Complete Mean-Variance Preferences

Transcription

1 Complete Mean-Variance Preferences W. Henry Chiu School of Economic Studies The University of Manchester Manchester, M13 9PL, U.K. (Preliminary draft. Please do not cite.) May 1, 2007 Abstract Assuming that preferences over distributions are utility-representable, we show that the preferences can be represented by a differentiable mean-variance utility function if and only if the preference functional is Frechet differentiable and the local utility function is quadratic for all distributions. Assuming the conditions for such a MV utility function, we further identify easily interpretable necessary and sufficient conditions on the preferences for each of properties that the MV utility function is commonly assumed to exhibit in applications of the MV approach. Key Words: Mean-Variance analysis, Borch paradox,. JEL Classification: D81. 1

2 1 Introduction Since the pioneering work of Markowitz (1952) and Tobin (1958), the mean-variance (MV) approach the approach to decision-making under risk based on the assumption that decision makers preferences over alternative distributions are determined solely by the means and variances of the distributions has been widely employed in both economic and financial theories. 1 In particular, it plays a central role in the development of modern theories of portfolio selection and asset pricing. More generally, its analytical tractibility makes possible the analysis of an economic agent s decision in the face of a large number of sources of risks, which explains its continuing use in mainstream theoretical and empirical studies (e.g., Mayer and Smith (1983) Coyle (1999)). Justifying the assumption of mean-variance preferences, however, has been a challenge from the inception of the approach. Tobin (1958) argued that the approach can be consistent with Expected Utility maximization if either the choice set of distribution functions is restricted to just normal distributions or the Von Neumann-Morgenstern (VNM) utility function is quadratic. Mean-variance analysis of portfolio choice indeed works well if all the assets s returns are normally distributed since not only are normal distributions completely determined by their first two moments but portfolios (i.e., linear combinations) of normal distributions are also normal distributions. Chamberlain (1983) further identifies the necessary and sufficient condition on the set of distributions for the distribution of every portfolio to be determined by its mean and variance only. 2 Nevertheless, distributions satisfying Chamberlain s (1983) condition, including of course the normal distribution, are necessarily symmetric and thus do not describe well most real-world asset returns. The assumption of a quadratic VNM utility function, on the other hand, is no less problematic. While it permits mean-variance analysis of portfolio choice without restricting the set of distributions, Arrow (1971) observes that even where a quadratic utility function is increasing, 1 Rothschild and Stigltiz (1970) remarks that The method most frequently used for comparing uncertain prospects has been mean-variance analysis. 2 Meyer (1987) shows that in a wide range of economic models, the MV approach is consistent with EU maximization since the random variables in these models differ only by their location and scale. The location-scale condition admits a larger variety of distributions than Chamberlain s condition. However linear combinations of random variables satisfying the location-scale condition are clearly linearly dependent, which renders the set of distributions uninteresting for portfolio analysis. 2

