MULTIVARIATE RISK MEASURES: A CONSTRUCTIVE APPROACH BASED ON
††The authors are grateful to Leonid Hanin for his advice on duality
results in Lipschitz spaces. IM acknowledges the hospitality of the
University Carlos III de Madrid. IM has benefited from discussions
with Qiyu Li, Michael Schmutz and Irina Sikharulidze at various stages
of this work. The second version of this preprint was greatly inspired
by insightful comments of Birgit Rudloff concerning her recent work on
multivariate risk measures. The authors are grateful to the Associate
Editor and the referees for thoughtful comments and encouragement that
led to a greatly improved paper.
This version corrects Lemma 7.1 and Lemma 7.2 in Section 7.
Universidad Carlos III de Madrid
University of Bern ††IC supported by the Spanish Ministry of Science and Innovation Grants No. MTM2011-22993 and ECO2011-25706. IM supported by the Chair of Excellence Programme of the Universidad Carlos III de Madrid and Banco Santander and the Swiss National Foundation Grant No. 200021-137527. ††Address correspondence to Ilya Molchanov, Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland; e-mail: firstname.lastname@example.org.
Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set-valued portfolios. This situation includes the classical Kabanov’s transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints.
We suggest a definition of the risk measure based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov’s exchange cone model, it is shown how the selection risk measure relates to the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010), and Hamel, Heyde and Rudloff (2013).
Key Words: exchange cone, random set, selection, set-valued portfolio, set-valued risk measure, transaction costs.
Since the seminal papers by *art:del:eber:99 and Delbaen (2002), most studies of risk measures deal with the univariate case, where the gains or liabilities are expressed by a random variable. We refer to Föllmer and Schied (2004) and *mcn:frey:emb05 for a thorough treatment of univariate risk measures and to Acciaio and Penner (2011) for a recent survey of the dynamic univariate setting.
Multiasset portfolios in practice are often represented by their total monetary value in a fixed currency with the subsequent calculation of univariate risk measures that can be used to determine the overall capital requirements. The main emphasis is put on the dependency structure of the various components of the portfolio, see Burgert and Rüschendorf (2006); Embrechts and Puccetti (2006). Numerical risk measures for a multivariate portfolio have been also studied in *ekel:gal:hen12 and Rüschendorf (2006). The key idea is to consider the expected scalar product of with a random vector and take supremum over all random vectors that share the same distribution with and possibly over a family of random vectors . *far:koc:mun13 defined the scalar risk as the infimum of the payoff associated with a random vector that being added to the portfolio renders it acceptable. A vector-valued variant of the value-at-risk has been suggested in Cousin and Di Bernardino (2013).
However, in many natural applications it is necessary to assess the risk of a vector in whose components represent different currencies or gains from various business lines, where profits/losses from one line or currency cannot be used directly to offset the position in a different one. Even in the absence of transaction costs, the exchange rates fluctuate and so may influence the overall risk assessment. Also the regulatory requirements may be very different for different lines (e.g. in the case of several states within the same currency area), and moving assets may be subject to transaction costs, taxes or other restrictions. For such cases, it is important to determine the necessary reserves that should be allocated in each line (currency or component) of in order to make the overall position acceptable. The simplest solution would be to treat each component separately and allocate reserves accordingly, which is not in the interest of (financial) agents who might want to use profits from one line to compensate for eventual losses in other ones. Thus, in addition of assessing the risk of the original vector , one can also evaluate the risk of any other portfolio that may be obtained from by allowed transactions. In view of this, it is natural to assume that the acceptability may be achieved by several (and not directly comparable) choices of capital requirements that form a set of possible values for the risk measure. This suggests the idea of working with set-valued risk measures.
