Bidual Representation of Expectiles

: Downside risk measures play a very interesting role in risk management problems. In particular, the value at risk (VaR) and the conditional value at risk (CVaR) have become very important instruments to address problems such as risk optimization, capital requirements, portfolio selection, pricing and hedging issues, risk transference, risk sharing, etc. In contrast, expectile risk measures are not as widely used, even though they are both coherent and elicitable. This paper addresses the bidual representation of expectiles in order to prove further important properties of these risk measures. Indeed, the bidual representation of expectiles enables us to estimate and optimize them by linear programming methods, deal with optimization problems involving expectile-linked constraints, relate expectiles with VaR and CVaR by means of both equalities and inequalities, give VaR and CVaR hyperbolic upper bounds beyond the level of confidence, and analyze whether co-monotonic additivity holds for expectiles. Illustrative applications are presented.


Introduction
Downside risk measures have been used in actuarial science, mathematical finance, and more general risk management problems.The value at risk (VaR) and the conditional value at risk (CVaR) are probably the most famous downside risk measures, since they are very intuitive and easy to interpret in practice.Expectile risk measures are much less commonly used, perhaps because their practical interpretation is not so obvious.This lower interest in expectiles is observed even in the Basel I I I (banking) and Solvency I I (insurance) regulatory systems.Nevertheless, expectile risk measures reflect very interesting properties.They are coherent, in the sense of Artzner et al. (1999), and elicitable (Bellini et al. 2014).In particular, their elicitability has important implications in backtesting (Bellini and Di Bernardino 2017) and other important applications (Embrechts et al. 2021).Both coherence (which fails for VaR) and elicitability (which fails for CVaR) may justify the use of expectiles.Moreover, Zou (2014), Bellini and Di Bernardino (2017), and Tadese and Drapeau (2020), among others, have shown close relationships between expectiles and CVaR, making it easier to interpret expectiles as downside risks.
Dual representation is another cornerstone in downside risk (Artzner et al. 1999;Föllmer and Schied 2002, etc.).Indeed, dual representations have played a critical role in estimating, managing, and optimizing downside risks in practice.Since dual representation frequently implies that a downside risk measure is the optimal value of a linear optimization problem, it makes sense to study the dual optimization problem of the dual representation, that is, the bidual representation of a downside risk measure.That is exactly the main objective of this paper, with a special focus on expectiles.In other words, the leitmotif of this paper is the use of bidual representation and several properties of the duality theory of linear programming as a unified methodology to address important issues affecting expectiles.
The paper outline is as follows.Section 2 begins by synthesizing the most important ideas related to VaR, CVaR, and expectiles.Their usual definitions and properties are presented despite the fact that they are not new; however, the main purpose is to facilitate the reading of the paper.This approach allows us to introduce the new Theorem 5 and its proof in a natural manner.Theorem 5 is one of the most important results of this paper, since it provides us with the bidual representation of the expectile and a new way to relate expectiles and linear programming methods.Theorem 5 leads to Theorems 6 and 7, which are the most important results of Section 3. Indeed, Theorem 6 shows that the optimization of expectiles can be reduced to a linear (convex) problem if the constraints are linear (convex).Analogously, Theorem 7 shows that optimization problems involving expectile-linked constraints can be linearized as well.Needless to say, the linearization of risk optimization problems has been a very important question in the risk analysis literature (Rockafellar and Uryasev 2000;Konno et al. 2005;Balbás and Charron 2019, etc.).
Important relationships between CVaR and expectiles have been addressed in Delbaen (2013), Bellini and Di Bernardino (2017), and Tadese and Drapeau (2020), to name a few.Section 4 is devoted to showing that these relationships may be also addressed by means of the bidual representation given in Theorem 5.In particular, by computing the expectation and the expectile of an arbitrary random gain, you will have an upper bound of the CVaR given by a simple hyperbolic function of the CVaR confidence level.Every confidence level may be involved in this simple formula.The other important finding of this section is Corollary 6, since it allows us to characterize whether co-monotonic additivity holds for expectiles.In general, the co-monotonic additivity of a downside risk measure is required in many practical applications (Dhaene et al. 2002;Bellini et al. 2021;Balbás et al. 2022, etc.), but it may fail for expectiles (Delbaen 2013).
The relationships in Section 4 are inequalities, and an obvious question is whether they can be improved.This problem is addressed in Section 5, where it is shown that they often become equalities, i.e., they cannot be improved (Theorems 10 and 11 and Corollary 9).Moreover, when they are equalities, they provide new ways to estimate different risk measures in practical applications, with special focus on the value at risk.It is known that topics related to the practical estimation of risks are very important in real-world applications (Buch et al. 2023;Dacorogna 2023, etc.).
Section 6 is devoted to illustrating all the above findings in a very popular actuarial example, namely, the (maybe optimal) combination of reinsurance contracts and financial investments.To the best of our knowledge, this is the first paper that simultaneously involves reinsurance contracts, financial markets, VaR, CVaR, and expectiles.A very general approach would significantly increase the paper length and therefore is beyond the scope of this work, so several simplifications are incorporated.In any case, the simplifications are sufficient to illustrate the main results of the previous sections, which are the major purpose.In fact, the bidual representation of expectiles enables us to estimate and optimize them by means of linear programming methods, relate them with VaR and CVaR by means of both equalities and inequalities, give VaR and CVaR upper bounds beyond the level of confidence, and analyze whether co-monotonic additivity holds for expectiles.Section 7 presents both a general discussion about contributions/limitations and the main conclusion.

