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Robust Distortion Risk Measures

The robustness of risk measures to changes in underlying loss distributions (distributional uncertainty) is of crucial importance in making well-informed decisions. In this paper, we quantify, for the class of distortion risk measures with an absolutely continuous distortion function, its robustness to distributional uncertainty by deriving its largest (smallest) value when the underlying loss distribution has a known mean and variance and, furthermore, lies within a ball - specified through the Wasserstein distance - around a reference distribution. We employ the technique of isotonic projections to provide for these distortion risk measures a complete characterisation of sharp bounds on their value, and we obtain quasi-explicit bounds in the case of Value-at-Risk and Range-Value-at-Risk. We extend our results to account for uncertainty in the first two moments and provide applications to portfolio optimisation and to model risk assessment.


Introduction
Many decisions in financial risk management are based on evaluations of so-called risk or performance measures, e.g., Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) (EIOPA 2009, BCBS 2013, which are limit cases of Range Value-at-Risk (RVaR) (Cont et al. 2010). Risk managers rely on such measures, as they provide a classification of risk severities by associating to every loss distribution a real number (Artzner et al. 1999). Many risk measures are, however, highly sensitive to changes in the underlying loss distribution, and this vulnerability can contribute to adverse decisions; see Cont et al. (2010), Kou et al. (2013), Embrechts et al. (2015), and Pesenti et al. (2016).
Here, we quantify distributional uncertainty via bounds on the values of risk measures. Specifically, we study worst-case and best-case values of risk measures -that is, the largest (resp. smallest) value a risk measure can attain when the loss distribution belongs to a set of plausible alternative distributions.
The literature on risk bounds presents two main approaches. The first studies risk bounds for the aggregate portfolio loss S = X 1 + . . . + X n given that the components have known marginal distributions but unknown interdependence; see, among others, Denuit et al. (1999), Embrechts and Puccetti (2006), Wang and Wang (2011), and Embrechts et al. (2013). Although explicit results can be derived in certain cases of interest, the bounds obtained are typically too wide to be practically useful. Specifically, the upper bound on a distortion risk measure is always at least as large as its value under the assumption of a comonotonic dependence, a situation that is arguably very extreme. Papers that study bounds with additional dependence constraints or by imposing modelling structures include Bernard et al. (2017), Lux and Rüschendorf (2019), and Wang et al. (2019). Another stream of the literature is concerned with deriving risk bounds of the aggregate loss S under partial knowledge of its moments. Since only information on the moments of S is required, these risk bounds apply to non-linear loss models, such as re-insurance portfolios, and to loss models for which the explicit aggregation is not known analytically, e.g., when S is obtained via simulation models. As (higher) moments of the aggregate loss S depend both on the marginal distributions and their interdependence, one thus retains some aspects of the marginal distributions and dependence information. Relevant papers include Hürlimann (2002) for the case of VaR and Cornilly et al. (2018) and Zhu and Shao (2018) for the case of (concave) distortion risk measures.
Risk bounds that use additional refined information, such as uni-modality or symmetry, can be found in, e.g., Li et al. (2018). This paper is situated in the second stream, in that we consider risk bounds of an aggregate loss S under knowledge of its first two moments. However, we additionally impose a probability distance constraint on S, specified via the Wasserstein distance. Specifically, all alternative aggregate loss distributions, over which the worst-and best-cases are sought -the uncertainty set -lie within a tolerance distance from a reference loss distribution. Thus, the size of the tolerance distance determines the uncertainty around the reference distribution, which may also entail uncertainties on its components. Indeed, if the tolerance is small enough, the uncertainty set contains only the reference distribution, whereas if the tolerance distance increases to infinity, we recover bounds with moments constraints only. Therefore, the tolerance distance allows for a refined notion of model risk, which results in a continuous spectrum of risk bounds interpolating from the risk of the reference distribution to the bounds on the aggregate loss under the knowledge of the first two moments only.
Worst-(best-) case values under probability distance constraints have been considered by Glasserman and Xu (2014) and Lam (2016), who use the Kullback-Leibler divergence, as well as by Blanchet and Murthy (2019), who utilise distances stemming from mass transportation. These papers, however, consider the expected value of a function of the loss random variable, whereas here we study the class of distortion risk measures. Moreover, we use a distance that allows for comparison of distributions on differing support, which is important in our context as it is not clear a priori whether the worst-case distribution shares the same support as the reference distribution.
For example, a discrete reference distribution, together with a particular choice of distortion risk measure (e.g., inverse-S shaped), will result in a continuous worst-case distribution. In this work, we quantify the distance between the alternative distribution function and the reference distribution via the Wasserstein distance of order 2. The Wasserstein distance has been widely applied to model distributional uncertainty in financial contexts (see e.g., Pflug and Wozabal (2007), Bartl et al. (2020) and Chen and Xie (2021)), and as a distance from reference model (see e.g., Pesenti (2021) and Pesenti and Jaimungal (2021)).
A related stream of literature studies distributionally robust optimisation, leading to an optimum in the worst case. Relevant literature, in which the worst-case value of a distortion risk measure is considered includes Cai et al. (2020) and Pesenti et al. (2020). Cai et al. (2020) consider distortion risk measures under the assumption that the uncertainty set of the components X 1 , . . . , X n is characterised by the knowledge of their moments, the components having compact support and additional convex constraints. Pesenti et al. (2020) consider signed Choquet Integrals (see Section 2 for a definition) and uncertainty sets characterised by so-called closedness-under-concentration.
Neither of these works considers the Wasserstein distance, and the closedness-under-concentration condition is not compatible with uncertainty sets that are √ ε-Wasserstein balls around an (arbitrary) reference distribution.
In this paper, we derive for any given distortion risk measure (with absolutely continuous distortion function) its worst-and best-case values when the loss distribution has known first two moments and lies within a tolerance distance from a reference distribution, specified through the Wasserstein distance. To derive these bounds, we propose a novel approach that builds on the technique of isotonic projections (Németh (2003)). As a result, we obtain quasi analytical bestand worst-case bounds that significantly improve existing moment bounds. We find that for small Wasserstein tolerance distances, the worst-case distribution function (the distribution function attaining the worst-case value) is close (in a probabilistic sense) to the reference distribution, whereas for large tolerance distances the worst-case distribution function is no longer affected by the reference distribution. Thus, when the Wasserstein tolerance distance goes to infinity, the reference distribution becomes irrelevant, and we recover, as corollaries to our results, known moment bounds on distortion risk measures; see Li (2018) and Zhu and Shao (2018). Indeed, Li (2018) considers worst-case concave distortion risk measures with given mean and standard deviation, and Zhu and Shao (2018) consider the best-case and worst-case of the entire class of distortion risk measure under the knowledge of the first two moments. Neither considers a Wasserstein constraint. We further apply our results to obtain quasi-explicit bounds on the RVaR and on the VaR as limiting cases. We find that for small Wasserstein tolerance distances, the worst-case distribution functions for RVaR and VaR are no longer two-point distributions, thus making the bounds attractive for financial risk management applications.
This paper is structured as follows: Section 2 introduces the necessary notation and formulates the problem. The main results -the quasi-analytical best-and worst-case values of distortion risk measures under a Wasserstein distance constraint -are presented in Section 3. Section 4 contains two extensions. In Section 4.1, we calculate bounds of VaR. In Section 4.2, we extend our results to complete moment uncertainty -that is, when the uncertainty set is a √ ε-Wasserstein ball around a reference distribution without any moment constraints. We provide applications of the risk bounds to portfolio optimisation and to model risk assessment of an insurance portfolio, including rationales for choosing the Wasserstein distance, in Section 5. All proofs are relegated to the appendix.