3 it implies increasing risk aversion. 3 More generally, the expected utility with a quadratic utility function is a mean-variance utility function of a certain form, which precludes many desirable properties of a mean-variance utility function often assumed in applications. Other proponents of the MV approach, such as Markowitz (1959), Levy and Markowitz (1979), take the view that a mean-variance utility approximates the expected utility with acceptable accuracy. While considerable effort has been devoted to justifying the MV approach on the basis of its consistency with the EU model, the descriptive power of the EU model itself has been questioned in the face of mounting contradicting experimental evidence, such as what is known as the Allais paradox, which led to the development of generalizations of the EU model, known as non-expected utility theories. Using the generalized expected utility model introduced by Machina (1982), Epstein (1985) shows that some strong forms of decreasing absolute risk aversion (DARA), which necessarily violate the independence axiom of the EU model, imply mean-variance preferences. Perhaps more importantly, the results in Epstein (1985) point to the possibility and advantages of treating the MV model and the EU model as two distinct approaches to choice under risk a general MV model not conforming to the independent axiom may not exhibit the undesirable propoerties implied by a quadratic VNM utility function and can be consistent with experimental regularities such as the Allais paradox. As with a quadratic VNM utility function in the EU model, however, each of Epstein s (1985) strong DARA postulates implies not just MV preferences but an MV utility function exhibiting a prescribed set of properties, some of which highly restrictive. 4 Thus, the ground-breaking insights present in Epstein s (1985) results notwithstanding, they offer axiomatizations of MV utility functions exhibiting properties implied by different forms of decreasing risk aversion but not a general MV utility function per se. 5 The question therefore remains of whether a general MV utility function can represent a consistent preference ordering over distributions. Tobin himself remarked that Indifference curves 3 See Baron (1977) for a discussion of other undesirable implications of a quadratic VNM utility function. 4 Epstein s (1985) DARA 4 postulates that individuals become less risk aversion if their background wealth improves in the sense of second-degree stochastic dominance and implies linear MV indifference curves. His DARA 5, on the other hand, is necessary and sufficient for an MV utility function exhibiting many properties commonly assumed in applications but the postulate itself is more difficult to interpret. 5 Intuitively, there is no explanation for any conceptual equivalence between MV preferences and any notions of decreasing risk aversion. 3

4 between [mean] and [variance] do not necessarily exist. (Tobin (1958, p.74)). On the basis of a simple example where it is shown that for any two arbitrary points on any purported MV indifference curve, two Bernoulli distributions with the required means and variances can always be constructed so that one of the distributions clearly first-degree stochastically dominates the other, Borch (1969) maintains that it is impossible to draw indifference curves in the mean-variance plane. In this paper, building on the results of Epstein (1985), we first show that utility-presentable preferences over distributions can be represented by a differentiable mean-variance utility function if and only if the preference functional is Frechet differentiable and the local utility function is quadratic for all distributions. That is, we show that any differentiable utility function of mean and variance, equivalently any well-behaved set of MV indifference curves, can be justified by a Frechet differentiable preference functional whose local utility function is quadratic for all distributions. Assuming the conditions for such a MV utility function, we further identify easily interpretable necessary and sufficient conditions on the preferences for each of properties that the MV utility function is commonly assumed to exhibit in applications of the MV approach. We then turn, in the light of our characterizations, to Borch s (1969) criticism of the MV approach and show that an MV utility function can represent a consistent preference ordering over the set of all probability distributions whose supports are contained in a specified finite real interval and the inconsistencies demonstrated in Borch s examples occur only when distributions whose supports lie outside the interval are involved. 2 The Model Let Ω = {ω} be the set of states of nature. Random variables, denoted by x, ỹ, etc., are functions from Ω to an interval [0,M] R and the associated (cumulative) distribution functions are denoted by F x, Fỹ etc. Letting D[0,M] be the set of (cumulative) distribution functions over the interval [0,M], for i =1, 2,..., individual i s preference ordering on D[0,M], denoted by º i, is a complete and transitive binary relation. We further assume that º i is continuous in distribution, i.e., the sets {G D[0,M]:F º i G} and {G D[0,M]:G º i F } are closed (in the topology of convergence in distribution). It follows from Debreu (1964) that º i is representable by a continuous real-valued 4