Since the first work on multivariate risk measures (Jouini, Meddeb, and Touzi, 2004), by now it is accepted that multiasset risk measures can be naturally considered as taking values in the space of sets, see *bent:lep13,cas:mol07,ham:hey10,ham:hey:rud11,kul08. The risk measures of random vectors are mostly considered in relation to Kabanov’s transaction costs model, whose main ingredient is a cone of portfolios available at price zero at the chosen time horizon (also called the exchange cone), while the central symmetric variant of is a solvency cone. If is the terminal gain, then each random vector with values in is possible to obtain by converting following the rules determined by . In other words, instead of measuring the risk of we consider the whole family of random vectors taking values in . In relation to this, note that families of random vectors representing attainable gains are often considered in the financial studies of transaction costs models, see e.g. Schachermayer (2004).
The set-valued portfolios framework can be related to the classical setting by replacing a univariate random gain with half-line and measuring risks of all random variables dominated by . The monotonicity property of a chosen risk measure implies that all these risks build the set . If is subadditive, then . Furthermore, if and only if contains the origin. While in the univariate case this construction leads to half-lines, in the multivariate situation it naturally gives rise to so-called upper convex sets, see Hamel and Heyde (2010); Hamel et al. (2011); Kulikov (2008). A portfolio is acceptable if its risk measure contains the origin and the value of the risk measure is the set of all such that the portfolio becomes acceptable if capital is added to it.
Note that in the setting of real-valued risk measures adapted in Ekeland et al. (2012), the family of all that make acceptable is a half-space, which apparently only partially reflects the nature of cone-based transaction costs models. Farkas et al. (2014) establish relation between families of real-valued risks and set-valued risks from Hamel et al. (2011). The setting of Riesz spaces (partially ordered linear spaces), in particular Fréchet lattices and Orlicz spaces, has become already common in the theory of risk measures, see Biagini and Fritelli (2008); Cheridito and Li (2009). However, these spaces are mostly used to describe the arguments of risk measures whose values belong to the (extended) real line. Furthermore, the space of sets is no longer a Riesz space — while the addition is well defined, the matching subtraction does not exist. The recent study of risk preferences (Drapeau and Kupper, 2013) also concentrates on the case of vector spaces for arguments of risk measures.
The dual representation for risk measures of random vectors in the case of a deterministic exchange cone is obtained in Hamel and Heyde (2010) and for the random case in Hamel et al. (2011), see also Kulikov (2008) who considers both deterministic and random exchange cones. However in the case of a random exchange cone, it does not produce law-invariant risk measures — the risk measure in Hamel and Heyde (2010); Hamel et al. (2011, 2013); Kulikov (2008) is defined as a function of a random vector representing the gain, while identically distributed gains might exhibit different properties in relation to the random exchange cone. Although the dual representations from Hamel and Heyde (2010); Hamel et al. (2011); Kulikov (2008) are general, they are rather difficult to use in order to calculate risks for given portfolios, since they are given as intersections of half-spaces determined by a rather rich family of random vectors from the dual space. Recent advances in vector optimisation have led to a substantial progress in computation of set-valued risk measures, see Hamel et al. (2013). However, the dual approximation also does not explicitly yield the relevant trading (or exchange) strategy that determines transactions suitable to compensate for risks. The construction of set-valued risk measures from Cascos and Molchanov (2007) is based on the concept of the depth-trimmed region, and their values are easy to calculate numerically or analytically, but it only applies for deterministic exchange cones and often results in marginalised risks (so that the risk measure is a translate of the solvency cone).
In order to come up with a law invariant risk measure and also cover the case of random exchange cones, we assume that the argument of a risk measure is a random closed set that consists of all attainable portfolios. This random set may be the sum of a random vector and the exchange cone (which has been the most important example so far) or may be defined otherwise. For instance, if only a linear space of portfolios is available for compensation, then the set of attainable portfolios is the intersection of with . In any such case we speak about a set-valued portfolio . This guiding idea makes it possible to work out the law-invariance property of risk measures and naturally arrive at set-valued risks.
In this paper we suggest a rather simple and intuitive way to measure risks for set-valued portfolios based on considering the family of all terminal gains that may be attained after some exchanges are performed. The crucial step is to consider all random vectors taking values in a random set (selections of ) as possible gains and regard the random set acceptable if it possesses a selection with all acceptable components. In view of this, we do not only determine the necessary capital reserves, but also the way of converting the terminal value of the portfolio into an acceptable one.
