The covariance sign of transformed random variables ...

IMA Journal of Management Mathematics (2011) 22, 291−300 doi:10.1093/imaman/dpq012 Advance Access publication on September 7, 2010

The covariance sign of transformed random variables with applications to economics and finance M ART´I N E GOZCUE Department of Economics, University of Montevideo, Montevideo 11600, Uruguay and Accounting and Finance Department, Norte Construcciones, Punta del Este, Maldonado 20100, Uruguay [email protected]

W ING -K EUNG W ONG Department of Economics and the Institute for Computational Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong [email protected] AND

ˇ R I CARDAS Z ITIKIS∗ Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario N6A 5B7, Canada ∗ Corresponding author: [email protected] [Received on 3 December 2009; accepted on 1 August 2010] A number of problems in economics, finance and insurance rely on determining the sign of the covariance of two transformations of a random variable. The classical Chebyshev’s inequality offers a powerful tool for solving the problem, but it assumes that the transformations are monotonic, which is not always the case in applications. For this reason, in the present paper, we establish new results for determining the covariance sign and provide further insights into the area. Unlike many previous works, our method of analysis, which is probabilistic in its nature, does not rely on the classical H¨offding’s representation of the covariance or on any of its numerous extensions and generalizations. We motivate our research with several problems arising in economics, finance and insurance. Keywords: Chebyshev’s inequality; covariance inequality; decision under risk.

1. Introduction The sign of the covariance Cov[α(X ), β(X )] of two real-valued transformations (i.e. functions) α and β of a random variable X plays an important role in economics, finance, insurance and, generally, in decision making under uncertainty. Chebyshev’s integral inequality and its various extensions and c The authors 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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L UIS F UENTES G ARC´I A Departamento de M´etodos Matem´aticos e de Representaci´on, Escola T´ecnica Superior de Enxe˜neiros de Cami˜nos, Canais e Portos, Universidade da Coru˜na, 15001 A Coru˜na, Spain [email protected]

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generalizations (see, e.g. Mitrinovi´c et al., 1993, and references therein) have been fundamental technical tools in the area. Specifically, Chebyshev’s inequality states that if in addition to α and β, we also have a non-negative and integrable function f on [a, b], then the quantity Cα,β [ f ], defined by Cα,β [ f ] =

Z

a

b

α(x)β(x) f (x)dx ×

Z

a

b

f (x)dx −

Z

a

b

α(x) f (x)dx ×

Z

b

β(x) f (x)dx

a

a

Rb where X 0 has the probability density function y 7→ f (y)/ a f (x)dx. Hence, the sign of Cα,β [ f ] is same as the sign of the covariance Cov[α(X 0 ), β(X 0 )]. The latter observation shows that Chebyshev’s inequality can be viewed as a special case of the general problem of determining the sign of the covariance Cov[α(X ), β(X )]. In a number of applications, at least one of the two functions α and β is not monotonic. Egozcue et al. (2009) have established results concerning the covariance sign in such non-monotonic situations. Their proofs rely on Cuadras’s (2002) generalization of the following covariance representation (H¨offding, 1940): ZZ Cov[Y, Z ] = (P[Y 6 y, Z 6 z] − P[Y 6 y]P[Z 6 z])dy dz. This representation has played a pivotal role in numerous research areas. For generalizations and extensions, we refer to Mardia (1967), Mardia & Thompson (1972), Sen (1994), Lehmann (1966), and Cuadras (2002). In particular, (Cuadras, 2002) has proved that ZZ Cov[α(Y ), β(Z )] = (P[Y 6 y, Z 6 z] − P[Y 6 y] P[Z 6 z]) dα(y) dβ(z), (1.1)

where α and β are real-valued functions of bounded variation. To see how equation (1.1) can be utilized in determining the sign of Cov[α(X ), β(X )], we note that when both Y and Z are equal to same random variable X , then P[Y 6 y, Z 6 z] − P[Y 6 y] P[Z 6 z] is non-negative for all y and z. Hence, it now becomes obvious from the right-hand side of equation (1.1) that Cov[α(X ), β(X )] is non-negative when α and β are comonotonic and non-positive when the two functions are anti-monotonic. When at least one of the two functions α and β is not monotonic, then the use of equation (1.1) becomes complex, as the proofs and results by Egozcue et al. (2009) demonstrate. In this paper, we depart from the approach of Egozcue et al. (2009), which is based on Cuadras’s (2002) representation (1.1) and which also assumes bounded supports of the underlying random variables. We suggest a new and direct route for analysing the covariance sign, which is also based on