Dual and Bidual Representations
As indicated in the introduction, this section is devoted to synthesizing several important and well-known properties of VaR, CVaR, and expectiles.They are given without proofs, but their presence will facilitate the reading of the paper.Furthermore, a new representation theorem for expectiles is given (Theorem 5), which will be vital in future sections.

VaR and CVaR
Consider the probability space (Ω, F , P) composed of the set of states of nature Ω, the σ-algebra F reflecting the information available at a future date T, and the probability measure P. Denote by IE(y) the mathematical expectation of every R-valued random variable y defined on Ω.Unless the opposite is indicated, IE(y) exists and is finite for every random variable in this paper.In other words, every random variable belongs to the classical space L 1 (or L 1 (Ω, F , P), if necessary).
Fix µ * ∈ (0, 1).If F y (x) = IP(y ≤ x) is the cumulative distribution function of the random variable y, 1 then the value at risk of y with the level of confidence 1 − µ * ∈ (0, 1) is given by and the conditional value at risk of y with the same confidence level is given by Previous papers provided us with representation theorems for both VaR and CVaR.
Theorem 1 provides us with a bidual representation of VaR.There exists a dual representation as well (Koenker 2005), but it is not needed in this paper.Theorem 1 was first proved in Balbás et al. (2017), and later Balbás and Charron (2019) showed that similar methods may allow us to prove extensions and/or slight modifications of Theorem 1.A particular case is Corollary 2 below.
Theorem 2 (CVaR dual representation, Rockafellar et al. 2006).CVaR 1−µ * (y) is the optimal value of the bounded and solvable problem where z * ∈ L 1 is the decision variable.

Expectiles
Fix µ ∈ (0, 1/2).There exits a unique solution to the equation where x ∈ R is the unknown (Bellini and Di Bernardino 2017).This solution is denoted by E µ (y) and is said to be the expectile of y at level µ. 3 The expectile risk measure at level µ is defined by E µ (y) := −E µ (y).Bearing in mind the equality (y − x) + + x = y + (x − y) + , it is easy to see that x ∈ R solves (7) if and only if x solves so (8) also characterizes the expectile risk measure.Furthermore, several authors have shown that E µ is a continuous, coherent (in the sense of Artzner et al. (1999)), expectationbounded (in the sense of Rockafellar et al. (2006)), and law-invariant risk measure (Ziegel 2016).In other words, E µ is continuous, sub-additive (E µ (y . Notice that there are some redundancies in the latter sentence.Firstly, positively homogeneous and decreasing imply continuous.Secondly, coherent and law invariant imply expectation bounded (mean dominating).Anyway, we have preferred to give an exhaustive list of properties.Recall that CVaR 1−µ * also satisties all the properties above, whereas VaR 1−µ * is neither continuous, nor sub-additive, nor mean dominating.
Theorem 4 (Dual representation of expectiles, Delbaen 2013).E µ (y) is the optimal value of the bounded and solvable problem where (ξ, z) ∈ R×L 1 is the decision variable.

