Prudence with Multiplicative Risk

Transcription

1 EDHEC-isk Institute promenade des Anglais Nice Cedex 3 Tel.: +33 (0) research@edhec-risk.com Web:.edhec-risk.com Prudence ith Multiplicative isk December 2008 Octave Jokung Professor, EDHEC Business School

2 Abstract In this paper e analyze the conditions under hich the presence of a multiplicative background risk induces a more prudent behavior. We sho that the results from Kimball (1990) concerning the convexity of the marginal utility are no longer sufficient ith multiplicative risk. An agent is multiplicative risk prudent hen the coefficient of relative prudence is greater than to. We introduce the concept of quintessence in order to guarantee the decrease of relative temperance. Both decreasing relative prudence and decreasing relative temperance are sufficient to guarantee more prudent behavior ith a multiplicative risk-prudent agent. Nevertheless, the convexity of the relative prudence combined ith the decrease (respectively increase) of the relative prudence implies more prudent behavior hen the agent is multiplicative risk prudent (respectively risk imprudent). Surprisingly, the derived utility function inherits some properties of the direct utility function and the presence of the multiplicative background risk preserves comparative riskprudent behavior under certain conditions. Our results apply to savings and portfolio selection. For example, a risk-prudent agent ill loer his or her illingness to hold risky assets hen he or she faces a multiplicative background risk if his or her relative risk aversion is convex and decreases. Keyords: additive background risk, multiplicative background risk, risk vulnerability, risk aversion, prudence, temperance JEL: D81, G11 EDHEC is one of the top five business schools in France. Its reputation is built on the high quality of its faculty and the privileged relationship ith professionals that the school has cultivated since its establishment in EDHEC Business School has decided to dra on its extensive knoledge of the professional environment and has therefore focused its research on themes that satisfy the needs of professionals. 2 EDHEC pursues an active research policy in the field of finance. The EDHEC-isk Institute carries out numerous research programmes in the areas of asset allocation and risk management in both the traditional and alternative investment universes. Copyright 2009 EDHEC

3 1. Introduction This paper addresses the question of hether the presence of a multiplicative background risk induces more risk-prudent behavior. What is the effect of a unitary-mean multiplicative background risk on the feeling of prudence? Intuitively, the presence of a background risk ill affect behavior. In fact, Franke, Stapelton and Schlesinger (2006) develop the concept of multiplicative risk aversion and they sho that the effect of the background risk depends on the type of risk ith hich the individuals are confronted. Attitudes toards risk differ depending on hether the agent faces additive risk or multiplicative risk. Hoever, they undertake their analysis by focusing on the alteration of risk aversion. Here, e deal ith risk prudence. That is, e focus on situations, such as savings or insurance, in hich marginal effects are great. Prudence is related to three concepts: precautionary savings, donside risk and preference for positive skeness. They has been special attention on additive risks in multi-risk problems that have been studied only recently. Kihlstrom, omer and Williams (1981), Pratt and Zeckhauser (1987) and Pratt (1988) make significant contributions. Kimball (1990) defines the concept of prudence or additive risk prudence, hich implies that the presence of some additive background risk increases the optimal savings rate. This concept is equivalent to the non-negativity of the third derivative of the utility function or to the convexity of the marginal utility. Kimball (1993) defines, in an expected utility frameork, the property of standard risk-aversion hich implies that the presence of some background risk reduces the optimal investment in a risky security ith an independent return. Eeckhoudt, Gollier and Schlesinger (1996) study the effect of independent exogenous risk on the optimal risk-taking behavior toards an endogenous risk. They examine conditions on preferences under hich some changes in the distribution of the background ealth entail more risk-averse behavior toards endogenous risk. Eeckhoudt and Kimball (1992) study ho a background risk affects optimal demand for insurance against an independent or dependent insurable risk. Eeckhoudt, Gollier and Schlesinger (1996), Gollier and Pratt (1996), Kimball (1993) and Pratt and Zeckhauser (1987), among others, sho that, under fairly general conditions on the utility function, investors reduce their holdings of risky assets hen facing an increase in the background risk. All these studies implement the derivatives of higher order of the utility function. Caballé and Pomansky (1996) present the notion of mixed risk aversion in order to sign the derivatives of the utility function and their results allo us to reconsider the relationship beteen those derivatives and the moments (variance, skeness, kurtosis). Eeckhoudt and Schlesinger (2006) introduce a ne but a more general concept, the concept of risk apportionment of any order. This ne concept allos us to recover all the former concepts in risk theory and also permits us to characterize those concepts ith preference relation over specified lotteries hich consist of to basic dislikes: a certain reduction in ealth adding a zero-mean independent noise random variable to the distribution of ealth. All these papers deal ith additive risk. They pay little attention to multiplicative risk. Franke, Stapelton and Schlesinger (2006) are the first to focus on the effect of a multiplicative risk but they restrict themselves to the feeling of risk aversion. They find out conditions under hich the presence of the multiplicative background risk leads to more cautious behavior. Our aim in this paper is to analyze the more prudent behavior of an agent facing a multiplicative risk. Under hich conditions does the agent behave more prudently ay in the presence of the multiplicative background risk? And hat are the implications on savings decision and risky assets holdings? Dealing ith multiplicative risk, e sho that the convexity of the marginal utility function is no longer sufficient to guarantee risk-prudent behavior in the presence of a multiplicative background risk. We must compare the coefficient of relative prudence ith a benchmark value equal to to. More precisely, an agent is said to be multiplicative risk prudent if he/she increases his/her savings hen facing multiplicative background risk. We also introduce the concept of risk apportionment of order five hich e call quintessence to guarantee the decrease of the feeling of temperance. Quintessence is to temperance as temperance is to prudence. It appears that the results obtained ith additive risks are ill suited to multiplicative risks. 3

