Variance Vulnerability, Background Risks, and Mean-Variance Preferences

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1 The Geneva Papers on Risk and Insurance Theory, 28: , 2003 c 2003 The Geneva Association Variance Vulnerability, Background Risks, and Mean-Variance Preferences THOMAS EICHNER VWL IV, FB 5, University of Siegen, Hölderlinstr. 3, Siegen, Germany eichner@vwl.wiwi.uni-siegen.de ANDREAS WAGENER andreas.wagener@univie.ac.at Department of Economics, University of Vienna, Hohenstaufengasse 9, 1010 Vienna, Austria Abstract An agent with two-parameter, mean-variance preferences is called variance vulnerable if an increase in the variance of an exogenous, independent background risk induces the agent to choose a lower level of risky activities. Variance vulnerability resembles the notion of risk vulnerability in the expected utility (EU) framework. First, we characterize variance vulnerability in terms of two-parameter utility functions. Second, we identify the multivariate normal as the only distribution such that EU- and two-parameter approach are compatible when independent background risks prevail. Third, presupposing normality, we show that analogously to risk vulnerability temperance is a necessary, and standardness and convex risk aversion are sufficient conditions for variance vulnerability. Key words: mean-variance preferences, background risk, variance vulnerability JEL Classification No.: D81, D21 1. Introduction In this paper we investigate the effect of an increase in a mean-zero background risk on an agent s willingness to take another independent risk when the agent has two-parameter, mean-standard deviation preferences. While this question has quite found extensive coverage in the expected-utility (EU) approach (see, among others, Gollier and Pratt [1996]; Kimball [1993]; Pratt and Zeckhauser [1987]), it has so far not been fully analyzed in the context of mean-standard deviation preferences. Lajeri-Chaherli [2002] recently studied the addition of independent undesirable risks to an already risky situation and transferred the concept of proper risk aversion, originally due to Pratt and Zeckhauser [1987], to the two-parameter framework. Our focus here is on increases in background risks. Inspired by Gollier and Pratt [1996] s notion of risk vulnerability, we propose and characterize the concept of variance vulnerability to formally capture the idea that an agent reduces his risky activities when being confronted with the increase in the variance of an independent background risk. Two-parameter, mean-standard deviation analysis has a long and productive history both in theoretical research and in applied work on decision making under uncertainty. Due to

2 174 EICHNER AND WAGENER its simplicity and ease of interpretation it continues to be used although it is well-known to be consistent with EU rankings only under specific circumstances. The least restrictive and, for economic applications, most useful condition for compatibility of both approaches is the location-scale assumption under which all lotteries an agent might choose from differ only in location and scale parameters (Sinn [1983]; Meyer [1987]). This assumption will be satisfied whenever the interaction between the agent s choices and a univariate source of uncertainty is linear, as it is, e.g., the case in Sandmo [1971] s analysis of a competitive firm under price uncertainty, Fishburn and Porter [1976] s portfolio choice problem with one risky and one risk-free asset and Ehrlich and Becker [1972] s study of insurance demand. As two-parameter and the EU approach ought, in general, to be regarded as two distinct models of choice under risk (Ormiston and Schlee [2001], we first develop the concept of variance vulnerability in a pure two-parameter setting without reference to the EU approach (Sections 2 through 4). We derive necessary and sufficient conditions on mean-standard deviation utility functions such that variance vulnerability prevails and background risks have a tempering effect on risk taking (Proposition 1). Since two-parameter and EU approach at least partially overlap it is, then, natural to ask for the precise connection