Higher-Order Risk Attitudes toward Correlation Jingyuan Li Department of Finane and Insurane, Lingnan University, Hong Kong E-mail: jingyuanli@ln.edu.hk August 10, 2013 Abstrat Higher-order risk attitudes other than risk aversion (e.g., prudene and temperane) play vital roles both in theoretial and empirial work. While the literature has mainly foused on how they entail a preferene for ombining good outomes with bad outomes, We onsider here an alternative approah whih relates higher-order risk attitudes to the sign of orrelation. The theoretial result in this paper proposes new insights into eonomi and finanial appliations suh as risk aversion in the presene of another risk, bivariate stohasti dominane and justifying the first-order approah to moral hazard prinipal-agent problems. Key words: higher-order risk attitudes; stohasti dominane dependene; orrelation; ovariane JEL lassifiation: D81.
1 Introdution Covariane is perhaps the most ommon notion of dependene between two random variables. Consider two random variables x and ỹ valued in the intervals [a, b] and [, d] respetively. It is well known that one of the most used formula in the eonomis of unertainty is the ovariane rule: Cov( x, ỹ) = E xỹ E xeỹ, (1) whih implies that E xỹ E xeỹ if and only if x and ỹ o-vary positively. A diret appliation of above ovariane rule is to sign the equity premium: Denote u as the bivariate utility of the representative agent, x as the GDP per apita and ỹ as the bakground risk. From Gollier (2001, page 65-68), we know that the equity premium, φ, an be written as 1 : φ = E xeu(1,0) ( x, ỹ) E xu (1,0) ( x, ỹ) 1. (2) If there is full information on the agent s preferene (e.g., u(x, y) = log(x + y)) and the distribution of ( x, ỹ) (e.g., ( x, ỹ) is joint-normal distributed), then we an simply sign φ. Generally, however, we have only partial information on preferenes (e.g., risk aversion) and distributions of random variables (e.g., affiliation). The rules of ovariane between funtions employ partial information on the agent s preferenes and the random variables and, therefore, they sign the orrelation. We reall a well-developed onept for ovariane between monotoni funtions. Definition (Esary et al. 1967, p1466) ( x, ỹ) is said to be assoiated if for all funtions α, β whih are inreasing in eah omponent, Cov(α( x, ỹ), β( x, ỹ)) 0. (3) Suppose that all that we know is that the agent risk averse in x (u (2,0) < 0) and orrelation averse (u (1,1) 0) 2, then the above definition and (2) imply that, the equity premium, φ, is positive when ( x, ỹ) is assoiated. Thus, we have partial information on the sign of the equity premium. Reently, prudene and temperane, as well as even higher-order risk attitudes, have beome important both in theoretial and empirial work. Eekhoudt and Shlesinger (2006) examine 1 u (k 1,k 2 ) denotes the (k 1, k 2 )th ross derivative of, that is, u (k 1,k 2 ) = k 1 +k 2 u(x, y). x k 1 y k 2 2 For the interpretation of orrelation averse, please see Rihard (1975), Epstein and Tanny (1980), Eekhoudt et al. (2007). 1
