Set-valued Risk Measures Research Articles

Set-valued dynamic risk measures for processes and for vectors

The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast to scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilise this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multi-portfolio time-consistency between set-valued risk measures for processes and vectors is provided. To accomplish this, an augmented definition for multi-portfolio time-consistency of set-valued risk measures for processes is proposed.

Set-Valued T-Translative Functions and Their Applications in Finance

A theory for set-valued functions is developed, which are translative with respect to a linear operator. It is shown that such functions cover a wide range of applications, from projections in Hilbert spaces, set-valued quantiles for vector-valued random variables, to scalar or set-valued risk measures in finance with defaultable or nondefaultable securities. Primal, dual, and scalar representation results are given, among them an infimal convolution representation, which is not so well known even in the scalar case. Along the way, new concepts of set-valued lower/upper expectations are introduced and dual representation results are formulated using such expectations. An extension to random sets is discussed at the end. The principal methodology consisted of applying the complete lattice framework of set optimization.

Time consistency for scalar multivariate risk measures

Abstract In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).

Multi-utility representations of incomplete preferences induced by set-valued risk measures

We establish a variety of numerical representations of preference relations induced by set-valued risk measures. Because of the general incompleteness of such preferences, we have to deal with multi-utility representations. We look for representations that are both parsimonious (the family of representing functionals is indexed by a tractable set of parameters) and well behaved (the representing functionals satisfy nice regularity properties with respect to the structure of the underlying space of alternatives). The key to our results is a general dual representation of set-valued risk measures that unifies the existing dual representations in the literature and highlights their link with duality results for scalar risk measures.

Set-valued risk measures as backward stochastic difference inclusions and equations

Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time.

Set optimization of set-valued risk measures

A new approach to optimizing or hedging a portfolio of financial instruments to reduce risk is presented. Central to this approach are concepts and tools of set-optimization theory. It focuses on the problem of minimizing set-valued risk measures applied to portfolios. We present sufficient conditions for the existence of solutions of a set-valued risk minimization problem under some semi-continuity assumption. The methodology is applied to the optimization of set-valued Value at Risk and Average Value at Risk. Two examples at the end illustrate various features of the theoretical construction, among them the geometry of the image sets.

Qualitative robustness of set-valued value-at-risk

Risk measures are defined as functionals of the portfolio loss distribution, thus implicitly assuming the knowledge of such a distribution. However, in practical applications, the need for estimation arises and with it the need to study the effects of mis-specification errors, as well as estimation errors on the final conclusion. In this paper we focus on the qualitative robustness of a sequence of estimators for set-valued risk measures. These properties are studied in detail for two well-known examples of set-valued risk measures: the value-at-risk and the maximum average value-at-risk. Our results illustrate, in particular, that estimation of set-valued value-at-risk can be given in terms of random sets. Moreover, we observe that historical set-valued value-at-risk, while failing to be sub-additive, leads to a more robust procedure than alternatives such as the maximum likelihood average value at-risk.

CAPITAL ALLOCATION FOR SET-VALUED RISK MEASURES

We introduce the definition of set-valued capital allocation rule, in the context of set-valued risk measures. In analogy to some well known methods for the scalar case based on the idea of marginal contribution and hence on the notion of gradient and sub-gradient of a risk measure, and under some reasonable assumptions, we define some set-valued capital allocation rules relying on the representation theorems for coherent and convex set-valued risk measures and investigate their link with the notion of sub-differential for set-valued functions. We also introduce and study the set-valued analogous of some properties of classical capital allocation rules, such as the one of no undercut. Furthermore, we compare these rules with some of those mostly used for univariate (single-valued) risk measures. Examples and comparisons with the scalar case are provided at the end.

SET-VALUED LAW INVARIANT COHERENT AND CONVEX RISK MEASURES

In this paper, we investigate representation results for set-valued law invariant coherent and convex risk measures, which can be considered as a set-valued extension of the multivariate scalar law invariant coherent and convex risk measures studied in the literature. We further introduce a new class of set-valued risk measures, named set-valued distortion risk measures, which can be considered as a set-valued version of multivariate scalar distortion risk measures introduced in the literature. The relationship between set-valued distortion risk measures and set-valued weighted value at risk is also given.

Capital Allocation for Set-Valued Risk Measures

We introduce the definition of set-valued capital allocation rule, in the context of set- valued risk measures. In analogy to some well known methods for the scalar case based on the idea of marginal contribution and hence on the notion of gradient and sub-gradient of a risk measure, and under some reasonable assumptions, we define some set-valued capital allocation rules relying on the representation theorems for coherent and convex set-valued risk measures and investigate their link with the notion of sub-differential for set-valued functions. We also introduce and study the set-valued analogous of some properties of classical capital allocation rules, such as the one of no undercut. Furthermore, we compare these rules with some of those mostly used for univariate (single-valued) risk measures. Examples and comparisons with the scalar case are provided at the end.

