Abstract Stanar Risk Aversion an the Deman for Risky Assets in the Presence of Backgroun Risk We consier the eman for state contingent claims in the p

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1 Stanar Risk Aversion an the Deman for Risky Assets in the Presence of Backgroun Risk Günter Franke 1, Richar C. Stapleton 2, an Marti G. Subrahmanyam. 3 November Fakultät für Wirtschaftswissenschaften un Statistik, University of Konstanz guenter.franke@uni-konstanz.e 2 University of Strathclye rcstaplet.emon.co.uk 3 Stern School of Business, New York University m.subrahmtern.nyu.eu We thank Louis Eeckhout, Christian Gollier an Harris Schlesinger for helpful comments.

2 Abstract Stanar Risk Aversion an the Deman for Risky Assets in the Presence of Backgroun Risk We consier the eman for state contingent claims in the presence of a zero-mean, nonhegeable backgroun risk. An agent is efine to be generalize risk averse if he/she reacts to an increase in backgroun risk by choosing a eman function for contingent claims with a smaller slope. We show that the conitions for stanar risk aversion: positive, eclining absolute risk aversion an pruence are necessary an sufficient for generalize risk aversion. We also erive a necessary an sufficient conition for the agent's erive risk aversion to increase with a simple increase in backgroun risk. "Journal of Economic Literature Classification Numbers: D52, D81, G11."

3 The Deman for Risky Assets 1 1 Introuction Recent avances in the theory of risk bearing have concentrate on the effect of a nontraeable backgroun risk on the risk aversion of an agent to a secon inepenent risk. For example, Gollier an Pratt (1996) efine a rather general class of utility functions such that risk-averse iniviuals become even more risk averse towar a risk, when a secon, inepenent, unfair backgroun risk is ae. They compare the risk aversion of an agent with no backgroun risk to that of an agent who faces the backgroun risk. They term the set of functions uner which the agent becomes more risk averse, the class of "riskvulnerable" utility functions. The set of risk-vulnerable functions is larger than the set of proper risk averse functions introuce earlier by Pratt an Zeckhauser (1987), who consier utility functions such thatthe expecte utility of an unesirable risk is ecrease by the presence of an inepenent, unesirable risk. Kimball (1993) has consiere the effect of the [even larger] set of expecte marginal utility increasing backgroun risks. This le him to efine the more restrictive class of stanar risk averse utility functions. Stana r risk aversion characterises those functions where the iniviual respon to an expecte marginal utility increasing backgroun risk by reucing the eman for a markete risk. Kimball shows that stanar risk averse functions are characterize by positive, ecreasing absolute risk aversion an absolute pruence. The set of stanar risk averse functions is a subset of the set of proper risk averse functions, which, in turn, are a subset of the risk vulnerable functions, as iscusse by Gollier an Pratt (1996, pp ). In a relate paper, Eeckhout, Gollier an Schlesinger (1996) exten this analysis by consiering a rather general set of changes in backgroun risk, which take the form of first or secon orer stochastic ominance changes. They establish a set of very restrictive conitions on the utility function such that agents become more risk averse when backgroun risk increases in this sense. The purpose of this paper is twofol. First, we consier a smaller set of increases in backgroun risk than Eeckhout, Gollier an Schlesinger (1996) an erive less restrictive conitions for an increase in backgroun risk to increase the erive risk aversion of agents. We restrict the set of increases in the risk of backgroun income y, with E(y) = 0, to simple increases (see also Eeckhout, Gollier an Schlesinger (1995)). A simple increase in backgroun risk is a change to y such that» [=][ ]0 for y < [=] > y 0 for some y 0 an E( ) = 0. We erive a necessary an sufficient conition on the utility function for a simple increase in backgroun risk to make the agent more risk averse. We show that stanar risk aversion is sufficient, but not necessary for a simple increase in backgroun risk to increase erive risk aversion. The secon an the main purpose of the paper is to investigate restrictions on utility