these higher-order risk attitudes toward partiular lasses of lottery pairs. They show how higher-order risk attitudes an be fully haraterized by a preferene relation over these lotteries. They all suh preferene as risk apportionment and show that, if preferenes are defined in an expeted utility framework with differentiable utility, the diretion of preferene for a partiular lass of lottery pairs is equivalent to signing the higher-order derivative of the utility funtion. Sine then, the onept of risk apportionment has been extended by Eekhoudt et al. (2007), Tsetlin and Winkler (2009), Jokung (2011) and Denuit et al. (2013) to higher orders of multivariate risk attitudes. Eekhoudt (2012) and Eekhoudt and Shlesinger (2012) provide exellent surveys on this line of researh. The agents with higher-order risk attitudes might be somewhat similar when they sign the orrelation of their utility funtions with some types of random variables. However, there is relatively little theory on how to sign ovariane between funtions beyond monotoniity. This paper onentrates on ovariane rule that an be applied to higher-order risk attitudes. Although assoiation is one of the basi onditions that desribe positive dependene, when we sign orrelation, finding other onditions of dependene whih are weaker than assoiation is still useful. Generally speaking, as we shall see in this paper, the more information we know for the agent s preferene, the weaker dependene of random variables we need to sign ovariane. We start by building a theory of ovariane between funtions with higher-order derivatives. Then there are three appliations to illustrate it, inluding risk aversion in the presene of another risk, bivariate stohasti orderings and justifying the first-order approah to moral hazard prinipal-agent problems. The paper is organized as follows. Setions 2 proposes a ovariane rule between funtions with higher-order derivatives. Setion 3 ontains three appliations of the ovariane rule in an expeted-utility framework. Setion 4 onludes the paper. 2 A Sign for Covariane of funtions with higher-order derivatives Eekhoudt and Kimball (1992) propose a positive dependene onept to study the demand for insurane in the presene of a bakground risk: The distribution of bakground risk onditional upon a given level of insurable loss deteriorates in the sense of third-order stohasti dominane as the amount of insurable loss inreases. Denote by F (y x) the onditional distribution of 2
ỹ given x = x. The following dependent struture extends Eekhoudt and Kimball (1992) s onept to N th -order stohasti dominane. Definition Define F 1 (y x) = F (y x) and F n+1 (y x) = y F n (t x)dt. We say that ỹ is N th -order stohasti dominane dependent on x (N th SDD(ỹ x)) if (i) F N (y x ) F N (y x) for all y and x x; (ii) F n (d x ) F n (d x) for all y, x x and n = 1,...N 1. F SDD(ỹ x) (N=1) states that x inreases ỹ via first-order stohasti dominane (FSD). SSDD(ỹ x) (N=2) means that x inreases ỹ via seond-order stohasti dominane (SSD) whih inludes a mean-preserving inrease in risk as defined by Rothshild and Stiglitz (1970). T SDD(ỹ x) (N=3) implies that x inreases ỹ via third-order stohasti dominane (TSD). Combing any SSD shift with any inrease in downside risk as defined by Menezes et al. (1980) yields a TSD. Eekhoudt and Shlesinger (2008) explain how the different metris in the extant empirial literature an be put into a stohasti dominane framework. Now we are going to sign Cov(α( x, ỹ), β( x, ỹ)) by N th SDD(ỹ x) and the signs of higherorder partial derivatives of α and β. Denuit et al. (1999) introdue the lass U (s1,s 2 ) iv of the regular (s 1, s 2 )-inreasing onave funtions defined as the lass of all the funtions u, for s 1 and s 2 are positive integers, suh that ( 1) k 1+k 2 +1 