SET-VALUED CASH SUB-ADDITIVE RISK MEASURES

In this paper, we introduce a new class of set-valued risk measures which satisfies cash sub-additivity. Dual representation for them is provided. Moreover, we also investigate dynamic set-valued cash sub-additive risk measures and discuss the corresponding multi-portfolio time consistency. The equivalent characterization of the multi-portfolio time consistency is given. Finally, an example is also given to illustrate the introduction of set-valued cash sub-additive risk measures. The present paper can be considered as a set-valued extension of scalar cash sub-additive risk measures introduced by El Karouii and Ravanelli [8].

Set-valued loss-based risk measures

In this paper, we introduce a new class of set-valued risk measures, named set-valued convex loss-based risk measures. Representation results are provided. This new class can be considered as a set-valued extension of those introduced by Cont et al. (Stat Risk Model Appl Finance Insur 30(2):133–167, 2013) and Chen et al. (Positivity, 2017). Finally, examples are also given to illustrate the set-valued convex loss-based risk measures.

SET-VALUED SHORTFALL AND DIVERGENCE RISK MEASURES

Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued versions of the entropic risk measure and the average value at risk. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved.

Set-Valued Haezendonck-Goovaerts Risk Measure and Its Properties

We propose a new set-valued risk measure, which is called set-valued Haezendonck-Goovaerts risk measure. First, we construct the set-valued Haezendonck-Goovaerts risk measure and then provide an equivalent representation. The properties of the set-valued Haezendonck-Goovaerts risk measure are investigated, which show that the set-valued Haezendonck-Goovaerts risk measure is coherent. Finally, an example of set-valued Haezendonck-Goovaerts risk measure is given, which exhibits the fact that the set-valued average value at risk is a particular case of the set-valued Haezendonck-Goovaerts risk measures.

A general solution for robust linear programs with distortion risk constraints

Linear optimization problems are investigated that have random parameters in their $$m\ge 1$$ constraints. In constructing a robust solution $${\mathbf {x}}\in \mathbb {R}^d$$ , we control the risk arising from violations of the constraints. This risk is measured by set-valued risk measures, which extend the usual univariate coherent distortion (=spectral) risk measures to the multivariate case. To obtain a robust solution in $$d$$ variables, the linear goal function is optimized under the restrictions holding uniformly for all parameters in a $$d$$ -variate uncertainty set. This set is built from uncertainty sets of the single constraints, each of which is a weighted-mean trimmed region in $$\mathbb {R}^d$$ and can be efficiently calculated. Furthermore, a possible substitution of violations between different constraints is investigated by means of the admissable set of the multivariate risk measure. In the case of no substitution, we give an exact geometric algorithm, which possesses a worst-case polynomial complexity. We extend the algorithm to the general substitutability case, that is, to robust polyhedral optimization. The consistency of the approach is shown for generally distributed parameters. Finally, an application of the model to supervised machine learning is discussed.

Multi-portfolio time consistency for set-valued convex and coherent risk measures

Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on \(L^{p}({\varOmega,\mathcal{F},\mathbb{P}; \mathbb{R}^{d}})\) with image space in the power set of \(L^{p}({\varOmega,\mathcal{F}_{t},\mathbb{P}; \mathbb{R}^{d}})\). In the convex case, multi-portfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multi-portfolio time consistency is equivalent to a generalized version of stability of the dual variables. As examples, the set-valued entropic risk measure with constant risk aversion coefficient is shown to satisfy the cocycle condition for its minimal penalty functions; the set of superhedging portfolios is shown to have in markets with proportional transaction costs the stability property and to satisfy in markets with convex transaction costs the composed cocycle condition; and a multi-portfolio time-consistent version of the set-valued average value at risk, the composed AV@R, is given, and its dual representation deduced.

Measuring risk with multiple eligible assets

The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.

Risk Measures and Capital Requirements with Multiple Eligible Assets

The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.

Monetary risk measures and generalized prices relevant to set-valued risk measures

Monetary risk measures and generalized prices relevant to set-valued risk measures

Representations of set-valued risk measures defined on the $$l$$ l -tensor product of Banach lattices

We obtain a representation for set-valued risk measures which are defined on the completed \(l\)-tensor product \(E\widetilde{\otimes }_l G\) of Banach lattices \(E\) and \(G\). This representation extends known representations for set-valued risk measures defined on Bochner spaces \(L^p(\mathbb {P}, \mathbb {R}^d)\) of \(p\)-integrable functions with values in \(\mathbb {R}^d\).