4 The Deman for Risky Assets 2 functions which guarantee a more risk averse behaviour in the presence of an increase, inepenentbackgroun risk, when the agent facesachoice between state-contingent claims. In this setting, changes in risk-averse behaviour are reflecte in the slope of the eman function for contingent claims. 1 Gollier (2000) consiers a moel where the agent can buy state-contingent claims on consumption, given no backgroun risk. Let ffi be the probability eflate price of obtaining one unit of consumption if a state occurs an nothing otherwise. Then, in this moel, the higher is ffi for a given state, the lower is the agent's eman for claims on that state, w. In other wor, the eman function, w(ffi), that relates the consumption in a state to the price, is ownwar sloping. Gollier [Proposition 51] shows that, if two agents with utility functions u 1 an u 2 have the same enowment, an if u 1 is more risk averse than u 2, then the eman function of agent 1, w 1 (ffi), `single-crosses from below' the eman function of agent 2,w 2 (ffi). This single-crossover property is illustrate in Figure 1. Gollier goes on to conclue that risk-vulnerable investors will select a safer consumption plan", when they face backgroun risk. Our results, showing the effect of a simple increase in backgroun risk on risk aversion, therefore imply that an agent facingan increase in backgroun risk will respon by choosing a eman function similar to investor 1 rather than that chosen by investor 2, in Figure 1. w 6 w Figure 1: Deman curve 1 is less steep than eman curve 2every- where ffi Figure 2: Deman curve 1 is less steep than eman curve 2 in some range an steeper in other ranges ffi 1 In a state-contingent claims moel, risk-averse behaviour can be characterize by the slope of the eman curve for contingent claims. In the case of extreme risk aversion, the agent buys an equal amount of claims on each state, espite the higher prices of the claims on some of the states. A less risk-averse agent buys a scheule of claims more weighte towar claims that are relatively cheap. In the Pratt (1964) characterization of risk aversion, the more risk averse agent buys less of a single risky asset an more of a risk-free asset. This also has the effect of proucing a eman curve with a lower slope. The equating of `less risk-averse behavior' with a smaller slope of the eman function for contingent claims is therefore a natural generalization of Pratt's characterization of risk aversion.

5 The Deman for Risky Assets 3 However, Gollier's analysis highlights a problem. Even though agent 1 is more risk averse, he coul have a eman function that has a smaller slope at the crossover point, but has a greater slope over some range of ffi, as Figure 2 illustrates. This means that the more risk-averse investor actually exhibits less risk-averse behaviour over some range. As Gollier notes, the single-crossover property only throws light on local risk-taking behaviour in the range aroun the crossover point. In this paper, we wish to look at local risk-taking behaviour over all ranges, hence we employ a stricter efinition of more risk-averse behaviour in the contingent claims moel. We efine an agent 1tobehave in a more risk averse manner than an agent 2, if his eman function has a smaller (absolute) slope than that of agent 2 everywhere. We then consier how the slope of the eman function changes as backgroun risk increases. If the agent respon to an increase in backgroun risk by choosing a eman function with a smaller slope everywhere, we say that the agent isgeneralize risk averse. This concept of generalize risk aversion relates closely to the previously iscusse concepts of `risk vulnerability' an `stanar risk aversion'. In the case of `risk vulnerability', an agent respon to the introuction of backgroun risk by reucing his eman for a single risky asset. In the case of stanar risk aversion, an agent respon similarly to a marginal utility-increasing backgroun risk. In the case of generalize risk aversion the iea of the response of risk-taking behaviour to an increase in backgroun risk is extene to the case of state-contingent claims. 2 We consier the effect of an inepenent backgroun risk on the eman for state-contingent claims, using an extension of the analysis of Back an Dybvig (1993), who establish conitions for the optimality of an agent's eman. We investigate the set of [restrictions on] utility functions such that the agent respon to monotonic increases in zero-mean backgroun risk by choosing a eman function that has a smaller slope at all price levels. In the context of this choice problem, we nee to further restrict the set of changes in backgroun risk that are consiere to the set of monotonic increases. A monotonic increase in backgroun risk y is efine as a change in y, =@y 0; 8y, an where E( ) = 0. Hence, a monotonic increase in backgroun risk is a change,, that itself increases with y. The simplest example of a monotonic increase is a proportionate increase where is proportionate to y. Ass uming monotonic increases in backgroun risk, we fin that the set of generalize risk-averse utility functions is the stanar risk-averse class of Kimball (1993). Hence, risk vulnerability is not sufficient for backgroun risk to reuce the slope of the eman function for state-contingent claims. 2 Various papers have analyse the impact of certain types of increases in backgroun risk on the eman for insurance, where the amount of insurance is measure by the coinsurance rate an the euctible, see, for example Eeckhout an Kimball (1992) an Meyer an Meyer (1998). While these papers show that stanar risk aversion is sufficient to guarantee a higher eman for insurance, we erive here necessary an sufficient conitions on preferences to yiel generalize risk averse behaviour.

