Multidimensional risk aversion: the cardinal sin

Attitudes towards multidimensional risk depend both on the shape of the indifference map under certainty and on the degree of concavity of the utility function representing preferences under risk. A decomposition of the risk premium is built on the new notion of “compensated risk aversion”. The balance between the two components is shown to depend on the association of the risks. Several applications are also presented, including the intertemporal model.


Introduction
It is well-known in the risk literature (see Kihlstrom & Mirman 1974) that when lotteries are defined on many attributes, the properties of the Von Neumann-Morgenstern (VNM) utility function can be confused with changes in the degree of substitutability between goods, which is an ordinal property of individual preferences. In fact, in the Expected Utility (EU) theory, risk attitudes are modeled through a utility function. Utility should describe an intrinsic appreciation of wealth, but risk attitudes consist of more than just this appreciation. Even if the analysis of multidimensional risk aversion is limited to the case of agents with identical indifference curves under certainty, it remains difficult to separate the consequences of the ordinal properties of preferences under certainty from those depending on risk attitudes. This issue is especially relevant for empirical purposes, raising questions about a possible bridge between the assessment of the degree of substitutability between determinants of individual well-being, the elicitation of risk aversion and the measurement of the strength of preferences. 1 When choices under risk involve two or more dimensions, the VNM utility functions used to represent individual preferences are defined on bundles of attributes varying across two or more states of the world. The nature of the substitutability/complementarity of the goods and the association between good and bad outcomes in the different dimensions over states of the world then become crucial for understanding the effects of risk on individual utility (Epstein & Tanny, 1980;Eeckhoudt et al., 2007;Denuit et al., 2010;Li & Ziemba, 1993). A more comprehensive interpretation of the utility function is then required, which is relevant for risky as well as riskless applications.
This paper develops the Pratt (1964) comparative risk aversion analysis in the multidimensional case and shows that the ordinal properties of the utility function can be disentangled from risk attitudes by a suitable decomposition of the risk premium into two components: the first one reflects the degree of substitutability between attributes in a certainty environment; the second one accounts for the sensitivity of the decision-maker to possible risks associated with each single attribute and for her willingness to compress or spread positive and negative outcomes across different states of the world. Therefore, the premium incorporates components describing both risk attitude and the willingness to pay for substituting goods. The main intuition is simple: It is well-known that concave transformations of the utility function modify the utility level attached to the indifference curves (and then risk attitudes) by leaving unchanged their shape. Then, if a risk moves an initial bundle along a direction tangent to its indifference curve, the effect on an agent's utility mainly depends on the degree of curvature of the indifference curve, which is invariant to concave transformations and cardinalization. The more divergent the risk direction is from the slope of the indifference curve, the higher is the effect of concave transformations of the utility function and then the weight of the cardinal component in the risk premium will increase.
Local results on comparative risk aversion are eventually linked with submodularity properties of the utility functions that guarantee an increase of the EU in presence of risks arising along general directions.
This analysis is then applied to assess the strength of preferences. Building on the "intrinsic risk aversion" and the "relative risk aversion" concepts due to Bell and Raiffa (1979) and Dyer and Sarin (1982), respectively, a cardinal value function is adopted to measure differences in utility levels in a certainty environment, while a VNM utility function accounts for preferences towards risk. In this setting, the informational content of the utility function is extremely rich. However, the degree of curvature of the indifference curves -a distinctly ordinal property of individual preferences -plays a crucial role both in the representation of the strength of preferences and in the assessment of risk aversion, which are two cardinal concepts. 2 The analysis of intertemporal risk aversion and further examples discussed in the last part of the paper suggest a variety of potential applications of the main results.
To simplify the exposition, the presentation focuses on the bidimensional case. A generalization to the case of many attributes and non-binary lotteries is provided in the Appendix.