Problem Formulation
We consider an atomless probability space (Ω, A, P) and let L 2 = L 2 (Ω, A, P) be the set of all square integrable random variables on that space. We denote by M 2 = { G(x) = P(X ≤ x) | X ∈ L 2 } the corresponding space of distribution functions with finite second moment. The (left-continuous) inverse, also called the quantile function, of a distribution function G ∈ M 2 is G −1 (u) = inf{ y ∈ R | G(y) ≥ u }, and we denote its right-continuous inverse by G −1,+ (u) = inf{ y ∈ R | G(y) > u }, 0 < u < 1. Throughout the exposition, we write U ∼ U(0, 1) for a standard uniform random variable on

Distortion Risk Measures
In this section, we recall the class of distortion risk measures that contains risk measures commonly encountered in financial applications, such as VaR and TVaR. A distortion risk measure, evaluated at a distribution function G ∈ M 2 , is defined via the Choquet integral whenever at least one of the two integrals is finite. The function g : [0, 1] → [0, 1] refers to a distortion function -that is, a non-decreasing function satisfying g(0) = 0 and g(1) = 1. If g is absolutely continuous, then the distortion risk measure H g has the following representation (Dhaene et al. 2012) with weight function γ(u) = ∂ − g(x)| x=1−u , 0 < u < 1, which satisfies 1 0 γ(u)du = 1 and where ∂ − denotes the derivative from the left. In what follows, we may sometimes write H g (G −1 ) instead of H g (G).

Modelling Distributional Uncertainty
In financial risk management, distributional uncertainty is prevalent and bounds on the value of a  Villani (2008) for a discussion of the Wasserstein distance.
Definition 1 (Wasserstein distance of order 2). The Wasserstein distance of order 2 between G 1 , G 2 ∈ M 2 is (Villani 2008) where the infimum is taken over all bivariate distributions with marginals G 1 and G 2 .
For distributions on the real line, the Wasserstein distance admits the following well-known repre- Hence, the Wasserstein distance between G 1 and G 2 is uniquely determined by their corresponding quantile functions, and we may write We denote by F ∈ M 2 the reference distribution and its first two moments by x dF (x) = µ F ∈ R and x 2 dF (x) = µ 2 F + σ 2 F , σ F > 0, respectively. Throughout, we fix the reference distribution F and consider distributional uncertainty sets of the type where µ ∈ R, σ > 0, and 0 ≤ ε ≤ +∞. The set M ε (µ, σ) thus contains all distribution functions whose first two moments are µ and µ 2 + σ 2 , respectively, and that lie within a √ ε -Wasserstein ball around the reference distribution F . The ε-constraint in the definition of M ε (µ, σ) is redundant when ε = +∞, and in this special case we denote the uncertainty set M ∞ (µ, σ) by M(µ, σ). Note that in applications, one may often encounter that (µ, σ) = (µ F , σ F ), but our set-up allows them to be different, which could for example be useful when their values are estimated using different data sources.
The best-and worst-case values of a distortion risk measure H g over the distributional uncer- In addition to the best-and worst-case values, we also study best-case and worst-case distribution functions if they exist -that is, the distribution functions attaining (5a) and (5b), respectively arg min We impose two additional conditions that are necessary in order to ensure that the considered optimisation problems are relevant and well-posed.
Assumption 3. We assume that M ε (µ, σ) contains at least two elements, and hence infinitely many; otherwise problems (5) and (6) are either ill-posed or trivial.
In this regard, Assumption 3 is equivalent to assuming that ε > (µ F − µ) 2 + (σ F − σ) 2 . To see this, note that for any G ∈ M ε (µ, σ) is a singleton, containing only one distribution with quantile function Moreover, in this special case, G coincides with the reference distribution F if and only if µ = µ F and σ = σ F . Thus, from now on, we assume that ε > (µ F − µ) 2 + (σ F − σ) 2 .