6 The Deman for Risky Assets 4 The conitions for stanar risk aversion - positive, eclining absolute risk aversion, an positive, eclining absolute pruence - are sufficient for a monotonic increase in backgroun risk to increase erive risk aversion. They are also sufficient for the slope of the eman function for contingent claims to become smaller everywhere. What is more surprising is that these conitions are also necessary for generalize risk aversion. Necessity arises from the fact that the slope of the eman function for contingent claims must become less steep at all possible values of ffi. As Kimball argues, eclining absolute risk aversion an eclining absolute pruence are natural attributes of the utility function. They are share, also, by the HARA class of functions with an exponent less than one. The larger set of risk-vulnerable utility functions, use by Gollier an Pratt, is not restrictive enough, when we consier the effect of backgroun risk on the slope of the eman function. Our result a to the case for the stanar risk-averse functions to be the natural class of functions to use when analysing the impact of backgroun risk. In section 2, we look again at the effect of an increase in backgroun risk on the risk aversion of the erive utility function. Here we are concerne, as were Eeckhout, Gollier an Schlesinger (1996) with changes in backgroun risk. However, in orer to avoi the restrictive conitions on utility they foun, we restrict the analysis to simple increases in backgroun risk. In section 3, we then introuce the problem of analysing the slope of the eman function for contingent claims. We then present our main result: agents choosing state-contingent claims become more risk averse in their choice, if an only if they are stanar risk averse, i.e. positive an eclining absolute risk aversion an pruence is the necessary an sufficient conition for generalize risk aversion. 2 The Effect of an Increase in Backgroun Risk on Derive Risk Aversion We consier an iniviual agent who can buy a set of contingent claims on future consumption an faces backgroun risk. The agent's total income at the en of the perio, W, is therefore compose of an income from traeable claims, w, plus the backgroun risk income y, i.e. W = w + y. We assume that backgroun risk, y, has a zero mean, an is boune from below, y a. Moreover we assume that y is istribute inepenently of w. A state of the worl etermines both the agent's income from traeable claims an the backgroun risk income. Let (Ω; F; P) be the probability space on which the ranom variables are efine. The agent's utility function is u(w ). We assume that the utility function is state-inepenent,

7 The Deman for Risky Assets 5 strictly increasing, strictly concave, an four times ifferentiable on W"(W ; 1), where W is the lower boun of W. We assume that there exist integrable functions on!"ω, u 0 an u 1 such that u 0 (!)» u(w )» u 1 (!) We also assume that similar conitions hol for the erivatives u 0 (W ), u 00 (W )anu 000 (W ). The agent's expecte utility, conitional on w, is given by the erive utility function, as efine by Kihlstrom et al. (1981) an Nachman (1982): ν(w) =E y [u(w )] E[u(w + y) j w] (1) where E y inicates an expectation taken over ifferent outcomes of y. Thus, the agent with backgroun risk an a von Neumann-Morgenstern concave utility function u(w ) acts like an iniviual without backgroun risk an a concave utility function ν(w). 3 The coefficient of absolute risk aversion is efine as r(w )= u 00 (W )=u 0 (W ) an the coefficient of absolute pruence as p(w ) = u 000 (W )=u 00 (W ). From Kimball (1993), the agent is stanar risk averse if an only if r(w ) an p(w ) are both positive an eclining. The absolute risk aversion of the agent's erive utility function is efine as the negative of the ratio of the secon erivative to the first erivative of the erive utility function with respect to w, i.e., ^r(w) = ν00 (w) ν 0 (w) = E y[u 00 (W )] E y [u 0 (W )] (2) We first investigate the question of how an agent's erive risk aversion is affecte by a simple increase" in backgroun risk. A simple increase in backgroun risk, which Eeckhout, Gollier an Schlesinger (1995) term 'a simple sprea across y 0 ', is efine as a change in y,,such that for a given y 0,» [=][ ]0; if y<[=][>]y 0 an E( ) = 0: Not surprisingly, the conition for an agent's erive risk aversion to increase, when there is a marginal increase in zero-mean backgroun risk, is stronger than the conition of Gollier an Pratt(1996). This is because the risk vulnerability" conition of Gollier an Pratt only consiers changes in backgroun risk from zero to a finite level, whereas we consier any changes in backgroun risk. 3 See, for example, Eeckhout, Gollier an Schlesinger (1996), p. 684.