5 preference functional V i ( ) on D[0,M]. µ x and σ2 x denote respectively the mean and variance of a random variable x, andµ F and σ 2 F denote respectively the mean and the variance of F D[0,M]. Let D[0,M]={λ(G F ) F, G D[0,M],λ R}. Then D[0,M] is a linear subspace of L 1 [0,M] (the space of all measurable functions f on [0,M] such that R M 0 f(x) dx <, endowed with the L 1 norm kfk = R M 0 f(x) dx). Definition 1 V is L 1 -Frechet differentiable at F D[0,M] if there exists a continuous linear functional dv (,F) defined on D[0,M] such that V (G) V (F ) dv (G F, F) lim =0 kg F k 0 kg F k V is said to be L 1 -Frechet differentiable if it is L 1 -Frechet differentiable at every F D[0,M]. Machina (1982) shows that if V is L 1 -Frechet differentiable, then there exists at each distribution F in D[0,M] a local utility function U(,F)over[0,M] such that for any other distribution G in D[0,M], V (G) V (F )= R U(w, F)[dG(w) df (w)] + o(kg F k). where o( ) denotes a function of higher order than its argument. Theorem 1 V (F ) is Frechet differentiable and the local utility function U(x, F ) is quadratic in x for all F if and only if V canbeexpressedintheformv (F )=v(µ F,σF 2 ) for some differentiable function v with domain {(µ F,σF 2 ):F D[0,M]}. The local utility can be written as U(x, F )=[v 1 (µ F,σ 2 F ) 2v 2 (µ F,σ 2 F )µ F ]x + v 2 (µ F,σ 2 F )x 2 (1) 5

6 3 Behavioral Characterizations in the Mean-Variance Model The standard concepts of first-degree stochastic dominance and a mean-preserving spread is defined as follows. Definition 2 (i) Fỹ first-degree stochastically dominates (FSD) F x if Fỹ(x) F x (x) for all x [0,M] where the inequality is strict for some subinterval(s). (ii) Fỹ is a mean-preserving spread (MPS) of F x if R x 0 [F ỹ(y) F x (y)]dy 0 for all x [0,M] where the inequality is strict for some subinterval(s) and R M 0 [F ỹ(y) F x (y)]dy =0. 6 Following Schmeidler (1989) and Ormiston and Quiggin (1994) respectively, we define the concepts of comonotonic random variables and a monotone mean-preserving spread. Definition 3 (i) Two random variables x and z are comonotonic if for any ω, ω 0 {ω}, ( x(ω) x(ω 0 ))( z(ω) z(ω 0 ) 0. (ii) Fỹ is a monotone mean-preserving spread (MMPS) of F x if there exists such that Fỹ = F x+ and x and are comonotonic. MMPS generalizes the concept of increasing risk in Yaari (1969) and is considerably stronger than MPS. Discussion of the concept and its properties and alternative characterizations can be found in Quiggin (1992) and Cohen (1994). Definition 4 (i) º i exhibits monotonicity if Fỹ FSD F x implies Fỹ  i F x. (ii) º i exhibits mean-monotonicity if F ( x+ µ)  i F x, for all µ >0. Definition 5 (Risk Aversion) (i) º i exhibits strong risk aversion if Fỹ being an MPS of F x implies F x  i Fỹ. (ii) º i exhibits monotone risk aversion if Fỹ being an MMPS of F x implies F x  i Fỹ. (iii) º i exhibits variance aversion if µỹ = µ x and σỹ 2 >σ2 x implies F x  i Fỹ. Definition 6 (Comparative Risk Aversion) (i) º 2 is more Ross-risk-averse than º 1 if F x 1 F x+ implies F x  2 F x+ for all x and such that 6 Following numerous other authors, an MPS is treated as equivalent to a mean-preserving increase in risk defined in Rothschild and Stiglitz (1970). 6