Basic concepts
In the one-dimensional case, an individual is risk averse if she dislikes all zero-mean risks at any initial wealth level x 0 : (2.1) An individual (with utility) v is more risk averse than an individual (with utility) u if v dislikes all lotteries that u dislikes, for any common level of initial wealth. Pratt (1964) characterized comparative risk aversion through absolute and relative measures based on the curvature of the utility function and demonstrated that any increasing and concave transformation w = f (u(x)) raises the degree of risk aversion and the amount of the risk premium (see Gollier 2001 for details).
In the bivariate case, let u : R 2 → R be a componentwise increasing and concave VNM utility function, differentiable as many times as are necessary, defined on two attributes x 1 , x 2 . Let u i denote the marginal utility of u with respect to x i and u i j the second derivatives with respect to x i and x j , for i, j = 1, 2. Richard (1975, p. 13) studied multivariate risk aversion in terms of the preferences of an agent for mixing bad and good outcomes and provided a characterization based on the cross derivative of the utility function. This definition confounds two different aspects: risk aversion and correlation aversion. Kihlstrom and Mirman (1974) defined a risk in terms of a share of the initial vector of the attributes, scaling all the components in the same way. The resulting definition of a risk premium does not fully generalize inequality (2.1) and for this reason what follows refers to the approach of Karni (1979). Karni proposed to complement the Pratt absolute risk aversion coefficient on a single dimension i, defined as: by using a matrix with entries −u i j /u i (see also (Duncan, 1977;Courbage, 2001)). The "coefficient of absolute correlation aversion" for attribute i with respect to attribute j is: The use of the cross derivative and the related notion of aversion to positive (or negative) correlation provides an incomplete view of the decision maker's preferences specification: on one side, risk aversion implies that one prefers with certainty (5, 5) to an equally probable lottery {(0, 0), (10, 10)}, regardless of the sign of the cross derivatives. On the other side, contrasting the latter with the equally probable lottery {(0, 10), (10, 0)}, the degree of substituability/ complementarity becomes crucial in the assessment of the risk premium.

The risk premium
Let (ε,δ) be a random vector, whereε is a zero-mean risk on x 1 andδ is a zero-mean risk on x 2 . Let V = [σ i j ] denote the corresponding symmetric positive semi-definite variancecovariance matrix. If the two risks are simultaneous, then: (2.4) Assuming small risks, the second order approximation of this expression around (x 1 , x 2 ) leads to: Notice that with negative correlation between the two risks and a positive cross derivative of u, (2.5) is always satisfied and the agent is risk averse for any (x 1 , x 2 ). The corresponding definition of the risk premium π 1 (ε,δ) that must be paid in one dimension, for instance x 1 , to get rid of risk in both dimensions is: The risk premium is positive if and only if the utility function is concave (Theorem 1, Karni 1979). 3 By a second order approximation of equation (2.6), it follows that: (2.7) Richard (1975) studied the effects of a concave transformation applied to the utility function on the magnitude of the risk premium. Kihlstrom and Mirman (1974) extended the Pratt approach to the bidimensional case characterizing "more risk aversion" in terms of "more concave" utility functions (p. 366). Under a specific definition of a risk premium (directional and multiplicative, see Proposition 1, p. 368), they also generalize the Pratt result that a higher risk premium is paid by a more risk averse agent. However, such transformations only change the utility values attached to the different indifference curves without affecting their shape, which contains the ordinal information about preferences under certainty. In what follows, the ordinal component of individual preferences is shown to contribute to determining the risk premium through a separable component.

Compensated risk aversion
To simplify the presentation we consider here the special case of binary equiprobable lotteries, while a more general setting is shown in the Appendix. 4 The risk premium π 1 is defined by: or equivalently, (2.10) The effect of the cross derivative of the utility function on the risk premium depends on the correlation between the outcomes of the risks.
The risk premium locally depends on the sum of: (i) the risk aversion coefficient of the reference good, (ii) the risk aversion coefficient of the other good weighted by the marginal rate of substitution and (iii) the correlation aversion index. A positive C A (u 12 < 0) means that a lottery that substitutes to some extent one good with another reduces the total riskiness. The effect on the risk premium of a concave transformation of the utility function u are now investigated. Considering negatively correlated risks, from equation (2.9), a concave transformation raises the risk aversion coefficient in each dimension and the risk premium. However, this effect might be balanced by a decrease in the cross derivative. As noted by Kannai (1980), for any utility function u, ALEP 5 complements can become substitutes by applying a sufficiently concave transformation f to u. 6 To study this "compensative effect" , we consider the relative variation of the total differential of u i with respect to dx i : If this approximation is computed at a fixed level of utility, then dx j dx i = − u i u j and we get: In the one-dimensional case, (2.11) reduces to the well-known Pratt absolute coefficient. With two dimensions, the relative change in the marginal utility resulting from the Pratt coefficient is fully compensated for by adding the indirect effect on the other variable. The Compensated Risk Aversion (CRA) coefficients are then defined as: A central property of the CRA coefficients is now stated.