between them in a setting with independent background risks. We elaborate on this in Section 5. We start from Chamberlain [1983] s observation that EUand two-parameter approach are compatible in multivariate settings if and only if risks are jointly elliptically symmetric distributed. Under this distributional assumption, all economic models where the interactions between an agent s choices and the multivariate sources of uncertainty are linear still satisfy the location-scale assumption àlasinn [1983] and Meyer [1987]. Among the elliptically symmetric distributions, however, the multivariate normal is the only one that allows for stochastic independence of risks as it is required in the generic background risk problem. Hence, we conclude that EU- and two-parameter approach are compatible in settings with independent background risks if and only if stochastics are Gaussian (Proposition 3). Presupposing normality, we find in Section 6 that temperance (Kimball [1992]) is a necessary condition, and that Kimball [1993] s property of standardness (i.e., the combination of decreasing absolute risk aversion and decreasing absolute prudence) or convexity of risk aversion (Gollier and Pratt [1996]) are sufficient restrictions on risk preferences to display variance vulnerability (Propositions 4 and 5). The same restrictions emerge as, respectively, necessary and sufficient conditions for risk vulnerability in the original setting by Gollier and Pratt [1996]. Thus, variance vulnerability and risk vulnerability are closely related concepts. At this point it is worth emphasizing that a distinction should be made between adding a (new) background risk and increasing a (pre-existing) background risk. While most analyses in the EU-framework (with the exception of Eeckhoudt, Gollier and Schlesinger [1996]) deal with the addition of a background risk and risk vulnerability applies to this setting only, mean-variance preferences allow for a simple way to capture and characterize increases in the background risk. However, in case where mean-variance approach and EU-approach are perfect substitutes, our necessary and sufficient conditions are coupled with a loss of generality, namely with the restriction to multivariate normal random variables.

3 VARIANCE VULNERABILITY, BACKGROUND RISKS, AND PREFERENCES Preferences Individual preferences over lotteries are represented by a two-parameter preference function V : R + R R, V = V (σ y,µ y ), (1) where y denotes random final wealth and σ y and µ y are, respectively, the standard deviation and the expected value of y.weassume that the function V is at least four times continuously differentiable, increasing in µ y, decreasing in σ y (risk aversion), and strictly concave (such that (σ y,µ y )-indifference curves are strictly convex). Denoting partial derivatives with subscripts, we thus require V µ (σ y,µ y ) > 0; V σ (σ y,µ y ) < 0; V µµ (σ y,µ y ) < 0, V σσ (σ y,µ y ) < 0, and V µµ (σ y,µ y )V σσ (σ y,µ y ) Vσµ 2 (σ y,µ y ) > 0 (2a) (2b) (2c) for all (σ y,µ y ) R ++ R. Denote by α(σ y,µ y ):= V σ (σ y,µ y ) V µ (σ y,µ y ) the marginal rate of substitution (MRS) between σ y and µ y. Graphically, α measures the slope of the indifference curve in (σ y,µ y )-space. It has been identified as the twoparameter analogue of the Arrow-Pratt concept of absolute risk aversion (Ormiston and Schlee [2001]). 1 Following Tobin [1958, p. 13] and Sinn [1983, pp. 112f.] we impose that (σ y,µ y )-indifference curves enter the µ y -axis with slope zero, i.e.: α(0,µ y ) = V σ (0,µ y ) = 0 for all µ y R. (2d) Furthermore, indifference curve slopes are assumed to be decreasing in µ y : α µ (σ y,µ y ) < 0 for all (σ y,µ y ) R + R. (3) Ormiston and Schlee [2001] identify property (3) as the formal equivalent of the concept of decreasing absolute risk aversion (DARA) in the two-parameter framework. 3. A linear problem The particular class of models considered in the present paper is of the form y(a) = z a + f (a) + ɛ. (4)