Bounds on Distortion Risk Measures
For ease of exposition, we first study worst-case values (5b) and worst-case distribution functions (6b) for the class of concave distortion risk measures that are distortion risk measures with concave distortion function g. The corresponding results for the case of general distortion risk measures are presented in Section 3.2 and those for best-case values (5a) and best-case distributions (6a) in Section 3.3.

Upper Bounds on Concave Distortion Risk Measures
Concave distortion risk measures are a subset of the class of distortion risk measures that are coherent and include the widely used TVaR. Furthermore, concave distortion risk measures span the class of law-invariant, coherent, and comonotone additive risk measures (Kusuoka 2001). For the special case of concave distortion risk measures, we obtain analytic solutions to problems (5b) and (6b), which are presented in the following theorem.
Theorem 1 (Worst-Case Values and Quantiles of Concave Distortions). Let Assumptions 1, 2, and 3 be fulfilled. Further, let g be a concave distortion function and denote by c 0 = corr(F −1 (U ), γ(U )). 1 Then, the following statements hold: is unique and has a quantile function given by where λ > 0 denotes the unique positive solution to d W (F −1 , h λ ) = √ ε, which is explicitly given by where V = var(γ(U )), C γ,F = cov (F −1 (U ), γ(U )), and K = ≥ 0 and where a λ = E(γ(U ) + λF −1 (U )) and b λ = std(γ(U ) + λF −1 (U )). The corresponding worst-case value, i.e., the solution to (5b), is The worst-case value provided in Theorem 1 is sharp for all (µ F − µ) 2 + (σ F − σ) 2 < ε ≤ +∞ and attained by the worst-case quantile function h λ . We observe that h λ is a weighted average of the reference quantile function F −1 and the (non-decreasing) weight function γ from the distortion risk measure. As the worst-case value H g is non-decreasing in the tolerance distance ε, we obtain that corr (γ(U ), γ(U ) + λF −1 (U )) is non-decreasing in ε, which in turn implies that λ is non-increasing in ε. Hence, we obtain that the influence of the reference distribution on the worst-case quantile function h λ diminishes with increasing tolerance distance ε. Furthermore, for ε sufficiently large, i.e., under case 2. of Theorem 1, λ is zero and the worst-case quantile function is independent of the reference distribution. Note that when ε = (µ F − µ) 2 + (σ F − σ) 2 + 2σσ F (1 − c 0 ), the expression of λ in (7) is still valid and simplifies to λ = 0.
As a direct corollary we obtain the worst-case values of TVaR. The proof is omitted as it follows by direct calculations from Theorem 1.
Corollary 1 (Worst-case Value of TVaR). Let Assumption 3 be fulfilled. Then, the worstcase value of the TVaR α is given by then λ is given by equation (7) in Theorem 1, Theorem 1 generalises results that first appeared in Li (2018). This author derived sharp upper bounds on H g under the knowledge of the first two moments, i.e., without considering an √ ε-Wasserstein distance constraint or a reference distribution. Applying Theorem 1 with ε = +∞, we obtain his result, which is stated here for completeness; see also Cornilly et al. (2018) and Zhu and Shao (2018).
The bound is sharp and obtained by a unique maximising distribution function with quantile function h 0 defined in Theorem 1.
Example 1. We illustrate the worst-case values of Theorem 1 for three concave distortion risk measures: the dual power distortion, the Wang transform, and TVaR defined in (2), (3), and (4), respectively. The reference distribution F is chosen to be standard normal, and we further set µ = µ F = 0 and σ = σ F = 1. In the top panels of Figure 1, we observe that the upper bounds are indeed non-decreasing and continuous functions of ε. The turquoise coloured vertical lines  The bottom panels of Figure 1 display the worst-case quantile functions for selected values of ε. We observe that for ε = 0, the worst-case quantile functions are equal to the reference quantile functions (blue solid line). When the Wasserstein distance ε increases, the influence of the reference distribution diminishes and, if ε > 2 (1 − c 0 ), the worst-case quantile function (violet dash-dotted line) is independent of the reference distribution. This can be seen clearly for TVaR, where the violet dash-dotted quantile function corresponds to a two-point distribution.

Upper Bounds on General Distortion Risk Measures
To characterise the worst-case values and quantile functions of general distortion risk measures we introduce the concept of isotonic projections; see e.g., Németh (2003). For this, we denote the space of square-integrable, non-decreasing, and left-continuous functions on (0, 1) by Definition 2. For λ ≥ 0, denote by k ↑ λ the isotonic projection of γ + λF −1 onto the space of square-integrable non-decreasing and left-continuous functions on (0, 1) -that is, the unique solution to where || · || denotes the L 2 norm. Whenever The isotonic projection differs from the isotonic regression (Barlow et al. 1972), which is a metric projection onto the set of finite dimensional and component-wise increasing vectors. We refer to Appendix A for further details and properties of isotonic projections.
We further define Before stating the worst-case value of general distortion risk measures, we need a further assumption.
Note that Assumption 4 is a requirement on the weight function and the reference distribution.
Assumption 4 is satisfied, for instance, when γ is bounded and the reference distribution has a quantile function that is unbounded to the right, i.e. F −1 (u) → +∞ for u → +∞.
Theorem 2 (Worst-Case Values and Quantiles of General Distortions). Let Assumptions 1, 2, 3, and 4 be fulfilled. Further, let g be a distortion function and denote c 0 = . Then, the following statements hold: , then a solution to (6b) is unique and is given by the quantile function h ↑ λ defined in (10), where λ > 0 is the unique positive Furthermore, the worst-case value, i.e., the solution to (5b), is If γ is non-increasing, equivalently the distortion function g is convex, then its isotonic projection is constant and equal to 1. Thus, for ε that is sufficiently large, case 2. of Theorem 2 implies that the upper bound is equal to µ and not attained. If γ is non-decreasing, equivalently g is concave, then γ + λF −1 is non-decreasing and thus equal to its isotonic projection. Hence, Theorem 2 reduces to Theorem 1 in the case of concave distortion risk measures. Theorem 2 generalises results by Zhu and Shao (2018), who analyse problem (5b) in the special case in which the Wasserstein constraint is redundant, which we state in the subsequent corollary.
Corollary 3 (Worst-Case Values for ε = +∞; Zhu and Shao (2018)). Let Assumptions 1, 2, and 4 be fulfilled. Further, let g be a distortion function; then, the following statements hold: The bound is sharp and attained by a distribution with quantile function h ↑ 0 defined in (10).

