MULTIPLICATIVE BACKGROUND RISK *

Transcription

1 May 003 MULTIPLICATIVE BACKGROUND RISK * Günter Franke, University of Konstanz, Germany (GenterFranke@ni-konstanzde) Harris Schlesinger, University of Alabama, USA (hschlesi@cbaaed) Richard C Stapleton, University of Manchester, England (richardstapleton1@btinternetcom) Abstract Althogh there has been mch attention in recent years on the effects of additive backgrond risks, the same is not tre for its mltiplicative conterpart We consider random wealth of the mltiplicative form xy, where x and independent random variables We assme that bt that y y are statistically x is endogenos to the economic agent, is an exogenos and nontradable backgrond risk, which represents a type of market incompleteness Or main focs is on how the presence of the mltiplicative backgrond risk y affects risk-taking behavior for decisions on the choice of characterize conditions on preferences that lead to more catios behavior x We Keywords: mltiplicative risks, backgrond risk, incomplete markets, standard risk aversion, affiliated tility fnction, mltiplicative risk vlnerability JEL Classification No: D81 * The athors thank seminar participants at Dke University, the Wissenschaftszentrm-Berlin, the Universities of Alabama, Konstanz, Melborne and Pennsylvania, as well as participants at the Risk Theory Seminar in Champaign, Illinois, the SIRIF Conference on Strategic Asset Allocation in Glasgow and the Meeting of the Eropean Grop of Risk & Insrance Economists in Nottingham for helpfl comments We also are gratefl to Axel Adam-Müller, Lois Eeckhodt, Bob Na, Lisa Tibiletti and Nicolas Treich for sefl discssions

2 Mltiplicative Backgrond Risk 1 Introdction Consider a risk-averse economic agent whose preferences can be represented within an expected-tility framework via the continosly differentiable tility fnction The agent mst decide pon choice parameters for a random variable representing final wealth, x For example, x might represent wealth from an individal s portfolio of financial assets, or x might represent random corporate profits based on management decisions within the firm A fair amont of attention in recent years has examined how decisions on x might be affected by the addition of an additive risk ~ ε, where ~ ε and x are statistically independent Ths, final wealth or profits can be written as x + ε The market is assmed to be incomplete in that ~ ε is not directly insrable For example, ~ ε might represent ftre wage income sbject to hman-capital risks; or ~ ε might represent an exogenos pension portfolio provided by one s employer Althogh it is interesting to examine the interdependence between x and ~ ε, the case of independence is of special interest and provides for many interesting observations In order to focs on the risk effects, rather than wealth effects, it is often assmed that E ~ ε = 0, where E denotes the expectation operator In sch a case, ~ ε is often called a backgrond risk Since any non-zero mean for ~ ε can be added to the x term, this assmption does not redce the applicability of the model Or prpose in the present paper is to examine the effects of introdcing a mltiplicative backgrond risk into the individal s final wealth distribtion The modern literatre on additive backgrond risk stems from the papers of Kihlstrom, et al (1981), Ross (1981) and Nachman (198) These papers focs on interpersonal behavior comparisons, mainly addressing the qestion: If I am willing to 1

3 Mltiplicative Backgrond Risk pay more than yo to rid myself of any fair lottery, wold I still be willing to do so in the presence of an additive backgrond risk? Doherty and Schlesinger (1983) incorporated the analysis into intrapersonal models of decision making nder ncertainty, focsing on differences in optimal behavior with vs withot a backgrond risk The literatre nderwent somewhat of a renaissance in the 1990 s thanks to new theoretical tools provided by Pratt and Zeckhaser (1987), Kimball (1990) and Gollier and Pratt (1996) One canonical hypothesis concerning additive backgrond risk is that the riskiness of ~ ε leads to a more catios behavior towards decisions on x For example, Giso, et al (1996) se Italian srvey data to show that individals with a riskier perception of their (exogenosly managed) pension wealth react by investing relatively more in bonds in their personal acconts However, this conclsion need not always be the case in theory, nless particlar restrictions on preferences are met Eeckhodt and Kimball (199) first examined this direction of research Rather than review the large body literatre for the case of additive backgrond risks, we refer the reader to the excellent comprehensive presentation of this material in Gollier (001) Srprisingly, very little attention has been given to the case where the backgrond risk is mltiplicative Indeed, if one were to ask the reader to think of possible types of backgrond risks, we believe that examples with mltiplicative types of backgrond risk wold be at least as prevalent as additive ones Or goal in this paper is to provide a theoretical fondation for models with a mltiplicative backgrond risk Under what conditions on preferences will the presence of a mltiplicative backgrond risk compel the agent to behave more catiosly in making decisions abot the endogenos wealth variable x? To this end, let y be a random variable on a positive spport that is statistically independent of x We consider final wealth to be given by the prodct xy The random variable y is considered to be exogenos to the individal and is not insrable Nmeros examples of sch mltiplicative risks inclde the following:

4 Mltiplicative Backgrond Risk 1 Let x be the pre-tax profits of a firm and let y represent the firm s retention rate net of taxes, where tax rates are random de to tax-legislation ncertainty Let x be the random wealth in an individal s financial portfolio in period one, and let y denote the retrn on a mandatory (and exogenosly managed) annity accont that ses proceeds from x in period two 3 Let x denote nominal wealth or profit and let y denote an end-of-period price deflator 4 Let x denote profit in some foreign crrency for which forward contracts or options are not available and let y denote the end-of-period exchange rate 5 Let x denote the random qantity of otpt for a farm commodity and let y denote an exogenos random per-nit profit In order to isolate the risk effects of y, we will assme that Ey =1 throghot this paper For the case where y has a mean that differs from one, we can incorporate this mean into x via a deterministic scaling effect 1 Since xy = x + x (y 1), the assmption that Ey =1, together with the independence of x and y, garantees that xy is riskier than x alone in the sense of Rothschild and Stiglitz (1970) We will refer to y, defined in this manner with Ey =1, as a mltiplicative backgrond risk We shold point ot at the otset that the reslts for the mltiplicative case do not simply mirror those of the additive case For instance, consider a simple portfolio example with an allocative choice between risky stocks and risk-free bonds The 1 Ths, for instance, in or first example above we can let ~ x represent after-tax profits based on the expected tax rates and let ~ y represent a deviation from the expected after-tax retention rates Or, in the second example let x ~ denote wealth inclding expected annity retrns and let y ~ denote a mltiplicative excess-retrn adjstment 3