Lemma 2.1 The C R A i measures are invariant to any strictly monotonic differentiable transformation of the utility function.
Proof By considering the transformation w = f • u, from the definition of absolute risk and correlation aversion we get: The two CRA coefficients then describe an ordinal feature of individual preferences. They are invariant to concave transformations of the utility function because the increase of the absolute risk aversion coefficient on each single variable is perfectly balanced by an increase in the correlation aversion −u i j /u i . The previous lemma, the definition of the risk premium stated in equation (2.9), together with equations (2.13) and (2.14), prove the following proposition.
Proposition 2.1 Given the risks (ε,δ), the risk premium π 1 (ε,δ) can be decomposed as (2.16) The P and ρ components are invariant to strictly monotonic transformations of the utility function, while C A 2 is sensitive to non-linear transformations of the utility function.
In the Appendix, the result of Proposition 2.1 is generalized to the n -dimensional case with non-binary random risks.
Proposition 2.1 allows one to disentangle the ordinal component, which represents individual preferences under certainty, from individual preferences under risk. The magnitude of these components depends on the direction of the risk. For instance, when ε/δ = −u 2 /u 1 (the risk follows exactly the direction of the MRS) then π 1 = P and the risk premium remains invariant to concave transformations of the utility function. As the risk direction moves away from the MRS, a more concave utility function generates a stronger impact on the risk premium. This feature is illustrated by the following example.

Example 2.1 Consider a risk averse agent with utility function
(2.17) The cases with γ = 0, 1, 2 correspond to a correlation prone, neutral or averse agent, respectively: Given the initial allocationx 1 = 100 andx 2 = 5, the left panel represents an equiprobable lottery with risks along the normalized directions ε = 1 √ 1.09 . Using (2.16) and (2.9), the risk premia π P 1 and π N 1 and the values of the components P, ρ P C A 2 and ρ N C A 2 are computed in the case of positively and negatively correlated risks, respectively (Table 1).
As expected, the risk premium increases with γ regardless of the type of risk. To illustrate the effect of correlation aversion, consider first the correlation neutral case γ = 1 when the utility is additively separable. The risk premium then coincides with the ordinal component. If the agent is correlation prone (γ = 0), she has a lower risk premium when the risks are positively correlated: π N 1 > π P 1 . Conversely, π N 1 < π P 1 when the agent is correlation averse (γ = 2).
To show the consequences of a change in the direction of the risk, in the right panel, the direction ε = is chosen, which coincides with the direction of the MRS. For negatively correlated risks, the risk premium π N 1 collapses to the ordinal component and ρ N is equal to 0 . For positively correlated risks, the risk premium π P 1 = 0 when γ = 0 and the ordinal component P is exactly compensated for by ρ P C A 2 .
Further, notice that empirical evidence shows that it is possible to be risk seeking with a diminishing marginal utility of wealth (Chateauneuf & Cohen, 1994), but in the unidimensional EU theory this is not allowed. While, in the multidimensional case, both these aspects can be considered, because the impact of the P component of π 1 can be reduced by ρC A 2 if C A 2 < 0, reflecting a more risk seeking behavior due to correlation proneness.
The ordinal component P depends on the elasticity of substitution. The elasticity of substitution is always equal to 1 for Cobb-Douglas utility functions. A suitable property for the class of the CES utility functions is stated next. 7 Remark 2.1 Given a CES utility function: (2.18) the "relative" compensated risk aversion x i C R A i is equal to the reciprocal of the elasticity of substitution. From (2.13) and (2.14), it follows that C R A i = 1/sx i , ∀i = 1, 2 where s is the elasticity of substitution. From (2.16), if s → +∞ (perfect substitutability), then P → 0, and the ordinal component of the risk premium disappears.
Next, we study the effects of having different directions of the change in risk.