4 Our paper contributes to the literature devoted to the attitude toard risk and the decision making in presence of risks in several ays. First, e provide a comprehensible definition of prudence hen facing a multiplicative risk to complement the ell knon definition of Kimball (1990). Second, e give the conditions under hich in the presence of a multiplicative background risk an agent behaves in a more risk-prudent manner. That is, e compare the degrees of relative prudence in the presence and in the absence of the multiplicative risk. Third, e sho the link beteen decreasing relative prudence of the derived utility function and decreasing relative prudence of the direct utility function. Fourth, e point out conditions under hich the comparative risk prudence order is preserved. Fifth, e consider savings demand and portfolio selection and e apply our results. The paper is organized as follos. Section 2 presents the concepts governing the attitudes toard risk. Section 3 is devoted to the definition of multiplicative risk prudence. In section 4 e introduce the affiliated utility function. The next section gives the conditions under hich an agent facing a multiplicative background risk behaves in a more risk-prudent manner. Section 6 deals ith the preservation of the decrease of relative prudence. In section 7 e analyze the preservation of the ordering by risk prudence ith multiplicative background risk. Sections 8 and 9 apply our result to savings and portfolios. The last section concludes. 2. From risk aversion to quintessence Modelling the behavior of investors facing uncertainty is generally done ith the concept of risk aversion, hich is related to the concavity of the utility function. The degree of risk aversion is given by: ʹʹ A u () = uʹ u () An individual ill be risk averse if the second derivative of the utility function is negative ( uʹʹ 0 ). Hoever, the concept of risk aversion has some limits and it is not appropriate hen dealing ith decision-making under uncertainty hen the uncertainty affects marginal utility rather than utility. To solve this problem, Kimball (1990) uses the notion of absolute prudence to measure the propensity to prepare and to forearm oneself for uncertainty. The degree of absolute prudence is given by: P A u () = uʹʹ ʹʹʹ u () An individual is prudent or additively risk-prudent hen the third derivative of his or her utility function is positive ( uʹʹʹ 0 ). These to notions of absolute risk aversion and absolute prudence are very rich. When taken together, they correspond to the notion of standardness, hich is sufficient and necessary to see an increase in the demand for a second independent risk due to a risk that increases the marginal utility function. Decreasing absolute prudence makes it necessary for temperance to be higher than prudence, here the coefficient of absolute temperance is given by: T A u () = u (4) () uʹʹʹ An individual is temperate hen the fourth derivative of his or her utility function is negative ( u (4) 0 ). Temperance describes the desire to totally reduce the risk exposure. That is, the agent reduces an endogenous risk hen an exogenous risk increases. 4 Nevertheless, absolute risk aversion and absolute prudence sometimes fail to explain the behavior of an investor facing several sources of risk and in the event of a deterioration of those risks. Other notions, such as proper risk aversion and risk vulnerability can be used. Pratt and Zeckhauser

5 (1987) devise the concept of proper risk aversion to characterize utility functions such that an undesirable risk can never be made desirable by the presence of an independent undesirable risk. A risk is said to be undesirable if it decreases the expected utility. The utility functions hose successive derivatives alternate in sign exhibit proper risk aversion. Gollier and Pratt (1996) introduce the concept of risk vulnerability to model the fact that an investor reduces his or her risky assets hen facing an increase in an independent unfair background risk. That is, an undesirable risk can never be made desirable by an independent unfair risk. In our study, the coefficient of absolute temperance ill be made to decrease ith ealth. This condition ill imply that the coefficient of absolute temperance is loer than a ne coefficient, hich e call quintessence given as follos: Q A u () = u (5) () (4) u An individual exhibits quintessence hen the fifth derivative of his or her utility function is positive (u (5) 0 ). Eeckhoudt and Schlesinger (2006) call this coefficient risk apportionment of order 5. We call this coefficient quintessence because e are dealing ith the fifth derivative of the utility function and quinte means five and essence substantial nature. It is precious and e ant to protect it by any means. ecall that this ord comes from Latin quinta essentia hich means literally fifth essence. It is the fifth element after earth, fire, ater and air. Therefore, it ill come after non-satiety ( u ʹ 0 ), risk aversion ( u ʹʹ 0 ), prudence ( u ʹʹʹ 0 ) and temperance ( u (4) 0 ). Temperance is to quintessence as prudence is to temperance. We can notice that quintessence corresponds to the prudence of the second derivative of the utility function: Q A u () = ( uʹʹ) ʹʹʹ () = P A ( uʹʹ) ʹʹ uʹʹ () We can also say that quintessence represents the temperance of the opposite of the marginal utility: Q A u () = ( u ʹ)(4) A () = T ( uʹ) ʹʹʹ uʹ We get the relative counterpart of each coefficient by multiplying it by ealth. This is important because dealing ith multiplicative risk induces the use of relative coefficients: u () = uʹ ʹʹ u () ; P u () = uʹʹ ʹʹʹ u () ; T u () = u (4) () and Q u () = u (5) () (4) uʹʹʹ u We begin by the fact that relative prudence and relative temperance are generally both decreasing and e analyze the main consequences. Property 1: Decreasing relative prudence implies that relative temperance exceeds relative prudence plus one and decreasing relative temperance implies that relative quintessence exceeds relative temperance plus one. Proof. The derivative of relative prudence ith respect to ealth is: d uʹʹ ʹʹʹ u () = u ʹʹʹ() 1+ u (4) () d u ʹʹ() u ʹʹʹ() u ʹʹʹ() u ʹʹ() = P A u 1 T u + P u ( ) Therefore, decreasing relative prudence is equivalent to P A u 1 T ( u + P u ) 0. 5

6 Finally, an increase in ealth leads to a decrease in relative prudence if and only if T u 1+ P u. The same approach yields the result concerning the decrease of the coefficient of relative temperance. If e consider the decrease of the coefficients of relative risk aversion, relative prudence and relative temperance e end up ith the folloing property: Property 2: Decreasing relative risk aversion, decreasing relative prudence and decreasing relative temperance imply: (1) Q u T u P u u (2) P u 2, Tu 3 and Q u 4 if u 1 Proof. Decreasing relative risk aversion implies P u 1+ u, then P u u. Decreasing relative prudence implies T u 1+ P u and therefore T u P u. Finally, decreasing relative temperance implies Q u 1+ T u and therefore Q u T u. We get Q u T u P u u. Assume that u 1 and knoing that P u 1+ u, thus P u 2. And adding the fact that T u 1+ P u gives T u 3. Therefore, knoing that Q u 1+ T u leads to Q u 4. It is orth noting that the former property uses the benchmark values of our four coefficients: one for relative risk aversion, to for relative prudence, three for relative temperance and four for relative quintessence. That leads to sixteen classes of utility functions according to the position of the coefficients ith respect to their respective benchmark: Class(λ 1,λ 2,λ 3,λ 4 ) = { u such that ( u 1)λ 1 0;(P u 2)λ 2 0;(T u 3)λ 3 0;(Q u 4)λ 4 0} here λ i { 1,1 } For example, any element of Class(1,1,1,1) is such that: u 1, P u 2, T u 3 and Q u 4. That is, any element of this class exhibits simultaneously higher feelings of relative risk aversion, relative prudence, relative temperance and relative quintessence. With these sixteen classes, e break don the set of ell behaved utility functions according to their high or lo feeling into the four feelings. High means higher than the benchmark, hereas lo means loer than the same benchmark. 3. Multiplicative derived utility function In this section, e present the general model. Let us consider an individual facing a multiplicative risk 1+ x (or x ) such that E( x ) = 0 and 1+ x 0 hich affects his or her entire ealth and ith a ell behaved utility function u ( ( u ʹ () 0, u ʹʹ() 0, u ʹʹʹ 0) ). We define the multiplicative derived utility function associated ith u as follos: [ ] = Eu (1+ x ) û() The marginal utility derived from the situation is given by: { [ ]} ˆʹ u () = E (1+ x ) uʹ (1+ x ) The multiplicative risk positively affects the behavior of the agent if any variation of ealth has a greater effect in the presence of the risk or if the marginal utility is higher in the presence of the multiplicative background risk. We have the folloing definition: 6