8 The Deman for Risky Assets 6 It is worth noting that, in the absence of backgroun risk, ^r(w) is equal to r(w ), the coefficient of absolute risk aversion of the original utility function. In the proposition that follows, we characterize the behavior of ^r(w) in relation to r(w ), an explore the properties of erive risk aversion in the presence of increasing zero-mean backgroun risk. We will procee by proving a proposition about the conition uner which any marginal increase in backgroun risk raises erive risk aversion. Since the conition hol for any marginal increase in backgroun risk, the same conition must hol for a finite increase to raise erive risk aversion. It is convenient to efine an inex of backgroun risk, sfflr +, where s = 0 if no backgroun risk exists. A marginal increase in backgroun risk is represente by a marginal increase in s. We assume that the backgroun risk income y is ifferentiable in s. 4 Proposition 1 (Derive Risk Aversion an Simple Increases in Backgroun Risk) If u 0 (W ) > 0 an u 00 (W ) < 0, > [=][<]0; 8(w; s) () u 000 (W 2 ) u 000 (W 1 ) < [=][>] r(w )[u 00 (W 2 ) u 00 (W 1 )]; 8(W;W 1 ;W 2 );W 1» W» W 2 Proof: See Appenix 1. In orer to interpret the necessary an sufficient conition uner which an increase in a zero-mean, backgroun risk will raise the risk aversion of the erive utility function, first consier the special case in which backgroun risk changes from zero to a small positive level. This is the case analyse previously by Gollier an Pratt (1996). In this case, we have Corollary 1 In the case of small risks, Proposition 1 becomes ^r(w) > [=][<]r(w < [=][>]0; 8W 4 This assumption in no way restricts the type of backgroun risk increases assume in the analysis. Consier, for example, jumps in backgroun risk. These can be analyse as sums of small increases. Our proof erives conitions for the erive risk aversion to change in a certain manner, given a small increment in backgroun risk. The same conitions assure that erive risk aversion changes in a similar manner in response to jumps in backgroun risk.

9 The Deman for Risky Assets 7 where (W ) u 000 (W )=u 0 (W ). Proof: Let W 2 W 1! W. In this case, u 000 (W 2 ) u 000 (W 1 )! u 0000 (W )W. Similarly u 00 (W 2 ) u 00 (W 1 )! u 000 (W )W. Hence, the conition in Proposition 1 yiel, in this case, u 0000 (W ) < [=][>] r(w )u 000 (W ). This is equivalent to@ =@W < [=][>]0; 8W.2 In Corollary 1, we efine an aitional characteristic of the utility function (W )=u 000 (W )=u 0 (W ) as a combine pruence/risk aversion measure. This measure is efine by the prouct of the coefficient of absolute pruence an the coefficient of absolute risk aversion. The corollary says that for a small backgroun risk erive risk aversion excee [is equal to] [is smaller than] risk aversion if an only if (W ) ecreases [stays constant] [increases] with W. Hence, it is significant that neither ecreasing pruence nor ecreasing absolute risk aversion is necessary for erive risk aversion to excee risk aversion. However, the combination of these conitions is sufficient for the result to hol, since the requirement is that the prouct of the two must be ecreasing. The conition is thus weaker than stanar risk aversion, which requires that both absolute risk aversion an absolute pruence shoul be positive an ecreasing. Note that the conition in this case is the same as the 'local risk vulnerability' conition erive by Gollier an Pratt (1996). Local risk vulnerability is r 00 > 2rr 0,whichisequivalent to 0 < 0. We now apply Proposition 1 to show that stanar risk aversion is a sufficient, but not a necessary conition, for an increase in backgroun risk to cause an increase in the erive risk aversion [see also Kimball (1993)]. We state this as Corollary 2 Stanar risk aversion is a sufficient, but not necessary, conition for erive risk aversion to increase with a simple increase in backgroun risk. Proof: Stanar risk aversion requires both positive, ecreasing absolute risk aversion an positive ecreasing absolute pruence. Further, r 0 (W ) < 0 ) p(w ) >r(w ). Also, stanar risk aversion requires u 000 (W ) > 0. It follows that the conition in Proposition 1 for an increase in the erive risk aversion can be written as 5 u 000 (W 2 ) u 000 (W 1 ) u 00 (W 2 ) u 00 (W 1 ) < r(w 1) 5 Note that whenever r 0 (W ) has the same sign for all W, the three-state conition in Proposition 1 (i.e. the conition on W, W 1, an W 2) can be replace by atwo-state conition (a conition on W 1 an W 2).