5 Mltiplicative Backgrond Risk individal has an initial wealth of 100 and the risk-free rate is assmed to be r f = 005 The retrn on the stock portfolio is assmed to be log-binomial with an expected retrn of Er = 011 and a standard deviation of σ = 00 (implying that, in a binomial model, stocks either retrn abot 33% or lose abot 10%, each with an eqally likely chance) Utility is assmed to belong to the HARA class with x ( ) = ( x+ ) 1 a, where a is a constant chosen sch that x+a remains positive over relevant wealth levels We note that preferences satisfy decreasing absolte risk aversion (DARA) for any choice of a, whereas relative risk aversion will be increasing [decreasing, constant] whenever a is positive [negative, zero] We examine the addition of two alternative sorces of backgrond risk The first is an additive backgrond risk, for which final wealth is either increased or decreased by 30, each with probability one-half The second is a mltiplicative backgrond risk, for which wealth is either increased or decreased by 30 percent, each with a probability one-half The optimal portfolio choices are illstrated in the following table TABLE 1: Bond Proportions: Mltiplicative vs Additive Backgrond Risk 1 (All tility is DARA within the HARA class, x ( ) = ( x+ a), initial wealth = 100) (Relative risk aversion is constant for a=0, increasing for a=+5 and decreasing for a=-5) Utility Backgrond Risk Proportion in Bonds a = 0 None 55% Additive 66% Mltiplicative 55% a = +5 None 45% Additive 54% Mltiplicative 41% a = -5 None 66% Additive 78% Mltiplicative 70% 4

6 Mltiplicative Backgrond Risk In each case in the above example, the proportion of wealth invested in risk-free bonds increases when an additive backgrond risk is inclded Since DARA inside of the HARA class of preferences also implies standard risk aversion (Kimball 1993), we know that bond proportions will always increase with an additive backgrond risk However, as the example shows, a mltiplicative backgrond risk might case the bond proportion to shrink In particlar, when a = 5, so that we have both DARA and increasing relative risk aversion hardly considered nsal cases we then have a lower proportion of wealth invested in the risk-free bond That is, the investor reacts to the mltiplicative backgrond risk by taking a more aggressive position in stocks Or paper will show how each of the sitations in the example above can be determined qalitatively (ie whether more or fewer bonds are prchased in the presence of a backgrond risk) before calclating the optimal portfolios The fact that the qalitative effects might be predetermined by the parameters of the model implies that care mst be taken when modeling varios economic and/or financial phenomena For example, seemingly innocos assmptions made abot preferences might actally predispose a model to achieve particlar reslts We begin in the next section by introdcing the basic framework We next examine some conditions on preferences that lead to more (or less) catios behavior towards x in the presence of a mltiplicative backgrond risk y In section 4, we derive rather technical necessary and sfficient conditions on preferences sch that a mltiplicative backgrond risk will always lead to more catios behavior: a condition that we label mltiplicative risk vlnerability In section 5, we define the affiliated tility fnction as the composite of tility with the exponential fnction This allows s to translate several reslts from the case of additive backgrond risk to or model with Note that, even for the cases with no backgrond risk, since relative risk aversion is decreasing in a, we have the bond proportion falls as a rises Or point in the table, however, is to compare the levels of bonds between varios types of backgrond risk for a fixed vale of a 5

7 Mltiplicative Backgrond Risk mltiplicative backgrond risk In particlar, necessary and sfficient conditions on the affiliated tility are presented sch that a mltiplicative backgrond risk will always lead to more catios behavior Section 6 extends the seflness of the conditions placed on the affiliated tility fnction by determining eqivalent properties of the individal s actal tility fnction Section 7 briefly looks at comparative risk aversion, before we offer some conclding remarks The Basic Model Consider a risk-averse economic agent with tility fnction We wish to determine how the addition of a mltiplicative backgrond risk ~ y affects decision making on ~ x Both ~ x and ~ y are assmed to be strictly positive as Let F and G denote the (cmlative) distribtion fnctions associated with the random variables ~ x and ~ y respectively Since ~ x and ~ y are independent, we can write expected tility as the iterated integral (1) E( xy ) = ( xy) dg( y) df E [ E ( xy )] 0 0 Define the derived tility fnction, see Nachman (198) 3, as the interior integral given in eqation (1) That is, F G () v ( xy) dg( y) = E ( xy ) G 0 G 3 Actally, Nachman considers a more general relationship between x ~ and y ~ We adapt his measre to the case of mltiplicative risks The derived tility fnction for the additive case is described earlier by Kihlstrom, et al (1981) 6

8 Mltiplicative Backgrond Risk Trivially, as v G (x) is increasing and concave since is Ths, eqation (1) can be written E( xy = E v ( x ) Decisions on ~ x made in the presence of the mltiplicative risk ~ y ) F G nder tility are isomorphic to decisions made on x ~ in isolation nder the risk-averse tility v G (x) Let Γ( x ) denote the set of positive random variables ~ y sch that ~ y is statistically independent from x ~ and Ey =1 Or focs here is in determining conditions on the tility fnction sch that the derived tility fnction, v G (x), is more risk averse than for all y~ Γ ( x ) In other words, we wish to determine conditions on that will garantee that (3) v x E xy y x v' E [ '( xy ) y ] ' " G( ) G[ "( ) ] "( ) G G x 4 write v(x) and To avoid excessive notation, we will dispense with the sbscripts and simply E( xy ), where we assme y ~ is an arbitrary member of Γ( x ~ ) We will let r v (x) and r (x) denote the measre of absolte risk aversion for v and respectively, ie the left-hand-side and right-hand-side of ineqality (3) respectively Since we are involved with a mltiplicative backgrond risk, it is often convenient to consider the corresponding measres of relative risk aversion, R v (x) xr v (x) and R (x) xr(x) Obviosly, for any positive wealth level x, R v R r r For arbitrary x, straightforward maniplation of (3) shows that v if and only if (4) '( ) ( ) [ ( ) xy y R x = E R xy ] R (xy)dη x (y) v E [ '( xy) y] 0 4 In order to keep the mathematics simple, we will take more risk averse to be in the weak sense of Pratt (1964) 7