In the case of exchange cones, we relate our construction to the dual representation from Hamel and Heyde (2010) and Kulikov (2008). Throughout the paper we concentrate on the coherent case and one-period setting, but occasionally comment on non-coherent generalisations. Feinstein and Rudloff (2014a) thoroughly analyse and compare various approaches, including one from this paper, in view of defining multiperiod set-valued risks, see also Feinstein and Rudloff (2014b).
Let represent terminal gains on two business lines expressed in two different currencies. Assume that and are i.i.d. normally distributed with mean and variance . Assume that the exchanges between currencies are free from transaction costs with the initial exchange rate (number of units of the second currency to buy one unit of the first one), the terminal exchange rate is lognormal with mean and volatility , and is independent of . The set-valued portfolio is a half-plane with the boundary passing through and normal .
Assume that necessary capital reserves are determined using the expected shortfall at level , see Acerbi and Tasche (2002). If compensation between business lines is not allowed, the necessary capital reserves at time zero are given by (all numbers are given in the units of the first currency). If the terminal gains are transferred to one currency, then the needed reserves are given by and respectively in the case of transfers to the first and the second currency. These values correspond to evaluating the risks of selections of located at the points of intersection of the boundary of with coordinate axes.
However, it is possible to choose a selection that further reduces the required capital requirements. After transferring to the second asset units of the first currency, we arrive at the selection of given by
obtained by projecting the origin onto the boundary of . In this case the needed reserves are . The situation of random frictionless exchanges is also considered analytically in Example 5.1 and numerically in Examples 8.3 and 8.4.
Assume that now liquidity restrictions are imposed meaning that at most one unit of each currency may be obtained after conversion from the other. This framework corresponds to dealing with a non-conical set-valued portfolio . A reasonable strategy would be to use selection from (1) if and and otherwise choose the nearest point to from the extreme points of . The corresponding selection is given by
and the needed reserves are , which is higher than those corresponding to the choice of in view of imposed liquidity restrictions.
Before describing the structure of the paper, we would like to point out that our approach is constructive in the sense that instead of starting with an axiomatic definition of a set-valued risk measure we explicitly construct one based on selections of a random portfolio and univariate marginal risk measures. Then we show that the constructed risk measure indeed satisfies the desired properties of set-valued risk measures, in particular, the coherency and the Fatou properties, instead of imposing them. We show how to approximate the values of the risk measure from below (which is the aim of the market regulator) and from above (as the agent would aim to do). The suggested bounds provide a feasible alternative to exact calculations of risk. Furthermore, the computational burden is passed to the agent who aims to increase the family of selections in order to obtain a tighter approximation from above and so reduce the capital requirements, quite differently to the dual constructions of Hamel et al. (2011) and Kulikov (2008), where the market regulator faces the task of making the acceptance criterion more stringent by approximating from below the exact value of the risk measure. It should be noted our approach constitutes just one possible way to construct multivariate risk measures (and the corresponding acceptance sets) that satisfy the axioms of set-valued coherent risk measures from Hamel et al. (2011).
Section 2 introduces the concept of set-valued portfolios and the definition of set-valued risk measures for set-valued portfolios adapted from Hamel et al. (2011), where the conical setting was considered. Section 3 defines the selection risk measure, which relies on univariate risk measures applied to the components of selections for a set-valued portfolio. In particular, the coherency of the selection risk measure is established in Theorem 3.4. While throughout the paper we work with coherent risk measures defined on spaces with , the construction can be also based on convex non-coherent and non-convex univariate risk measures, so that it yields their non-coherent set-valued analogues, such as the value-at-risk.
Section 4 derives lower and upper bounds for risk measures. Section 5 is devoted to the setting of exchange cones (or conical market models) that has been in the centre of attention in all other works on multiasset risks. It is shown that, for the exchange cones setting, the lower bound corresponds to the dual representation of risk measures from Hamel and Heyde (2010); Hamel et al. (2011) and Kulikov (2008). For deterministic exchange cones the bounds become even simpler and in the case of comonotonic portfolios the risk measure admits an easy expression.