If γ ↑ is constant, then
sup G∈M(µ,σ) H g (G) = µ and the bound cannot be attained.

Lower Bounds on General Distortion Risk Measures
In a similar way as in Section 3.2, we first introduce the notation and assumptions needed to derive the best-case values and quantile functions of general distortion risk measures.
Definition 3. For λ ≥ 0, denote by k ↓ λ the isotonic projection of γ − λF −1 onto the set of square-integrable non-increasing functions and left-continuous -that is, the unique solution to Whenever λ = 0, we write γ ↓ instead of k ↓ 0 , as k ↓ 0 is indeed the isotonic projection of γ onto the non-increasing functions.
The next theorem states the best-case values and quantile functions of general distortion risk measures.
Theorem 3 (Best-Case Values and Quantiles of General Distortions). Let Assumptions 1, 2, 3, and 5 be fulfilled. Further, let g be a distortion function and denote c 0 = lim λ 0 corr(F −1 (U ), −k ↓ λ (U )). Then, the following statements hold: , then a solution to (6a) is unique and is given by the quantile function h ↓ λ defined in (13), where λ > 0 is the unique positive The corresponding best-case value, i.e., the solution to (5a), is If γ ↑ is constant, then the infimum is equal to µ but cannot be attained.
Theorem 3 implies that for ε that is sufficiently large, i.e., in case 2., the lower bound of any concave distortion risk measure is equal to µ and not attained. The next corollary collects the results for ε = +∞.
Corollary 4 (Best-Case Values for ε = +∞). Let Assumptions 1, 2, and 5 be fulfilled. Further, let g be a distortion function; then, the following statements hold: The bound is sharp and is attained by a distribution with quantile function h ↓ 0 defined in (13).
and the bound cannot be attained.

Remark 1. Distortion risk measures have an alternative representation that makes it possible
to write minimisation problems in terms of maximisation problems. Specifically, whereḡ is the dual distortion of g, given byḡ(x) = 1−g(1−x), and G X , G −X denote the distribution functions of the random variables X and −X, respectively. Equation (14) follows from the fact that H g (G X ) = −Hḡ(G −X ); see e.g., Dhaene et al. (2012). This observation makes it possible to obtain Corollary 4, the statements on lower bounds, from Corollary 3, the corresponding statements on upper bounds, in a more direct manner. Such reasoning, however, cannot be extended when dealing with the Wasserstein distance constraints considered in this paper.
Remark 2. Theorems 2 and 3 can be generalised to signed Choquet integrals. A signed Choquet integrals generalise the class of distortion risk measures in that the distortion function g can be decreasing. For absolutely continuous g, the signed Choquet integral admits a representation as in (1), see Wang et al. (2020), thus, Theorems 2 and 3 can be extended to signed Choquet integrals.
Example 2 (Continued). As all three risk measures are concave, their lower bounds -for ε sufficiently large (case 2. in Theorem 3) -are all equal to µ = 0 but not attained. That the lower bounds are not attained can be seen in the lower panels, where the corresponding best-case quantile functions are displayed.
The best-case quantile functions become flatter for larger ε, and for ε sufficiently large the quantile functions are no longer defined.

Extensions
In this section we consider two extensions. First, we derive explicit bounds on RVaR -a risk measure that includes VaR and TVaR as limiting cases -and derive the bounds on VaR as limiting cases of those of RVaR. Second, we extend the results to the uncertainty sets solely described by the Wasserstein distance.