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is non-negative when both α and β are non-decreasing or when both α and β are non-increasing (in both cases, the functions α and β are called comonotonic). If, on the other hand, one of the two functions α and β is non-decreasing but the other one is non-increasing (such functions are called anti-monotonic), then Cα,β [ f ] 6 0. Note that in probabilistic terms, the quantity Cα,β [ f ] can be expressed, up to a constant, as the covariance Cov[α(X 0 ), β(X 0 )] with a specially defined random variable X 0 . Namely, we have the equation Z b 2 f (x)dx Cov[α(X 0 ), β(X 0 )], Cα,β [ f ] =

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assumptions that naturally manifest in a variety of applications. In particular, throughout the present paper, the underlying random variables can be arbitrary, except of course some moment type or other requirements that are natural in this context. In Section 2, we provide several illustrative examples of such applications. Section 3 contains main results and techniques for determining the covariance sign. Section 4 concludes with potential extensions and additional thoughts on the topic. 2. Illustrative examples Here, we present three examples that will naturally lead to our main results developed in Section 3. Namely, three special covariances will emerge in this section and the signs of the covariances will be of importance. We note at the outset that the covariances will be of the form Cov[α(X ), β(X )] with α(x) = x, which is of course an increasing function, but the function β may not, in general, be monotonic.

πw [X ] =

E[X w(X )] , E[w(X )]

where w is a non-negative ‘weight’ function. The premium has been initiated and explored by Furman & Zitikis (2008); see also Furman & Zitikis (2009) for an overview and further references. The weighted premium can be expressed as follows: πw [X ] = E[X ] +

Cov[X, w(X )] , E[w(X )]

where the ratio on the right-hand side is called the ‘loading’, which needs to be non-negative because the premium πw [X ] has to be larger than the net premium E[X ] for the insurer to remain solvent. The non-negativity of the loading is of course equivalent to the non-negativity of the covariance Cov[X, w(X )],

(2.1)

which is known to be such when w is non-decreasing (see, e.g. Lehmann, 1966). In many examples that appear in the actuarial literature, the weight function w is indeed non-decreasing, but in general it may not be monotonic. E XAMPLE 2.2 Another application of the covariance sign concerns the so-called indifference curve in the two-moment expected utility theory. For details and references on the curve, we refer to, e.g. Eichner (2000), Eichner & Wagener (2004, 2005, 2009), Sinn (1989, 1990), Wagener (2003, 2006) and Wong (2006). Specifically, let u be a (non-decreasing) utility function. Assume that the function is continuously differentiable. Furthermore, let V (μ, σ ) denote the expected utility E[u(μ + σ X )], where X is a ‘seed’ random variable with zero mean and unit variance and μ ∈ R and σ > 0 are constants. The indifference curve μ = μ(σ ), drawn on the (σ, μ) plane for a fixed utility level ν, is given by Cν = {(σ, μ) | V (σ, μ) = ν} . The slope of the indifference curve is S(μ, σ ) = ∂μ/∂σ , which is usually expressed (see, e.g. Wong, 2006, and references therein) using two auxiliary first-order derivatives: Vμ (μ, σ ) = ∂ V (μ, σ )/∂μ and

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E XAMPLE 2.1 A number of well-known insurance premiums, such as the Esscher premium (B¨uhlmann, 1980, 1984), Kamps premium (Kamps, 1998) and Wang or distortion premium (Wang, 1995, 1998), can be viewed as special cases of the ‘weighted premium’

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Vσ (μ, σ ) = ∂ V (μ, σ )/∂σ . With the notation Yμ,σ = μ + σ X , the expression is S(μ, σ ) =

−Cov[Yμ,σ , u 0 (Yμ,σ )] . σ E[u 0 (Yμ,σ )]

Therefore, the sign of S(μ, σ ) and, consequently, the monotonicity of the indifference curve are determined by the sign of the covariance Cov[Yμ,σ , u 0 (Yμ,σ )].