7 E[ x] =µ. (ii) º 2 is more monotone-risk-averse than º 1 if F x 1 F x+ implies F x  2 F x+ for all comonotonic x and. (iii) º 2 is more variance-averse than º 1 if F x 1 F x+ implies F x  2 F x+ for all x and such that σ2 x+ >σ2 x. Definition 7 (Decreasing Risk Aversion) (i) º i exhibits Ross decreasing absolute risk aversion (Ross-DARA) if F x+ i F x implies F x+ +w º i F x+w for all w>0 and E[ x] =µ. (ii) º i exhibits monotone decreasing absolute risk aversion (Monotone-DARA) if F x+ i F x implies F x+ +w º i F x+w, for all w>0, and x and are comontonic. (iii) º i exhibits decreasing variance aversion if F x+ i F x implies F x+ +w º i F x+w for all w>0 and x and such that σ2 x+ >σ2 x. Ross-DARA is the concept of decreasing absolute risk aversion (DARA) introduced and characterized in the EU setting by Ross (1981) while the weaker monotone DARA is equivalent to what is used in Yaari (1969). The original and still weaker concept of DARA in Arrow (1974) and Pratt (1964) is the weaker form of either of the definitions above with x assumed degenerate. The distinction between these concepts of DARA are paralleled by the different concepts of comparative risk aversion defined below. Definition 8 (Properness) (i) º exhibits properness if F x i F x+ 1 and F x i F x+ 2 imply F x º i F x , given that x, 1, and 2 are independent. (ii) º i exhibits strong properness if F x i F x+ 1 and F x i F x+ 2 imply F x º i F x ,given E[ 1 x] =µ 1 > 0 and E[ 2 x] =µ 2 > 0. (iii) º exhibits monotone properness if F x i F x+ 1 and F x i F x+ 2 imply F x º i F x ,given that x, 1,and 2 are comonotonic. (iv) º i exhibits proper variance aversion if F x i F x+ 1 and F x i F x+ 2 imply F x º i F x , given σ 2 x+ 1 >σ 2 x, σ2 x+ 2 >σ 2 x,andσ2 x >σ 2 x+ 1. 7

8 Definition 9 (Risk Vulnerability) (i) º i exhibits risk vulnerability if F x i F x+ 1 implies F x+ 2 º i F x , given that x, 1,and 2 are independent and E[ 2 ]=0. (ii) º i exhibits strong risk vulnerability if F x i F x+ 1 implies F x+ 2 º i F x ,givenE[ 1 x] = µ 1 > 0 and E[ 2 x] =0. (iii) º i exhibits monotone risk vulnerability if F x i F x+ 1 implies F x+ 2 º i F x , given that x, 1,and 2 are comonotonic and E[ 2 ]=0. (iv) º i exhibits variance vulnerability if F x i F x+ 1 implies F x+ 2 º i F x , given σ 2 x+ 1 >σ 2 x, µ 2 =0,andσ 2 x >σ 2 x+ 1, Proposition 1 Suppose U i (x, F ) is quadratic in x for all F D[0,M]. (i) º i exhibits monotonicity if and only if U1 i (x, F ) > 0 for all F D[0,M]. (ii) v i (µ, σ 2 ) is increasing in µ if and only if º i exhibits mean-monotonicity. Proof. (i) is proved in Machina (1982). (ii) For any F x D[0,M]andany x >0 such that F x+ x D[0,M], mean-monotonicity implies F x+ x  i F x. Equivalently v i (µ x + x, σ 2 x ) >vi (µ x,σ 2 x ). That is, vi (µ, σ 2 )isincreasinginµ. Conversely, if there exist F x D[0,M]and x >0 such that F x+ x D[0,M]andF x º i F x+ x, then we have v i (µ x + x, σ 2 x ) vi (µ x,σ 2 x ) and hence vi (µ, σ 2 ) is not increasing in µ. 2 Proposition 2 Assuming U i (x, F ) is quadratic in x for all F D[0,M], the following are equivalent, (i) U11 i (x, F ) < 0 for all x [0,M] and F D[0,M]; (ii) v i (µ, σ 2 ) is decreasing in σ 2 ; (iii) º i exhibits variance aversion; (iv) º i exhibits strong risk aversion; (v) º i exhibits monotone risk aversion. Proof. We first show that (v) (ii): Suppose there exist ( µ, σ 2 ) [0,M] [0,M 2 ]andδ>0such that v i ( µ, σ 2 +δ) >v i ( µ, σ 2 ). Then we can find x and comonotonic be such that (µ x,σ 2 x )=( µ, σ2 ), 8