9 Mltiplicative Backgrond Risk where η x (y) '(xt)tdg(t) E [ '( ) ] G xy y y 0 Note that η (y) is itself a well-defined probability distribtion We define Ê to x denote the expectation operator based on the probability distribtion (y) η x, which is a type of risk-adjsted probability measre 5 Ths, we see that relative risk aversion for v is a weighted average of relative risk aversion for, namely R = Eˆ [ R ( xy )] v x 3 Risk Aversion Properties From eqation (4), it follows trivially that v inherits constant relative aversion (CRRA), whenever exhibits CRRA More explicitly, if R = γ x, then R v = γ x as well Since it then also follows that r r x, we see that and v are eqivalent tility representations nder CRRA This is not srprising, since any optimal choice of an endogenos = x~ also will be optimal for x~ y, for every constant v positive level of y nder CRRA preferences From eqation (4), we also see that (x) R v will be everywhere greater than [less than] one if (x) R is everywhere greater than [less than] one This reslt is more than jst a technicality Since many reslts in the literatre on choice nder ncertainty specify a global condition that either R (x) >1 or R (x) <1, sch reslts also will hold in ) the presence of a mltiplicative backgrond risk, since R v (x also will satisfy the ) appropriate property More generally, eqation (4) provides bonds for (x R v, sch that given any y Γ( x), with distribtion fnction G, { } { } inf R ( xy) R sp R ( xy) y Spp( G) v 5 If we have a representative agent model, and if we confine orselves to a fixed vale of x, this measre is simply the risk-netral probability measre The random variable [ ' ( xy ) y ]/ E[ '( xy ) y ] in eqation (4) is the Radon-Nikodym derivative of this measre with respect to G, again conditional on a fixed vale of x To simplify notation below, we will write simply Ê, since the x sbscript shold be nderstood 8

10 Mltiplicative Backgrond Risk We next wish to examine conditions nder which (3) holds ~ y Γ( ~ x), ie, we want to know when v is more risk averse than We may consider conditions for which this holds locally, with x, by applying r r, by examining the eqivalent condition v R R Or approach is to consider this last ineqality for a particlar vale of v Rv R η x as in eqation (4) If the vale of x chosen is arbitrary, so that x, then we are done In the rest of this section, we extend eqation (4) to directly obtain sfficient conditions for which v is more (or less) risk averse than In the following sections of the paper, we introdce two additional approaches to the problem Sppose that R (x) is (not necessarily strictly) convex Since η (y) is a probability distribtion, it follows from Jensen s ineqality and eqation (4) that x (5) R ER ˆ ( xy ) R ( xey ˆ ), v where (6) ˆ '( xy) y Ey = ydη ( y) = y dg( y) E [ '( xy ) y ] x 0 0 Next, note that (7) xy ( ) = [ '( xy) y] = '( xy)[1 R ( xy)] xy y The sign of (7) tells s whether increases in the level of y will increase or decrease the marginal tility of x The derivative in (7) will be everywhere positive [negative] if R ( xy) < [ > ] 1 y in the spport of G This implies that increases in y redce the 9

11 Mltiplicative Backgrond Risk marginal tility of x whenever whenever R < 1 Since R > 1, and increases in y increase the marginal tility of x '( xy ) y E =1, we see from (6) and (7), for example, that E [ '( xy ) y ] R > 1 everywhere implies that the probability measre η x (y) pts relatively more weight on lower vales of y than does the tre probability measre G(y) The opposite is tre if R < 1 We ths obtain the following reslt from (6) and (7) Lemma 1: Êy Ey = 1 if R ( xy) 1 y Spp( G) We are now ready to prove the following reslt: Proposition 1: Sppose that (x) R is convex and that one of the following conditions holds ( x, y) Spp( F) Spp(G) : (i) R ( xy >1 and R is decreasing, or (ii) R ( xy <1 and R is increasing ) ) Then v is more risk averse than Proof: Since R (x) is convex, it follows from eqation (4) that R ( ) ( ˆ v x R xey ) by Jensen s ineqality If R 1 ( ˆ >, then Êy <1 from Lemma 1 Hence, R xey ) R( x ) nder the assmption of decreasing relative risk aversion (DRRA) If R < 1, then it follows from Lemma 1 that Êy >1 Hence, R ( xey ˆ ) R ( x nder the assmption of ) increasing relative risk aversion (IRRA) Ths we have condition (i) or (ii) holds R v (x) R whenever Interestingly, if we have CRRA preferences, we have already seen that and v are eqivalent regardless of whether or not relative risk aversion exceeds one If relative risk aversion is increasing in wealth, as originally postlated by Arrow (1971) and empirically 10

12 Mltiplicative Backgrond Risk spported by mch literatre, most recently by Giso and Paiella (001), then v will be more risk averse than whenever R is convex and less than 1 If R is everywhere greater than 1 and exhibits increasing relative risk aversion, we cannot se Proposition 1 to verify that v is more risk averse than Indeed, if we have R >1 and if R is (not necessarily strictly) concave, it is easy to show that v is then less risk averse than Indeed, the following two cases are easy to show Proposition : Sppose that (x) R is concave and that one of the following conditions holds ( x, y) Spp( F) Spp(G) : (i) R ( xy) >1 and R is increasing, or (ii) R ( xy) <1 and R is decreasing Then v is less risk averse than Proof: The proof is similar to Proposition 1 and left to the reader Of corse, whether risk aversion exhibits constant-, increasing-, or decreasing relative risk aversion, or none of these, is an empirical qestion Certainly constant relative risk aversion is very common in eqilibrim asset-pricing models Bt empirical spport also exists for both increasing relative risk aversion (eg Giso and Paiella (001)) and for decreasing relative risk aversion (eg Ogaki and Zhang (001)) Whether relative risk aversion might be concave or convex in wealth has not received mch attention at all ntil fairly recently For example, Aït-Sahalia and Lo (000) examine S&P 500 option prices to find some evidence of an oscillating level of relative risk aversion, althogh they do find R to be decreasing and convex at relatively low levels of wealth 6 Aït-Sahalia and Lo (000) also review mch of the literatre examining whether 6 See also Jackwerth (000) 11