(2.2)

E XAMPLE 2.3 Consider the following model of a competitive company under price uncertainty (cf., e.g. Holthausen, 1979; Feder et al., 1980; Hey, 1981; Meyer & Robison, 1988). Let Q denote the amount of output of the company produced at cost C(Q). Assume for the sake of simplicity that Q and C(Q) are known. The output can be sold either at a random market price P or hedged in the forward market at a fixed price P0 . Let H be the amount to be hedged, which we want to determine. Namely, we want H to be such that the expected profit ρ(H ) = E[u(Π (H ))] is maximal, where u is a utility (or value) function and the profit Π(H ) is given by the formula P(Q − H ) + P0 H − C(Q). The hedged amount H can be any real number: if a part or the entire output is hedged without speculation, then H ∈ [0, Q], but if speculation is involved, then we have either H < 0 or H > Q. From the mathematical point of view, we need to find critical points Hc of the function ρ(H ), which are solutions in H to the equation (∂/∂ H )ρ(H ) = 0, and then we need to determine those Hc that maximize ρ(H ). Assuming that the utility function u is differentiable, we rewrite the equation (∂/∂ H )ρ(H ) = 0 as follows: E[Pu 0 (Π (Hc ))] = P0 , E[u 0 (Π (Hc ))]

(2.3)

where we of course assume E[u 0 (Π(Hc ))] > 0. Equation (2.3) already tells us a remarkable story. Indeed, since (P − E[P])(Q − Hc ) = Π(Hc ) − E[Π(Hc )], we rewrite equation (2.3) as follows: (P0 − E[P])(Q − Hc ) =

Cov[Π(Hc ), u 0 (Π (Hc ))] . E[u 0 (Π(Hc ))]

Consequently, the sign of the covariance Cov[Π(Hc ), u 0 (Π(Hc ))]

(2.4)

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Determining the sign of the covariance might be a complex task, especially when the function u 0 is not monotonic. Indeed, u 0 may or may not be monotonic, as argued by Friedman & Savage (1948), Markowitz (1952), and many others. Within the classical von Neumann and Morgenstern expected utility theory, the utility function u is of course concave and thus u 0 is monotonic, but within the prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992), the utility function u is S-shaped and thus u 0 is not monotonic: it increases before a reference point, which is usually set to 0, and decreases after the reference point.

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determines the sign of the product (P0 − E[P])(Q − Hc ). This is useful, and to see a reason, assume for the sake of simplicity that the utility function u is concave, which holds within the classical von Neumann and Morgenstern expected utility theory. Hence, the first derivative u 0 is non-increasing and thus Cov[Π(Hc ), u 0 (Π(Hc ))] 6 0 (see, e.g. Lehmann, 1966), implying the following statements (cf. Hey, 1981): • If P0 < E[P], then Hc 6 Q (speculation if Hc < 0, and no speculation if 0 6 Hc 6 Q). Likewise, if Hc < Q, then P0 6 E[P] (normal backwardation). • If P0 > E[P], then Hc > Q (speculation if Hc > Q). Likewise, if Hc > Q, then P0 > E[P] (contango).

3. Results To elucidate our general considerations, we first look at the covariance Cov[Y, r (Y )], whose special cases have appeared in Examples 2.1–2.3. We have already noted that when the function r is monotonic, then determining the sign of the covariance can be accomplished using Chebyshev’s inequality (see, e.g. Mitrinovi´c et al., 1993) or, more generally, Lehmann’s (1966) results. Hence, we concentrate on the case when r may not be monotonic. As Examples 2.2 and 2.3 show, this happens when the utility function u is S-shaped, and thus its first derivative r (x) = u 0 (x) is non-decreasing for x 6 0 and non-increasing for x > 0. E XAMPLE 3.1 Let u(x) = Φ(x) − 1/2, where Φ is the standard normal distribution function, which is an S-shaped function. Then r (x) = φ(x), where φ = Φ 0 is the standard normal density. Let the random variable Y be symmetric, and let μ denote its mean. Then Cov[Y, r (Y )] = Cov[α0 (X ), β0 (X )] with the centred random variable X = Y − μ and the functions α0 (x) = x and β0 (x) = r (x + μ). The random variable X is symmetric around 0. The function α0 is odd, i.e. α0 (x) = −α0 (−x) for all x ∈ R. Furthermore, the function α0 is non-negative on the positive real half-line, that is, α0 (x) > 0 for all x > 0. As to the function β0 , we first observe that when μ = 0, then the function is symmetric. In general, we have that • when μ 6 0, then β0 (x) > β0 (−x) for all x > 0 and • when μ > 0, then β0 (x) 6 β0 (−x) for all x > 0.