Optimization Problems Involving Expectiles
Since Rockafellar and Uryasev (2000) proved that the CVaR minimization may often be addressed by means of linear programming methods, many authors have extended the analysis and dealt with other risk measures (Konno et al. 2005; Balbás and Charron 2019, for instance).Let us show that the optimization of expectiles may be also linearized.Accordingly, consider a functional α : L 1 −→ R, an arbitrary set X, a function β : X −→ L 1 , and the optimization problem Taking into account Theorems 3 and 5, the proof of Theorem 6 below becomes simple and therefore omitted.
Notice that both ( 13) and ( 14) may inherit several properties of the set X and the function β.In particular, if X is given by linear (convex) constraints and β is linear, then ( 13) and ( 14) become linear (convex) optimization problems.In other words, Theorem 6b may play a critical role in linearizing the minimization of expectiles, and Theorem 6a presents a way to linearize the CVaR minimization.
Expectile-linked constraints can be also linearized by the application of Theorem 5. Indeed, consider a subset Y ⊂ L 1 , a real-valued function f : Y −→ R, a real number k ∈ R, and the optimization problem where "Opt" applies for both "Max" or "Min".Then, one has the following.

Theorem 7. Consider the optimization problem
Opt f (y) with (λ, y, λ m , λ M ) ∈ R × Y × L 1 2 being the decision variable.y solves (15) if and only if there Proof.Theorem 5 implies that E µ (y) ≤ k holds if and only if there exists (λ, 16) again inherits the properties of Y and f .In particular, if Y is given by linear (convex, concave) constraints and f is linear (convex, concave), then ( 16) is a linear (convex, concave) problem.

Linking CVaR and Expectiles
Several authors have pointed out the existence of inequalities involving VaR, CVaR, and expectiles (Delbaen 2013;Bellini and Di Bernardino 2017;Tadese and Drapeau 2020, to name a few).Let us show that this type of relationship may also be addressed by means of the bidual approach.Some of the inequalities below are quite similar to others proved in Tadese and Drapeau (2020).Nevertheless, the use of the bidual representation in Theorem 5 may simplify the proofs.Needless to say, the simplification of proofs may deserve the interest of many researchers (Herdegen and Munari 2023, for instance).Moreover, bidual representation will allow us to study the potential co-monotonic additivity of expectiles, as well as to verify in Section 5 whether the given inequalities may become exact equalities.
First of all, the expectile risk measure E µ will be the envelope of a family of continuous, coherent, expectation-bounded, law-invariant and co-monotonically additive risk measures.4 In order to show that, let us fix the subsets A and B of R 2 given by (17) and for (µ, ξ) ∈ A and µ * ∈ (0, 1), let us consider the sets Let us prove an instrumental lemma.

Lemma 1. (a)
The function given by is well-defined and a one-to-one bijection whose inverse If µ * is given by ( 19), then the function is well-defined and a one-to-one bijection whose inverse is given by is well-defined and a one-to-one bijection whose inverse is given by Proof.In order to see that ( 19) is well-defined, one must show that 0 < µ * < 1.The first inequality is evident, so let us see the second one.Since the one-to-one bijection is strictly decreasing, and µ/(1 − µ) < ξ, one has that and ( 19) leads to Additionally, trivial manipulations of ( 19) lead to (20), so let us see that, for (µ * , ξ) ∈ B = (0, 1) 2 , (20) implies that 0 < µ < 1/2.The first inequality is obvious, so let us prove the second one.One has that and therefore µ < 1/2 (see ( 20)).Lastly, it only remains to see that µ/(1 − µ) < ξ holds.
Bearing in mind (20), it is equivalent to (c) IE(z * ) = 1 and 0 ≤ z * are evident, so let us see that z * ≤ 1/µ * and therefore z * ∈ D µ * .Indeed, (19) shows that one must prove that which must hold because z ∈ C (µ,ξ) .Additionally, the equivalence is obvious, so it only remains to prove that z and (20) shows that so the given equivalence becomes evident.□ Theorem 8. Consider y ∈ L 1 , (µ, ξ) ∈ A, and the optimization problems where z ∈ L 1 is the decision variable, and where (λ, λ m , λ M ) ∈ R× L 1 2 is the decision variable.Equations ( 23) and ( 24) are bounded and solvable, and the optimal value of both problems coincide.If z is (23)-feasible and (λ, λ m , λ M ) is (24)-feasible, then they solve the corresponding problem if and only if Proof.The (23)-feasible set is included in the space L ∞ of essentially bounded random variables.Moreover, this feasible set is σ L ∞ , L 1 -compact owing to Alaoglu's theorem (Zeidler 1995).Since the objective function of ( 23) is σ L ∞ , L 1 -continuous (Zeidler 1995), Weierstrass' theorem implies that ( 23) is bounded and solvable.Following Anderson and Nash (1987), ( 24) is the dual problem of (23), although one must show the constraint (λ m , λ M ) ∈ L 1 2 because the dual space of L ∞ is larger than L 1 .Nevertheless, this constraint may be proved with the method used in Balbás et al. (2021) to relate (2) and (3).Since ( 24) and ( 23) are infinite-dimensional problems, the so-called duality gap between them could arise.However, it may be proved that this duality gap vanishes by using a method similar to that in Balbás et al. (2021) for ( 2) and (3).The absence of a duality gap guarantees that (3) is bounded and solvable, and (25) reflects necessary and sufficient optimality conditions (Anderson and Nash 1987).□ Definition 1.Consider y ∈ L 1 and (µ, ξ) ∈ A. The sub-expectile E (µ,ξ) (y) of y will be given by the optimal value of ( 23) or (24).
M is obviously (10)-feasible for y = y 1 + y 2 .If this is a solution to the problem, then E µ y j = λ (j for j = 1, 2 leads to E µ (y and the equality holds if and only if there exists z ∈ L 1 such that (ξ, z) solves (9).
so the fulfillment of ( 29) and (30) as equalities and the fulfillment of (36) are evident.Hence, the open problem presented in Example 1 may be addressed for P(y = IE(y)) < 1.