9 µ =0andσ2 x+ σ2 x = δ. And under the assumption that U i (x, F ) is quadratic in x for all F D[0,M], v i ( µ, σ 2 + δ) >v i ( µ, σ 2 )implies x + Â i x. That is, º i does not exhibit monotone risk aversion. Since we clearly have (ii) (iii), (iii) (iv), and (iv) (v), and the equivalence between (i) and (ii) is immediate from (1), we have the equivalence of (i)-(v). 2 Since variance aversion implies and is implied by a mean-variance utility function that is decreasing in variance, mean-variance preferences (or quadratic local utility function) coupled with a fairly weak notion of risk aversion such as monoton risk aversion is equivalent to variance aversion. Define S i (µ, σ 2 )= vi 2 (µ, σ2 ) v i 1 (µ, σ2 ) S i (µ, σ 2 )isthustheslopeoftheµ σ 2 indifference curve at (µ, σ 2 ). We next show that it measures individuals degree of risk aversion. To that end and for characterizing other properties of the µ σ 2 indifference curves, we further define µ i (µ, σ 2 ; δ) by the solution to U i (µ + µ, σ 2 + δ) =U(µ, σ 2 ) for µ. Then µ i (µ, σ 2 ; δ) lim = S i (µ, σ 2 ) δ 0 δ Proposition 3 Assuming U 1 (x, F ) and U 2 (x, F ) are quadratic in x for all F D[0,M] and º 1 and º 2 exhibit mean-monotonicity and strong risk aversion,thefollowingareequivalent, (i) S 2 (µ, σ 2 ) >S 1 (µ, σ 2 ) for all (µ, σ 2 ) [0,M] [0,M 2 ]; (ii) º 2 is more monotone-risk-averse than º 1 ; (iii) º 2 is more Ross-risk-averse than º 1 ; (iv) º 2 is more variance-averse than º 1. Proof. We first show that (ii) (i): For any (µ, σ 2 ) [0,M] [0,M 2 ], let x be such that (µ x,σ 2 x )=(µ, σ2 )and and x be comonotonic and F x 1 F x+. Then º 2 being more monotone risk averse than º 1 implies that F x º 2 F x+. Equivalently (under the assumption that U 1 (x, F )and U 2 (x, F ) are quadratic in x for all F D[0,M]), letting δ = σ 2 x+ σ2 x,wehaveu1 (µ x,σ2 x )=U(µ x+ µ 1 (µ x,σ 2 x ; δ),σ2 x + δ) implyingu 2 (µ x,σ 2 x ) U 2 (µ x + µ 1 (µ x,σ 2 x ; δ),σ2 x + δ). But U 2 (µ x,σ2 x )= 9

10 U 2 (µ x + µ 2 (µ x,σ 2 x ; δ),σ2 x + δ). Hence µ1 (µ x,σ 2 x ; δ) µ2 (µ x,σ 2 x ; δ). That is S2 (µ, σ 2 ) > S 1 (µ, σ 2 ) for all (µ, σ 2 ) [0,M] [0,M 2 ]. 2 Since we clearly have (iii) (ii), (iv) (iii), and (i) (iv), we have the equivalence of (i)-(iv). Proposition 4 Assuming U i (x, F ) is quadratic in x for all F D[0,M] and º i exhibits meanmonotonicity and strong risk aversion, the following are equivalent, (i) S i (µ, σ 2 ) is decreasing in µ; (ii) º i exhibits Ross-DARA; (iii) º i exhibits monotone-dara; (iv) º i exhibits decreasing variance aversion. Proposition 5 Assuming U i (x, F ) is quadratic in x for all F D[0,M] and º i exhibits meanmonotonicity and strong risk aversion, the following are equivalent, (i) U i (µ, σ 2 ) is quasi-concave; (ii) º i exhibits properness; (iii) º i exhibits strong properness; (iv) º i exhibits monotone-properness; (v) º i exhibits proper variance aversion. Proposition 6 Assuming U i (x, F ) is quadratic in x for all F D[0,M] and º i exhibits meanmonotonicity and strong risk aversion, the following are equivalent, (i) S i (µ, σ 2 ) is increasing in σ 2 ; (ii) º i exhibits risk vulnerability; (iii) º i exhibits strong risk vulnerability; (iv) º i exhibits monotone risk vulnerability; (v) º i exhibits variance vulnerability. 10