10 The Deman for Risky Assets 8 or, alternatively,»» p(w 1 ) 1 u000 (W 2 ) = 1 u00 (W 2 ) >r(w [u 000 (W 1 ) [u 00 1 ) (W 1 ) Since p(w 1 ) >r(w 1 ), a sufficient conition is that the term in the square bracket excee 1. This, in turn, follows from ecreasing absolute pruence, p 0 (W ) < 0. Hence, stanar risk aversion is a sufficient conition. To establish that stanar risk aversion is not necessary, consier a case that is not stanar risk averse. Suppose, in particular, that u 000 (W ) < 0;u 0000 (W ) < 0, that is, the utility function exhibits increasing risk aversion an negative pruence. 6 In this case, it follows from Proposition 1 > 0; 8(w; s). 2 In orer to obtain more insight into the meaning of the conition in Proposition 1, consier the case where the increase in backgroun risk raises erive risk aversion. Defining y 0 ^r(w) =E y " # u0 (W ) E y [u 0 (W )] r(w ) = E y " # " u0 (W ) E y [u 0 (W )] r0 (W )y 0 + E y r(w " # # u0 (W ) y E y [u 0 (W )] (3) As shown in appenix 1, it suffices to consier a three-point istribution of backgroun risk with y 1 < 0;y 2 > 0;y 1 <y 0 <y 2 an y 0 0 =0;y 0 1 < 0;y0 2 > 0. The first term in equation (3) is positive whenever r is eclining an convex. This follows since E(y 0 ) = 0 an y 0 2 > y0 1 implies that E[r 0 (W )y 0 ] 0. Since u 0 (W ) is eclining, it follows that the first term in (3) is positive. Now consier the secon 0 (W )=E y [u 0 (W )]]=@y is positive for y 1 an negative for y 2 an has zero expectation. Therefore a eclining r implies that the secon term is positive. Hence a sufficient conition 0 is a eclining an convex r. 7 The first term is higher, the more convex is r. 0 is also possible for an increasing r, if convexity is sufficiently high. Therefore, there are utility functions with 6 As an example, consier the utility function u(w )= 1 fl fl» A + W fl ; where fl 2 (1; 2);W <A(fl 1) 1 fl This utility function exhibits increasing risk aversion an negative pruence. Still, (W ) ecreases with wealth even in this case an the erive risk aversion increases with backgroun risk. 7 See also Corollary 1 of Gollier an Pratt (1996).

11 The Deman for Risky Assets 9 increasing risk aversion which still imply that simple increases in zero-mean backgroun risk raise the erive risk aversion. 3 The Effect of Changes in Backgroun Risk on the Optimal Deman Function for Contingent Claims In the previous section we erive the conition uner which an increase in backgroun risk increases the agent's erive risk aversion. As will be shown, this conition is not sufficient to guarantee that the increase in backgroun risk reuces the slope of the agent's eman curve for state-contingent claims, everywhere, i.e., it is not sufficient for generalize risk aversion. In this section we erive the necessary an sufficient conition for the utility function to exhibit generalize risk aversion. We assume that the capital market is perfect. A state of nature etermines both the agent's traeable income w an his backgroun income y. We partition the state space into subsets of states that iffer only in the backgroun income, y. We call these subsets trae states" since they represent states on which state-contingent claims can be trae. We assume there is a continuum of such states an, for convenience, we label these states by a continuous variable xfflr +. We assume the market, in the trae states, is complete. We also assume that there exists a pricing kernel, ffi = ffi(x) with the property ffi > 0, where ffi(x) is a continuous function. 8 Let w = g(x) be the agent's income from the purchase of state-contingent claims. The agent chooses w = g(x), subject to the constraint that the cost of acquiring this set of claims is equal to his/her initial enowment. The agent's consumption at the en of the single perio, W, is equal to the chosen markete claim, w, plus an inepenent, zero-mean backgroun risk y, i.e. W = w + y. The backgroun risk y affects his/her choice of the function w = g(x). We assume that the agent has sufficient enowment to ensure that w can be chosen to obtain W W in all trae states. We also assume certain properties of 8 The market is complete in the sense of Nachman (1988). The agent can buy a igital option which pays one unit of consumption, if x k, an 0 otherwise, 8kfflR +. The price of such anoptionis Z 1 ffi(x)f(x)(x); k where ffi(x) is the pricing kernel an the probability ensity function is f(x). A contingent claim is a contract (a portfolio of igital options) paying one unit of consumption if xffl[k; k + ) an nothing otherwise, for positive, infinitely small.

12 The Deman for Risky Assets 10 the utility function. First, the marginal utility has the limits: u 0 (W )!1if W! W ; u 0 (W )! 0ifW!1: Secon, the risk aversion goes to zero at high levels of income, i.e. r(w )! 0ifW!1: These reasonable restrictions are satisfie, for example, by the HARA class with an exponent lessthan1. The agent solves the following maximization problem: max E x [ν(w)] = E x [ν(g(x))] (4) w=g(x) i s.t. E x h(g(x) g 0 (x))ffi(x) =0 In the buget constraint, w 0 = g 0 (x) is the agent's enowment of claims. ffi(x), the pricing kernel, is given exogenously. The maximisation problem (4) is a stanar state-preference maximisation problem. The expectation, E x (:), is taken only over the trae states. Note that the backgroun income, y, has only an inirect impact on problem (4) through its effect on the erive utility function. This is efine by equation (1) as the expecte value of utility over ifferent outcomes of y, given the trae income w. The first orer conition for a maximum is or simply ν 0 (g(x)) = ffi(x); ν 0 (w) = ffi; (5) where is a positive Lagrange multiplier which reflects the tightness of the buget constraint. Equation (5) hol as an equality since, by assumption, u 0 (W )!1for W! W an u 0 (W )! 0 for W! 1. The eman for claims in equation (5) can be shown to be optimal an unique uner some further finiteness restrictions. 9 This follows from the results of Back an Dybvig (1993). 9 E[wffi] < 1 for any >0 an each w satisfying (5) is assume.