We briefly comment on scalarisation issues in Section 6, i.e. explain relationships to univariate risk measures constructed for set-valued portfolios, which in the case of a deterministic exchange cone are related to those considered in Ekeland et al. (2012); Rüschendorf (2006) and Farkas et al. (2014).
Section 7 establishes the dual representation of the selection risk measures. While the idea is to handle set-valued portfolios through their support functions, the key difficulty consists in dealing with possibly unbounded values of the support functions. For this, we introduce the Lipschitz space of random sets and specify the weak-star convergence in this space in order to come up with a general dual representation for set-valued risk measures with the Fatou property. Theorems 7.4, 7.5, and 7.7 establish the Fatou property of the selection risk measure under some conditions and so yield the closedness of its values and the validity of the dual representation. In the deterministic exchange cone model and for random exchange cones with , the selection risk measure has the same dual representation as in Hamel and Heyde (2010); Kulikov (2008).
Section 8 presents several numerical examples of set-valued risk measures covering the exchange cone setting, the frictionless case, and liquidity restrictions. The algorithms used to approximate risk measures are transparent and easy to implement in comparison with a considerably more sophisticated set-optimisation approach from Löhne (2011) used in Hamel et al. (2013) in order to come up with exact values of set-valued risk measures in conical models. A particular computational advantage is due to the use of the primal representation of selection risk measures in order to compute upper bounds, while utilising the dual representation to arrive at lower bounds.
2 Set-valued portfolios and risk measures
2.1 Operations with sets
In order to handle set-valued portfolios, we need to define several important operations with sets in . The closure of a set is denoted by . Further,
denotes the centrally symmetric set to . The sum of two (deterministic) sets and in a linear space is defined as the set . If one of the summands is compact and the other is closed, the set of pairwise sums is also closed. In particular, the sum of a point and a set is given by . For instance, is the set of points dominated by , where . Denote .
The norm of a set is defined as , where is the Euclidean norm of . A set is said to be upper, if and imply that , where all inequalities between vectors are understood coordinatewisely. Inclusions of sets are always understood in the non-strict sense, i.e. allows for the equality .
The -envelope of a closed set is defined as the set of all points such that the distance between and the nearest point of is at most . The Hausdorff distance between two closed sets and in is the smallest such that and . The Hausdorff distance metrises the family of compact sets, while it can be infinite for unbounded sets.
The support function (see (Schneider, 1993, Sec. 1.7)) of a set in is defined as
where denotes the scalar product. The support function may take infinite values if is not bounded. Denote by
the effective domain of the support function of . The set is always a convex cone in . If is a cone in , then equals the dual cone to defined as
2.2 Set-valued portfolios
Let be an almost surely non-empty random closed convex set in (shortly called random set) that represents all feasible terminal gains on assets expressed in physical units. The random set is called set-valued portfolio. Assume that is defined on a complete non-atomic probability space . Any attainable terminal gain is a random vector that almost surely takes values from , i.e. a.s., and such is called a selection of . We refer to Molchanov (2005) for the modern mathematical theory of random sets.
Since the free disposal of assets is allowed, with each point , the set also contains all points dominated by coordinatewisely and so is a lower set in . The efficient part of is the set of all points such that no other point of dominates in the coordinatewise order. While itself is never bounded, is called quasi-bounded if is a.s. bounded.
Fix and consider the space of -integrable random vectors in defined on . The reciprocal is defined from . The -norm of is denoted by . Furthermore, the family of -integrable selections of is denoted by , and is the family of all essentially bounded selections.
In the following we assume that is -integrable, i.e. possesses at least one -integrable selection. A random closed set is called -integrably bounded if its norm if -integrable (a.s. bounded if ). In the case of set-valued portfolios, this property is considered for .
The closed sum is defined as the random set being the closure of for . It is shown in (Hiai and Umegaki, 1977, Th. 1.4) (see also (Molchanov, 2005, Prop. 2.1.4)) that the set of -integrable selections of coincides with the norm closure of if . If , then is a norm closed set that contains the closed sum .