13 Mltiplicative Backgrond Risk relative risk aversion is greater- or less-than one, with most spport these days finding R>1 To illstrate Proposition 1 and, consider the following examples: kx Example 1: Let = e where k > 0 This is the case of constant absolte risk aversion (CARA) In this case R '( x ) = k and R ''( x ) = 0 Ths, is increasing and is R both convex and concave If we consider ( F ) Spp (G Spp However, if ~ x and ~ y sch that xy < 1/ k ( x, y) ), then R ( xy) < 1 and v is more risk averse than by Proposition 1 xy > 1/ k ( x, y) Spp( F ) Spp(G), then R ( xy) > and v is less risk averse than by Proposition 1 Example : Let = x kx where > 0 We restrict 1 k k x < so that marginal tility is positive This is the case of qadratic tility It is straightforward to show that 1 R ) = kx(1 kx and that R is both strictly increasing and convex Moreover, 1 R ( xy) < 1 if xy< 4 k ( x, y) Spp( F ) Spp( G), so that v is more risk averse than by Proposition 1 In other words, v is more risk averse than over the first half of the relevant (pward-sloping) range of the qadratic tility fnction On the other hand, if k < xy< k ( x, y) Spp F G, then R ( xy) >1, bt we cannot ( ) Spp ( ) apply Proposition 1 (since is increasing) or Proposition (since R is convex) R Both tility fnctions above belong to the so-called HARA class of tility, as does CRRA tility 7 Since we already showed that and v are eqivalent nder CRRA, we see that no general reslts seem to apply to the HARA class of tility However, we have x 1 γ x more tractability in the shape of R nder HARA Let x ( ) = ξη ( + ), where η + > 0 and ξ(1 γ ) γ > 0 Straightforward calclations show that γ x = ηη+ and that R ' ( γ ) γ 7 Utility belongs to the HARA class if [r(x)] -1 is linear in x 1

14 Mltiplicative Backgrond Risk x 1 R" = [ γ ( η + γ) ] R '( x ) Ths, for the case of constant absolte risk aversion (γ ), we obtain R '( x ) = R "( x ) > 0 k and R ''( x ) = 0, as in Example 1 If we have increasing absolte risk aversion, then we mst have γ < 0 and η > 0 It follows that, so that we mst have R increasing and convex, as is the case with or aversion (DARA), then γ > 0 Hence, sgn R " = sgn R '( x ) Conseqently, we mst R '( x ) > 0 qadratic tility in Example On the other hand, if we have decreasing absolte risk have R either (i) constant, (ii) decreasing and convex, or (iii) increasing and concave Conseqently, if preferences are DARA within the class of HARA tility fnctions, it follows that we might have v either more risk averse than, less risk averse than or eqally as risk-averse as In particlar, corresponding to cases (i) - (iii) above: and (i) (ii) If satisfies CRRA, then v and are eqivalent If R > 1, as well as decreasing, then v is more risk averse than by Proposition 1 (iii) If R >1, as well as increasing, then v is less risk averse than by Proposition Note that R > 1 in or example in the introdction of this paper (see Table 1) and that conditions (i), (ii) and (iii) above apply to the three cases considered in or example, with a=0, a=-5 and a=+5 respectively 4 Mltiplicative Risk Vlnerability Let the spport for y be contained in some positive interval [a,b] As a direct analoge to Gollier and Pratt (1996), who examine the case of additive risks, we define preferences as being mltiplicatively risk vlnerable if for every positive wealth variable 13

15 Mltiplicative Backgrond Risk x and every y Γ( x ), that is for every y independent of x with Ey =1, the derived tility fnction v is more risk averse than In other words, any (independent) mltiplicative backgrond risk with a mean eqal to one always cases an individal with mltiplicatively risk-vlnerable preferences to behave in a more catios manner towards risk x 8 In this section, we present a necessary and sfficient condition for tility to be mltiplicatively risk vlnerable Since this condition is rather complex, we trn in the next section to some sfficient conditions on preferences to garantee mltiplicative risk vlnerability We also show how or condition relates to the Gollier and Pratt conditions for the case of additive backgrond risks Before proceeding, we reqire the following Theorem, which is de to Gollier and Kimball (1996) A proof of this Theorem also can be fond in Gollier (001) Diffidence Theorem (Gollier and Kimball): Let Λ denote the set of all random variables with spport contained in the interval [a,b] and let f and g be two real-valed fnctions The following two conditions are eqivalent: (i) For any y Λ, Ef( y ) = 0 Eg( y ) 0 (ii) m sch that g( y) mf( y) y [ a, b] We now are ready to show the following reslt Proposition 3: Utility is mltiplicatively risk vlnerable if and only if for every x>0 and every y [ a, b], (8) '( xy) y[ R ( xy ) R ] ( y 1) ' xr ' 0 8 Althogh aesthetically nappealing, the limitation to bonded spports is not particlarly restrictive We already limit x and y to be positive, and for any ε>0, we can always find a vale for b sch that the probability that y > b is less than ε 14

16 Mltiplicative Backgrond Risk Proof: From the definition of mltiplicative risk vlnerability, we need to examine properties on preferences sch that E [ ''( xyxy ) ] (9) Rv = R( x ) x, y with Ey =1 E [ '( xy ) y ] This is eqivalent to finding conditions on sch that (10) Ey = 1 E [ ''( xyxy ) ] R( xe ) [ '( xy ) y ] 0 By the Diffidence Theorem, this statement is eqivalent to finding a scalar m, sch that (11) ''( xy) xy R '( xy) y m( y 1) y, or eqivalently, (1) Considering (13) R( xy) R [sgn( y 1)] '( xy) y [sgn( y 1)] y 1 m y 1, we see that the only candidate for m is dr ( xy) m = [ '( xy) y ] ' xr ' dy = y = 1 Replacing m in (11) above completes the proof Note that the steps in the proof of Proposition 3 wold also follow if we reversed the initial ineqality in (9) above We ths immediately have the following reslt, showing a necessary and sfficient condition for v to be less risk averse than Corollary to Proposition 3: Derived tility v is less risk averse than, where y Γ ( x), if and only if for every x>0 and every y [ a, b], (14) '( xy) y[ R ( xy) R ] ( y 1) ' xr ' 0 15