4 176 EICHNER AND WAGENER Here a is the individual s one-dimensional choice variable, y(a) is random final wealth depending on the choice variable a, f (a) isareal-valued function which is assumed to be concave in a, and z and ɛ are random variables. Denoting by µ χ and σ χ, respectively, the mean and the standard deviation of random variable χ,weassume: µ z > 0, σ z > 0, µ ɛ = 0, and σ ɛ > 0. We further suppose that the risks z and ɛ are independently distributed. Thus, with respect to the choice variable a, the risk over ɛ is an exogenous and undesirable background risk. In terms of the (σ y,µ y )-approach the decision problem of the risk-averse individual is then given by max a V (σ y (a),µ y (a)) with σ y (a) = σ 2 z a2 + σ 2 ɛ, µ y (a) = µ z a + f (a) + µ ɛ. (5) Decision problem (5) can be interpreted as portfolio choice of a single asset (Fishburn and Porter [1976]), as output choice of a competitive firm under price uncertainty (Sandmo [1971]) or as insurance demand (Ehrlich and Becker [1972]). Restricting attention to interior solutions to this program, the optimal level a > 0isdetermined by 2 where ( [µ z + f (a σy (a ) ) 1 )] α(σ y (a ),µ y (a )) = 0 (6) a σ y (a ) a = a σz 2 > 0. (7) σ y (a ) The second-order condition for a is satisfied due to the concavity 3 of V, the concavity of f, and the independence of z and ɛ. 4. Comparative statics and variance vulnerability In this section we present a comparative static proposition with respect to changes in the standard deviation σ ɛ of the background risk. 4 We will say that an agent with (σ y,µ y )- preferences is variance-vulnerable if his willingness to bear risks is vulnerable to the introduction or an increase in another independent risk. Formally, an agent is called variancevulnerable if he optimally reduces his risky activity a in response to an increase in the variability of the background risk: a σ ɛ < 0. (8) The notion of variance vulnerability is inspired by the concept of risk vulnerability, coined by Gollier and Pratt [1996]. In the EU approach, risk vulnerability captures the idea that

5 VARIANCE VULNERABILITY, BACKGROUND RISKS, AND PREFERENCES 177 agents become more risk averse when they are confronted with additional, exogenous and independent background risks. From Pratt [1964] it is known that the behavioural equivalent of increased risk aversion in the EU approach is a lower level of risk taking. Thus, riskvulnerable agents choose a lower level of risky activities when they encounter background uncertainty. In terms of the (σ y,µ y )-approach, such behaviour is captured by (8). The following result presents the properties which a (σ y,µ y )-utility function must possess in order to generate variance-vulnerable behaviour: Proposition 1: (a) An agent exhibits variance vulnerability in the sense of (8) if and only if α σσ (σ y,µ y ) > 0. (9) (b) With DARA preferences (i.e., if α µ (σ y,µ y ) < 0), a necessary condition for variance vulnerability is given by V σσσ (σ y,µ y ) < 0. (10) Proof. Implicit differentiation of (6) with respect to σ ɛ yields, after some algebraic rearrangements: sgn a σ ɛ = sgn [α(σ y,µ y ) σ y α σ (σ y,µ y )]. (11) (a) Given that α(0,µ y ) = 0 for all µ y R, α σ α/σ y is equivalent to α σσ 0. This leads to item (a) of the assertion. (b) Calculate that α σ y α σ is equal in sign to V µ (V σ σ y V σσ ) σy V σ V σµ.if preferences satisfy DARA, i.e., if 0 > sgn α µ = sgn [V σµ V µ V µµ V σ ], we must have V σµ > 0 due to (2a) (2c). Hence, a / σ ɛ can only be non-positive if V σ >σ y V σσ. Using the assumption V σ (0,µ y ) = 0, this is equivalent to V σσσ < 0. While condition (10) is relatively easy to verify or to reject for a given two-parameter utility function, the necessary and sufficient condition (9) is quite cumbersome to check. This suggests that variance vulnerability, in spite of its intuitive appeal, is not an evident or trivial property Mean-variance preferences and EU-approach 5.1. General remarks The two-parameter approach, on which we have focussed until now, is well-known to be a perfect substitute for the expected-utility (EU) approach under certain conditions. The least restrictive and, for economic applications, most useful of these conditions is the locationscale assumption which presupposes that the interaction between an agent s choice variable and the source of uncertainty is linear (Meyer [1987]; Sinn [1983]).