u (k 1,k 2 ) 0 for all k 1 = 0, 1,..., s 1, k 2 = 0, 1,..., s 2 with k 1 + k 2 1. Many higher-order risk attitudes are in U (s1,s 2 ) iv lass. For more details, we refer the interested readers to Eekhoudt et al. (2007). The following proposition extends Esary et al. s result (1967, Theorem 4.3) from monotoni funtions to funtions with higher-order derivatives. Proposition 2.1 The following statements are equivalent. (i) Cov(α( x, ỹ), β( x, ỹ)) 0 (4) for all α and β suh that α (1,0) 0, β (1,0) 0, α U (0,I) iv and β U (0,J) iv ; (ii) N th SDD(ỹ x) where N = min(i, J). Proof See appendix. Q.E.D. When x is the wealth, ỹ is another risk that annot be hedged and β(x, y) = u(x, y) is the agent s utility funtion, Cov( x, u( x, ỹ)) 0 implies wealth and utility go in the same diretion. 3
For example, Proposition 2.1 shows that: (i) when ( x, ỹ) is F SDD(ỹ x), for a monotoni agent (u (1,0) 0 and u (0,1) 0), wealth and utility move in the same diretion; (ii) when ( x, ỹ) is SSDD(ỹ x), for an agent who is monotoni in x and y, and risk averse in y (u (1,0) 0, u (0,1) 0 and u (0,2) 0), wealth and utility go into the same diretion; (iii) when ( x, ỹ) is T SDD(ỹ x), for an agent who is monotoni in x and y, risk averse and prudent in y (u (1,0) 0, u (0,1) 0, u (0,2) 0 and u (0,3) 0), wealth and utility move together. When x is the GDP per apita, ỹ is the bakground risk, and β(x, y) = u (1,0) (x, y) is the agent s marginal utility funtion. Cov( x, u (1,0) ( x, ỹ)) 0 implies GDP per apita and marginal utility move in the opposite diretions. Proposition 2.1 and (2) show that, if one of the following onditions is satisfied, then the equity premium is positive: (i) F SDD(ỹ x), the agent is risk averse in x and orrelation averse (u (2,0) < 0 and u (1,1) 0); (ii) SSDD(ỹ x), the agent is risk aversion in x, orrelation averse and ross-prudent in x (u (2,0) < 0, u (1,1) 0 and u (1,2) 0); (iii) T SDD(ỹ x), the agent is risk aversion in x, orrelation averse, ross-prudent and rosstemperate in x (u (2,0) < 0, u (1,1) 0, u (1,2) 0 and u (1,3) 0). For more disussions of orrelation averse, ross-prudent and ross-temperate, please see Eekhoudt et al. (2007). To lose this setion, we note that Proposition 2.1 also adds knowledge of omparing stohasti dependene strutures. From Proposition 2.1 and the definition of assoiation, we know that ( x, ỹ) is assoiated implies N th SDD(ỹ x). Proposition 2.1 an also help us to show that N th SDD(ỹ x) implies x is positive N th ED on ỹ (The disussion of the relationship between N th SDD(ỹ x) and N th ED is given in Appendix). Now we an state formally the relationships between the various onepts of bivariate dependene: ( x, ỹ) is assoiated (5) N th SDD(ỹ x) x is positive N th ED on ỹ ( x, ỹ) is positive orrelated. 3 Some eonomi onsequenes This part of the paper demonstrates the usefulness of Proposition 2.1 for eonomi and finanial appliations suh as risk aversion in the presene of another risk, bivariate stohasti dominane and justifying the first-order approah to moral hazard prinipal-agent problems. 4
