initial wealth invested in the risky asset. W 0 Then, dr R d dr R d dr R d < 0 = 0 > 0 da dw 0 da dw 0 da dw 0 > 0 = 0 < 0 Suppose that an individual has a Bernoulli utility unction that ehibits decreasing relative risk aversion then i the agent becomes richer then the proportion o the initial wealth invested in the risky asset increases.

23 .3 Stochastic dominance Suppose that there are two risky assets. The ollowing question is addressed in this part: Under what conditions can we say that an individual will preer one asset to another when the only inormation we have about preerences is that the utility unction is increasing or is concave? To answer this question we introduce the concepts o stochastic dominance that are useul to compare random variables..3. First degree stochastic dominance Deinition: y u(. increasing and continuous E( u( E( u( y. FDSD Remark: We ask continuity in order to take epectations. We do not ask dierentiability because a unction that is continuous and increasing is dierentiable almost everywhere. Property: y y E( E( FDSD Remark: This property is useul in the sense that i we have ( E( Proo: Consider u(z=z. E <, then y FDSD 3

24 The opposite implication is not true in general. A counter-eample is the ollowing: Characterizations: Let F ( F ( and denote the cumulative distribution o and y, respectively. y. y F ( z F ( z z [ a,b] FDSD Intuition: y,. Proo: 4

25 . y y = d + ε, with ε 0. FDSD Proo: This part is omitted because is very technical. E u y = E u + ε E u ( ( ( ( ( (.3. Second degree stochastic dominance Deinition: y u(. concave whose irst derivative is continuous E( u( E( u( y (ecept on a countable set. y y is riskier than in the Rothschild-Stiglitz sense SDSD Properties: SDSD Proo: y u(. concave whose irst derivative is continuous E( u( E( u( y (ecept on a countable set. In particular, u( z = z E y ( E( u( z = z E y ( E( y E( E( E = ( E( y u( z = ( ( z E z var( var( y var( var( y The opposite implication is not true. In the Laont s book there is a numerical counter-eample: 5

26 Characterizations: Let F ( F ( and denote the cumulative distribution o and y, respectively. y. t y i F ( z F ( z ( y dz, t [ a,b] a ( F z F z dz = 0 SDSD ii ( ( Intuition: b a y 0. 6

27 Proo: 7

28 . y y = d + ε SDSD E ε., with ( = 0 y is called a mean-preserving spread o. Intuition: (I you add noise the new density is more disperse Proo: This part is omitted because is very technical. E( u( y = E( u( + ε = E ( E ( u( E ( u( E ( E ( u( E( u( + ε + ε = =.3.3 Second degree stochastic monotonic dominance Deinition: ε. y u(. increasing and concave E( u( E( u( y SDSMD Property: y y E( E( SDSMD ε Characterizations: Let F ( F ( and denote the cumulative distribution o and y, respectively. y. t y i F ( z F ( z ( y dz, t [ a,b] a ( F z F z dz 0 SDSMD ii ( ( b a. y y = d + ε SDSMD, with ( 0 y E ε. 0. 8

29 Application o the concept o stochastic dominance to the simplest portolio selection problem (Rothschild and Stiglitz (97 Consider a risk averse investor with an initial wealth W 0 to be invested in a risky asset, asset, with a gross return R, and a riskless asset with a gross return R. Let A denote the optimal amount invested in the risky asset. Suppose that A is characterized by the FOC, that is ( u ( R W ( R R A ( R R 0 E. ' 0 + = Imagine now the ollowing situation. There eists another risky asset, asset, that is riskier than asset in the Rothschild and Stiglitz sense, R R. Question: SDSD I the same individual can invest in the risky asset and the riskless asset, is he going to invest less in the risky asset because now the risky asset is more risky? Let A denote the optimal amount invested in the risky asset. Suppose that A is characterized by the FOC, that is ( u ( R W ( R R A ( R R 0 E. ' 0 + = Notice that a suicient condition or having A A is E u' R W R R A R R 0 = E u' R W + R R A R R. ( ( 0 + ( ( ( ( 0 ( ( Let g u' ( R W + ( R A ( R. I ( holds. = 0 Eercise: Prove that R concave. g is concave, then this inequality dr drr A R, 0 and 0 implies that g is d d 9

30 A utility unction that satisies the tree conditions is σ z u z =, with 0 < σ < σ (. 30