17 Mltiplicative Backgrond Risk Propositions 1 and can be derived directly from Proposition 3 and its Corollary While we do not wish to re-prove or earlier reslts, we nevertheless illstrate one of the proofs here, since it helps to nderstand the necessary and sfficient conditions above For the sake of concreteness, let s consider the case where relative risk aversion is convex as in Proposition 1 In addition, assme that y>1 The case where y<1 is similar Under the assmptions in Proposition 1(i) or (ii), it follows that (15) R( xy) R ' R ' R '( ) xy x x '( xy) y The first ineqality above follows from the convexity of relative risk aversion The second ineqality follows from (7) in two particlar cases: it follows if relative risk aversion is greater than one and decreasing, or if relative risk aversion is less than one and increasing Bt (8) is easily seen to follow from (15), so that Proposition 1 follows In a similar manner, it is easy to derive Proposition from the Corollary to Proposition 3 Before contining frther, we shold point ot that decreasing relative risk aversion is not necessary for Proposition 3 to hold, whereas Gollier and Pratt (1996) provide a condition similar to (8), that together with decreasing absolte risk aversion is necessary and sfficient for (additive) risk vlnerability 9 The sorce of this discrepancy is that Gollier and Pratt consider additive backgrond risks with nonpositive means In particlar, their (additive) risk vlnerability is defined as the condition on preferences sch that the derived tility fnction wx ( ) Ex ( + ε ) is more risk averse than ( x ) for any independent additive backgrond risk ε with Eε 0 The fact that the mean of the backgrond risk can be negative is what reqires decreasing absolte risk aversion in their model In or mltiplicative analoge, we cold consider mltiplicative backgrond 9 The additional Gollier and Pratt condition for the additive case, with independent backgrond risk ε, E ε = 0, can be written as '( x + ε)[ r( x+ ε) r] ε' r ' 0 x, ε, which is seen to be similar to or condition (8) This condition alone (withot assming decreasing absolte risk aversion) is both necessary and sfficient if we restrict orselves to zero-mean additive backgrond risks, as the athors point ot 16

18 Mltiplicative Backgrond Risk risks y for which Ey 1; that is, all niversally ndesirable mltiplicative backgrond risks In this case, it follows trivially that we wold need to add decreasing relative risk aversion to condition (8) to ensre that all of these backgrond risks lead to a more riskaverse behavior 5 The Affiliated Utility Fnction In this section, we obtain additional reslts by considering ln( xy) = ln x+ ln y This allows s to adapt several reslts from the case of additive backgrond risks In order to accomplish this, we introdce the affiliated tility fnction, ˆ, which we define sch that x ( ) = ˆ (ln x), for all x > 0 Eqivalently, we can sbstitte θ = ln x to write ˆ( θ) ( e θ ) θ In other words, ˆ is the composite of with the exponential fnction Althogh û is increasing, it need not be concave Since ( xy ) = ˆ(ln x + ln y), we will examine the additive risks ln ~ x + ln ~ y in this section Let r ˆ( θ ) denote the absolte risk aversion for ˆ( θ), ie rˆ ( θ) = ˆ ( θ) / ˆ ( θ) Straightforward calclations show that (16) ˆ (ln x) R ˆ = 1 = 1 + r(ln x) ˆ (ln x) Note that R (x) <1 implies that rˆ (ln x) < 0 Ths, if < 1 x < 0, then R û exhibits risk-loving behavior and is convex This is not srprising given the constrction of the affiliated tility fnction If is more concave than the natral logarithm fnction, û will be concave That is, ˆ will be everywhere risk averse if and only if is everywhere more risk averse 17

19 Mltiplicative Backgrond Risk than log tility 10 If = ln x, then ˆ is risk netral Note that ˆ does not represent anyone s tility of wealth, however To refer to ˆ as risk averse, risk loving or risk netral is only a technical convenience, since in all cases, we are assming that tre preferences are risk averse Still, by examining the natre of rˆ, we will be able to adapt several existing reslts on additive backgrond risk to the mltiplicative case A few examples can help to illstrate the relationship between tility fnctions and the corresponding affiliated tility fnctions: (i) If (x) = x, so that preferences are risk netral, then θ û( θ) = e, which is risk loving with constant absolte risk aversion 1 1 γ (ii) If = x 1 γ, γ > 0, γ 1, so that preferences exhibit constant relative (1 γ ) θ risk aversion with = γ, then ˆ( θ) = e Note that affiliated tility R fnctions exhibit constant absolte risk aversion of degree γ-1, which is risk averse only if γ>1 1 1 γ 1 (iii) If = x bx, b > 0, x < b, so that tility is qadratic, then ˆ (θ) = e θ be θ (iv) The above examples are all special cases of HARA tility Let = ξ( η + ) x 1 γ γ, η + x γ > 0, ξ( 1 γ ) γ 1 γ e θ > 0 Then ˆ( θ) = ξ η+ γ 10 If we consider backgrond risks for which Eln y = 0, an increase in the riskiness of y will case the mean of y to increase Sch an increase will represent a mean-tility preserving increase in risk for someone with logarithmic tility As a reslt, the change in backgrond risk will be detrimental [beneficial] to someone with R > 1[ R < 1] See Diamond and Stiglitz (1974) 18