3.1 Risk aversion in the presene of another risk An agent is risk averse if she prefers the expeted value of a random amount of money than the random amount of money. Finkelshtain et al. (1999) study the question of risk aversion in the presene of bakground risks in other argument of the utility funtion. They propose the following result 3 ; Theorem 3.1 (Finkelshtain et al. 1999, part (a) and (b) of Theorem 2) The following statements are equivalent. (i) Eu( x, ỹ) Eu(E x, ỹ) for ( x, ỹ) suh that E( x ỹ) is inreasing in y; (ii) u (2,0) 0 and u (1,1) 0. Proposition 2.1 an extend risk aversion in the presene of bakground risks to higher-order risk attitudes. We propose the following result. Proposition 3.2 Suppose N th SDD(ỹ x), u (2,0) 0 and u (1,0) U (0,N) iv, then Eu( x, ỹ) Eu(E x, ỹ). (6) Proof See appendix. Q.E.D. The above proposition shows that, if one of the following onditions is satisfied, then an agent is risk averse for x in the presene of ỹ: (i) she is risk averse in x (u (2,0) 0), orrelation averse (u (1,1) 0) and F SDD(ỹ x); (ii) she is risk averse in x (u (2,0) 0), orrelation averse u (1,1) 0, ross-prudent in x ( u (1,2) 0) and SSDD(ỹ x); (iii) she is risk averse in x (u (2,0) 0), orrelation averse u (1,1) 0, ross-prudent and ross-temperate in x (u (1,2) 0 and u (1,3) 0), and T SDD(ỹ x). Propositions 3.1 and 3.2 study risk aversion in the presene of bakground risks for different stohasti dependene strutures. In the following two examples we will apply Proposition 3.2 to optimal of savings and health investments problems studied by Denuit et al. (2011). The appliations onsider an agent faing a finanial risk in the presene of a non-hedged bakground risk suh as health or environmental risk. A deision has to be made about the amount of an investment (in the finanial dimension) resulting in a future benefit either in the same dimension (savings) or in the other dimension (environmental quality or health improvement). The stohasti dependene strutures proposed here are different to Denuit et al.(2011) s. 3 In statistis literature, the similar result is alled as Bakhtin s Lemma (Bulinski and Shashkin 2007, Lemma 1.11; Rao 2012, Theorem 1.2.31). 5
Example The agent s objetive is to selet the optimal amount of savings (s) to be transferred from period 0 to period 1. The hoie s is made in order to maximize total utility U defined as U(s) = u 0 (x 0 s, h 0 ) + 1 1 + ρ Eu 1(s(1 + r) + x, h) (7) where ρ is the subjetive disount rate and r is the rate of return. savings s is determined by The optimal amount of u (1,0) 0 (x 0 s, h 0 ) = 1 + r 1 + ρ Eu(1,0) 1 (s (1 + r) + x, h) (8) Define s x,e as the solution of (8) with ( x, E h) substituted for ( x, h), and s E,h as the solution of (8) with (E x, h) substituted for ( x, h). Proposition 3.2 implies the following result. Proposition 3.3 (i) u (2,0) 0 0, u (2,0) 1 0, u (1,2) 1 0, u (1,1) 1 U (N,0) iv and N th SDD( x h) s s x,e ; (ii) u (2,0) 0 0, u (3,0) 1 0, u (2,0) 1 U (0,N) iv and N th SDD( h x) s s E,h. Proof See appendix. Q.E.D. The above proposition proposes the stohasti dependene strutures between the two risks and higher-risk attitudes whih imply the inrease of optimal amount of savings. Example The agent hooses how muh of resoures x 0 is to be devoted to an investment a that will improve his future health by an amount ma, where m represents the produtivity of the urrent monetary sarifie expressed in units of the other attribute. The hoie a is made in order to maximize total utility V defined as V (a) = u 0 (x 0 a, h 0 ) + 1 1 + ρ Eu 1( x, h + a) (9) The optimal amount of investment a is determined by u (1,0) 0 (x 0 a, h 0 ) = m 1 + ρ Eu(0,1) 1 ( x, h + ma ) (10) Define a x,e as the solution of (10) with ( x, E h) substituted for ( x, h), and a E,h as the solution of (10) with (E x, h) substituted for ( x, h). Proposition 3.4 (i) u (2,0) 0 0, u (0,3) 1 0, u (0,2) 1 U (N,0) iv and N th SDD( x h) a a x,e ; (ii) u (2,0) 0 0, u (0,2) 1 0, u (2,1) 1 0, u (1,1) 1 U (0,N) iv and N th SDD( h x) a a E,h. Proof See appendix. Q.E.D. Proposition 3.4 shows under whih onditions optimal investment for health (environmental) improvement is reahed. 6