2.3 Examples of set-valued portfolios
Example 2.1 (Univariate portfolios).
If , then is a half-line and the monotonicity of risks implies that it suffices to consider only its upper bound as in the classical theory of risk measures.
Example 2.2 (Exchange cones).
Let represent gains from assets. Furthermore, let be a convex (distinct from the whole space and possibly random) exchange cone representing the family of portfolios available at price zero. Its symmetric variant is the solvency cone, while the dual cone contains all consistent price systems, see Kabanov and Safarian (2009); Schachermayer (2004). Formally, is a random closed set with values being cones. Define , so that selections of correspond to portfolios that are possible to obtain from following the exchange rules determined by .
If does not contain any line (and is called a proper cone), then the market has an efficient friction. Otherwise, some exchanges are free from transaction costs. In this case different random vectors yield the same portfolio , which is also an argument in favour of working directly with set-valued portfolios. The cone is a half-space if and only if all exchanges do not involve transaction costs. In difference to Ben Tahar and Lépinette (2013), our setting does not require that the exchange cone is proper.
If is deterministic, then we denote it by . If , no exchanges are allowed.
Example 2.3 (Cones generated by bid-ask matrix).
In the case of currencies, the cone is usually generated by a bid-ask matrix, as in Kabanov’s transaction costs model, see Kabanov (1999); Schachermayer (2004). Let be a (possibly random) matrix of exchange rates, so that is the number of units of currency needed to buy one unit of currency . It is assumed that the elements of are positive, the diagonal elements are all one and meaning that a direct exchange is always cheaper than a chain of exchanges. The cone describes the family of portfolios available at price zero, so that is spanned by vectors and for , where are standard basis vectors in . If the gain contains derivatives drawn on the exchange rates, then we arrive at the situation when and the exchange cone are dependent.
Example 2.4 (Conical setting with constraints).
It is possible to modify the conical setting by requiring that all positions acquired after trading are subject to some linear or other constraints, see e.g. Farkas et al. (2014). This amounts to considering the intersection of with a linear subspace or a more general subset of , that (if a.s. non-empty) results in a possibly non-conical set-valued portfolio and provides another motivation for working with set-valued portfolios.
Pennanen and Penner (2010) study in depth not necessarily conical transaction costs models in view of the no-arbitrage property, see also Kaval and Molchanov (2006). One of the most important examples is the model of currency markets with liquidity costs or exchange constraints.
The following examples describe several non-conical models that yield quasi-bounded set-valued portfolios. Despite the fact that some of them are generated by random vectors, it is essential to treat these portfolios as random sets, e.g. for possible diversification effects. The latter means that a sum of such set-valued portfolios is not necessarily equal to the set-valued portfolio generated by the sum of the generating random vectors.
Example 2.5 (Restricted liquidity).
Let , where . Then the exchanges up to the unit volume are at the unit rate free from transaction costs while other exchanges are not allowed. A similar example with transaction costs and a random exchange cone can be constructed as for some . A more general variant from (Pennanen and Penner, 2010, Ex. 2.4) models liquidity costs depending on the transaction’s volume.
Let be random vectors in that represent terminal gains in lines (e.g. currencies) of investments. The random set is defined as the set of all points in dominated by at least one convex combination of the gains. In other words, is the sum of and the convex hull of .
If and for and a bivariate random vector , then describes an arbitrary profit allocation between two different lines without transaction costs up to the amount .
Assume that , where is the ball of fixed radius centred at the origin. This model corresponds to the case, when infinitesimally small transactions are free to exchange at the rate that depends on the balance between the portfolio components.
Example 2.8 (Transactions maintaining solvency).
Let be an exchange cone from Example 2.2 and let be the value of a portfolio. Define to be the set of points coordinatewisely dominated by a point from if belongs to the solvency cone , and if . In this case no transactions are allowed in the non-solvent case and otherwise all transactions should maintain the solvency of the portfolio.
2.4 Set-valued risk measure
A function defined on -integrable set-valued portfolios is called a set-valued coherent risk measure if it takes values being upper convex sets and satisfies the following conditions.
for all (cash invariance).