Combining Actuarial and Financial Risks
Let us illustrate the ideas of Sections 2-5 by dealing with an important actuarial problem, that is, the optimal combination of reinsurance contracts and financial instruments.The particular problems are the selection of the optimal reinsurance, which arises if one imposes that the selected financial strategy must equal zero, and the portfolio choice problem, which arises when there is no actuarial risk involved.Both the optimal reinsurance problem and the portfolio selection one have been addressed by dealing with downside risk measures, 9 and a recent line of research integrates both problems into a single one, which is our first focus in this section, namely, the optimal combination.Though there are several interesting perspectives, we will focus on that of Balbás et al. (2023), since it is very general and properly fits the illustrative objective of this section. 10 Suppose that Y ≥ 0 reflects the global indemnification to be paid by a direct insurer within the time interval [0, T].There is a reinsurance market, and Y can be divided according to Y = Y c + Y r , where Y c represents the ceded risk, and Y r represents the retained one.In order to guarantee that Y, Y c , and Y r are co-monotonic, a typical requirement to prevent the moral hazard, let us deal with the marginal retained indemnification rather than the retained indemnification itself.Accordingly, consider the interval (0, +∞), its Borel σ-algebra B, and the Lebesgue measure L. For every R-valued, essentially bounded measurable function the retained indemnification will be given by for ω ∈ Ω.If Y has a finite expectation, then, according to Balbás et al. (2023), ( 44) and ( 45) lead to a random variable Y r with a finite expectation such that Y, Y r , and Y c = Y − Y r are co-monotonic.There is also a financial market, and the insurer may focus on an international stock index whose stochastic behavior is conducted by a geometric Brownian motion (GBM).Hence, the evolution {S t ; 0 ≤ t ≤ T} of the index quotation is given by dS t = S t ((r * − γ)dt + σdB t ), where r * is the index drift, γ is the index dividend yield, and σ is the index volatility.There are future contracts whose underlying asset is the index above.If r 0 denotes the riskless rate, it is known that the future quotation {F t ; 0 ≤ t ≤ T} is another GBM and evolves according to dF t = F t (rdt + σdB t ), where r = r * − r 0 is the index excess return.Since the Black-Scholes-Merton (BSM) model is complete, given δ ∈ L ∞ ((0, ∞), B, L), the European-style derivative security for ω ∈ Ω may be replicated by means of a self-financing stochastic strategy combining the future contract and the riskless security.It has been shown by Balbás et al. (2023) that δ is the usual delta-Greek (sensitivity, or first-order mathematical derivative) at T of the derivative J F T (δ) with respect to F T .If the insurer selects the marginal retained indemnity x and the financial Greek δ, then its random wealth at T will become where P is the global premium paid by insureds, the random variable Y r = J Y (x) is given by ( 45), Π(J Y (x)) is the reinsurance price, and the random pay-off J F T (δ) is given by ( 46).The insurer problem may be the risk minimization under a minimum expected value R of y and a maximum Greek ∆ ∈ L ∞ ((0, ∞), B, L), 11 where the risk is going to be measured by means of E µ for some µ ∈ (0, 1/2).Thus, bearing in mind that E µ is translation-invariant, the insurer problem becomes where (x, δ) ∈ L ∞ ((0, ∞), B, L) 2 is the decision variable and R 0 = R − P. Theorem 6 enables us to transform (48) into the equivalent problem 2 being the decision variable (see ( 14)).The premium principle Π is frequently convex (Pichler 2014), and therefore (49) is a convex problem.Furthermore, (49) is linear if the reinsurer premium principle Π is linear too, and in particular, under the expected value premium principle K ≥ 0 being the reinsurer loading rate.
Equations ( 48) and (49) present an illustrative example showing that Theorems 5 and 6 may be useful in practice in order to address the minimization of the expectile risk measure by linear programming methods.As already mentioned, this is just an illustrative section, and a complete solution to (49) is beyond the scope and would significantly increase the paper length.Moreover, Balbás et al. (2023) have presented an exhaustive methodology to solve (49), which does not need to be repeated here.Nevertheless, if (x, δ, λ, λ m , λ M ) solves (49) and y is given by (47), then (30) and (32) yield upper bounds for the insurer CVaR, applying to every confidence level.
Also consider the insurer's final wealth Then, (41) and (42) enable us to verify that (30) and (32) cannot be improved, in the sense that, for every µ * < 0.48, there exists µ ∈ (0, 1/2) such that they are satisfied as equalities.
Table 2 below shows a selected sample for µ * .In all cases, µ and E µ (y) have been rounded, 13 and the equality VaR 1−µ * (y) = E µ (y) is implied by Theorem 6 or Corollary 9. Since IE(y) = 0.07, the obtained upper bounds become for every µ * ∈ (0, 1), with equality in those cases presented in Table 2. To sum up, Sections 6.1 and 6.2 have illustrated that the minimization of actuarial/financial risks given by expectiles may be frequently linearized, that this minimization may permit us to control other downside risks beyond any parameter/confidence level, and that the inequalities connecting VaR and expectiles or CVaR and expectiles may often become equalities.Moreover, to the best of our knowledge, this is the first study combining reinsurance contracts, financial markets, and expectiles.