6 178 EICHNER AND WAGENER To be more specific, we follow Meyer [1987] and consider a choice set Y in which all random variables y Y only differ from one another by location and scale parameters. Let x be the random variable obtained by normalization of an arbitrary y Y. Then all y Y are equal in distribution to µ y + σ y x, where µ y and σ y are the mean and the standard deviation of the respective y Y. Given a von-neumann/morgenstern (VNM) utility index u : R R, the expected utility from the distribution of y can be written in terms of the mean and the standard deviation of y: Eu(y) = b a u(µ y + σ y x)df(x) =: V (σ y,µ y ). (12) The interval (a, b) with a < b is the support of x and F is its distribution. As has been shown by Meyer [1987], identity (12) gives rise to various equivalences between EU- and two-parameter preferences. In particular, u (y) > 0 V µ (σ y,µ y ) > 0; u (y) < 0 V σ (σ y,µ y ) < 0; V (σ y,µ y )isstrictly concave; V σ (0,µ y ) = 0. (13a) (13b) (13c) (13d) Observe that these four equivalences match well with assumptions (2a) to (2d) which we imposed on V previously. For the VNM index u(y) define A(y) := u (y) u (y) and B(y) := u (y) u (y) as, respectively, the Arrow-Pratt index of absolute risk aversion and Kimball [1990] s index of absolute prudence at wealth level y. Asshown by Meyer [1987] and Lajeri and Nielsen [2000], A (y) < 0 α µ (σ y,µ y ) < 0. (14) Hence, DARA in the EU-framework is, under the location-scale assumption, indeed equivalent to assumption (3) on (σ y,µ y )-preferences The case of multiple risks While many economic models (in particular, those referred to in the introduction and in Section 3) with a univariate source of uncertainty meet the location-scale assumption under which two-parameter and EU-approach are consistent, models with both a direct and a background risk do obviously not belong to such univariate settings. Chamberlain [1983] shows, however, that multivariate distributions that lead to consistency of EU- and twoparameter approach have the common attribute of elliptical symmetry, a generalization of spherical symmetry. 6 Elliptically [spherically] symmetric random vectors have contours of

7 VARIANCE VULNERABILITY, BACKGROUND RISKS, AND PREFERENCES 179 equal density that are ellipsoids [hyper-spheres] (see Fang, Kotz and Ng [1990], for extensive coverage). If a n 1 random vector Z = (Z 1,...,Z n ) has an elliptical distribution, then it is uniquely determined by the mean vector of Z, E(Z) = M R n, the covariance matrix of Z, = Cov(Z) (where R n n is symmetric and positive semi-definite), and knowledge of the type of distribution, represented by characteristic function φ. Hence, we can write: Z E n (M,,φ). A prominent feature of elliptical distributions is that every linear combination of an elliptically distributed random vector is also elliptically distributed with the same characteristic generator φ.inparticular, for all m n matrices A and m 1 vectors b, Z E n (M,,φ) = A Z + b E m (A M + b, A A,φ) (15) (Fang, Kotz and Ng [1990, Theorem 2.16]). An immediate corollary of (15) is that with elliptical symmetry all economic models where the interactions between an agent s choices and the multivariate sources of uncertainty are linear satisfy the location-scale assumption: For all vectors d R n and all b R, ifz is elliptically symmetric, Z E n (M,,φ), then Y = d Z + b has expected value EY = d M + b and variance Var(Y ) = d d. Hence, EU- and two-parameter approach are equivalent in the sense of (12). As an example, consider our model in Section 3: Suppose that initial risk and background risk (z,ɛ) are jointly elliptically distributed, (z,ɛ) E 2 ((µ z,µ ɛ ),,φ) with ( ) σ 2 = z σ zɛ, σ ɛz σ 2 ɛ where σ zɛ = Cov(z,ɛ)isthe covariance of (z,ɛ). Setting d = (a, 1) and b = f (a), final wealths y(a) = z a + f (a) + ɛ satisfy the location-scale assumption and the (σ y,µ y )- approach