20 Mltiplicative Backgrond Risk Define vˆ(ln x ) Eˆ(ln x+ ln y ) From the definition of v(x) in () and of ˆ above, it follows that vˆ(ln x) = v In a manner analogos to eqation (16) we can derive (17) Eˆ ''(ln x + ln y ) R ( ) 1 1 ˆ v x = + rv(ln x ) Eˆ '(ln x + ln y ) From (16) and (17), we easily obtain the following reslt Lemma : (i) R R if and only if rˆ(ln x) rˆ (ln x), v and (ii) Rt ( x ) is decreasing if and only if v rˆ (ln x) t is decreasing, t =,v Eqivalent to (i) above, rv r if and only if rˆ ˆ v (ln x) r (ln x) For the case where R <1, so that û is risk loving, we can still interpret rˆ > ˆ as meaning vˆ is v r more risk averse than ˆ, bt in the sense of being less risk loving We are now ready to establish an eqivalence between the additive risk vlnerability of the affiliated tility û and the mltiplicative risk vlnerability of Consider the set of ~ y Γ( ~ x ), so that E ~ y = 1 We define û as being additively risk vlnerable if ˆ (ln x) ˆ (ln x) for every x and for every ~ y Γ( ~ x ) In other words, ˆ r v r is additively risk vlnerable if vˆ(ln x) Eˆ(ln x+ ln y ) is more risk averse than ˆ(ln x) for any y with Ey = 1 Note that, nlike Gollier and Pratt (1996), we do not reqire that û be concave The fact that risk aversion of ˆ is not reqired becomes important here, since the affiliated tility fnction ˆ is convex whenever be additively risk vlnerable even in this case From Lemma, the following reslt is immediate: R <1 In other words, ˆ may 19

21 Mltiplicative Backgrond Risk Proposition 4: Preferences are mltiplicatively risk vlnerable if and only if the affiliated tility fnction û is additively risk vlnerable Using Propositions 3 and 4, we can characterize additive risk vlnerability of ˆ by simply translating the ineqality in (8) to properties of û Corollary to Proposition 4: Preferences are mltiplicatively risk vlnerable if and only if (18) ˆ'(ln xy)[ rˆ (ln xy) rˆ (ln x)] ( y 1) ˆ'(ln x) rˆ '(ln x) 0 x, y [ a, b] If we wish to extend reslts from the literatre on additive backgrond risks to the case of mltiplicative ones, we need to relate or setting to that of Gollier and Pratt (1996) In addition to not reqiring risk aversion, or definition of additive risk vlnerability differs from Gollier and Pratt in that we restrict orselves to additive backgrond risks ln y for which Ey =1, whereas Gollier and Pratt consider backgrond risks sch that Eln y 0, sing or notation 11 Since Ey = 1 E ln y 0, it follows that û will satisfy or condition for additive risk vlnerability, whenever ˆ is risk vlnerable in the sense of Gollier and Pratt In other words, mltiplicative risk vlnerability follows whenever û is risk vlnerable in the sense of Gollier and Pratt 1 Since risk vlnerability, and in particlar ineqality (18), is not an easy trait to verify, Gollier and Pratt offer s several sefl sfficient conditions for risk vlnerability, which they define exclsively for the case where preferences are risk averse If we restrict tility sch that R > 1, so that the affiliated tility fnction û is risk averse, we 11 Perhaps srprisingly, Gollier and Pratt s proof of their necessary and sfficient conditions for their risk vlnerability does not reqire risk aversion It reqires only that tility be strictly increasing This might not seem srprising if we note that or proof of Proposition 3 also does not reqire risk aversion to hold 1 This can be seen more formally as follows Assme, as do Gollier and Pratt, that r is decreasing Then their necessary and sfficient condition for ˆv to be more risk averse than û, for all y with Eln y 0 is ˆ'(ln xy)[ rˆ ˆ ˆ ˆ (ln xy) r(ln x)] (ln y) '(ln x) r '( ln x) 0 x, y [ a, b] This is eqivalent to the condition in footnote 9 above Bt since y 1 ln y for all y, ineqality (18) follows whenever r ˆ ' 0 ˆ < 0

22 Mltiplicative Backgrond Risk may apply some of the Gollier and Pratt reslts to ˆ This leads to the following two sfficient conditions on the affiliated tility fnction mltiplicatively risk vlnerable û to ensre that preferences are Proposition 5: Sppose that R > 1 x Then is mltiplicatively risk vlnerable if either (i) rˆ is decreasing and convex, or (ii) û exhibits standard risk aversion (see Kimball, 1993, and below) In some instances, we might be able to check the conditions on the affiliated tility fnction ˆ in Proposition 5 directly However, we typically will find it easier to deal with properties of directly, rather than properties of û We address this isse in the next section 6 Properties of Utility and Affiliated Utility In this section, we examine conditions on the tility fnction that mst hold if its affiliated tility fnction ˆ is additively risk vlnerable In particlar, we first show that R (x) is decreasing and convex, whenever r ˆ (ln x) is decreasing and convex We then show how there is a close relationship between standard absolte risk aversion of the affiliated tility fnction û and standard relative risk aversion of We have already established in Lemma that (x) R is decreasing whenever rˆ (ln x) is decreasing From eqation (16), it follows that (19) 1 R '( x ) = r ˆ '(ln x ) x and 1

23 Mltiplicative Backgrond Risk 1 (0) R "( x ) = [ r ˆ "(ln x ) r ˆ '(ln x )] x If rˆ (ln x) is decreasing and convex, it follows from eqation (0) that (x) is also convex As a conseqence, the conditions holding in Proposition 5(i) imply those of Proposition 1(i), so that Proposition 1 also might be thoght of as a corollary to Proposition 5(i) The property of standard risk aversion, as presented in Kimball (1993), has become an integral part of the literatre on behavior nder ncertainty as based pon the expected-tility paradigm It is especially sefl since it is easily characterized by decreasing absolte risk aversion and decreasing absolte prdence, where absolte ''' prdence is measred as px ( ) = '' If ' '' > 0, preferences are said to be prdent If the affiliated tility fnction is standard risk averse, which by definition implies that it mst be risk averse, we may apply Proposition 5(ii) to conclde that v is more risk averse than 13 We first obtain a preliminary reslt that will prove sefl Straightforward calclations show that R () '' ' x''' ' + x[ ''] R ' = = r ( )[1 ( ) ( )] x P x + R x, [ '] where P x'''( x) denotes the measre of relative prdence Conseqently, we ''( x) directly obtain the following reslt 13 Frther properties of standard risk aversion as well as a discssion of mch of the literatre applying this property can be fond in Gollier (001)