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To check whether Hc gives the maximal expected utility, we need to verify the condition E[(P − P0 )2 u 00 (Π(Hc ))] < 0, assuming of course that u is twice differentiable. This condition is easily verifiable when u is concave, because the second derivative is non-positive. If, however, u is more complexly shaped, like in the case of the prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992), then verifying the condition can be challenging (see Broll et al., 2010, for details in a special case). Determining the sign of covariance (2.4) can also be challenging within the prospect theory because S-shaped utility functions do not possess monotonic derivatives u 0 and thus neither Chebyshev’s inequality (see, e.g. Mitrinovi´c et al., 1993) nor Lehmann’s (1966) results are applicable. We tackle this problem in Section 3.

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We shall see later in this section that the above two conditions on β0 , and thus the sign of the mean μ, determine the sign of the covariance Cov[Y, r (Y )]. This concludes Example 3.1. The properties of the functions α0 and β0 specified in Example 3.1 provide us with hints as to what assumptions are reasonable to assume when dealing with general functions α and β. With this in mind, we next formulate a general proposition. P ROPOSITION 3.1 Let X be symmetric around 0, and let the function α be odd, i.e. α(x) = −α(−x) for all x ∈ R. If α(x) > 0 and β(x) > β(−x) for all x > 0, then Cov[α(X ), β(X )] > 0. Proof. We easily check that when the random variable X is symmetric around 0 and the function α is odd, then Cov[α(X ), β(X )] = E[α(X )(β(X ) − β(−X ))1{X > 0}]

(3.1)

C OROLLARY 3.1 If μ 6 0, then Cov[Y, r (Y )] > 0, but if μ > 0, then Cov[Y, r (Y )] 6 0. Note that Corollary 3.1 is in sharp contrast with the case when the utility function u is concave. This is due to the fact that under the latter (concavity) assumption, the utility-rate function r = u 0 is non-increasing and thus Cov[Y, r (Y )] 6 0 (cf. Lehmann, 1966) irrespectively of the sign of the mean μ. The following corollary follows from Proposition 3.1 and generalizes Theorem 2.2 of Egozcue et al. (2009) to arbitrary random variables, thus making it more widely applicable for solving problems in economics, finance, insurance and other areas of application. C OROLLARY 3.2 Let X be symmetric around 0, and let the function α be odd. If α(x) > 0 for all x > 0 and β is non-decreasing on R, then Cov[α(X ), β(X )] > 0. When comparing Proposition 3.1 and Corollary 3.2, we see that the conditions on α are same but the condition on β is stricter in Corollary 3.2 than in Proposition 3.1. This suggests exploring a possibility of weakening the condition on α in Corollary 3.2 though still staying within the class of odd functions α. We achieve this balancing of conditions on α and β in the next proposition, where we use the notation Tα (x) = E[α(X )1{X > x}].

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(cf. Lemma 3.1 at the end of this section for technical details and further generalizations). Applying the assumptions of Proposition 3.1 on the right-hand side of equation (3.1), we conclude that Cov[α(X ), β(X )] > 0.  Proposition 3.1 actually covers more ground than we see in its formulation per se. For example, note that when α is odd, then −α is also odd. Hence, Proposition 3.1 is applicable for the latter function and therefore implies that if α(x) 6 0 and β(x) > β(−x) for all x > 0, then Cov[α(X ), β(X )] 6 0. Furthermore, replacing β by −β, we have from Proposition 3.1 that if α(x) > 0 and β(x) 6 β(−x) for all x > 0, then Cov[α(X ), β(X )] 6 0. Likewise, if α(x) 6 0 and β(x) 6 β(−x) for all x > 0, then Cov[α(X ), β(X )] > 0. Proposition 3.1 is a substantial generalization of Theorem 2.3 of Egozcue et al. (2009) because the latter theorem covers only random variables with compact supports. However, when modelling various phenomena in economics, finance and insurance, we frequently use random variables that have infinite supports: the positive real half-line or the entire real line, depending on the problem at hand. In view of Proposition 3.1 and related notes, we have the following corollary concerning the earlier (see Example 3.1) noted covariance Cov[Y, r (Y )] with the utility-rate function r (x) = φ(x) and the symmetric random variable Y , whose mean is μ.