is applicable. In particular, µ y (a) = µ z a + f (a) + µ ɛ and σ y (a) = σ 2 z a2 + σ 2 ɛ + 2 a σ zɛ Multiple risks, location-scale framework, and normal distribution In the analysis of variance vulnerability in Section 4 we assumed that initial and background risks are independently distributed. This assumption, which is common in the EUframework (for an exception see Gollier and Schlee [1999]), has a far-reaching implication for our previous discussion: EU- and two-parameter framework are only compatible in a setting with independent primary and background risks if both risks are (jointly and severally) normally distributed. This is due to the fact that, within the class of elliptical distributions, the multivariate normal is the only distribution of independent random variables: Proposition 2 (Fang, Kotz and Ng [1990, Theorem 4.11]): where = diag(σ 11,...,σ nn ) is a diagonal matrix. 7 Then: Assume Z E n (M,,φ) Zisnormally distributed The components of Z are independent. (16)

8 180 EICHNER AND WAGENER Given Chamberlain [1983] s result that in multivariate settings EU- and two-parameter approach are compatible if and only if risks are jointly elliptically distributed, the above proposition yields the following Proposition 3: EU- and two-parameter approach are compatible in settings with multiple but independent risks if and only if the distribution of risks is the multivariate normal. 6. Variance vulnerability and normal distributions In this section we will combine Propositions 1 and 3. Let us suppose that both the primary and the background risk are normally distributed. So, then, will be final wealth y. We will now relate the necessary and sufficient restrictions on (σ y,µ y )-preferences which we obtained for variance vulnerability from Proposition 1 to properties of VNM utility functions employed in the EU-framework Necessary condition: Temperance Item (b) of Proposition 1 identifies V σσσ < 0asanecessary condition such that an agent reduces risk-taking upon an increase in an independent background risk. Suppose that (12) holds (i.e., EU- and two-parameter approach are compatible). Use integration by parts to calculate that for any symmetrically distributed x (i.e., for Ex 3 = 0 which especially holds if x N(0, 1)), V σσσ (σ y,µ y ) = x 3 u (µ y + σ y x)df(x) = ( x ) u (4) (σ y,µ y ) z 3 f (z)dz dx. This is negative if and only if u (4) (y) < 0 for all y. The property u (4) (y) < 0iscalled temperance, and it has been shown by Kimball [1993], Eeckhoudt, Gollier and Schlesinger [1996], and Gollier and Pratt [1996] to be one of the necessary conditions in the EU-framework to have agents reducing their demand for a risky asset when independent background risks increase in certain specific senses. Proposition 4: If EU- and two-parameter approach are compatible, variance vulnerability necessitates temperance Sufficient conditions: Standardness or convex risk aversion For Gaussian random variables, Chipman ([1973, Theorem 1]) has shown that the (σ y,µ y )- utility function obeys the differential equation V σ (σ y,µ y ) = σ y V µµ (σ y,µ y ). (17)

9 VARIANCE VULNERABILITY, BACKGROUND RISKS, AND PREFERENCES 181 Further differentiation and subsequent rearrangement yields that, in the case of normally distributed stochastics, (α σ y α σ )isequal in sign to φ(σ y,µ y ):= V µµµµ (σ y,µ y ) V µ (σ y,µ y ) V µµ (σ y,µ y ) V µµµ (σ y,µ y ). (18) Using this, we obtain Proposition 5: Suppose that stochastics are (multivariate) Gaussian. Then variance vulnerability in the sense of (8) will prevail if the utility function u exhibits decreasing absolute risk aversion and decreasing absolute prudence: A (y) < 0 and B (y) < 0 for all y; or if absolute risk aversion is decreasing and convex: A (y) < 0 A (y) for all y. Proof. In the following chain of transformations, the argument of u is always (µ y + σ y x). Calculate: φ(σ y,µ y ) = u