3.2 A lass of bivariate stohasti orderings Stohasti dominane is a very useful tool in various areas of eonomis and finane. It has been studied extensively in the univariate ase (e.g. Hadar Russell (1969) and Hanoh and Levy (1969)). Denuit et al. (2013) extend the univariate ase to multivariate N th -degree onave (onvex) stohasti dominane and N th -degree risk. In this setion we use a stohasti order that an be related to harateristis suh as higher-order risk attitudes. We study bivariate stohasti dominane for this stohasti order. Let ( x 1, ỹ 1 ) and ( x 2, ỹ 2 ) be two 2-dimensional random vetors with density funtions f and g. Sine Eu( x 1, ỹ 1 ) = = b d a b d a u(x, y)f(x, y)dxdy (11) f(x, y) u(x, y) g(x, y)dxdy g(x, y) = E[u( x 2, ỹ 2 ) f( x 2, ỹ 2 ) g( x 2, ỹ 2 ) ] = E[u( x 2, ỹ 2 )] + Cov[u( x 2, ỹ 2 ), f( x 2, ỹ 2 ) g( x 2, ỹ 2 ) ], we have the following onlusion: If ( x 2, ỹ 2 ) is assoiated and f g is inreasing in x and y, then Eu( x 1, ỹ 1 ) E[u( x 2, ỹ 2 )] (see e.g., Shaked and Shanthikumar 2007, Theorem 6.B.8). We an apply Proposition 2.1 to (11) to get: Proposition 3.5 The following statements are equivalent. (i) Eu( x 1, ỹ 1 ) E[u( x 2, ỹ 2 )] for all u, f and g suh that u (1,0) 0, ( f g )(1,0) 0, u U (0,I) iv, ( f g ) U (0,J) iv; (ii) N th SDD(ỹ 2 x 2 ) where N = min(i, J). The advantage of above the proposition is that it extends the bivariate stohasti ordering from assoiated random variables to N th SDD random variables, while the ost is that it requires more restritions on the higher-order partial derivatives of u and f g. For example, when I = J = 2, Proposition 3.5 implies that, Eu( x 1, ỹ 1 ) E[u( x 2, ỹ 2 )] if u is monotoni (u (1,0) 0 and u (0,1) 0) and risk averse in y (u (0,2) 0), f g is inreasing in x and y, onave in y, and SSDD(ỹ x). In the following two examples, we re-examine the optimal of savings and health investments problems again by Proposition 3.5. We suppose f and g are density funtions of ( x, h) and ( x, h ) respetively 7
Example We onsider (7) again. Define s as the solution of (8) with ( x, h ) substituted for ( x, h). Proposition 3.5 implies the following result. Proposition 3.6 u (2,0) 0 0, u (2,0) 1 0, ( f g )(1,0) 0, u (1,0) 1 U (0,I) iv, ( f g ) U (0,J) iv and N th SDD( h x ) where N = min(i, J) s s. Proof See appendix. Q.E.D. Example We onsider (9) again. Define a as the solution of (10) with ( x, h ) substituted for ( x, h). We obtain the following result from Proposition 3.5. Proposition 3.7 u (2,0) 0 0, u (0,2) 1 0, u (1,1) 1 0, ( f g )(1,0) 0, u (0,1) 1 U (0,I) iv, ( f g ) U (0,J) iv and N th SDD( h x ) a a. Proof See appendix. Q.E.D. The above two examples demonstrate the usefulness of Proposition 3.5 for deriving omparative stati effets of hanges of risk. 3.3 Justify the first-order approah to bi-signal prinipal-agent problems Suppose an agent hooses an effort level a 0, and she has a von NeumannVMorgenstern utility funtion u(s) a, where s is the agents monetary payoff and u(.) is a stritly inreasing funtion. Let s = s(x, y) be the funtion, hosen