13 The Deman for Risky Assets 11 >From the first orer conition (5), it follows that we can efine a function w = w(ffi) = ν 0( 1) ( ffi). Hence, given the erive utility function an the initial enowment, the eman for claims contingent on a trae state x epen only on ffi(x). Thus w(ffi) is a eterministic function relating the eman for state-contingent claims to the pricing kernel. It follows from our assumptions that w(ffi) isatwice ifferentiable function of ffi. 10 Our aim is to fin the necessary an sufficient conitions on the utility function, which guarantee that the agent's eman function becomes less steep when backgroun risk increases. First we efine Definition 1 An agent is generalize risk-averse if the absolute value of the slope of his/her eman function for state-contingent claims w(ffi) becomes smaller for all ffi, given an increase in backgroun risk. Differentiating equation (5) with respect to ffi, for a given level of backgroun risk, an iviing by ffi, yiel the slope of the eman = 1=ffi ; 8ffi (6) ^r(w) Suppose that backgroun risk increases the erive risk aversion of the agent, ^r(w). It follows from equation (6) that the backgroun risk affects the slope of the eman function. We now consier the effect of changes in the level of backgroun risk, assuming that the pricing function ffi is given. From equation (6) it appears at first sight that the slope of the eman function becomes less steep whenever the increase in backgroun risk increases the agent's erive risk aversion. In fact, it follows from Gollier (2000, Proposition 51) that: Proposition 2 Suppose that an increase in backgroun risk raises the agent's erive risk 10 Consier the function F (w; ffi; s) =ν 0 (w) ffi =0: The partial erivative F w exists an is continuous, since the utility function u(w + y) an its first three erivatives are assume to exist an to be integrable. Also F w 6= 0for w < 1. Hence, by theimplicit function theorem, the function w = w(ffi) is ifferentiable with =@ffi = F ffi F w : Also, since y = y(s) is ifferentiable, an since F ffi an F w are ifferentiable in y, thenf ffi an F w ifferentiable in s. It follows 2 w=@ffi also exists. are also

14 The Deman for Risky Assets 12 aversion, everywhere. Then the new eman curve for contingent claims intersects the original one once from below. Proof: At anintersection of the new eman curve, w 1 (ffi), an the original eman curve, w 0 (ffi), w 1 = w 0 so that, by equation (6), 1 =@ffi > 0 =@ffi follows from ^r 1 > ^r 0. A secon intersection woul require 1 =@ffi < 0 =@ffi, which contraicts (6). Also, at least one intersection must exist, in orer for the buget constraint to be satisfie.2 However, as note by Gollier (2000), the one-intersection property oes not imply that the new eman curve is less steep than the original one, everywhere. This is because a change in backgroun risk, affects ^r(w) both irectly an through the inuce change in w. This is state in the following proposition. Proposition 3 For the slope of the eman function for contingent claims to become smaller with an increase in backgroun risk (generalize risk aversion), it is necessary, but not sufficient for the absolute risk aversion of the erive utility function to increase with backgroun risk. That is 0 0; but oes not 0 Proof: Totally ifferentiating equation (6) with respect to s yiel, since 1=ffi is given, Since = 1=ffi [^r(w)] 2 : (8) 1=ffi [^r(w)] 2 > 0; 0, ^r(w) + 0: (9)

15 The Deman for Risky Assets 13 Given the buget constraint, = has to be positive in some trae states an negative in others. It follows immeiately 0 is not sufficient to ensure that Now to establish necessity, suppose that for all ffi, then since the sign epen on the sign of =, which can be positive ornegative, 2 0 0: Having shown that increase erive risk aversion is a necessary, but not sufficient conition for generalize risk aversion, we can now establish our main result. In orer to analyse the impact of backgroun risk on the slope of the agent's eman function for contingent claims we nee to make stronger assumptions. Regaring the backgroun risk we now assume monotonic changes in backgroun risk. This is a somewhat stronger than the previous assumption of simple increases in backgroun risk. First we efine monotonic increases in backgroun risk. Definition 2 (Monotonic Increases in Backgroun Risk) Let y i (s) enote a realisation i = 1; :::; j of backgroun risk income, given the inex of backgroun risk, s. Suppose that y 1 (s)» y 2 (s)» :::» y i (s)» :::» y j (s) with y i (0) = 0; 8i. Then, increases in backgroun risk are monotonic, if for any s>s 0, y 1 (s) y 1 (s)» y 2 (s) y 2 (s)» :::y i (s) y i (s)» :::» y j (s) y j (s) The effect of assuming monotonic increases in backgroun risk is that the rank orer of the outcomes y 1 ;y 2 ; ::: is preserve uner a monotonic increase in backgroun risk. The main result of the paper is Proposition (4).