Bounds on Range Value-at-Risk, Value-at-Risk, and Tail Value-at-Risk
In this section we provide bounds on the risk measures RVaR, TVaR, and VaR. Specifically, we first calculate the best-and worst-case values of RVaR and then derive bounds on VaR and TVaR as a limiting case. The RVaR α,β at levels 0 < α < β ≤ 1 is defined by (Cont et al. 2010) By letting β 1, we obtain TVaR; that is, for any G ∈ M 2 it holds that Next, recall that VaR α (G) = G −1 (α) and denote VaR + α (G) = G −1,+ (α). Both VaR and VaR + are dis- (2012)); however, they do not belong to the class of distortion risk measures considered in this paper as they do not satisfy Assumption 1. Furthermore, it holds that for any G ∈ M 2 The worst-case values of these risk measures under constraints on the first two moments but without a Wasserstein constraint have been extensively studied; indicatively see Kaas and Goovaerts (1986), Lo (1987), Grundy (1991), Hürlimann (2005), and De Schepper and Heijnen (2010)  Corollary 5 (Worst-Case Values for ε = +∞). For 0 < α < β ≤ 1, the worst-case values of VaR α , VaR + α , RVaR α,β , and TVaR α coincide; that is, While the worst-case value of VaR α cannot be attained, the worst-case distribution of VaR + α , RVaR α,β , and TVaR α is a two-point distribution with quantile function Corollary 6 (Best-Case Values for ε = +∞). For 0 < α < β ≤ 1, the best-case values of While the best-case values of VaR + α and T V aR α,β cannot be attained, the best-case distribution corresponding to the best-case VaR α has a quantile function given by (15) whereas for the best-case RV aR α,β it is a two-point distribution with quantile function Next, we study the lower and upper bound on RVaR when, in addition to the moment constraints, the distributions in the uncertainty set lie within an √ ε-Wasserstein ball of the reference distribution F .
Proposition 1 (Worst-Case Quantiles of RVaR). Under the assumptions of Theorem 2 case 1., a solution to (6b) with RVaR α,β is unique and is given by and w 0 , w 1 , and c, with α ≤ w 0 ≤ β ≤ w 1 , c < +∞, satisfy Proposition 2 (Best-Case Quantiles of RVaR). Under the assumptions of Theorem 3 case 1., a solution to (6a) with RVaR α,β is unique and is given by and z 0 , z 1 , and c, with We verified with numerical experiments that the closed-form expressions in (16) and (17) match well with numerically obtained isotonic and antitonic projections. 2 The subsequent corollaries provide the best-and worst-case quantile functions of VaR and TVaR, as limiting cases of RVaR. The results are also summarised in Table 1.
Corollary 7 (Best-and Worst-Case Quantiles of TVaR). Under the assumptions of case 1. in Theorems 2 and 3, respectively, it holds that the worst-case quantile function of TVaR α is given by Proposition 1 with β = 1 and w 0 = w 1 = 1. The corresponding worst-case value of TVaR α is given in Corollary 1. The best-case quantile function of TVaR α is given by Proposition 2 with β = 1.
. It holds that the worst-case quantile function of VaR + α is given by Proposition 1 with w 0 = α. The worst-case value of VaR α is equal to that of VaR + α but not attained. The best-case quantile function of VaR α is given by Proposition 2 with z 1 = β. The best-case value of VaR + α is equal to that of VaR α but is not attained.  Figure 4 displays the corresponding graphs for the best-case quantile functions.
The left plot of Figure 5 displays the lower and upper bounds on VaR α , RVaR α,β , and TVaR α as a function of the Wasserstein distance ε. When ε is sufficiently large (case 2. in Theorems 2 and 3), the Wasserstein distance no longer affects the bounds; these bounds coincide with the well-known Cantelli bounds. The normalised lengths of the bounds on VaR α , RVaR α,β , and TVaR α as a function of ε are displayed in the right plot of Figure 5. The normalised length of the bounds are the differences between (5b) and (5a) divided by the risk measure evaluated at the reference distribution and have been introduced to assess model risk by Barrieu and Scandolo (2015). The length of the bounds on For the risk measure RVaR α,β , we plot γ(u) + λF −1 (u), u ∈ (0, 1) and its isotonic projection k ↑ λ onto the set of non-decreasing functions (derived in Proposition 1). The reference distribution F is standard normal with µ = µF = 0 and σ = σF = 1, and we set ε = 0.2.

Wasserstein Uncertainty Set
In this section we consider worst-case concave distortion risk measures under complete moment uncertainty. Specifically, we consider an uncertainty set that contains all distribution functions that lie within an √ ε-Wasserstein ball around the reference distribution, i.e., without a first and second moment constraint. Thus, the optimisation problem becomes arg max where M ε = {G ∈ M 2 | d W (F, G) ≤ √ ε} is the set of all distribution functions that lie within a √ ε-Wasserstein ball around the reference distribution F .
Theorem 4 (Wasserstein Uncertainty). Let Assumptions 1 and 2 be fulfilled. Further, let g be a concave distortion function; then, the solution to problem (18) exists, is unique, and has quantile function given by where a λ , b λ , V , and C γ,F are given in Theorem 1 and λ > 0 denotes the unique positive solution to which is explicitly given in Theorem 1. The corresponding worst-case value is Note that if ε > 0, then the mean and standard deviation of the worst-case distribution fulfil µ * > µ F and σ * > σ F (recall that C γ,F ≥ 0). Thus, the worst-case distribution when the uncertainty set is only characterised by a Wasserstein ball has a larger mean and standard deviation than the benchmark. Thus, the uncertainty set M ε (µ, σ) with fixed µ and σ results in a different worst-case distribution compared to the uncertainty set characterised solely by the Wasserstein distance.

Applications
The tolerance distance ε, i.e., the degree of the uncertainty, should be adequately chosen. Pesenti and Jaimungal (2021) study optimal portfolio choice and assume that the investor is prepared to accept terminal wealth distributions that stay within an ε-Wasserstein ball around some benchmark distribution. In this case, the Wasserstein tolerance distance ε reflects the investor's tolerance of deviating from the benchmark portfolio strategy and its value is thus driven by the investor's risk preferences. In most applications, however, the uncertainty set is constructed to contain all "plausible" distributions, i.e., the uncertainty set should be large enough to contain with high probability the true data-generating distribution and at the same time small enough to exclude pathological distributions, which would incentivise overly conservative decisions (Esfahani and Kuhn (2018)). In these settings, the tolerance distance should be driven by data considerations rather than being exogenously specified. In this regard, in various (data) contexts and under various assumptions Wozabal (2014) chooses ε via cross-validation whereas Blanchet et al. (2021) choose ε such that the true distribution function lies with a given probability within a Wasserstein ball around the empirical reference distribution. Additionally, including expert opinion in constructing the uncertainty set may provide valuable information, particularly in the case of limited available data (Clemen and Winkler (1999)). In the applications hereafter we do not pursue the estimation of ε in great length, but rather provide some guidelines for choosing it.