7 Definition 1: u is multiplicative risk-prudent if and only if the marginal utility of û is greater than that of u. This formulation is analogous to that of Kimball (1990), ho defines the feeling of prudence in the presence of an additive risk as follos: v() = Eu( + ε) and E( ε) = 0 According to Kimball (1990), prudence is equivalent to the convexity of the marginal utility in order to guarantee that v ʹ() = Eu ʹ( + ε) u ʹ(). Therefore, an agent is prudent if adding a zeromean background risk to his or her second-period income leads to an increase in savings. This notion of prudence ill be called additive prudence. An agent ill be multiplicatively risk-prudent if multiplying his or her second period ealth by a unitary-mean background risk increases savings. In this situation, the marginal utility of û is greater than that of u if and only if ˆʹ u () = E {(1+ x ) uʹ[ (1+ x )]} u ʹ() or if and only if x u ʹ (x ) is convex in x. The last condition is equivalent to P u () = u ʹʹʹ() 2. Here, e deal ith multiplicative u ʹʹ() prudence. Let us consider the definition of prudence given by Eeckhoudt and Schlesinger (2006): Definition 2: An individual is said to be prudent if the lottery B 3 = [ k; ε ] is preferred to the lottery A 3 = [ 0; ε k ], here all outcomes of the lotteries have equal probability, for all initial ealth level, for all k and for all (zero-mean random variable) ε. In our frameork, e consider multiplicative risk rather than additive risk. Therefore, the former definition becomes: Definition 3: An individual is said to be (multiplicatively) prudent if the lottery B = [ k;(1+ x )] is preferred to the lottery A = [( k )(1+ x ); ], here all outcomes of the lotteries have equal probability, for all initial ealth level, for all k and for all unitary-mean random variable 1+ x In the expected utility frameork, the definition becomes: 1 2 Eu( + x ) Eu( k ) > 1 2 Eu( k + ( k ) x ) Eu() Or equivalently: Eu( + x ) Eu( k + ( k ) x ) > u() u( k ) We kno that: u( + x ) u( k + ( k )x ) = [ u(t + xt )] k and u() u( k ) = [ u(t )] k = u ʹ(t )dt k Therefore, the definition can be expressed as follos: E k (1+ x ) u ʹ(t + xt )dt > E = (1+ x ) u ʹ(t + xt )dt k k u ʹ (t )dt [ ] > ʹ [ ] is convex in t. That is the Thus, E (1+ x ) u ʹ(t + t x ) u (t ), hich is true if and only if (1+ x ) u ʹ (1+ x )t case hen relative prudence is greater than to. We can say that a multiplicatively risk-prudent individual prefers to attach the multiplicative risk to the better outcome. The definition from Eeckhoudt and Schlesinger (2006) is equivalent to our definition if e replace the additive risk by a multiplicative risk and the non-negativity of the third derivative of the utility function by the fact that the coefficient of relative prudence is greater than to. 7

8 This equivalence helps us to point out conditions under hich the derived utility function û is multiplicatively risk-prudent knoing that the direct utility function u exhibits multiplicative risk prudence. Property 3: Consider a ell behaved utility function u hich exhibits multiplicative risk prudence, then the derived utility function is multiplicatively risk-prudent if and only if E(φ( Z 2 ) > E(φ( Z )) Z here is convex. Proof. û is multiplicatively risk-prudent if the lottery B = [ k;(1+ x )] is preferred to the lottery A = [( k )(1+ x ); ]. That is, Eû( + x ) Eû( k + ( k ) x ) > û() û( k ),k, x Or: Eu (1+ x ) 2 Eu ( k )(1+ x ) 2 > Eu[ (1+ x )] Eu [( k )(1+ x )] The last condition yields: The condition becomes: E(φ( Z 2 ) > E(φ( Z )) Z, here φ(z) = zu ʹ(z). E (1+ x ) uʹ t(1+ x ) 2 dt > E (1+ x ) uʹ[ t(1+ x )]dt k k The derivative of φ ith respect to z is given by: φ ʹ(z) = u ʹ(z) + zu ʹʹ(z) Therefore, the second derivative of φ ill be: φ ʹʹ(z) = 2 u ʹʹ(z) + 2 zu ʹʹʹ(z) = u ʹʹ(z) u ʹʹʹ(z) 2 + (z) u ʹʹ(z) u ʹʹʹ(z) And φ is convex if and only if (z) 2. That is, the direct utility function is multiplicatively u ʹʹ(z) risk-prudent. We no present some properties associated ith the derived utility function and relative to its n th derivatives. We define the nth order coefficient of relative risk aversion as follos: n () = u(n+1) () u (n) () Property 4: Consider a ell behaved utility function u such that u (2p), u (2p+1), u (2p+2) and u (2p+3) exist, then: (1) û (2p) u (2p) iff 2p+1 2p 4p 2p + 2p(2p 1) 0 hen u (2p) 0 (2) û (2p+1) iff 2p+1 2p+2 2(2p + 1) 2p+1 + 2p(2p + 1) 0 hen u (2p+1) 0 Proof. (1) û (2p) () = E (1+ x ) 2p u (2p) [ (1+ x )] in x. x 2p u (2p) (x ) is concave if and only if { } is less than u (2p) () if and only if x 2p u (2p) (x ) is concave x 2p 2 (u (2p) (x )) x u (2p+1) u (2p) x u (2p+2) u (2p+1) 4p x u (2p+1) u (2p) + 2p(2p 1) 0 Or equivalently if and only if x 2p 2 ( u (2p) (x )) 2p+1 2p 4p 2p + 2p(2p 1) 0 here 2p () = u (2p+1) () u (2p) () is the relative risk coefficient of order 2p. 8