If a.s., then (monotonicity).
for all (homogeneity).
The risk measure is said to be closed-valued if its values are closed sets. Furthermore, is said to be a convex set-valued risk measure if the homogeneity and subadditivity conditions are replaced by
The names for the subadditivity and convexity properties are justified by the fact that sets can be ordered by the reverse inclusion; we follow Hamel and Heyde (2010); Hamel et al. (2011) in this respect. Definition 2.9 appears in Kulikov (2008) and Hamel and Heyde (2010); Hamel et al. (2011), with the argument of being a random vector and for a fixed exchange cone , which in our formulation means that the argument of is the random set .
The set-valued portfolio is acceptable if . The subadditivity of means that the acceptability of and entails the acceptability of , as in the classical case of coherent risk measures. In the univariate case, and for a coherent risk measure , so that is acceptable if and only if . If a.s., then is acceptable under any closed-valued coherent risk measure. Indeed, then , while contains the origin by the homogeneity and subadditivity properties and the closedness of the values for . For a non-coherent , it is sensible to extra impose the normalisation condition for each deterministic exchange cone .
Example 2.3 (cont.) (Capital requirements in the exchange cones setting).
The value of the risk measure determines capital amounts that make acceptable. The necessary capital should be allocated at time zero, when the exchange rules are determined by a non-random exchange cone . Thus, the initial capital should be chosen so that intersects , and
is the family of all possible initial capital requirements. Optimal capital requirements are given by the extremal points from in the order generated by the cone . If is not random, . If is a half-space, meaning that the initial exchanges are free from transaction costs, then is a half-space too. In this case, the sensible initial capital is given by the tangent point to in direction of the normal to , see Example 1.1.
If is the whole space, which might be the case, for instance, if and are two different half-spaces, then it is possible to release an infinite capital from the position, and this situation should be excluded for the modelling purposes.
3 Selection risk measure for set-valued portfolios
3.1 Acceptability of set-valued portfolios
Below we explicitly construct set-valued risk measures based on selections of . Let be law invariant coherent risk measures defined on the space with values in . Furthermore, assume that each satisfies the Fatou property, which for follows from the law invariance (Jouini et al., 2006) and for is always the case if takes only finite values, see (Kaina and Rüschendorf, 2009, Th. 3.1). For a random vector write
Random vector is said to be acceptable if , i.e. for all . This is exactly the case if portfolio is acceptable with respect to the set-valued measure
It is a special case of the regulator risk measure considered in (Hamel et al., 2013). The following definition suggests a possible acceptability criterion for set-valued portfolios that leads to a risk measure satisfying the axioms from Definition 2.9.
A -integrable set-valued portfolio is said to be selection acceptable (in the following simply called acceptable) if for at least one selection .
The monotonicity property of univariate risk measures implies that is acceptable if and only if its efficient part admits an acceptable selection.
Example 2.3 (cont.).
The acceptability of means that it is possible to transfer the assets given by the components of according to the exchange rules determined by , so that the resulting random vector with has all acceptable components.
Remark 3.2 (Generalisations).
The acceptability of selections can be judged using any other multivariate coherent risk measure, e.g. considered in Hamel et al. (2011), that are not necessarily of point plus cone type, or numerical multivariate risk measures from Burgert and Rüschendorf (2006); Ekeland et al. (2012) and Farkas et al. (2014). Furthermore, it is possible to consider acceptability of a general convex subset of that might not be interpreted as the family of selections of a set-valued portfolio. This may be of advantage in the dynamic setting (Feinstein and Rudloff, 2014a) or for considering uncertainty models as in (Bion-Nadal and Kervarec, 2012).
3.2 Coherency of selection risk measure
The selection risk measure of is defined as the set of deterministic portfolios that make acceptable, i.e.
Its closed-valued variant is .
The selection risk measure defined by (4) and its closed-valued variant are law invariant set-valued coherent risk measures, and
We show first that is an upper set. Let and . If is acceptable for some , then also is acceptable because of the monotonicity of the components of . Hence . If and , then for a sequence . Consider any , then for sufficiently large , so that . Letting decrease to yields that , so that is an upper set.