Portfolio Optimisation
We consider a portfolio optimisation problem in which an investor aims to construct a robust portfolio x = (x 1 , . . . , x n ) ∈ P among returns R = (R 1 , . . . , R n ), n ∈ N, in which P is a suitable polyhedral set. We denote the multivariate distribution function of R by G R and assume that it is subject to uncertainty in that only its mean vector µ and covariance matrix Σ are known. For each portfolio x, we denote by G x the (unknown) distribution function of the aggregate portfolio loss −x R having mean −µ x = −x µ and variance σ 2 There is a benchmark model under which the investor assumes 3 that the aggregate portfolio loss has distribution function F x belonging to a location-scale family, i.e., there is a distribution function F such that Then, for fixed x ∈ P, the Wasserstein distance between F x and G x is given by The investor now aims to find the portfolio x that has the smallest risk -measured via a concave distortion risk measure -among all possible portfolios whose aggregate loss lie within a Wasserstein distance of the benchmark portfolio. Specifically, we consider the optimisation problem where M(µ, Σ) is the set of all n-dimensional distribution functions with mean vector µ and covariance matrix Σ, and where we allow the tolerance distance to depend on x, i.e., ε x ≥ 0, x ∈ P.
Note that problem formulation (20) is general in that the tolerances ε x may depend on the portfolio x ∈ P at hand. As a consequence, it is not always possible to obtain the minimum in (21) in explicit form, and numerical procedures may be required.
From (19), however, we observe that the Wasserstein distance scales linearly with the standard deviation of the portfolio loss. Hence, it appears natural to take the tolerance ε x of the form 4 ε x = 2σ 2 x A for 0 ≤ A ≤ 1. The next result considers this case. Its proof follows from Proposition 3 via direct calculation and is omitted.
Then, the following holds: In all cases of Corollary 9, the robust portfolio optimisation problem reduces to solving a second order cone program. Specifically, for A = 0, there is no ambiguity and the investor aims to obtain 4 If historical asset returns are available the value of A can be determined as follows: using the historical returns we estimate for a given portfolio x = (x1, . . . , xn) ∈ P, the empirical distributionĜx of the portfolio loss, its standard deviationσx, and the Wasserstein distancedx betweenĜx and the (discretised version of the) reference distribution Fx. By doing so for a set of portfolios x (i) ∈ P, i ∈ {1, . . . , N }, one can estimate A byÂ = max{d x 2σ 2 x | x (i) ∈ P, i = 1, . . . , N }. This procedure in particular guarantees that for each portfolio x (i) , i ∈ {1, . . . , N } its corresponding empirical distributionĜ x (i) lies in the uncertainty set.
the optimal portfolio under the assumption that for each portfolio x the portfolio (loss) has known quantile function F −1 x (u) = −µ x + σ x F −1 Z (u). The optimal choice x * is the portfolio 5 for which where we recall that √ V c 0 = H g (F ). When 1 − c 0 ≤ A ≤ 1, the Wasserstein distance becomes irrelevant and the optimal portfolio problem reduces to the one considered in Li (2018), i.e., to In this regard, √ V c 0 = H g (F ) ≤ √ V implies that in the case of no ambiguity (A = 0), more emphasis is placed on the expected return component than in the case in which the Wasserstein distance constraint is irrelevant (1 − c 0 ≤ A ≤ 1 .) Moreover, in this instance, the optimal terminal wealth has a quantile function that is linear in the (non-negative) weight function γ (see Corollary 2), and thus is bounded from below, a feature that might be considered as not very realistic. The cases 0 < A < (1 − c 0 ) deal with situations that are between these extreme scenarios. Observe that c 0 (1 − A) + A(2 − A) 1 − c 2 0 is increasing in A on the interval [0, 1 − c 0 ] taking value c 0 when A = 0, and value 1 when A = 1 − c 0 . As A and thus the degree of ambiguity increases, less emphasis is placed on the expected return leading to optimal portfolios that become more conservative.
Note that in all intermediate cases the optimal quantile function inherits structure of the reference quantile function F −1 x * (see Theorem 1).