9 Assume that u (2p) () 0, then the condition to be fulfilled is : 2p+1 2p 4p 2p + 2p(2p 1) 0 { } is greater than u (2p+1) () if and only if x 2p+1 u (2p+1) (x ) is (2) û (2p+1) () = E (1+ x ) 2p+1 u (2p+1) [ (1+ x )] convex in x. x 2p+1 u (2p+1) (x ) is convex in x if and only if x 2p 1 u (2p+1) (x ) x u (2p+3) u (2p+1) x u (2p+2) u (2p+1) u 2(2p + 1) x (2p+2) u (2p+1) + 2p(2p + 1) 0a Or if and only if x 2p 1 u (2p+1) (x )) 2p+1 2p+2 2(2p + 1) 2p+1 + 2p(2p + 1) 0 Assume that u (2p+1) () 0, then the condition to be fulfilled is: 2p+1 2p+2 2(2p + 1) 2p+1 + 2p(2p + 1) 0 The next property ill allos us to sign the former to inequalities. Property 5: Assume that u is such that n is decreasing for 1 n N, then: 1) 2p+1 2p 4p 2p + 2p(2p 1) 0 if N = 2p and 2) 2p+1 2p+2 2(2p + 1) 2p+1 + 2p(2p + 1) 0 if N = 2p + 1 Or N N +1 2N N + N(N 1) 0 Proof. 1) 2p 2p+1 4p 2p + 2p(2p 1) = 2p 2p+1 2p (2p) 2p (2p 1) but 2p+1 2p + 1 2p, 2p 2p 2p 1 and 2p+1 (2p) 2p (2p 1) The last inequality comes from the decrease of 2p+1 ( 2p+2 2p ). And the result follos. 2) 2p+1 2p+2 2(2p + 1) 2p+1 + 2p(2p + 1) = 2p+1 2p+2 (2p + 1) (2p + 1) 2p+1 2p but 2p+2 2p + 2 2p + 1, 2p+1 2p + 1 2p and 2p+2 (2p + 1) 2p+1 2p The last inequality is due to the decrease of 2p+1. And the result follos. 2p+1 2p 4p 2p + 2p(2p 1) 0 if N = 2p and 2p+1 2p+2 2(2p + 1) 2p+1 + 2p(2p + 1) 0 if N = 2p + 1 are equivalent to N N +1 2N N + N(N 1) 0 Property 5 bis: Assume that u is such that N is decreasing. If N N, then N N +1 2N N + N(N 1) 0 Proof. N N +1 2Nx N + N(N 1) = N N +1 N N N (N 1) But N N N 1, N N N + 1 N. And the result follos. For example, if u () = u ʹʹ() u ʹ() 1 and decreasing, then P u () = u ʹʹʹ() u ʹʹ() 2 and P u () u () 2 u () 0. 9

10 We have also the folloing result: if P u () 2 and decreasing, then P u ()T u () 4P u () T u () 3 and 4. Affiliated utility function Folloing Franke, Schlesinger and Stapleton (2006), e can use the concept of affiliated function to demonstrate the properties of the direct utility function. The affiliated utility function u is such that uʹ( ln() ) = u ʹ() or u ʹ(θ) = u ʹ(e θ ) here θ = ln(). Dealing ith multiplicative risk ith the direct utility function is equivalent to dealing ith additive risk hen taking the affiliated utility function: Eu[ (1+ x )] is replaced by Eu [ ln( (1+ x ))] = Eu ( ln() + ln(1+ x )) hich becomes Eu ( W + ε) ) here W = ln() and ε = ln(1+ x ). We can notice that ε = ln(1+ x ) is an unfair background risk because E( ε) = E [ ln(1+ x )] ln(1+ E( x )) = 0. Therefore, multiplying the ealth of the agent ith utility function u by 1+ x is equivalent to adding ε to the ealth of the agent ith utility function u. The first fifth derivatives of the direct utility function and that of the affiliated utility function are related as follos: i) u ʹ(ln) = u ʹ() ii) u ʹʹ (ln) = u ʹʹ () iii) u ʹʹʹ(ln) = u ʹʹ() + 2 u ʹʹʹ() iv) u (4) (ln) = u ʹʹ() u ʹʹʹ() + 3 u (4) () v) u (5) (ln) = u ʹʹ() u ʹʹʹ() u (4) () + 4 u (5) () We can sign the successive derivative of the affiliated utility function thanks to the coefficients of relative risk aversion, relative prudence, relative temperance and relative quintessence of the direct utility function as follos: ii) u ʹʹ(ln) 0 if and only if u is risk averse ( u () 0 ). iii) u ʹʹʹ(ln) 0 if and only if P u () 1 iv) u (4) (ln) 0 if and only if P u () T u () 2 P u () 1. Then, u (4) (ln) 0 if P u () 1 and T u () 2 v) u (5) (ln) 0 if and only if P u () T u ()( Q u () 3) 3( T u () 2) + P u () 1 0. Thus, u (5) (ln) 0 if P u () 1, T u () 3 and Q u () 1+ T u () (or T u () decreases) We have the folloing property because absolute risk aversion, absolute prudence, absolute temperance and absolute quintessence of the affiliated utility function are related to relative risk aversion, relative prudence, relative temperance and relative quintessence of the direct utility function. Property 6: Consider a ell behaved utility function u such that u ʹ, u ʹʹ, u ʹʹʹ, u (4) and u (5) exist, then: i) A u (ln) = u () ii) P u A (ln) = P u () 1 iii) T A u (ln) = P u () T u () 2 1 P u () 1 iv) Q A u (ln) = P u () T u ()Q u () 5T u () P u ()T u () 3P u () + 1 Proof. i) A u (ln) = u ʹʹ(θ) u ʹ(θ) = e θ u ʹʹ(e θ ) = u ʹʹ() = u ʹ(e θ ) u ʹ() u () 10