Changes in risk
Given an initial allocation, the effects of different zero-mean risks on the EU depend on the direction and the intensity of gains and losses in each dimension. The setting here is the same as in Eeckhoudt et al. (2007) with simple lotteries of two outcomes that have an equally likely chance of occurrence (see also the aforementioned paper of (Richard, 1975) (Boland & Proschan, 1988), who characterized correlation increasing functions in terms of Schur-convexity and L-superadditivity).
The link between the shape of the utility function and the directions of the change in risk that guarantee an EU improvement is well-known in the stochastic orders literature. Consider the points B = (x 1 , x 2 ) and C = (x 1 − k, x 2 − ), with k, > 0.
Definition 3.1 (Müller & Scarsini, 2012;Müller, 2013) Given a function u : R 2 + → R and two lotteries with equally probable outcomes B and C, or A and D, then: submodular changes in risk) 8 iff the function is submodular, that is u i j ≤ 0, for any i = j; iii) for any A, B, C, D in R 2 + such that C ≤ D ≤ B, C ≤ A ≤ B and A + D = C + B (directionally concave changes in risk) iff the function u is directionally concave 9 , that is u i j ≤ 0, for any i, j = 1, 2.
The changes in risk considered in the definition above are illustrated in Fig. 1. In the first panel, mean preserving contractions are preferred if and only if u is concave. 10 The second panel illustrates case ii), where a negative sign of the cross derivative u i j describes an individual who prefers a lottery where the shortage of one good is compensated for by an increase of the other good in each state of the world . 11 In case iii) the lottery over the best outcome B and the worst outcome C is compared to a lottery over A and D where gains and losses on both dimensions are allowed. Both in cases ii) and iii) the order structure of the vector space, joined with the monotonicity of u, guarantees that points B and C can be ranked in terms of all non-decreasing utility functions, while A and D guarantee a level of utility intermediate between u(B) and u(C) but can be ranked only specifying the functional form of u.
In the next paragraph the previous definition ii) is extended by allowing risk in any direction, such that u(B) ≥ u(D) ≥ u(C) and u(B) ≥ u(A) ≥ u(C) still applies.
Look at the lotteries' oucomes in Fig. 2: neither C, A nor D, B can be ranked by using the orthant order. To analyze this case, it is convenient to introduce a more general order structure in R 2 . The outcomes B = (x 1 , x 2 ) and C = (x 1 − k, x 2 − ), with k, > 0 can be ranked as before by the elementary order structure of R 2 and the monotonicity of u. But now, it is possible to rank also C, A and D, B in the space composed of all the couples (a, b), such that a = α 1 x 1 + α 2 x 2 and b = β 1 x 1 + β 2 x 2 . More precisely, in the space , we get: 12 13 The notion of submodularity is then applied for defining a function : → R, such that (a, b) ≡ u(x 1 , x 2 ). According to Definition 3.1, a function : → R is submodular if for any quadruple A, B, C, D in such that B = A ∨ D and C = A ∧ D, the following inequality holds: (3.1)