Insurance Portfolio of Risks
This section illustrates the best-and worst-case values of VaR on a simulated portfolio of dependent risks, that could e.g., arise from insurance activities. We consider the Pareto-Clayton model that offers a flexible way of modelling portfolios with dependent risks, see Oakes (1989)  for applications in finance. In the Pareto-Clayton model, the portfolio components X 1 , . . . , X d are, given Θ = θ > 0, independent and Exponentially distributed with parameter θ, that is drawn from a Gamma distribution, i.e., Θ ∼ Gamma(a, b). The aggregate portfolio risk S = d i=1 X i thus follows, conditionally on Θ = θ, a Gamma(d, θ) distribution. By Dubey (1970) Equations 1.1 and 1.3, the scaled aggregate portfolio risk S/b is a Beta distribution of the second kind. Thus, the quantile function of the aggregate portfolio risk S and the first five moments are explicitly given by where B(·, ·) denotes the Beta function and G −1 B the quantile function of the Beta distribution with parameters d and a.
We consider the situation where modellers or experts may not fully agree on the reference Pareto-Clayton model, as available data or estimation may induce uncertainties or the distribution of Θ might not seem appropriate. As alternative models for the distribution of the aggregate portfolio risk S, we consider two-parameter distributions with support in (0, +∞) that are commonly utilised for modelling insurance portfolios. Specifically, we consider a Lognormal model ( Table 2 and additional information is provided in Appendix C. For the numerical simulations of the Pareto-Clayton model we use a = 10, b = 1, and d = 100, as in Bernard et al. (2018). The parameters of the alternative models, reported in Table 2, are obtained by matching the first two moments to those of the reference Pareto-Clayton model, i.e., E(S) = 11.11 and E(S 2 ) = 140.28.
We illustrate the reference Pareto-Clayton model (solid black) and the alternative models in Figure   6 via their densities. It is apparent that the considered alternative models can be grouped into light-tailed models (W, IG, IΓ, IW) and heavy-tailed models (LN, Γ, LL).
Next, we discuss the choice of Wasserstein distance by means of model uncertainty. Specifically, we let ε be the maximum of all Wasserstein distances between the reference and a selection of alternative models. The constructed Wasserstein ball then includes all distribution functions whose Log-Logistic LL α LL = 10 β LL = 5.3 0.345 Table 2 Alternative models for the aggregate portfolio risk. The last column reports the Wasserstein distance between the Pareto-Clayton and the alternative model. Wasserstein distance is smaller or equal to the maximum of all Wasserstein distances between the reference and the considered alternative models and, in particular, all considered alternative models. We report the Wasserstein distances between the Pareto-Clayton model and all alternative models in   (Bernard et al. 2018). Note that the forth row in Table 3 (k = 2) corresponds to M(µ, σ), i.e., with Wasserstein tolerance ε = +∞. It is apparent that the bounds with a tolerance distance that contain the heavy-tailed alternative models (ε = 0.637) are significantly smaller than the ones corresponding to both heavy-tailed and light-tailed alternative models (ε = 3.868). Furthermore, we report in the last row of Table 3 the bounds on VaR for an uncertainty set with fixed mean, bounded standard deviation, and where the uncertainty set further contains only unimodal distributions (Bernard et al. 2020). We observe in Table 3, that the Wasserstein constraint significantly reduces the bounds and that information on higher moments only affects the bounds in a minor way. Moreover, when only heavy-tailed models are considered as alternatives (ε = 0.637), the length of the bounds is 18.8−12.8 21.6−10.7 = 55% of the length when both light-and heavy-tailed are valid alternative models (ε = 0.3868).

Conclusion
We derive quasi explicit best-and worst-case values of a large class of distortion risk measures when the underlying loss distribution has a given mean and variance and lies within a √ ε -Wasserstein ball around a given reference distribution. We find that for small Wasserstein tolerance distances the worst-case distribution function (the distribution function attaining the worst-case value) is probabilistically close to the reference distribution, whereas for large tolerance distances, the worstcase distribution function is no longer affected by the reference distribution. We also derive the worst-case distribution function for RVaR and show that for small Wasserstein tolerance distances, the worst-case distribution functions of VaR and TVaR are no longer two-point distributions, thus making the bounds attractive for risk management applications.
Our results generalise findings in Li (2018) and Zhu and Shao (2018), which correspond to a Wasserstein tolerance of ε = +∞. Furthermore, we illustrate the risk bounds on two applications, to portfolio optimisation and to model risk assessment of an insurance portfolio.
Appendix A: Isotonic Projection.
Here, we collect properties of the metric projection defined in (9). For this, we denote by N = { | : (0, 1) → R , 1 0 (u) 2 du < +∞} the space of square-integrable functions on (0, 1) and consider the following metric projection from N to K (set defined by (8)): A metric projection is called isotone if the projection preserves the order induced by K (Németh 2003). The partial order K on N induced by K is given, for any 1 , 2 ∈ N , by Proposition 4 (Theorem 3. Németh (2003)). The metric projection P K is isotone and subadditive; that is, P K fulfils for all 1 , 2 ∈ N : In this paper, we compare only projections of functions of the type γ + λF −1 , for fixed F and γ. Thus, we view the projection of γ + λF −1 as a function of λ and define k λ = P K (γ + λF −1 ), λ ≥ 0.
Proposition 5. For fixed γ and F , the metric projection k λ is an isotonic projection with the following properties: 1. Isotone: For λ 1 ≤ λ 2 , it holds that k λ 1 K k λ 2 .
Note that |e n |F −1 − e n F −1 ∈ K; thus, using first subadditivity and then isotonicity of P K in the second inequality, we obtain The next proposition collects a few properties of the isotonic projection P K and thus of k λ . For this we denote by ·, · the inner product on N -that is, 1 , 2 = 1 0 1 (u) 2 (u) du, for 1 , 2 ∈ N .
Proof of Proposition 6. We follow the arguments of the proof of Theorem 1.3.2 by Barlow et al. (1972).
Property 1. For ∈ N , f ∈ K, and w ∈ [0, 1] define the function (1 − w)P K ( ) + wf . By definition of the isotonic projection it holds that (1 − w)P K ( ) + wf ∈ K. Next, we consider the function which obtains its minimum in w = 0, by definition of the isotonic projection of P K ( ). The first order condition at w = 0, implies that Property 2. By the definition of an isotonic projection, it holds that P K ( ) − , P K ( ) − ≤ k − , k − for all , k ∈ N . Choosing k = c P K ( ) with c > 0, we obtain 2 (c − 1) P K ( ) , ≤ (c 2 − 1) P K ( ) , P K ( ) .
Thus, for 0 < c 1 < 1 and c 2 > 1 the above inequality becomes Taking limits with c 1 1 and c 2 1 concludes the statement of the property.
Property 3. For ∈ N and k ∈ K, we obtain, by first applying property 2. and then property 1.