by the prinipal, speifying her payment to the agent as a funtion of the signal ( x, ỹ) with probability density funtion f(x, y a). The agents expeted payoff is U(a) = b d a u(s(x, y))f(x, y a)dxdy a, (12) It is well known that, one way to guarantee that the First-order-approah (FOA) is valid is to show that the U(a) is onave, given the wage ontrat. The suffiient onditions for this requirement are the monotone likelihood ratio ondition (MLRC) and the onavity of the distribution funtion ondition (CDFC) (Rogerson, 1985). However, most of the distribution funtions do not have the CDFC property. Jewitt (see Conlon 2009 p274-275 ) has suggested another suffiient onditions for the onavity of U(a). Define H(x, y) = u(s(x, y)). We an obtain (Conlon 2009, 274-275): d 2 da 2 U(a) = Cov(H( x, ỹ)), f aa( x, ỹ a) ), (13) f( x, ỹ a) 8
whih implies that, ( x, ỹ) is affiliated, H(x, y) and faa(x,y a) f(x,y a) suffiient onditions for d2 da 2 U(a) 0. are inreasing funtions are Applying Proposition 2.1 to (13), we an propose a suffiient and neessary ondition for the onavity of U(a). Proposition 3.8 The following statements are equivalent. (i) d 2 U(a) 0 (14) da2 for all H(x, y) and f(x, y a) suh that H (1,0) 0 ( f aa f )(1,0) 0, H U (0,I) iv and f aa f U (0,J) iv ; (ii) N th SDD(ỹ x) where N = min(i, J). Sine ( x, ỹ) is affiliated (15) ( x, ỹ) is assoiated N th SDD(ỹ x), Proposition 3.8 extends our knowledge of the validity of the FOA in the prinipal-agent problems from affiliated bi-signal to N th SDD bi-signal. 4 Conlusion The rule of ovariane has been widely studied in the monotoni funtions ase. The monotoni funtions are onsistent with some basi preferene onditions suh as monotoniity. However, many higher-order risk attitudes, that are easy to explain to deision makers, are beyond the monotoni funtions. This paper fills this gap by linking higher-order risk attitudes to the sign of orrelation. We present some eonomi and finanial appliations that are useful in applying the result. 9
5 Appendix 5.1 Proof of Proposition 2.1 (ii) (i): From Esary et al. (1967, p1471), we know that Cov[α( x, ỹ), β( x, ỹ)] (16) = E{Cov[α( x, ỹ), β( x, ỹ) x]} + Cov{E[α( x, ỹ) x], E[β( x, ỹ) x]}. α (0,1) 0 and β (0,1) 0 imply Cov[α(x, ỹ), β(x, ỹ)] 0, and hene E{Cov[α( x, ỹ), β( x, ỹ) x]} 0. It is well known that if de[α( x,ỹ) x] dx 0, and de[β( x,ỹ) x] dx 0, then Cov{E[α( x, ỹ) x], E[β( x, ỹ) x]} 0, (17) From E[α( x, ỹ) x] = d α(x, y)df (y x) and note that F (d x) = 1 and F ( x) = 0 for all x F x (d x) = F x ( x) = 0, we have de[α( x, ỹ) x] dx = = d d d α (1,0) (x, y)df (y x) + α (1,0) (x, y)df (y x) d α(x, y)df x (y x) (18) α (0,1) (x, y)f x (y x)dy. Then, applying integration by parts to d α(0,1) (x, y)f x (y x)dy, we obtain, for I 1 (we define 0i=1 (.) = 0), = d I 1 i=1 α (0,1) (x, y)f x (y x)dy (19) ( 1) i+1 α (0,i) (x, d)f i+1 x d (d x) + ( 1) I+1 α (0,I) (x, y)fx I (y x)dy. So, α (1,0) 0, α U (0,I) iv and I th SDD(ỹ x) imply and hene de[α( x,ỹ) x] dx 0. d α (0,1) (x, y)f x (y x)dy 0, By the same approah we an show that, β (1,0) 0, β U (0,J) iv and J th SDD(ỹ x) imply de[β( x,ỹ) x] dx 0. Finally, we an onlude that, α (1,0) 0, β (1,0) 0, α U (0,I) iv, β U (0,J) iv and N th SDD(ỹ x) where N = min(i, J), imply (17). (i) (ii): We prove this by ontraditions. 10