11 ( ) 2 ʹʹʹ ii) P A u (ln) = u ʹʹʹ(θ) u ʹʹ(θ) = e θ u ʹʹ(e θ ) + e θ u (e θ ) = 1 e θ u ʹʹʹ(e θ ) = P e θ u ʹʹ(e θ ) u ʹʹ(e θ u () 1 ) ( ) 2 ʹʹʹ ( ) 3 u (4) (e θ ) ( ) 2 ʹʹʹ iii) T A u (ln) = u (4) (θ) = e θ u ʹʹ(e θ ) + 3 e θ u (e θ ) + e θ u ʹʹʹ(θ) e θ u ʹʹ(e θ ) + e θ u (e θ ) = 1 e θ uʹʹʹ 2 + e θ u (4) (e θ ) u ʹʹ (e θ ) uʹʹʹ 1+ e θ uʹʹʹ u ʹʹ (e θ ) iv) Q A u (ln) = u (5) (θ) u (4) (θ) = e θ u ʹʹ(e θ ) + 7 e θ e θ = 1+ P u (e θ ) 2 T u (e θ ) 1 P u (e θ ) = P u () T u () 2 P u () 1 ( ) 2 u ʹʹʹ(e θ ) + 6( e θ ) 3 u (4) (e θ ) + ( e θ ) 4 u (5) (e θ ) ( ) 2 u ʹʹʹ(e θ ) + ( e θ ) 3 u (4) (e θ ) u ʹʹ(e θ ) + 3 e θ = 1 4 ( e θ ) 2 u ʹʹʹ(e θ ) + 5( e θ ) 3 u (4) (e θ ) + ( e θ ) 4 u (5) (e θ ) e θ u ʹʹ(e θ ) + 3( e θ ) 2 u ʹʹʹ(e θ ) + ( e θ ) 3 u (4) (e θ ) = P u () T u ()Q u () 5T u () P u ()T u () 3P u () + 1 This property tells us that the affiliated utility function exhibits: additive risk aversion if and only if the direct utility function is multiplicatively risk averse, additive risk prudence if and only if the relative prudence of the direct utility function exceeds one, additive risk temperance if the relative temperance of the direct utility function is greater than four or if the relative prudence of the direct utility function is greater than one and the relative temperance of the direct utility function is greater than to, additive risk quintessence if the relative quintessence of the direct utility function is greater than six or if the relative prudence of the direct utility function is greater than one and the relative temperance of the direct utility function is decreasing and greater than three. The comparison beteen to utility functions ill be made ith the folloing property: Property 7: Consider to utility functions u and v, then: i) v () u () if and only if v A (ln) u A (ln) ii) P v () P u () if and only if P v A (ln) P u A (ln) iii) Assume that P v () P u (), then T v () T u () if T v A (ln) T u A (ln) Proof. i) Property 6 says that u A (ln) = u (). Therefore, v is relatively more risk averse than u if and only if v is absolutely more risk averse than u. ii) Property 6 tells us that v is relatively more risk prudent than u if and only if v is absolutely more risk prudent than u. iii) Thanks to 6, e have: T A u (ln) = P u () T u () 2 P u () 1 v is relatively more temperate than u if 1 or T u () = 2 + P u () 1 1+ T A P u (ln() u () P v () 1 1+ T A P v (ln() v () P u () 1 1+ T A P u (ln() u () 11

12 But the fact that v is relatively more prudent than u implies: P v () 1 P u () 1 P v () P u () And the result follos. That is, v is relatively more temperate than u hen v is relatively more prudent than u and v is absolutely more temperate than u. 5. More prudent behavior In this section, e provide sufficient conditions ith hich an agent facing a multiplicative background risk behaves in a more risk-prudent manner. That is, the coefficient of relative prudence associated ith the derived utility function is greater than that of the direct utility function. We take to approaches. The first uses the notion of risk vulnerability as soon as the opposite of the marginal utility is concerned and the second a risk-neutral probability. With the first approach, e get the folloing proposition hich gives the conditions under hich an agent behaves in a more risk-prudent manner hen facing a multiplicative background risk. Proposition 1: When both relative prudence and relative temperance are decreasing, an agent facing a multiplicative background risk behaves in a more risk-prudent manner compare to the case ithout the background risk if his/her relative prudence is greater than to. Proof. ecall that dealing ith multiplicative risk and a direct utility function is equivalent to dealing ith additive risk and affiliated utility function. The agent ill behave in a more prudent fashion hen facing multiplicative risk if and only the agent ith the affiliated utility function behaves in a more prudent manner hen facing additive risk. That is, uʹ is risk vulnerable. Gollier and Pratt (1996) give sufficient conditions for risk vulnerability, namely the coefficient of absolute risk aversion of u ʹ is decreasing and convex or both absolute risk aversion and absolute prudence of u ʹ are decreasing ( uʹ is said to be standard). These to conditions become: absolute prudence of the affiliated utility function is decreasing and convex absolute prudence and absolute temperance of the affiliated utility function decrease. Here, e consider the second condition, namely, the decrease of both absolute prudence and absolute temperance of the affiliated utility function. This turns out to be the decrease of both relative prudence and relative temperance of the direct utility function due to the folloing: First, P u A (ln) = P u () 1, thus d P u () A dp = u d d (ln) 1 And absolute prudence of the affiliated utility function and relative prudence of the direct utility function evolve in the same direction. The decrease of relative prudence of the direct utility function implies the decrease of the absolute prudence of the affiliated utility function and viceversa. Second, T A u (ln) = P u () T u () 2 P u () 1 ith T A u (ln) = P u () T u () 2 P u () 1 1= ϕ() φ() 1 1= ϕ() φ() 1 and φ() = P u () 1. 12