B.1. Proof of Theorem 1 and Corollary 2.
For the proof of Theorem 1 we need the following lemmas.
. We split this part into three steps. First, we show that the optimal quantile function has to be of the form h λ for some λ ≥ 0. Second, we show that the solution lies at the boundary of the Wasserstein ball. Third, we show uniqueness.
For the second step, let λ ≥ 0 be such that h λ satisfies d W (F −1 , h λ ) = √ ε. By construction, it holds that Furthermore, by Lemma 1 we have that γ + λF −1 , h λ ≥ γ + λF −1 , h λ 1 , which, together with (31), implies Therefore, we obtain that γ, h λ ≥ γ, h λ 1 , and hence that the solution lies at the boundary of the Wasserstein ball. Uniqueness follows from the uniqueness of h λ as the solution of Lemma 1. The worst-case value The explicit expression for λ then follows from Lemma 2 and is given in (23).
Proof of Corollary 2. This result follows directly from Theorem 1, case 2.

B.2. Proof of Theorem 2 and Corollary 3.
The following lemma is needed to prove Theorem 2.
. Similar to the proof of Theorem 1, we split the proof into three steps: first, the worst-case quantile function is of the form h λ , second, the solution lies at the boundary of the Wasserstein ball, and third we show that it is unique. The first two steps follow along the lines of the proof of Theorem 1 case 1., replacing h λ with h ↑ λ and using in the second step the properties of the isotonic projection h ↑ λ . Uniqueness follows from the uniqueness of h ↑ λ as the solution to (3) in Lemma 3. If γ ↑ is constant, then we find that for all h ∈ M −1 (µ, σ), γ, h ≤ γ ↑ , h = µ , where the first inequality follows from the properties of the isotonic projection (Proposition 6, property 3.) and the equality follows from the fact that γ ↑ = 1 as γ, 1 = γ ↑ , 1 (applying Proposition 6, property 3. with k ≡ 1 and k ≡ −1). Finally, we show that µ is indeed the supremum. Consider the case in which µ ≥ 0 and define t α = α µ γ ↑ + (1 − α)z α , for 0 < α < 1, and where z α is a non-negative non-decreasing function such that E(z α (U )) = µ and std(z α (U )) = σ/(1 − α). Hence, t α ∈ M −1 (µ, σ), for 0 < α < 1. Moreover, for every ε > 0 that are sufficiently small, there exists 0 < α < 1 such that For µ < 0 the proof is similar and is omitted.

B.3. Proof of Theorem 3.
The following lemma is needed for the proof of Theorem 3.
Proof of Proposition 1. By Theorem 2 case 1., we have to calculate the isotonic projection of 1 β−α 1 (α,β] + λF −1 onto the space of non-decreasing functions. According to Lemma 5.1 in Brighi and Chipot (1994), the isotonic projection has the form where J is a countable index set, and I j ⊂ [0, 1], j ∈ J , are disjoint intervals, and for any j ∈ J , I j = [0, 1]\I j denotes the complement of the set I j .
We first show that J contains only one element. For this, let k * be an optimal solution with representation given by (34). By definition k * minimises 1 β−α 1 (α,β] + λF −1 − k 2 2 over all non-decreasing functions k. Moreover, Case 1 : If I j ∩ (α, β] = ∅, then the j th summand in (35) can be improved by choosing k * (u) = min{ λF −1 (u) , k * (u + j ) }, for all u ∈ I j , and where u + j denotes the right endpoint of I j .
Proof of Corollary 8. We first consider the worst-case of VaR + α and second that of VaR α .
To show the worst-case value of VaR + α we split the proof into three steps.
Step 1: We show that for any η, δ > 0 and m ∈ R the unique quantile function attaining is given by where w 1 ∈ (0, 1] and c ∈ R are the unique solutions to First, we show that the solution to (37), denoted here by G * , lies at the boundary of the Wasserstein ball, i.e., d W (G * ,−1 , F −1 ) = √ δ. Assume by contradiction that G * lies in the interior of the Wasserstein ball; then which is a contradiction to the optimality of the solution. Next, we show that the optimal solution to (37) has to be of the form (38). For this, assume that G * is a solution with V aR + α (G * ) > V aR + α (H * η,δ,m ) = c. By non-decreasingness of G * it holds that G * ,−1 (u) > c ≥ ηF −1 (u) − m, for all u ∈ [α, w 1 ]. Thus, the Wasserstein distance has a lower bound given by which is a contradiction to the fact that the optimal solution lies at the boundary of the Wasserstein ball.
Uniqueness of the solution follows from the uniqueness of δ and w 1 .
The proof of the best-case values of VaR α and VaR + α follows similar steps to the worst-case values. Note that the best-case VaR α is attained while the best-case VaR + α is not.
Proof of Lemma 5. First, we define the set and show that N ε = M ε . Clearly, if G ∈ N ε , then G ∈ M ε . Conversely, let G ∈ M ε with mean µ G and standard deviation σ G > 0; then the Wasserstein distance between F and G fulfils (µ F − µ G ) 2 + (σ F − σ G ) 2 ≤ d W (F, G) 2 ≤ ε. Thus, G ∈ N ε and we indeed obtain that M ε = N ε . Applying Theorem 1 case 1. concludes the proof.
Proof of Proposition 3. The general projection property (Theorem 1) in Popescu (2007) shows that for every portfolio x ∈ P and every distribution function G x having mean −µ x = −x µ and variance σ 2 x = x Σx there exists a multivariate distribution function G R ∈ M(µ, Σ) such that the portfolio loss −x R is distributed with G x . As a consequence, optimisation (20) is equivalent to where M(−µ x , σ x ) denotes the set of univariate distributions with mean −µ x and standard deviation σ x .