Then a lottery on A, D is preferred to another on C, B if and only if
Observe that just as the partial derivatives are taken with respect to a change in one variable, the directional derivatives evaluate the rate at which the utility changes when both inputs move, with positive or negative dependence.
This result shows that all the second derivatives of u, combined with the directions of the risk, determine the effects of the risk on the EU.
Notice that, by the introduction of the function , all the standard cases presented in Definition 3.1 and represented in Fig. 1 can be covered: Case i) represents the concavity of the utility function. It is recovered when the second order derivative of is computed only with respect to a change in one variable. 14 Case ii) corresponds to u 12 < 0. It is obtained when changes in risk are independent, that is, for instance, a = α 1 x 1 and b = β 2 x 2 . 15 Case iii) requires that the slopes of a = α 1 x 1 + α 2 x 2 and b = β 1 x 1 + β 2 x 2 are both positive. 16 In order to separate ordinal from cardinal properties of individual preferences under risk, it is useful to recall the aforementioned work of Kannai (1980), which points out that the cross derivative of u used in the ALEP definition (or equivalently submodularity) is not appropriate to capture ordinal aspects, as substitutability or complementarity among goods. This is also true for the submodularity of . The sensitivity of the submodularity of to utility transformations, however, depends on the direction of the risk. For instance, if the risk changes the allocations along the direction of the MRS, then the submodularity of is not affected by the cardinalization of the utility function. If α 1 /α 2 = −β 1 /β where f is concave, it is easy to check that ab (a, b) and˜ ab (a, b) have the same sign. In fact, moving along the indifference curve (close to a starting position), does not change the preferences of the agent towards the goods, even after a concave transformation of the given utility function.
This observation suggests that the direction of the risk, the shape of the indifference curves and the cardinalization of the VNM utility concur in determining the risk attitude.
Proof Equation (3.2) determines the sign of the cross derivatives of on the basis of the derivatives of u, for a risk along general directions. Consider equation (2.8), for positively and negatively correlated risks, (ε 1 , δ 1 ) and (ε 2 , δ 2 ), respectively. Then, it holds: Comparing this equation with equation (3.1), with a = α 1 x 1 + α 2 x 2 and b = β 1 x 1 + β 2 x 2 , from the definition of risk premium and the monotonicity of u, the result follows.
As the rate of substitution of the goods shows how much one individual is willing to exchange one good with another under certainty and the cross derivative of u, given orthogonal variations of the goods, indicates preferences for joining or spread gains and losses across states of the world, the cross derivative of describes preferences for changes in the quantity of the goods across states of the world which are not orthogonal, but arise along general directions.
The next section investigates an application of our decomposition under the framework of cardinal value functions: risk aversion will be disentangled from the strength of preferences.
14 Recall that, for instance, aa = (β 2 2 u 11 + β 2 1 u 22 − 2β 1 β 2 u 12 )/(α 1 β 2 − α 2 β 1 ) 2 . It results to be negative if and only if the utility function u is concave by the definition of negative semidefinite Hessian matrix. 15 Then we obtain ∂ 2 ∂a∂b = α 1 β 2 u 12 (x 1 , x 2 ), which is negative if and only if u 12 is negative, with α 1 , β 2 > 0. 16 The only feasible cases are given by α i , α j and β i , β j of opposite sign with also α i , β i different in sign ∀i = 1, 2. Under these assumptions, if all the second order derivatives of u are negative, then ab is negative.

Applications
To show the potential of the previous analysis in different fields, we present here some suitable applications in the representation of the strength of preferences and in intertemporal decision-making. Further possible applications will be discussed in the conclusive section.