13 ( ) ( ) We have: d P u () d 3 T u () d T u () d P u () due to T u () 3 (The decrease of relative prudence implies that T u () 1+ P u () and e kno that P u () 2 ) ( ) ( ) ( ) or equivalently d P u () d d P u () d T u () 2 d T u () d P u () The last inequality is φ ʹ() ϕ ʹ () and it tells us that ϕ() decreases faster than ϕ(). Therefore ϕ() φ() decreases and T u A (ln) also decreases. Collecting the to conditions implies the decrease of the absolute temperance of the affiliated utility function. Finally, both absolute prudence and absolute temperance of the affiliated utility function are decreasing. That is, the opposite of the marginal affiliated utility function is standard. And therefore risk vulnerable. In other ords, the derived utility function is more prudent than the direct utility function or equivalently û is more prudent than or PûA () P u A, hich is equivalent to: Pû () = u ˆʹʹʹ() u ˆʹʹ() P u = u ʹʹʹ() u ʹʹ(). In the second approach, like Franke, Schlesinger and Stapleton (2006) and Jokung (2004), e can express the coefficient of relative prudence of the derived utility function ith a risk-neutral probability: Pû () = u ˆʹʹʹ() u ˆʹʹ() = E (1+ x )3 u ʹʹʹ( + x ) E (1+ x ) 2 u ʹʹ( + x ) Let us define the risk-neutral probability dλ u : dλ u (1+ x ) 2 u ʹʹ( + x ) (x ) = E (1+ x ) 2 u ʹʹ( + x ) Then the coefficient of relative prudence of the derived utility function is given by: u ˆʹʹʹ() u ˆʹʹ() = E (1+ x ) u ʹʹʹ( + x ) u ʹʹ( + x ) dλ (x ) = E u ʹʹʹ( + x ) u (1+ x ) λ u ʹʹ( + x ) Or equivalently: ( ) P û () = E u λ P u (1+ x ) The risk-neutral probability is interesting because e kno the sign of the derivative of dλ u (x ) ith respect to x. The sign of dλ u (x ) is the same as the sign of 2 P u ( + x ) because: dλ u (x ) x = (1+ x ) u ʹʹ( + x ) 2 P E (1+ x ) 2 u ( + x ) u ʹʹ( + x ) If P u () 2, then dλ u (x ) puts more eight on loer values of x than the original probability and one consequence ill be E λ u (1+ x ) E(1+ x ) = 1. If P u () 2, then dλ u (x ) puts more eight on higher values of x than the original probability and one consequence ill be E λ u (1+ x ) E(1+ x ) = 1. 13

14 We have the folloing proposition: Proposition 2: Assume P u () is convex. If one of the folloing conditions holds then û is more risk-prudent than u: (i) P u () 2 and P u () is decreasing (ii) P u () 2 and P u () is increasing Proof. The convexity of P u () implies that: P û () = E λ P u ( (1+ x )) P (1+ E ( x )) u λ (i) Assume that P u () 2, then + E λ ( x ) because E λ (1+ x ) E(1+ x ) = 1. Decreasing relative prudence leads to P u + E λ ( x ) P u (). Therefore, Pû () P u () and the result follos. (ii) Assume that P u () 2, then + E λ ( x ) because E λ (1+ x ) E(1+ x ) = 1. Increasing relative prudence leads to P u + E λ ( x ) P u (). Therefore, P û () P u () and the result follos. We have pointed out other conditions under hich an agent facing a multiplicative risk behaves in a more risk-prudent manner. These conditions are based on the convexity of the relative prudence rather than on the decrease of both relative prudence and relative temperance as in the first approach. It ill be of interest to kno ho these conditions are related. Nevertheless, e have the folloing proposition if relative prudence is concave: Proposition 3: Assume P u () is concave. If one of the folloing conditions holds then û is less risk-prudent than u: (i) P u () 2 and P u () is increasing (ii) P u () 2 and P u () is decreasing Proof. The proof is similar to that of proposition 1, so it is omitted. Here e have pointed out the conditions under hich an agent facing a multiplicative risk behaves in a more risk-prudent manner. The conditions are stronger than those obtained in the additive case, in hich the utility function needs to be standard. This result is ne because usually authors dealing ith background risk focus on the risk-averse behavior. If e consider the class of hyperbolic absolute risk averse functions (HAA), their relative temperance, their relative prudence and their relative risk aversion are related as follos: T u () = 3 γ 2 γ P u and P u () = 2 γ 1 γ u Because u() = 1 γ b + a 1 γ γ 1 γ ith b 0 and a > 0. ecall that: isk neutral utility function: γ 1 Quadratic utility function: γ = 2 Negative exponential utility function: γ and b = 1 Poer utility function: γ < 1 and b = 0 14 If γ < 1, relative prudence ill be convex. Therefore the former proposition reduces to the decrease of the coefficient of relative prudence if γ < 2. That is, decreasing relative prudence is sufficient for û to be more risk prudent than u hen γ < 2.

15 The condition reduces to the decrease of the coefficient of relative risk aversion if γ < 1. That is, decreasing relative risk aversion is sufficient for û to be more risk prudent than u hen γ < 1. Let us consider the case of constant relative prudence utility functions. A utility function exhibits constant relative prudence if and only if: u ʹʹʹ() u ʹʹ() = Θ + 1. Or if and only if u ʹʹʹ() + (Θ + 1) u ʹʹ() = 0. Therefore, u() = λ 1 Θ And the set of constant relative risk aversion (CAA) coincides ith the set of constant relative prudence (CP). 6. Preservation of decreasing relative prudence We have an important result concerning the decrease of the coefficients of relative prudence of the direct utility function and the derived utility function. That is, if the utility function exhibits decreasing relative prudence, the derived utility function also exhibits decreasing relative prudence. Decreasing relative prudence ith the direct utility function implies decreasing relative prudence ith the derived utility function. Proposition 4: If P u () decreases, then Pû () decreases Proof. We have to demonstrate that T u 1+ P u Tû 1+ Pû or equivalently that: u (4) () u ʹʹʹ() 1 u ʹʹʹ() u ʹʹ() E(1+ x )4 û (4) ( + x ) E(1+ x )3û ʹʹʹ( + x ) 1 E(1+ x ) 3 û ʹʹʹ( + x ) E(1+ x ) 2 û ʹʹ( + x ) Folloing Franke, Stapleton and Schlesinger (2006), e use the diffidence theorem (Gollier, 2001) and e must sho the folloing implication: E (1+ x )3 u ʹʹʹ( + x ) + (λ 1)(1+ x ) 2 u ʹʹ( + x ) = 0 E (1+ x )4 u (4) ( + x ) + λ(1+ x ) 3 u ʹʹʹ( + x ) 0 x Which holds if there is a real m such that: x Or: Or: (1+ x )4 u (4) ( + x ) + λ(1+ x ) 3 u ʹʹʹ( + x ) m (1+ x ) 3 u ʹʹʹ( + x ) + (λ 1)(1+ x ) 2 u ʹʹ( + x ) (1+ x ) 3 u ʹʹʹ( + x ) u (4) ( + x ) + λ uʹʹʹ m(1+ x ) 2 u ʹʹ( + x ) (1+ x ) uʹʹ ʹʹʹ ( + x ) + (λ 1) u (1+ x )3u ʹʹʹ( + x ) λ T u m(1+ x ) 2 u ʹʹ( + x ) λ 1 P u We can notice that because of the decrease of relative prudence, e have (1+ x )3u ʹʹʹ( + x ) λ T u (1+ x ) 3 u ʹʹʹ( + x ) λ 1 P u So, e ill get our result if e can find m such that: (1+ x )3u ʹʹʹ( + x ) λ 1 P u m(1+ x ) 2 u ʹʹ( + x ) λ 1 P u If e take m = 1 λ, then e have : (1+ x )3u ʹʹʹ( + x ) λ 1 P u 1 λ (1+ x )2 u ʹʹ( + x ) λ 1 P u 15