24 Mltiplicative Backgrond Risk Lemma 3: R ' 0 if and only if P 1 + R R We already know that the affiliated tility fnction û is risk averse whenever > 1 Lemma 4 shows a condition on the nderlying preferences that is eqivalent to the prdence of û Lemma 4: The affiliated tility fnction ˆ exhibits prdence, ˆ'''( θ) > 0 1 if P > 3 R θ, if and only Proof: Recall that ˆ(ln x) = ( x), so that we obtain the following by differentiating with respect to ln x : ˆ'(ln x) = x'( x) ˆ ''(ln x) = x'( x) + x ''( x) ˆ 3 '''(ln x) = x'( x) + 3x ''( x) + x '''( x) Ths, dividing ˆ'''(ln x) by x ''( x) > 0 we obtain x ''' ' ˆ '''(ln x) > 0 3 > 0 '' x'' 1 P > 3 R From Lemmata 3 and 4, we can easily now show the following Lemma 5: If exhibits decreasing relative risk aversion, the affiliated tility fnction ˆ exhibits prdence 3

25 Mltiplicative Backgrond Risk Proof: From Lemmata 3 and 4, the conclsion follows if 1 + R 3 [ R ] Since R (x) is positive, this is eqivalent to {[ R ] R + 1} = [ R 1] 0, which obviosly holds 1 We can se the derivatives in the proof of Lemma 4 to calclate the measre of absolte prdence for the affiliated tility fnction In particlar, we obtain (3) ˆ '''(ln x) x ''' + x'' P ˆ ''(ln x) 1 x'' + ' 1 ( R ) pˆ(ln x) = 1 = 1, where the last step follows from dividing both the nmerator and denominator in (3) by x' ' We are now ready to prove that standard relative risk aversion of is a necessary condition for ˆ to be standard: Proposition 6: Sppose that ˆ exhibits standard risk aversion Then > 1 R and exhibits standard relative risk aversion; that is, both P (x) and (x) are positive and decreasing R Proof: From eqation (16), we know that û risk averse implies that > 1 Since ˆ exhibits decreasing absolte risk aversion, it follows from Lemma that exhibits decreasing relative risk aversion Ths, we mst show that also exhibits positive and decreasing relative prdence That relative prdence is positive follows easily from Lemma 3 Differentiating eqation (3) with respect to ln x we obtain R 1 ˆ (ln x) x[1 ( R) ] P ' x[ P ]( R) R ' = 1 dln x [1 ( R ) ] dp 4

26 Mltiplicative Backgrond Risk Becase R > 1, it follows that [ R] R > 0 and, from Lemma 3, that dp(ln ˆ x) P > 0 Ths, it follows that is negative if and only if dln x P (4) P ' < R '( ) 0 x < [ R ] R From the proof of Proposition 6, we see that exhibiting standard relative risk aversion is necessary, bt not qite sfficient to imply that the affiliated tility fnction û is standard risk averse However, we do obtain the following reslt Corollary to Proposition 6: Let R > 1 If exhibits standard relative risk aversion and the ineqality in (4) holds, then the affiliated tility fnction averse û is standard risk Proof: Since R > 1, it follows from eqation (16) that rˆ( θ ) > 0 Standard relative risk aversion of implies, from Lemma, that û exhibits decreasing absolte risk aversion It also follows, from Lemma 5, that ˆ ''' > 0 Since (4) holding implies that ˆ also exhibits decreasing absolte prdence, the Corollary follows From Proposition 6 and its Corollary, it follows that whenever condition (4) holds, the following two conditions are eqivalent: (i) Utility is standard relative risk averse with R >1, and (ii) Affiliated tility ˆ is standard risk averse Since (4) might seem a bit opaqe, we provide an illstration of a case where it applies in the following example 5

27 Mltiplicative Backgrond Risk Example: Let belong to the HARA class of tility fnctions, sppose that γ > 1 Now 1 γ x ( ) = ξη ( + ) and x 1 γ γ ( ) x R x = Ths, it follows easily that exhibits η + x 1 decreasing relative risk aversion if and only if η < 0 Hence, x > η + x, so that R > 1 To see that exhibits standard relative risk aversion, note that P ( ) x = R ( x ) Ths, exhibits decreasing relative prdence if and only if exhibits 1+ γ γ decreasing relative risk aversion Ths, is standard relative risk averse and > 1 γ R We now wish to show that û is standard risk averse By the Corollary to Proposition 6, we wold be done if the ineqality in (4) holds Since both P '( x ) < 0 and R '( x ) < 0, ineqality (4) is eqivalent to 1+ γ ( γ ) R 1+ γ P > = γ R [ R 1] R [ R 1] γ R x R x > R x 1+γ ( )[ ( ) 1] ( ) ( ) 1 γ [ R 1] > 1 + γ This last ineqality follows, since γ>1 Hence, ˆ is standard risk averse It follows from Proposition 5(ii) that the derived tility fnction v is more risk averse than If nder HARA preferences we assme the property that follows that γ>1 mst hold Hence, the additional assmption of standard relative risk aversion of wold be both necessary and sfficient to garantee standard (absolte) risk aversion of û R ( ) 1 (0, ) x > x, it 6

28 Mltiplicative Backgrond Risk 7 Comparative Risk Aversion A key reslt in the literatre on additive backgrond risk is that the properties of constant absolte risk aversion and decreasing absolte risk aversion for tility are carried over to the derived tility fnction On the other hand, the property of increasing absolte risk aversion does not always carry over In this section we show the analogos reslts for relative risk aversion in the case of a mltiplicative backgrond risk We have already seen that v inherits constant relative risk aversion from Indeed, the level of constant risk aversion is identical To see that the same holds tre for decreasing relative risk aversion, we can apply the Diffidence Theorem once again It is important to note that Ey =1 is not reqired in Proposition 7 Proposition 7: Let ~ y have a bonded spport contained in [a,b] If exhibits nonincreasing relative risk aversion, then so does the derived tility fnction vx ( ) Exy ( ) Proof: It follows from Lemma 3, that we need to show that, x, (5) P 1 + R P 1 + R v v That is, we mst show that (6) E '''( xy) y x E ''( xy) y x E ''( xy ) y E '( xy ) y Ineqality (6) is eqivalent to the following, where λ denotes the vale of the righthand side in (6): (7) E [ ''( xy ) yx + ( λ 1) '( xy ) y ] = 0 E [ '''( xy ) yx 3 + λ''( xy ) y ] 0 7