11 4 Borch Paradox and the Limitations of the Mean-Variance Model Let [(y, p)(z,1 p)] denote a Bernoulli distribution that gives y with probability p and z with probability (1 p). Borch (1969) argues for any two points (µ F,σ F )and(µ G,σ G ) on a supposed µ σ indifference curve, if two Bernoulli distributions [(y F,p)(z, 1 p)] and [(y G,p)(z,1 p)] are constructed as follows z = σ F µ G σ G µ F σ F σ G, p = (µ F µ G ) 2 (µ F µ G ) 2 +(σ F σ G ) 2, y F = µ F +σ F σ F σ G µ F µ G, y G = µ G +σ G σ F σ G µ F µ G, then, for i = F, G the mean and standard deviation of [(y i,p)(z, 1 p)] are µ i and σ i. Since [(y F,p)(z,1 p)] is first-degree stochastically dominated by [(y G,p)(z, 1 p)] if y F <y G,thetwo distributions being on the same indifference curve implies y F = y G, which in turn implies µ F = µ G and σ F = σ G. Alternatively, if (µ F,σ F )and(µ G,σ G )aretwodistinctpointsonanupward-sloping indifference curve and µ F <µ G and σ F <σ G, then we must have y F <y G and hence the two distributions cannot be on the same indifference curve. Recall that µ σ 2 preferences exhibiting monotonicity is equivalent to U 1 (x, F ) > 0 for all x [0,M]andF D[0,M], which by (1) is true if and only if x< v 1(µ F,σ 2 F ) 2v 2(µ F,σ 2 F )µ F 2v 2 (µ F,σ 2 F ) = v 1(µ F,σ 2 F ) 2v 2 (µ F,σ 2 F ) + µ F Let ˆv(µ, σ) v(µ, σ 2 ). Then ˆv 1 (µ, σ) v 1 (µ, σ 2 )andˆv 2 (µ, σ) 2σv 2 (µ, σ 2 ). Hence v 1(µ F,σF 2 ) 2v 2 (µ F,σF 2 ) = σ ˆv 1 (µ F,σ F ) F ˆv 2 (µ F,σ F ) We now show that if (µ F,σ F )and(µ G,σ G ) are on an upward-sloping convex indifference curve and µ F <µ G and σ F <σ G, then we will necessarily have U 1 (y G,G) < 0: y G [ v 1(µ G,σ 2 G ) 2v 2 (µ G,σ 2 G ) + µ G]=µ G + σ G σ F σ G µ F µ G [ σ G ˆv 1 (µ G,σ G ) ˆv 2 (µ G,σ G ) + µ G] = σ G [ σ F σ G µ F µ G ( ˆv 1(µ G,σ G ) ˆv 2 (µ G,σ G ) )] > 0 11

12 where the last inequality is due to the fact that the convexity of the µ σ indifference curve implies that ˆv 2(µ G,σ G ) ˆv 1 (µ G,σ G ) ) > µ F µ G σ F σ G If (µ F,σ F )and(µ G,σ G ) are on an upward-sloping concave indifference curve (and µ F <µ G and σ F <σ G ), we can similarly show that y F > [ v 1(µ F,σ 2 F ) 2v 2 (µ F,σ 2 F ) + µ F ] and hence U 1 (y F,F) < 0.Inthecasewheretheindifference is neither convex nor concave and ˆv 2(µ F,σ F ) ˆv 1 (µ F,σ F ) ) < µ F µ G σ F σ G and ˆv 2(µ G,σ G ) ˆv 1 (µ G,σ G ) < µ F µ G σ F σ G as illustrated below, µ 6 indifference curves q (µ G,σ G ) q (µ F,σ F ) 0 - σ Figure 1: Indifference curves in µ σ space. Defining µ α = αµ F +(1 α)µ G and σ α = ασ F +(1 α)σ G, there clearly exists ˆα [0, 1] such that ˆv 2(µˆα,σˆα ) ˆv 1 (µˆα,σˆα ) > µ F µ G σ F σ G Hence given µ G >µˆα and σ G >σˆα,wehave σ F σ G ˆv 1 (µˆα,σˆα ) y G = µ G + σ G > σˆα µ F µ G ˆv 2 (µˆα,σˆα ) + µˆα = v 1(µˆα,σ 2ˆα ) + µˆα 2v 2 (µˆα,σ2ˆα ) 12