16 u ʹʹʹ( + x ) Or: (1+ x ) λ 1 P u ʹʹ( + x ) u (1 λ) λ 1 P u Which is equivalent to P u λ 1 P u (1 λ) λ 1 P u Or P u λ 1 P u + (1 λ) λ 1 P u 0 2 Which leads to λ 1 P u 0 And the result follos. This result is of interest because utility functions are expected to have decreasing relative prudence. Therefore, e ill have the same property ith the derived utility function as soon as the direct utility function exhibits decreasing relative prudence. 7. Comparative risk prudence under multiplicative background risk Let us consider to individuals ith respective utility functions u and v. Our objective is to point out conditions under hich given that the one ith utility function v is more risk-prudent ill remain more risk-prudent in the presence of the multiplicative background risk. Or if you prefer, under hich conditions vˆ is more risk-prudent than û knoing that v is more risk-prudent than u. Does a multiplicative background risk preserve the risk-prudence order? Proposition 5: Assume v is more risk-prudent than u. If there is χ such that P v () χ P u () then vˆ is more risk-prudent than û. Proof. If P v () χ P u (), then P v ( + x ) χ P u ( + x ), x. That is, P v ( + x ) χ and χ P u ( + x ) By taking the expectation ith respect to dλ v and dλ u respectively, one obtains: v = E v λ P v ( + x ) E v λ (χ) = χ v ith dλ Pˆ and E u λ (χ) = χ E u λ P u ( + x ) u = Pû ith dλ. And finally, Pˆ v () χ Pû () And the result follos. We have another proposition concerning the preservation of the risk-prudence order by the use of the theorem of Kihlstrom, omer and Williams (1981): Proposition 6: Assume v is more risk-prudent than u, relative prudence of u is greater than one and either u or v exhibits decreasing relative prudence, then vˆ is more risk-prudent than û. Proof. v s being more risk-prudent than u and either u or v s exhibition of decreasing relative prudence is equivalent to v s being more risk-prudent than u and either u or v s exhibition of decreasing absolute prudence because: a) v s being more risk-prudent than u means v ʹ is more risk-averse than uʹ b) A uʹ (ln) = P A u (ln) = P u () 1 c) u ʹ (ln + ln(1+ x )) = u ʹ((1+ x )) 16

17 Applying Kihlstrom, omer and Williams theorem leads to Ev ʹ[ ln + ln(1+ x )] being more riskaverse than Euʹ[ ln + ln(1+ x )] hich is equivalent to vˆʹ s being more risk-averse than uˆʹ because the effect of the multiplicative risk on the direct utility is equivalent to the effect of the additive risk on the affiliated utility function. Finally: ( vˆʹ ) ʹʹ ( () u ˆʹ ) ʹʹ () or ( vˆʹ ) ʹ ( uˆʹ ) ʹ vˆʹʹʹ () uˆʹʹʹ () vˆʹʹ uˆʹʹ And ˆ v is more risk-prudent than û. 8. The optimal demand for savings ith multiplicative background risk Consider a to-period life-cycle model ith an additively separable utility: U(c 1, c 2 ) = u(c 1 ) δ Eu( c 2 ) ith c 1 = y 1 ky 1 and c 2 = y 2 + ky 1 (1+ r f ) (1+ x ) and here: - u is a ell behaved utility function ( uʹ 0, uʹʹ 0 and u ʹʹʹ 0 ) - y 1 and y 2 are the present and future labor incomes - k is the rate of savings - r f is the risk-free rate - x is a zero-mean multiplicative background risk associated ith ealth or 1+ x a unitary-mean multiplicative background risk - δ is the time preference. The agent solves: Max u(y 1 ky 1 ) + 1 k 1+ δ Eu y 2 + ky 1 (1+ r f ) (1+ x ) { } The optimal savings rate is given by the first-order condition: y 1+ r 1 u ʹ(y 1 k y 1 ) + y f 1 1+ δ { } = 0 E (1+ x ) u ʹ y 2 + k y 1 (1+ r f ) (1+ x ) ecall the problem to solve in case of certainty. The optimal savings rate K o satisfies: Max k u(y 1 ky 1 ) δ u y 2 + ky 1 (1+ r f ) With the folloing first-order condition: u ʹ(y 1 k 0 y 1 ) + 1+ r f uʹ y 1+ δ 2 + k 0 y 1 (1+ r f ) = 0 Proposition 7: An individual increases his or her savings in the presence of a multiplicative risk if and only if Pu () = u ʹʹʹ() u ʹʹ() 2. Proof. The optimal savings rate in the presence of a multiplicative background risk ill be greater than that ithout uncertainty if and only if the first-order condition of the program ith the background risk expressed at K o the savings rate ithout uncertainty is positive: { } ʹ E (1+ x ) ʹ u y 2 + k 0 y 1 (1+ r f ) (1+ x ) u (y 2 + k 0 y 1 (1+ r f )) This condition is fulfilled if and only if Ω() = u ʹ(x ) is convex according to Jensen s inequality. Ω() is convex if and only if 2u ʹʹ(x ) + xu ʹʹʹ(x ) 0 or equivalently if and only if P u () = u ʹʹʹ() u ʹʹ() 2. 17