16 The Deman for Risky Assets 14 Proposition 4 (Generalize Risk Aversion) Assume any monotonic increase in an inepenent, zero-mean backgroun risk. Let u 0 (W ) > 0 an u 00 (W ) < 0, where W"(W ; 1). Suppose that u 0 (W )! 1 for W! W an that u 0 (W )! 0 an r(w )! 0, for W!1. Then 0; 8ffi; 8 probability istributions of () utility is stanar risk averse: We first establish three lemmas which are require in the proof. We have Lemma 1 Suppose that u 0 (W )!1for W! W, then r(w )!1an p(w )! 1 for W! W. Proof: u 0 (W )! 1, for W! W, 0 (W )=@W! 1, an hence r(w )! 1. Also, since for W! W, r 0 < 0;p >r, an hence p(w )!1. 2 The secon lemma establishes the equivalence of eclining risk aversion an eclining erive risk aversion. We have: Lemma 2 ^r 0 (w)» 0 for any backgroun risk, r 0 (W )» 0 Proof: Kihlstrom et. al. (1981) an Nachman (1982) have shown that eclining risk aversion implies eclining erive risk aversion. Conversely, eclining erive risk aversion implies eclining risk aversion of u(w ). This follows from the case of small backgroun risks.2 The thir lemma establishes a conition for eclining pruence, in the case of monotonic changes in backgroun risk: Lemma 3 For monotonic increases in backgroun (w)= 0, p 0 (W )» 0 0 (w)=

17 The Deman for Risky Assets 15 Proof: See Appenix 2. We now present the proof of Proposition (4). Proof of Proposition (4): Totally ifferentiating equation (5) with respect to s 0 (w) (w) Substituting from equation (5) then yiel = ffi: 0 (w) (w) = ln ν0 (w): Hence, the effect of the backgroun risk on the eman for claims is given by = ln The Proposition is concerne with the conitions uner which ffi 1 0 (w)= : (11) 0: We investigate these conitions by looking at the behaviour of the two terms in equation (11). Sufficiency of Stanar Risk Aversion: First, we show that the first term in (11) is negative, while the secon term is positive. In orer to satisfy the buget constraint, = has to be positive in some trae states an negative in others. Given positive 0 (w)= > 0, so that the secon term in (11) is positive. It follows that the first term must be negative. We cannowinvestigate ffi ; by taking the two terms in (11) one-by-one. First, the (negative) first term increases with @ffi

18 The Deman for Risky Assets 16 is positive. This follows from < 0 ( see equation (6)) 0(which in turn follows 0 an Lemma 2). Secon, the (positive) secon term increases in ffi, given eclining pruence (see Lemma 3). Hence is positive given stanar risk aversion. ffi Necessity of Stanar Risk Aversion: We establish necessity of stanar risk aversion by taking the special case of a small backgroun risk. Also, we assume ffi converges in probability to a egenerate istribution, ffi 0. By assuming w(ffi 0 ) is, in turn, large [small], we show that the first [secon] term in (11) ominates. For the first term in (11) to increase in ffi, eclining risk aversion is require. For the secon term in (11) to increase in ffi, eclining pruence is require. Hence, to cover both of these possibilities, stanar risk aversion is require. First, we consier the term ln =. We have from equation (5), E[ν 0 (w)] = E[u 0 (w + y)] = E( ffi) = an = E[u0 (w + y)] = E[u0 (w ψ)]; where ψ = ψ(w) is the precautionary premium as efine by Kimball(1990). Hence, ρ» = E u 00 ff : Assume that we start from a position of no backgroun risk. In this case, s = 0, ψ = 0, = 0. Since, for small backgroun risks with variance ff 2, the precautionary premium is 11 ψ = 1 2 p(w)ff2 ; it follows that ρ» = E u 00 ff ρ» = E u 00 (w) 1 ff 2 p(w)ff2 : 11 This follows by analogy with the Pratt-Arrow argument for the risk premium, since initially, s =0.