Intrinsic risk-aversion
Inspired by the works of Bell and Raiffa (1979) and Dyer and Sarin (1982), preferences under risk of an agent can be modeled by means of the utility function U (x) = u(v(x)), where u is a VNM utility and v : R 2 + → R + is a cardinal value function representing preferences under certainty. More precisely, the change in the strength of preferences over a good after small changes in the other good is measured through the coefficient 17 This measure can be interpreted as the bidimensional extension of the "value satiation" coefficient of Dyer and Sarin ((Dyer & Sarin, 1982), p. 877). The impact of the introduction of risk on the preferences of an agent "can then be thought of as a basic psychological (personality) trait of the individual" ( (Bell and Raiffa, 1979), p. 393) called the "intrinsic" risk aversion attitude ("relative" risk aversion in (Dyer & Sarin, 1982)), defined as Proposition 2.1 can be applied in this setting to separate the effect of the strength of preferences on the risk premium from that of risk aversion.
Proposition 4.1 Given the risks (ε,δ) and the utility function U (x) = u(v(x)), the risk premium π 1 can be decomposed as Proof Proceeding as in equations (2.8) and (2.9), the risk premium becomes that, rearranged, leads to (4.6) 17 In what follows, with an abuse of notation, the C A i and C R A i terms are also used for the value function.
The result is then obtained by applying Proposition 2.1 to the component − 1 Notice that P, ρ and C A v 2 depend only on the derivatives of the value function v, accounting for different aspects of the preferences of the agent under certainty. P depends on ordinal characteristics of the value function, ρ depends on the MRS (and on the characteristics of the risk).
The cross derivative of the utility funtion U determines the correlation aversion coefficient C A 2 (x) = −U 12 /U 2 , which can be rewritten as (4.7) The shape of the VNM utility u only affects the correlation aversion coefficient C A 2 (x). Equation (4.7) shows that C A 2 (x) is the sum of C A v 2 (which measures the effect of changes in the attributes under certainty) and R A v , the risk aversion coefficient, weighted by the marginal value function v 1 .
Under the intrinsic risk aversion setting, it is then possible to provide a finer decomposition of the risk premium, by distinguishing in a neat way the different constituents of individual preferences. This result is illustrated by the following extension of Example 2.1. . The risk premium can be decomposed as shown in Table 2. If γ = 0, the premium does not depend on the intrinsic risk aversion coefficient R A v v 1 . With γ = 1, the effect of the intrinsic risk aversion is exactly compensated for by the strength of preferences, as measured by C A v 2 . The agent is intrinsically risk averse but correlation neutral. Finally, when γ = 2, the agent is both risk averse and intrinsically risk averse because the R A v v 1 term dominates the C A v 2 term. In the following subsection our setting is applied to a consumption model of intertemporal risk aversion.

Intertemporal risk aversion
The preference based measures of intertemporal risk attitude can be applied to the intrinsic risk aversion framework. Multidimensional consumption bundles are independent of the consumption characteristics under observation, of monetary valuation, markets, and liquidity constraints.
Intertemporal risk aversion provides a new interpretation for risk attitudes and it is a tool for reinterpreting the assumptions on functional forms and for establishing comparability of risk attitude across agents.
In the model under consideration there are two time periods and an associated consumption bundle (x 1 , x 2 ), but actual consumption may be for a shorter duration. In this case, after the first period, the decision maker is sure to die (or a company will be liquidated). On the other hand, if the decision maker is sure to live for both the time periods, then the preferences are determined by a risk-free intertemporal rate of substitution between the two periods. At intermediate situations, when death between the first and the second time periods is possible but not certain, a combination of these two factors shall determine intertemporal decision making, and intertemporal discounting.
Our application is inspired by models of consumption under uncertainty (see, e.g., Dréze & Modigliani, 1972;Kihlstrom & Mirman, 1981;Traeger, 2014), but it hosts the theory of intrinsic risk aversion to provide new insights for the interpretation of these models.
Let us consider the cardinal measure v that evaluates the agent intertemporal preferences under certainty where ψ i is the cardinal utility assessing the value of consumption x i for the time period i. Risk preferences are then described by the VNM utility function: From the result of Proposition 4.1, given the risks (ε,δ) on present and future consumption, the risk premium paid by the decision maker is: (4.10) P is a weighted sum of the consumption value satiation coefficients intrinsic risk aversion of the decision maker, is multiplied by the marginal utility of the first period and by ρ, which accounts for the risk composition and the MRS between consumption in the two periods. This example also shows better the role of the cross derivative. In fact, in the case of additively-separable utilities, imply that there are no compensatory effects on the consumption value satiation coefficients and not even on the risk aversion coefficient. There is no added information about the willingness to exchange one risk with another one, because they are not correlated. In turn, only gives an evaluation of the rate of intertemporal substitution between the two periods.
While, in the general model of Proposition 4.1, where U (x) = u(v(x)), with v(x 1 , x 2 ) = ψ(x 1 , x 2 ), C A 2 (x) in equation (4.7) and C R A i in equation (4.3) deal with an intertemporal risk aversion attitude. In fact, both on the cross derivative v i j , that is the correlation aversion to exchange one risk with the other one. Note that the substitution effect of the rate ψ 2 /ψ 1 accounts for changes in x 1 and x 2 that let the decision maker lay on the same indifference curve. On the other hand, v i j implies a substitution effect between x 1 and x 2 when the decision maker goes from an indifference curve to another one, that is when risk evaluations matter.
Finally, it is possible to state that the risk premium represents how much the decision maker is willing to pay now to avoid the present and the future risk on consumption. From Proposition 4.1 and equation (4.3), this quantity depends on: i) the timeless risk attitudes R A i , ii) the intertemporal substitution of risks v i j , iii) the temporal uncertaint prospects R A v , iv) the intertemporal substitution of periods ψ 2 /ψ 1 and v) the risk characteristics ε/δ.