In order to confirm the convexity of , assume that with and and take any . The subadditivity of components of implies that
It remains to note that is a selection of in view of the convexity of . Then is convex as the closure of a convex set.
The law invariance property is not immediate, since identically distributed random closed sets might have rather different families of selections, see (Molchanov, 2005, p. 32). Denote by the -algebra generated by the random closed set , see (Molchanov, 2005, Def. 1.2.4). If is acceptable, then for some . The dilatation monotonicity of law invariant numerical coherent risk measures on a non-atomic probability space (see Cherny and Grigoriev (2007)) implies that
Therefore, the conditional expectation is also acceptable. The convexity of implies that is a -integrable -measurable selection of . Therefore, is acceptable if and only if it has an acceptable -measurable selection. It remains to note that two identically distributed random sets have the same families of selections which are measurable with respect to the minimal -algebras generated by these sets, see (Molchanov, 2005, Prop. 1.2.18). In particular, the intersections of these families with are identical. Thus, and are law invariant.
Representation (5) follows from the fact that is the union of for .
The first two properties of coherent risk measures follow directly from the definition of . The homogeneity and subadditivity follow from the fact that all acceptable random sets build a cone. Indeed, if is acceptable, then is an acceptable selection of and so is acceptable. If and are acceptable, then is acceptable because the components of are coherent risk measures and so contains an acceptable random vector. Thus, and by passing to the closure we arrive at the subadditivity property of . ∎
Remark 3.5 (Eligible portfolios).
In order to simplify the presentation it is assumed throughout that all portfolios can be used to offset the risk. Following the setting of Hamel and Heyde (2010); Hamel et al. (2011, 2013), it is possible to assume that the set of eligible portfolios is a proper linear subspace of . The corresponding set-valued risk measure is , which equals the union of for all .
3.3 Properties of selection risk measures
Conditions for closedness of set-valued risk measures in the exchange cone setting were obtained in Feinstein and Rudloff (2014b). The following result establishes the closedness of for portfolios with -integrably bounded essential part. Further results concerning closedness of are presented in Corollary 7.6 and Corollary 7.8.
If is -integrably bounded, then the selection risk measure is closed.
Let be acceptable and . Note that is weak compact for by (Molchanov, 2005, Th. 2.1.19) and for by its boundedness in view of the reflexivity of . By passing to subsequences we can assume that weakly converges to in . The dual representation of coherent risk measures yields that for a random variable , equals the supremum of over a family of random variables in . If weakly converges to in , then , so that . Applying this argument to the components of we obtain that
whence . If , then are uniformly bounded and so have a subsequence that converges in distribution and so can be realised on the same probability space as an almost surely convergent sequence. The Fatou property of the components of yields (6). ∎
Theorem 3.7 (Lipschitz property).
Assume that all components of take finite values on . Then there exists a constant such that for all -integrable set-valued portfolios and .
Example 3.8 (Selection expectation).
If is the expectation of , then
where is the selection expectation of , i.e. the closure of the set of expectations for all integrable selections , see (Molchanov, 2005, Sec. 2.1). Thus, the selection risk measure yields a subadditive generalisation of the selection expectation.
Assume that and all components of are given by . Then is acceptable if and only if is almost surely non-empty, and is the set of points that belong to with probability one.
If is an exchange cone, then is a deterministic convex cone that contains . If is deterministic, then .
Selection risk measures have a number of further properties.
I. Assume that has all identical components. If is acceptable, then its orthogonal projection on the linear subspace of generated by any is also acceptable, and so the risk of the projected contains the projection of .
II. The conditional expectation of random set with respect to a -algebra is defined as the closure of the set of conditional expectations for all its integrable selections, see (Molchanov, 2005, Sec. 2.1.6). The dilatation monotonicity property of components of implies that if is acceptable, then is acceptable. Therefore, is also dilatation monotone meaning that
In particular, . Therefore, the integrability of the support function for at least one provides an easy condition that guarantees that is not equal to the whole space.