19 The Deman for Risky Assets 17 Now we assume that ffi converges to the egenerate istribution ffi 0, in probability. Since we can write = E[f(ffi)]; where f is a continuous, uniformly integrable function, then it follows that! u00 (w 0 )» 1 2 p(w 0)ff 2 ; where w 0 = w(ffi 0 ), since 0 = = 0, for the case of the egenerate istribution, ffi 0. Diviing by = u 0 (w 0 ), ln! u00 (w 0 )» 1 u 0 (w 0 ) 2 p(w 0)ff 2 an hence ln Substituting in (11), we now have Starting with no backgroun risk, the term = 0. Hence, we can write Differentiating (12), we then have ffi 1! r(w 0 ) 2 p(w 0)ff 2 : 1 1! r(w 0) 2 p(w 0)ff 2 0 (w)= ( 0 ( )= = 1 2 p(w)ff2 ; 1 1! r(w 0) 2 p(w 0)ff 2 ^r(w) p(w)ff2 : (12) ρ 1! r(w 0 ) 2 p(w 0)ff 2 ^r0 (w) ^r(w) + 1 ff 2 2 p0 (w)ff Since =@ffi < 0, the conition for a smaller slope becomes r(w 0 )» 1 2 p(w 0)ff 2 ^r0 (w) ^r(w) p0 (w)ff 2» 0: (13)

20 The Deman for Risky Assets 18 To establish the necessity of eclining absolute risk aversion, we choose ffi 0 such thatw 0! W. By Lemma 1 hence r(w)! 1 an p(w 0 )! 1, for w! W. Therefore, ^r 0 (w) > 0 implies that the first term in equation (13)!1. Then, since the secon term in (13) is inepenent ofw 0 ; ^r 0» 0 an by Lemma 2, r 0» 0 is require for the conition (13) to hol. r 0» 0 also establishes the necessity of positive pruence, p>0. To establish necessity of eclining absolute pruence, we choose ffi 0 such that w 0! 1 an hence, by assumption r(w 0 )! 0. Then r 0 (w 0 ) = r(w 0 )[r(w 0 ) p(w 0 )]! 0 implies r(w 0 )p(w 0 )! 0. Hence the first term in equation (13)! 0. Then, since the secon term in (13) is inepenent ofw 0, p 0» 0 is require for the conition (13) to hol. Hence stanar risk aversion is a necessary conition for a smaller slope.2 Proposition (4) allows us to analyze the effect of a marginal increase in a zero-mean, inepenent backgroun risk, given that this increase has a negligible impact on the prices of state-contingent claims. Since a finite increase in backgroun risk is the sum of marginal increases, the conition in Proposition (4) also hol for finite increases in backgroun risk. Proposition (4) says that an increase in backgroun risk will reuce the steepness of the slope of this agent's eman function. As can be seen from Proposition (4), the agent reacts to a monotonic increase in backgroun risk by purchasing more claims in trae states for which the price ffi is high, financing the purchase by selling some claims in the trae states with low prices. Proposition (4) can also be interprete by comparing, within an equilibrium, the eman of agents, who iffer only in the size of their respective backgroun risks. Proposition (4) suggests that agents with higher backgroun risk will ajust their eman functions by buying state-contingent claims on high-price trae states an selling claims on low-price trae states. This is illustrate in Franke, Stapleton an Subrahmanyam (1998), for an economy in which all agents have the same type HARA-class utility function, exhibiting eclining absolute risk aversion. These functions are stanar risk averse an hence generalize risk averse. In this economy, agents with high backgroun risk buy options from those with relatively low backgroun risk. The latter agents sell portfolio insurance to the former with relatively high backgroun risk. 4 Conclusions The main conclusions regaring the effects of an increase in backgroun risk, on risk aversion an on the eman for contingent claims, are summarise in the four propositions of the paper. Proposition 1 provies a necessary an sufficient conition for simple increases in backgroun risk to increase the erive risk aversion of agents. The conition on utility is weaker than Kimball's stanar risk aversion, but stronger than Gollier an Pratt's risk

21 The Deman for Risky Assets 19 vulnerability. By consiering only the set of simple increases in backgroun risk, we fin a larger set of utility functions which satisfy the criterion of increase erive risk aversion, than those of Eeckhout, Gollier an Schlesinger. We then procee to examine the conition for 'generalize risk aversion', whereby agents react to increase backgroun risk by reucing the slope of the eman curve for state contingent claims. We fin in Proposition 3 that increase erive risk aversion is necessary, but not sufficient, for generalize risk aversion. The stronger requirement for generalize risk aversion is shown for the case of monotonic increases in backgroun risk in Proposition 4. Stanar risk aversion, i.e. positive, eclining absolute risk aversion an absolute pruence, is a necessary an sufficient conition for generalize risk aversion.