29 Mltiplicative Backgrond Risk By the Diffidence Theorem, (7) will hold if we can find a real nmber m, sch that (8) '''( xyyx ) 3 + λ''( xyy ) m [ ''( xyyx ) + ( λ 1) '( xyy ) ] y [ ab, ] The left-hand side of (8) can be written as xy ''( xy) '( xy) y xy '''( xy) '( xy) y '( xy) x ''( xy) x (9) + λ = R ( xy) [ λ P ( xy) ] Since P 1 R, it follows from (8) and (9) that + 3 '( xy) y + x (30) '''( xy) y x λ ''( xy) y R ( xy) [ λ 1 R ( xy) ] From (8) and (), we wold be done if we cold find an m, sch that (31) '( xy) y R( xy) [ λ 1 R( xy) ] m[ ''( xy) y x + ( λ 1) '( xy) y] x = m '( xy) y[ λ 1 R ( xy)] This follows by taking m = ( 1 λ) / x, since we then obtain (31) is eqivalent to (3) R xy [ λ R xy ] λ λ R xy λ R xy ( ) 1 ( ) + ( 1)[ 1 ( )] = [ 1 ( )] 0 Hence, (5) holds and v exhibits decreasing relative risk aversion We next trn briefly to examining some interpersonal characteristics of comparative risk aversion Kihlstrom, et al (1981) and Ross (1981) examined these for the case of an additive backgrond risk 14 Their reslts are special cases of more general 14 Actally, Ross considers the backgrond risk to be mean-independent, which is not as restrictive as the assmption of independence 8

30 Mltiplicative Backgrond Risk reslts fond in Nachman (198) Nachman is one of the few who considers the case of mltiplicative backgrond risks as a special case of his general reslts, albeit briefly The basic qestion we address is the following: If agent 1 is more risk averse than agent, will this property be preserved in the presence of a mltiplicative backgrond risk? That is, if 1 is more risk averse than, when will it follow that v is also more risk averse 1 than v? One reslt that is qite easy to obtain is the following: a b a Proposition 8: Let and be risk-averse tility fnctions sch that is more risk averse than b a b, ie R R a b then v is more risk averse than v a b x If λ sch that x R (x) λ R (x), Proof: Follows directly from eqation (4) The proof of Proposition 8 also follows directly from the following more general reslt, which is de to Nachman (198) We inclde the trivial proof above mostly to illstrate how or model can also generate these types of reslts We present Nachman s reslt below for the sake of completeness Proposition (Nachman): Let a b a and be risk-averse tility fnctions sch that is b c more risk averse than, ie x If there exists a fnction sch that R a R b R a c b R R x and R c a is nonincreasing, then v is more risk averse b than v a It follows easily from Nachman s reslt that v will be more risk averse than v a b if either of the tility fnctions, or, exhibits nonincreasing relative risk aversion This reslt is a direct conterpart to the reslt by Kihlstrom, et al in the case of additive backgrond risk b 9

31 Mltiplicative Backgrond Risk 8 Conclding Remarks The notion that markets are complete is a mathematical nicety that does not hold tre in practice Many types of political, hman-capital and social risks, as well as some financial risks, are not represented by direct contracts Obviosly, many of these risks might be hedged indirectly via so-called cross hedging However, even when sch backgrond risks are independent of other risks and cannot be hedged per se, they still may have an impact pon risk-taking strategies that are within the control of the economic agent Mch has been done over the past twenty years in examining the effects of additive backgrond risks Bt srprisingly little has been done to systematically stdy economic decision making in the presence of a mltiplicative backgrond risk This paper is a first step towards developing a comprehensive theory of backgrond risk in this direction As the examples in or introdction illstrate, models with sch mltiplicative backgrond risks are not hard to find within the literatre Whereas properties of absolte risk aversion play a key role in analyzing the effects of an additive backgrond risk, properties of relative risk aversion are more important in examining behavior in the presence of a mltiplicative backgrond risk However, reslts for the case of a mltiplicative backgrond risk do not simply mirror those for the case where the backgrond risk is additive An nderstanding of the basic concepts presented here hopeflly might help s nderstand a mltitde of reslts for which standard theories (in the absence of any backgrond risk) yield predictions that seem at odds with everyday observations of reality Since risk aversion captres all the essential information abot preferences within an expected-tility framework, or focs here has been on comparing risk aversion with and withot a mltiplicative backgrond risk As we learn more abot these inherent properties, we hopeflly will be able to find better models to se in the realm of positive theories 30

32 Mltiplicative Backgrond Risk References Aït-Sahalia, Y and AW Lo, (000), Nonparametric Risk Management and Implied Risk Aversion, Jornal of Econometrics 94, 9-51 Arrow, KJ, (1971), Essays in the Theory of Risk Bearing, Chicago: Markm Pblishing Company Diamond, P and J Stiglitz, (1974), Increases in Risk and in Risk Aversion, Jornal of Economic Theory 8, Doherty, N A, and H Schlesinger, (1983), Optimal Insrance in Incomplete Markets, Jornal of Political Economy, 91, Eeckhodt, L and MS Kimball, (199), Backgrond Risk, Prdence and the Demand for Insrance, in: Dionne, G (Ed), Contribtions to Insrance Economics, Boston: Klwer Academic Pblishers, pp Gollier, C, (001), The Economics of Risk and Time, Cambridge: MIT Press Gollier, C and M Kimball (1996), Towards a Systematic Approach to the Economics of Uncertainty: Characterizing Utility Fnctions, npblished working paper, University of Michigan Gollier, C, and J W Pratt, (1996), Risk Vlnerability and the Tempering Effect of Backgrond Risk, Econometrica, 64, Giso, L, T Japelli and D Terlizzese, (1996), Income Risk, Borrowing Constraints and Portfolio Choice, American Economic Review 86, Giso, L and M Paiella, (001), Risk Aversion, Wealth and Financial Market Imperfections, CEPR Discssion Paper No 78 Jackwerth, JC, (000), Recovering Risk Aversion from Option Prices and Realized Retrns, Review of Financial Stdies 13, Kihlstrom, R, D Romer and S Williams, (1981), Risk Aversion with Random Initial Wealth, Econometrica, 49, Kimball, MS, (1993), Standard Risk Aversion, Econometrica, 61, Nachman, D, (198), Preservation of More Risk Averse nder Expectations, Jornal of Economic Theory, 8,