In the setting of Example 2.2, written as a function of only becomes a centrally symmetric variant of the coherent utility function considered in (Kulikov, 2008, Def. 2.1). It should be noted that the utility function from Kulikov (2008) and risk measures from Hamel et al. (2011) depend on both and and on the dependency structure between them and so are not law invariant as function of only, if is random.
If the components of are convex risk measures, so that the homogeneity assumption is dropped, then is a convex set-valued risk measure, which is not necessarily homogeneous. If the components of are not law invariant, then is a possibly not law invariant set-valued risk measure. The following two examples mention non-convex risk measures, which are also defined using selections.
Assume that the components of are general cash invariant risk measures without imposing any convexity properties, e.g. values-at-risk at the level , bearing in mind that the resulting selection risk measure is no longer coherent and not necessarily law invariant. Then is acceptable if and only if there exists a selection of such that for all .
Let be a deterministic exchange cone and fix some acceptance level . Call random vector acceptable if and note that this condition differs from requiring that for all . Then a set-valued portfolio is acceptable if and only if . If and , then is sometimes termed a multivariate quantile or the value-at-risk of , see Embrechts and Puccetti (2006) and Hamel and Heyde (2010).
4 Bounds for selection risk measures
4.1 Upper bound
The family of selections of a random set is typically very rich. An upper bound for can be obtained by restricting the choice of possible selections. The convexity property of implies that it contains the convex hull of the union of for the chosen selections . This convex hull corresponds to the case of a higher risk than . Making the upper bound tighter by considering a larger family of selections is in the interest of the agent in order to reduce the capital requirements.
At first, it is possible to consider deterministic selections, also called fixed points of , i.e. the points which belong to with probability one. If a.s., then . However, this set of fixed points is typically rather poor to reflect essential features related to the variability of .
Another possibility would be to consider selections of of the form for a fixed random vector and a deterministic . If for a deterministic set (which always can be chosen to be convex in view of the convexity of ), then
It is possible to tighten the bound by taking the convex hull for the union of the right-hand side for several . The inclusion in (7) can be strict even if , since taking random selections of makes it possible to offset the risks as the following example shows.
Let , where is the unit ball and is the standard bivariate normal vector. Consider the risk measure with two identical components being expected shortfalls at level . Then is the upper set generated by the ball of radius one centred at . Consider the selection of given by . By numerical calculation of the risks, it is easily seen that , which does not belong to .
4.2 Lower bound
Below we describe a lower bound for , which is a superset of and is also a set-valued coherent risk measure itself. For , (resp. ) denote the vectors composed of pairwise products (resp. ratios) of the coordinates of and . If is a set in , then . By agreement, let and .
Let be a non-empty family of non-negative -integrable random vectors in . Recall that denotes the selection expectation of with the coordinates scaled according to the components of , see Example 3.8. It exists, since is assumed to possess at least one -integrable selection and so . Define the set-valued risk measure
which is similar to the classical dual representation of coherent risk measures, see Delbaen (2002) and also corresponds to the -representation of risk measure in conical models from (Hamel et al., 2011, Th. 4.2) that is similar to (9) from the following theorem.
Assume that is a non-empty family of non-negative -integrable random vectors. The functional is a closed-valued coherent risk measure, and
The closedness and convexity of follow from the fact that it is intersection of half-spaces; it is an upper set since the normals to these half-spaces belong to . It is evident that is monotonic, cash invariant and homogeneous. In order to check the subadditivity, note that
Recall that the support function of the expectation of a set equals the expected value of the support function. Since is an upper set,
and we arrive at (9) by replacing with . ∎
The components of admit the dual representations
where, for each , is the dual cone to the family of random variables such that , i.e. consists of non-negative -integrable random variables such that for all with , see Delbaen (2002); Föllmer and Schied (2004). Note that are the maximal families that provide the dual representation (10). Despite contains a.s. vanishing random variables, letting ensures the validity of (10).
Assume that the components of admit the dual representations (10). Then for any family of -integrable random vectors such that , .