df(x) u df(x) u df(x) u df(x) > < 0 (σ y,µ y ) u u u u df u u u u df u u u > u df < 0 (σ y,µ y ) u u u dg u u dg u dg > < 0 (σ y,µ y ) u u u u ( u u u dg (u ) 2 u dg u dg u ) > (u ) dg 2 < 0 (σ y,µ y ) E G [(u /u ) ] + Cov G (u /u, u /u ) > < 0 (σ y,µ y ). (19) The first of these equivalences comes from premultiplying with 1/( u df) 2 > 0. To obtain the second we used the distribution function G defined by dg = (u / u df)df; E G denotes the expectation operator with respect to G. The third follows from adding u u dg u u dg = 0 and rearranging terms. The fourth is by definition of expectation (u ) 2 (u ) 2 operator and covariance. Under DARA, u /u is an increasing function. Hence, φ is non-positive whenever [ u ] ) )] = [( u ( u u u u is non-positive (both the expected value and the covariance in (19) will be non-positive then). Given DARA, this will happen whenever u /u is non-increasing. This proves the first item. To see the second, check that A(y) B(y) = A 2 (y) A (y). If A is decreasing

10 182 EICHNER AND WAGENER and convex, A (y) < 0 A (y), then A(y) B(y) will be decreasing in y.now rewrite (19) as sgn φ(σ y,µ y ) = sgn {E G [(A(y) B(y)) ] Cov G (A(y) B(y), A(y))}. For A < 0 A, the expected value will be negative while the covariance will be positive. Hence, the whole expression will be negative then. Proposition 4 identifies two independent sets of conditions that are well-established as sufficient conditions for risk vulnerability in the EU-framework: standardness (Kimball [1993]) or decreasing and convex absolute risk aversion (Gollier and Pratt [1996]). Note that standardness (i.e., the combination of decreasing absolute risk aversion and decreasing absolute prudence) has an accessible (σ y,µ y )-counterpart in the location-scale framework: We know that α µ < 0isequivalent to decreasing absolute risk aversion. Furthermore, Lajeri and Nielsen [2000] define a (σ y,µ y )-index for absolute prudence as β(σ y,µ y ):= V µσ (σ y,µ y )/V µµ (σ y,µ y ) and they show that β µ < 0ifand only if B (y) < 0; also see Wagener [2002] and Eichner and Wagener [2003]. To check for variance vulnerability, the standardness requirement β µ < 0isoften easier to verify than the condition α σσ > 0 obtained in Proposition Concluding remarks Our work might give rise to future research in the mean-variance framework along several lines. One might wish to study dependent multiple risks, e.g. analyzing the comparative static effects of changing a dependent background risk, the covariance and correlation between initial and background risks. In the EU-approach first results with regard to comparative statics in models with dependent risks can be found in Gollier and Schlee [1999]. Moreover, one might wish to extend the analysis to decision problems with more than one choice variable, as it has been initiated in the EU-framework by Hadar and Seo [1990] or Meyer and Ormiston [1994]. Such undertakings would require further clarification of the connections between EU- and two-parameter decision models in multivariate frameworks, but might open the way for various interesting applications. Acknowledgments We thank Fatma Lajeri-Chaherli and two anonymous referees for helpful suggestions and criticism. The usual disclaimer applies. Notes 1. A referee pointed out that for normal distributions the relation between α and the Arrow-Pratt measure of absolute risk aversion already appeared in Lajeri and Nielsen [1995]. 2. This first-order condition, which states that the frontier of the opportunity set has to be equal to the indifference curve slope, would remain unaltered if the choice variable a in (4) was a vector. This hints at a further advantage