Discussion
As already indicated, expectiles are much less used in practice than VaR or CVaR, and this lower use is also reflected in the regulatory and supervisory systems.However, expectiles have very important analytical properties, and for this reason they have deserved the attention of many researchers.In particular, their coherence, elicitability, and relationships with CVaR have been extensively studied.This paper has presented a theoretical study based on the relationships between the dual representation and the bidual representation of expectiles.In this sense, the approach seems to be new, since the main instrument of analysis is the duality theory of linear programming.This methodology enables us to integrate under the same prism different problems affecting expectiles.Indeed, the methodology has allowed us to recover important inequalities relating CVaR and expectiles, but further issues have been addressed, including relationships between VaR and expectiles, potential improvements to the CVaR-linked inequalities, the potential co-monotonic additivity of expectiles, and the linearization of (actuarial, financial, or risk management) optimization problems involving risks in both the objective function and the constraints.This suggests that bidual representation and the duality theory of linear programming could also be a powerful tool for dealing with potential problems that may arise in the future.Additionally, since both the dual and the bidual representation of expectiles may lead to infinite-dimensional linear optimization problems, perhaps the most important practical limitation of this linear programming-linked approach is the lack of universal algorithms valid for every infinite-dimensional optimization problem.

Conclusions
The bidual representation of expectiles may be a powerful instrument to address important properties of these coherent and elicitable downside risk measures.In particular, this representation leads to new estimation and optimization methods by means of linear programming, new ways to analyze whether the co-monotonic additivity holds for expectiles, further relationships involving VaR, CVaR, and expectiles, and hyperbolic upper bounds of VaR and CVaR applying to every confidence level.Some theoretical findings have been illustrated in classical actuarial problems.

Table 1 .
Coefficients of the CVaR hyperbolic upper bound.

Table 2 .
Main Hyperbolic upper bounds become equalities.