13 That is U 1 (y G,H) < 0 for any H D[0,M] such that µ H = µˆα and σ H = σˆα. That is, given [0,M], monotonicity implies restrictions on the function v(µ, σ 2 )intheformof M v 1(µ F,σ 2 F ) 2v 2 (µ F,σ 2 F ) + µ F for any F D[0,M]. But if (µ F,σ F )and(µ G,σ G ) are two distinct points on an indifference curve implied by such an µ σ 2 utility function and µ F <µ G and σ F <σ G,thenGconstructed by Borch s method will necessarily not be in D[0,M]. To ensure the preferences exhibit monotonicity, we need to require M v 1(µ F,σ 2 F ) 2v 2(µ F,σ 2 F )µ F 2v 2 (µ F,σ 2 F ) = v 1(µ F,σ 2 F ) 2v 2 (µ F,σ 2 F ) + µ F where v 1(µ F,σ 2 F ) v 2 (µ F,σ 2 F ) is the inverse of the slope of the µ σ 2 indifference curve at (µ F,σF 2 ), Si (µ F,σF 2 ). The restriction is thus a restriction on the function v(µ, σ 2 ) or the shapes of the indifference curves. For example, assuming M is a large number, S i (0,σ 2 )mustbesufficiently small for all σ 2 so that M v 1(0,σ 2 ) 2v 2 (0,σ 2 ) = 1 2S i (0,σ 2 ) This implies that plungers as defined in Tobin (1958) will necessarily not be very risk-averse 5 Concluding Remarks The results of this paper thus, by clarifying the implications of using the MV approach, remove many reservations about using it. We of course cannot pretend that the approach can be taken as a general positive theory capable of explaining all empirical and experimental regularities 7 since its fundamental assumption that decision makers do not care about characteristics of distributions 7 No existing theory, however general and complex, can lay claim to such capability. 13

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17 Proposition 7 If U 1 (x, F ) > 0 for all x [0,M] and all F D[0,M], thenv 1 (µ F,σ 2 F ) > 0. Proof. U(x, F )=[v 1 (µ F,σ 2 F ) 2v 2(µ F,σ 2 F )µ F ]x + v 2 (µ F,σ 2 F )x2 U 1 (x, F )=[v 1 (µ F,σ 2 F ) 2v 2(µ F,σ 2 F )µ F ]+2v 2 (µ F,σ 2 F )x If v 2 (µ F,σ 2 F ) < 0, then U 1(x, F ) > 0isequivalentto x< v 1(µ F,σ 2 F ) 2v 2(µ F,σ 2 F )µ F 2v 2 (µ F,σ 2 F ) = v 1(µ F,σ 2 F ) 2v 2 (µ F,σ 2 F ) + µ F (2) If v 1 (µ F,σF 2 ) 0, then there exists no M > 0 such that (3) is true for all x [0,M]andall F D[0,M]. If v 2 (µ F,σ 2 F ) > 0, then U 1(x, F ) > 0implies x> v 1(µ F,σ 2 F ) 2v 2(µ F,σ 2 F )µ F 2v 2 (µ F,σ 2 F ) = v 1(µ F,σ 2 F ) 2v 2 (µ F,σ 2 F ) + µ F (3) If v 1 (µ F,σF 2 ) 0, then there exists no M > 0 such that (4) is true for all x [0,M]andall F D[0,M]. 17