Suppose that F N (y x 2 ) > F N (y x 1 ) for x 2 x 1. Due to the ontinuity of F N (y x), we have F N x (y x) > 0 for some x (x 1, x 2 ). Choose the following funtion: then d d d β(x, y) = dt 1 dt 2... 1 [x1,x 2 ](t N )dt N, (20) y t 1 t N 1 β (1,0) (x, y) = 0, (21) and d d d β (0,k 1) (x, y) = ( 1) k dt k dt k+1... 1 [x1,x 2 ](t N )dt N, for k = 2,..., N 1, (22) y t k t N 1 d β (0,N 1) (x, y) = ( 1) N 1 [x1,x 2 ](t N )dt N, (23) y β (0,N) (x, y) = ( 1) N+1 1 [x1,x 2 ](y). (24) Thus, ( 1) N+1 β(0,n) 0, and β (0,k 1) (x, d) = 0 for k = 2,..., N. From (18) and (19) we get de[ β( x, ỹ) x] dx d = ( 1) N+1 β (0,N) (x, y)fx N (y x)dy < 0, (25) and hene, form (16), we obtain Cov[ x, β( x, ỹ)] (26) = E{Cov[ x, β( x, ỹ) x]} + Cov{E[ x x], E[ β( x, ỹ) x]} = Cov{ x, E[ β( x, ỹ) x]} < 0 whih is a ontradition. Now suppose F j (d x 2 ) > F j (d x 1 ) for x 2 > x 1 and 1 j N 1. Due to the ontinuity of F j (d x), we have F j x(d x) > 0 for some x (x 1, x 2 ). Choose the following funtion: then β(x, y) = ( 1)j (j 1)! (y d)j 1 (27) β (1,0) (x, y) = 0, (28) and β (0,k 1) (x, y) = ( 1) k β(0,k 1) (x, y) = ( 1) j (j k)!(y d)j k k 0 k > j 1 (j k)!(d y)j k k = 2, 3,.., j 0 k > j, = 2, 3,.., j (29) (30) 11
and hene, ( 1) k β(0,k 1) (x, y) 0 and ( 1) k β(0,k 1) (x, d) = Therefore, (18) and (19) we get 0 k = 2, 3,.., j 1 1 k = j 0 k > j. (31) de[ β( x, ỹ) x] dx = F j x(d x) < 0, (32) and hene, form (16), we obtain Cov[ x, β( x, ỹ)] (33) = E{Cov[ x, β( x, ỹ) x]} + Cov{E[ x x], E[ β( x, ỹ) x]} = Cov{ x, E[ β( x, ỹ) x]} < 0 whih is a ontradition. 5.2 Various onepts of bivariate dependene The expetation dependene onept is introdued by Wright (1987). Li (2011) proposes the higher-order extensions. The definition is realled next. Define ED 1 ( x y) = [E x E( x ỹ y)] 0 for all y, (34) then x is positive first-degree expetation dependent on ỹ. Define ED 2 ( x y) = y ED 1( x t)f Y (t)dt, repeated integrals defined by ED N ( x y) = y Definition If ED K ( x d) 0, for k = 2,..., N 1 and ED N 1 ( x t)dt, for N 3. (35) ED N ( x y) 0 for all y [, d], (36) then x is positive N th -order expetation dependent (N th ED) on ỹ. Expetation dependene and its higher-order extensions have been shown to play a key role in many eonomi and finanial problems, suh as asset alloation (Wright, 1987; Hadar and Seo, 1988), demand for risky asset under bakground risk (Li, 2011), first-order risk aversion 12
(Dionne and Li 2011, 2013) and asset priing (Dionne et al., 2013). It is interesting to find the relationship between N th SDD(ỹ x) and ED N ( x y). Denuit et al. (2013, Equation (19)) propose the following result. 1 ED N ( x Y ) = Cov[ x, (Y ỹ)n 1 + ]. (37) (N 1)! Define g(x, y) = (Y y) N 1 +, then g (1,0) = 0 and ( 1) i g (0,i) 0 for i = 1, 2,..., and from Proposition 2.1 we obtain: Proposition 5.1 N th SDD(ỹ x) x is positive N th ED on ỹ Proposition 5.1 shows that, when ( x, ỹ) is N th SDD(ỹ x), the results derived in above appliations of N th ED (Wright, 1987; Hadar and Seo, 1988; Li, 2011; Dionne and Li 2011; Dionne and Li 2013; Dionne et al., 2013) still hold. 5.3 Proof of Proposition 3.2 We are going to use an approah motivated by Bulinski and Shashkin (2007, Lemma 1.11) and Rao (2012, Theorem 1.2.31). Let Then β(x, y) = u (1,0) (E x, y) u(x,y) u(e x,y) x E x x E x x = E x (38) β (1,0) (x, y) = u(1,0) (x, y)(x E x) [u(x, y) u(e x, y)] (x E x) 2 0 (sine u (2,0) (x, y) 0) (39) and, for i = 1,..., N, ( 1) i β (0,i) (x, y) = ( 1) i u(0,i) (x, y) u (0,i) (E x, y) x E x 0 (sine ( 1) i u (1,i) (x, y) 0). (40) Hene, from Proposition 2.1, we obtain Cov( x, β( x, ỹ)) 0 for N th SDD(ỹ x). Sine Cov( x, β( x, ỹ)) (41) = Cov( x E x, β( x, ỹ)) = E[β( x, ỹ)( x E x)] E[β( x, ỹ)]e[( x E x)] = E[u( x, ỹ) u(e x, ỹ)], we obtain E[u( x, ỹ) u(e x, ỹ)] 0. 13