11 VARIANCE VULNERABILITY, BACKGROUND RISKS, AND PREFERENCES 183 of applying mean-variance preferences, namely that two-moment decision models remain two-dimensional even when the choice variable is n-dimensional. 3. Observe that preferences over (σ y,µ y ) are ordinal, i.e., the function V is only unique up to a strictly increasing transformation. Since, under suitable restrictions, any quasi-concave function can be transformed into a concave function by a strictly increasing transformation (Takayama [1985, p. 113]), the assumption of concavity could be relaxed to quasi-concavity. However, in the following we stick to concavity for simplicity. 4. Of course, decision problem (5) may give rise to other comparative static problems as well. Ormiston and Schlee [2001] and Battermann, Broll and Wahl [2002] analyse the comparative static effects of changes in µ ɛ, µ z and σ z, albeit without background risk (i.e., for σ ɛ = 0). The introduction of background risks affects these comparative statics, but not dramatically so. It can be shown (proofs are available from the authors) that DARA which is necessary and sufficient for a / µ ɛ > 0 and sufficient for a / µ z > 0inthe absence of background uncertainty still suffices to yield the effects mentioned in the case with background risk. Similarly, an elasticity of risk aversion η := σ y α/α σ greater than unity which is the sufficient and necessary condition for a /σ z > 0 without background risk remains sufficient (but no longer necessary) for that effect in the presence of a background risk. 5. The same applies to risk vulnerability, which gave the model for the concept of variance vulnerability. The necessary and sufficient condition for risk vulnerability derived in Gollier and Pratt [1996] are complex and demanding. 6. A n 1 random vector Z = (Z 1,...,Z n ) has a spherical distribution, Z S n, if, for every orthogonal map S R n n, i.e., for every map that satisfies S S = I (the identity matrix), S Z is equal in distribution to Z (Fang, Kotz and Ng [1990, pp. 27ff]). A n 1 random vector Z = (Z 1,...,Z n ) has an elliptical distribution, Z E n,ifthere exists an affine map T : R n R n, x T (x) = A x + b with A R n n, b R n, such that T (Z) is spherically distributed. 7. If is not diagonal, the components of Z are obviously correlated and therefore stochastically dependent. References BATTERMANN, H., BROLL, U., and WAHL, J.E. [2002]: Insurance Demand and the Elasticity of Risk Aversion, OR Spectrum, 24, CHAMBERLAIN, G. [1983]: A Characterization of the Distributions that Imply Mean-Variance Utility Functions, Journal of Economic Theory, 29, CHIPMAN, J.S. [1973]: The Ordering of Portfolios in Terms of Mean and Variance, Review of Economic Studies, 40, EECKHOUDT, L., GOLLIER, C., and SCHLESINGER, H. [1996]: Changes in Background Risk and Risk Taking Behaviour, Econometrica, 64, EHRLICH, J. and BECKER, G.S. [1972]: Market Insurance, Self-Insurance and Self-Protection, Journal of Political Economy, 80, EICHNER, T. and WAGENER, A. [2003]: More on Parametric Characterizations of Risk Aversion and Prudence, Economic Theory, 21, FANG, K.-T., KOTZ, S., and NG, K.-W. [1990]: Symmetric Multivariate and Related Distributions. London/New York: Chapman and Hall. FISHBURN, P.C. and PORTER, R.B. [1976]: Optimal Portfolios with One Safe and One Risky Asset: Effects of Changes in Rate of Return and Risk, Management Science, 22, GOLLIER, C. and PRATT, J.W. [1996]: Risk Vulnerability and the Tempering Effect of Background Risk, Econometrica, 64, GOLLIER, C. and SCHLEE, E.E. [1999]: Increased Risk Taking with Multiple Risks, Mimeo, University of Toulouse. HADAR, J. and SEO, T. [1990]: The Effects of Shifts in a Return Distribution on Optimal Portfolios, International Economic Review, 31, KIMBALL, M.S. [1990]: Precautionary Saving in the Small and in the Large, Econometrica, 58, KIMBALL, M.S. [1992]: Precautionary Motives for Holding Assets, in The New Palgrave Dictionary of Money and Finance, P.Newman, M. Milgate, and J. Falwell (Eds.), London: MacMillan.

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