18 This proposition generalizes the results for Leland (1968) and Sandmo (1970) ho studied the case of additive risk to that of multiplicative risk. Therefore, it induces the folloing definition regarding the prudent behavior as do Wong and Chang (2005): Definition 4: An individual is said to be multiplicatively risk-prudent if he/she increases his/her savings in the presence of a multiplicative background risk. We can illustrate this result by considering poer utility functions: u() = 1 θ. 1 θ. The coefficient of relative prudence is equal to θ + 1. Suppose that the multiplicative risk increases or decreases ealth by 8% ith the same probability. To focus solely on the effect of the multiplicative background risk, e assume that the risk-free rate is null and that there is no time preference. We also assume that current income is 100 and final income is 50. In the event of certainty, the savings rate is 25%. Therefore, consumption is 75 today and tomorro. The optimal savings rates are given in the folloing table for three utility functions: Utility function Savings rate in the event of certainty Savings rate ith the background risk Coefficient of relative prudence u() = = 2 25% 24.93% 1.5 u() = ln() 25% 25% 2 u() = 1 1 = 1 25% 25.12% 3 The coefficients of absolute prudence of the three functions are positive but only one function leads to an increase in the savings rate in the presence of the multiplicative background risk. One of them implies a decrease in the savings rate! In this last case, a prudent individual in the sense of Kimball (1990) decreases his or her savings rate hen facing uncertainty ith a multiplicative background risk instead of an additive background risk! This result helps to explain hy an increase in the ealth risk can lead to a decrease in the savings rate. The result from Kimball is related to additive background risk. It explains the increase in the savings rate in the presence of income risk but it fails to explain the modification of the savings rate in the presence of ealth risk or capital risk. At that time, multiplicative risk prudence is necessary. 9. Portfolio selection ith multiplicative background risk Consider a to-period model in hich the agent receives a non random labor income in the first period. This initial ealth ill be invested in a risk-free asset r f and in a risky asset hich yields a random return r. Let α be the fraction of ealth invested in the risky asset. The agent faces a multiplicative background risk 1+ x such that E x [ ] = 0 and he or she chooses to maximize his or her expected utility function: Max α or Max α { } E u (1+ x ) (1+ r f ) + α(r r f ) { } E û (1+ r f ) + α(r r f ) 18

19 Where û(z) = Eu z(1+ x ) [ ] The problem yields the folloing first-order condition: E (r r f ) u ˆʹ( f ) = 0 here f = (1+ r f ) + α(r r f ) And the solution is αû = ArgMax α { } E û (1+ r f ) + α(r r f ) In the absence of the multiplicative background risk, the problem that should be solved is the folloing: Max α { } E u (1+ r f ) + α u (r r f ) The optimal proportion invested in the risky asset is given by: E (r r f ) u ʹ( f ) = 0 And the solution is α u = ArgMax α { } E u (1+ r f ) + α(r r f ) The difference beteen the to optimal proportions ( α u αû ) measures the effect of the presence of the multiplicative background risk on the portfolio decision. The next proposition gives the sign of this difference. Proposition 8: Assume u is convex. The fraction of initial ealth optimally invested in the risky asset decreases in the presence of the multiplicative background risk if one of the to folloing conditions holds: (1) u 1 and P u 1+ u (2) u 1 and P u 1+ u Proof. Franke, Schlesinger and Stapleton (2006) sho that the individual ith utility function û is more risk averse than the individual ith utility function u if u is convex and one of the to folloing conditions holds: (1) u 1 and u is decreasing or (2) u 1 and u is increasing. But P u 1+ u if and only if u is decreasing. Therefore û is more risk averse than u. And applying Arro (1971) leads to the fact that α u αû. We get the same result ith the folloing proposition: Proposition 9: When both u and P u are decreasing, the fraction of initial ealth optimally invested in the risky asset decreases in presence of the multiplicative background risk. Proof. 1) u is decreasing and P A u is also decreasing, then u A and P u are decreasing. That is, the affiliated utility function is standard or at least risk vulnerable. That the affiliated utility function is risk vulnerable implies that the derived utility function is more risk averse than the direct utility function. And the result follos. We can define the feeling of standardness in the presence of multiplicative risk by replacing absolute risk aversion and absolute prudence by relative risk aversion and relative prudence. When both relative risk aversion and relative prudence ill decrease, the agent ill be relatively standard or relatively risk-standard. Therefore, the last proposition implies that any risk-standard agent in the multiplicative sense behaves as e expect in portfolio selection. That is, he or she reduces his or her holdings of risky assets. 19

20 10. Conclusion This paper shos that the concept of prudence defined by Kimball (1990) corresponds to additive risk prudence, hich implies that the presence of some additive background risk increases the optimal savings rate. This concept is related to the convexity of the marginal utility function. This paper introduces the concept of multiplicative risk prudence, hich implies that the presence of a multiplicative background risk increases the optimal savings rate. It also introduces the notion of quintessence (the fifth element) to guarantee the decrease of the feeling of temperance. This paper describes to conditions under hich an individual facing a multiplicative risk behaves in a more risk-prudent manner. The first condition is given by the decrease of both relative prudence and relative temperance. The second assumes the convexity of the coefficient of relative prudence and either a decreasing relative prudence larger than to or an increasing relative prudence less than to. This paper likeise presents the conditions under hich the more risk-prudent individual in the absence of any multiplicative background risk remains more risk-prudent in the presence of a multiplicative background risk. Dealing ith savings demand, this paper shos that the demand for savings in the presence of some multiplicative background risk increases if and only if relative prudence is greater than to. When the results are applied to portfolio selection and it is assumed that relative risk aversion is convex, in the presence of the multiplicative background risk, individuals demand feer risky assets hen relative prudence is greater (less) than relative aversion plus one and relative risk aversion is greater (less) than one. The result is the same hen the agent is relatively standard. 11. eferences Arro K. J. (1971), Exposition of the Theory of Choice under Uncertainty in K.J. Arro, Essays in the Theory of isk Bearing, Elsevier Publishing Company Inc. J. Caballé and A. Pomanski (1996), Mixed isk Aversion, Journal of Economic Theory, 71, Cho G., I. Kim and A. Sno (2001), Comparative Statics Predictions for Changes in Uncertainty in the Portfolio and Saving Problem, Bulletin of Economic esearch, n 53, Eeckhoudt L. and M. Kimball (1992), Background isk, Prudence and Insurance Demand in Contributions to Insurance Economics, ed by G. Dionne, Kluer, Eeckhoudt L., C. Gollier and H. Schlesinger (1996), Changes in Background isk and isk Taking Behavior, Econometrica 64, n 3, Eeckhoudt L. et H. Schlesinger (2006), Putting isk in its Proper Place, American Economic evie 96, Elmendorf D. and M. Kimball (1999), Taxation of Labor Income and the Demand for isky Assets, International Economic evie 41, Franke G.,. C. Stapleton and H. Schlesinger (2006), Multiplicative Background isk, Management Science 52, n 1,

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