5.4 Proof of Proposition 3.3 (i) Beause U is onave in s, s s x,e if and only if U (s ), with ( x, E h) substituted for ( x, h), is negative. In other words, there will be a preautionary demand for savings if and only if U (s ) = u (1,0) 0 (x 0 s, h 0 ) + 1 + r 1 + ρ Eu(1,0) 1 (s (1 + r) + x, E h) (42) = 1 + r 1 + ρ [Eu(1,0) 1 (s (1 + r) + x, E h) Eu (1,0) 1 (s (1 + r) + x, h)] 0, where the seond equality is obtained by using Proposition 3.2. (ii) Beause U is onave in s, s s E,h if and only if U (s ), with (E x, h) substituted for ( x, h), is negative. In other words, there will be a preautionary demand for savings if and only if U (s ) = u (1,0) 0 (x 0 s, h 0 ) + 1 + r 1 + ρ Eu(1,0) 1 (s (1 + r) + E x, h) (43) = 1 + r 1 + ρ [Eu(1,0) 1 (s (1 + r) + E x, h) Eu (1,0) 1 (s (1 + r) + x, h)] 0, where the seond equality is obtained by using Proposition 3.2. 5.5 Proof of Proposition 3.4 (i) Beause V is onave in a, a a x,e if and only if V (a ), with ( x, E h) substituted for ( x, h), is negative. In other words, there will be a preautionary investment if and only if V (a ) = u (1,0) 0 (x 0 a, h 0 ) + m = 0, 1 + ρ Eu(0,1) 1 ( x, E h + ma ) (44) m 1 + ρ [Eu(0,1) 1 ( x, E h + ma ) Eu (0,1) 1 ( x, h + ma )] where the seond equality is obtained by using Proposition 3.2. (ii) Beause V is onave in a, a a E,h if and only if V (a ), with (E x, h) substituted for ( x, h), is negative. In other words, there will be a preautionary investment if and only if V (a ) = u (1,0) 0 (x 0 a, h 0 ) + m = 0, 1 + ρ Eu(0,1) 1 (E x, h + ma ) (45) m 1 + ρ [Eu(0,1) 1 (E x, h + ma ) Eu (0,1) 1 ( x, h + ma )] where the seond equality is obtained by using Proposition 3.2. 14
5.6 Proof of Proposition 3.6 Beause U is onave in s, s s if and only if U (s ), with ( x, h ) substituted for ( x, h), is positive. In other words, there will be a preautionary demand for savings if and only if U (s ) = u (1,0) 0 (x 0 s, h 0 ) + 1 + r 1 + ρ Eu(1,0) 1 (s (1 + r) + x, h ) (46) = 1 + r 1 + ρ [Eu(1,0) 1 (s (1 + r) + x, h ) Eu (1,0) 1 (s (1 + r) + x, h)] 0, where the seond equality is obtained by using Proposition 3.5. 5.7 Proof of Proposition 3.7 Beause V is onave in a, a a if and only if V (a ), with ( x, h ) substituted for ( x, h), is positive. In other words, there will be a preautionary investment if and only if V (a ) = u (1,0) 0 (x 0 a, h 0 ) + m = 0, 1 + ρ Eu(0,1) 1 ( x, h + ma ) (47) m 1 + ρ [Eu(0,1) 1 ( x, h + ma ) Eu (0,1) 1 ( x, h + ma )] where the seond equality is obtained by using Proposition 3.5. 6 Referenes Bulinski, A., Shashkin A., 2007, Limit Theorems for Assoiated Random Fields and Related, Systems. World Sientifi, Singapore. Conlon, J. R., 2009. Two new onditions supporting the first-order approah to multisignal prinipal-agent problems, Eonometria 77, 249-278. Cuadras, C.M. 2002, On the ovariane between funtions, Journal of Multivariate Analysis 81, 19-27. Denuit, M. M., Eekhoudt, L., Menegatti, M., 2011. Correlated risks, bivariate utility and optimal hoies. Eonomi Theory, 46, 39-54. Denuit, M., Eekhoudt, L., Tsetlin, I., Winkler, R., 2013. Multivariate onave and onvex stohasti dominane. In: Risk Measures and Attitudes, pp. 11-32. Springer-Verlag, London. 15
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