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P ROPOSITION 3.2 Let X be symmetric around 0. Furthermore, let the function α be odd and the function β be continuous. If Tα (x) > 0 for all x > 0 and the function β is non-decreasing on R, then Cov[α(X ), β(X )] > 0. Proof. By equation (3.1), we have that Z Cov[α(X ), β(X )] = E[α(X )( 1{−X < x < X }dβ(x))1{X > 0}] Z = E[α(X )1{−X < x < X }1{X > 0}]dβ(x) Z = Tα∗ (x)dβ(x),

(3.2)

where

The integration in equation (3.2) is over the entire real line. However, since Tα∗ (−x) = Tα∗ (x) for all x ∈ R, the sign of the function Tα∗ is determined by its sign on the positive half-line. Noting that  Tα∗ (x) = Tα (x) for all x > 0, we finish the proof of Proposition 3.2. We conclude this section with a general lemma, whose part (cf. equation (3.1)) has already played a fundamental role in our considerations above. L EMMA 3.1 Let X be symmetric around 0. Then we have the following two statements: 1. If the function α is odd, i.e. α(x) = −α(−x) for all x ∈ R, then we have that Cov[α(X ), β(X )] = E[α(X )(β(X ) − β(−X ))1{X > 0}].

(3.3)

2. If at least one of the two conditions (a) α(0) = 0 and β(0) = 0 or (b) P[X = 0] = 0 is satisfied, then we have that Cov[α(X ), β(X )] = E[α(X )(β(X ) − β(−X ))1{X > 0}] + E[(α(X ) + α(−X ))β(X )1{X > 0}] − E[(α(X ) + α(−X ))(β(X ) − β(−X ))1{X > 0}] − E[(α(X ) + α(−X ))1{X > 0}] E[(β(X ) − β(−X ))1{X > 0}].

(3.4)

N OTE 3.1 The second part of Lemma 3.1 is a general result that allows us to investigate the sign of the covariance Cov[α(X ), β(X )] under various combinations of the conditions α(x) 6 −α(−x), α(x) > −α(−x), β(x) 6 β(−x) and β(x) > β(−x). We have found these conditions natural in a number of applications as Example 3.1 has already illustrated. Proof of Lemma 3.1. We first express the covariance Cov[α(X ), β(X )] as the difference between E[α(X )β(X )] and E[α(X )] E[β(X )]. Using the symmetry of X , we have that E[α(X )β(X )] = E[α(X )β(X )1{X > 0}] + E[α(−X )β(−X )1{X > 0}] + α(0)β(0)P[X = 0].

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Tα∗ (x) = E[α(X )1{−X < x < X }1{X > 0}].

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When α is odd, then α(0) = 0. Hence, irrespectively of whether we work under the conditions of the first or second parts of Lemma 3.1, we have that E[α(X )β(X )] = E[α(X )(β(X ) − β(−X ))1{X > 0}]

+ E[(α(X ) + α(−X ))β(X )1{X > 0}] − E[(α(X ) + α(−X ))(β(X ) − β(−X ))1{X > 0}].

(3.5)

Under the condition of the first part, we have that E[α(X )β(X )] = E[α(X )(β(X ) − β(−X ))1{X > 0}], which implies equation (3.3) because E[α(X )] = 0. Indeed, the latter property follows from the equation: (3.6)

combined with the assumption that α is an odd function in which case we have α(x) + α(−x) ≡ 0 and thus α(0) = 0. This established the first part of Lemma 3.1. In view of equation (3.5), in order to establish the second part of Lemma 3.1, we only need to check that E[α(X )] E[β(X )] = E[(α(X ) + α(−X ))1{X > 0}] E[(β(X ) − β(−X ))1{X > 0}].

(3.7)

For this, in addition to equation (3.6), we also write the following one: E[β(X )] = E[(β(X ) − β(−X ))1{X > 0}] + β(0)P[X = 0].