Conclusions
A new decomposition of the risk premium is established. Three different features of individual preferences are shown to drive risk attitudes in the multidimensional case: the degree of substitutability among goods, the intensity of risk aversion on each single dimension and the degree of correlation aversion of the decision-maker. This suggests that empirical investigations on risk attitudes require integrating data of different nature: individual preferences on several goods in a risk-free framework, risk preferences elicited on single goods, and preferences for correlation. Several empirical studies suggest that risk measures built using a cardinal value function are more flexible compared to the traditional ones (Currim & Sarin, 1984;Pennings & Smidts, 2000;Qazi et al., 2018). New applications can be envisaged in the analysis of risk factors on health (see Mussard and Pi Alperin 2021), to model social discounting (Fleurbaey and Zuber, 2015) or to evaluate risk in unitary or collective household models of consumption or saving. In this framework, the risk premium accounts for both the timeless and temporal risk attitudes and for the intertemporal substitution effect (under certainty and under risk). 18 Given the bridge established between risk and inequality assessment in the economic literature (Gajdos & Weymark, 2012) our results could also be employed in social welfare analysis, for instance in the framework of prioritarian decision making. Prioritarianism is a normative view coherent with well-known axioms in normative economics: the Pigou-Dalton principle, the Pareto principle and separability. Under continuity, prioritarianism can be represented via composition of two functions: a concave function representing the preference of the social planner is applied to the agent utility (see Adler, 2012, pp. 311, 356 for further details). 19 Examples 2.1 and 4.1 could then describe a prioritarian decisionmaker being risk averse and assessing health measures and risks associated to two health dimensions, which could be correlated or not. In this case, the different components of the willingness to hedge the risks would be neatly disentangled.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The paper develops the case of equiprobable binary lotteries. In this Appendix, the results are generalized to the case of a risk represented by any n-dimensional random vector. Let x ∈ R n + andε be a random vector, whereε i is a zero-mean risk on x i , with i = 1, . . . , n, and let V = [σ i j ] denote the corresponding symmetric positive semi-definite variance-covariance matrix. With simultaneous risks, we know that: Eu(x +ε) ≤ u(x), ∀ε s.t. E(ε) = 0. (5.1) Assuming small risks, the second order approximation of this expression around x leads to: The definition of the risk premium π 1 (ε) that must be paid in the first dimension x 1 , to get rid of risk in all dimensions is: Eu(x +ε) = u(x 1 − π 1 (ε), x 2 , . . . , x n ), ∀ε s.t. E(ε) = 0. (5.3) Computing the second order approximation of equation (5.3), it follows: This equation can be equivalently written in terms of the indices of risk aversion and correlation aversion: The Compensated Risk Aversion (CRA) coefficients are defined as: As in the bidimensional case, the change in the marginal utility measured by the Pratt coefficient is fully compensated for by the indirect effect on the other variables. This indirect effect is measured by a mean of the correlation aversion coefficients. The central property of the CRA coefficients still holds.

Lemma 5.1 The C R A i measures are invariant to any strictly monotonic transformation of the utility function.
Proof By considering the transformation w = f • u, we get: (5.7) The decomposition of the premium can be derived from equations (5.4) and (5.6).
Proposition 5.1 Given the riskε, the risk premium π 1 (ε) can be decomposed as π 1 ≈ P + i j =i C A j ρ i j , with P = 1 2 i σ ii u i u 1 C R A i and ρ i j = 1 2 σ ii n − 1 u i u 1 + σ i j u j u 1 .
(5.8) P and ρ i j are invariant to strictly monotonic transformations of the utility function, while C A j is sensitive to non-linear transformations of the utility function.