(3.8)

Hence, when α(0) = 0 and β(0) = 0, which is assumed in part (1), then equations (3.6) and (3.8) imply equation (3.7). When P[X = 0] = 0, which is assumed in part (2), then the right-most summands of both equations (3.6) and (3.8) vanish, and we thus again arrive at equation (3.7). This completes the entire proof of Lemma 3.1.  4. Concluding notes In this paper, we have established new results about the sign of the covariance between two transformations of a random variable. We have illustrated applications of the results using examples from areas such as economics, finance and insurance. As a by-product, we have also suggested new and simple (if compared to those already available in the literature) techniques for analysing the covariance sign and in this way provided a convenient venue for further extensions and generalizations in the area. To illustrate possible extensions, we next discuss two scenarios. First, even though results of Section 3 give the impression that they are applicable only in the case of symmetric around 0 random variables X , the case of general symmetric random variables Y with means μ can be accommodated via the equation Cov[α(Y ), β(Y )] = Cov[α1 (X ), β1 (X )],

(4.1)

where X = Y − μ, α1 (x) = α(x + μ) and β1 (x) = β(x + μ). The covariance on the right-hand side of equation (4.1) is within the framework of Section 3. In a special case, the above centering at 0 technique has already been utilized in Example 3.1.

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E[α(X )] = E[(α(X ) + α(−X ))1{X > 0}] + α(0)P[X = 0]

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Next, if the distribution of Y is skewed, then we can still reduce the covariance to that based on a symmetric random variable, as follows: Cov[α(Y ), β(Y )] = Cov[α2 (U ), β2 (U )], where G is the distribution function of Y , U is a uniform on the interval [0, 1] random variable, which is symmetric around 1/2, and the new transformations α2 (x) = α(G −1 (x)) and β2 (x) = β(G −1 (x)), where G −1 is the quantile function corresponding to G, i.e. G −1 (u) = inf{x : G(x) > u} for all u ∈ (0, 1). If the uniform random variable U is not convenient to work with, which depends on a context, then we can replace it by any other symmetric random variable and redefine the transformations α2 and β2 accordingly. Results might be involved, which is natural because the skewness of the distribution function G and the non-monotonicity of at least one of the functions α and β may interact in a very complex manner.

We are grateful to the Editor, Phil Scarf, an anonymous associate editor and two anonymous referees for suggestions and constructive criticism that have led us to a significant improvement of the paper. The third author also thanks Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement. Funding Agencia Nacional de Investigaci´on e Inovaci´on (ANII), Uruguay; the Research Grants Council (RGC) of Hong Kong; the Natural Sciences and Engineering Research Council (NSERC) of Canada. R EFERENCES B ROLL , U., E GOZCUE , M., W ONG , W. K. & Z ITIKIS , R. (2010) Prospect theory, indifference curves, and hedging risks. Appl. Math. Res. Express (in press). ¨ , H. (1980) An economic premium principle. ASTIN Bull., 11, 52–60. B UHLMANN ¨ , H. (1984) The general economic premium principle. ASTIN Bull., 14, 13–21. B UHLMANN C UADRAS , C. M. (2002) On the covariance between functions. J. Multivar. Anal., 81, 19–27. E GOZCUE , M., F UENTES G ARCIA , L. & W ONG , W. K. (2009) On some covariance inequalities for monotonic and non-monotonic functions. J. Inequalities Pure Appl. Math., 10, 1–7. E ICHNER , T. (2000) A note on indifference curves in the (μ, σ )-space. OR Spectr., 22, 491–499. E ICHNER , T. & WAGENER , A. (2004) Relative risk aversion, relative prudence and comparative statics under uncertainty: the case of (μ, σ )-preferences. Bull. Econ. Res., 56, 159–170. E ICHNER , T. & WAGENER , A. (2005) Measures of risk attitude: correspondences between mean-variance and expected-utility approaches. Decis. Econ. Financ., 28, 53–65. E ICHNER , T. & WAGENER , A. (2009) Multiple risks and mean-variance preferences. Oper. Res., 57, 1142–1154. F EDER , G., J UST, R. E. & S CHMITZ , A. (1980) Futures markets and the theory of the firm under price uncertainty. Q. J. Econ, 94, 317–328. F RIEDMAN , M. & S AVAGE , L. J. (1948) The utility analysis of choices involving risk. J. Polit. Econ., 56, 279–304. F URMAN , E. & Z ITIKIS , R. (2008) Weighted premium calculation principles. Insur. Math. Econ., 42, 459–465. F URMAN , E. & Z ITIKIS , R. (2009) Weighted pricing functionals with applications to insurance: an overview. N. Amer. Actuar. J., 13, 483–496. H EY, J. D. (1981) Hedging and the competitive labor-managed firm under price uncertainty hedging and the competitive labor-managed firm under price uncertainty. Am. Econ. Rev., 71, 753–757.

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Acknowledgements

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