A theory for robust risk measures
In this paper, we address the uncertainty regarding the choice of a probability measure in the context of risk measures theory. To do that, we consider the notion of probability-based risk measurement and propose two general approaches to generate risk measures that are robust. The first approach is a worst-case situation, while the second one is based on weighted averaging. We develop results regarding issues on the theory of risk measures, such as financial, statistical, and continuity properties, as well dual representations. Furthermore, we explore similar results for deviation measures under our proposed framework.
Keywords: Risk measures, Robustness, Uncertainty, Worst case, Weighted averaging, Deviation measures.
Since the seminal paper of Artzner et al. (1999), the attention to risk measures from a theoretical point of view has been raised in mathematical finance, where a stream of literature has proposed and discussed distinct features, such as axiom sets, dual representations, statistical properties etc. Under this framework, the objective of risk measurement is to establish a functional on some vector space that summarizes risk in one number. For a comprehensive review of this theory, we recommend the books of Pflug and Römisch (2007), Delbaen (2012) and Föllmer and Schied (2016). For shorter reviews, see Föllmer and Knispel (2013) and Föllmer and Weber (2015). This classic theory for risk measures is developed under the assumption of a given probability measure that represents beliefs of the world. However, we frequently do not know if there is a correct probability measure. Typically, we have a set of candidate probability measures that can be understood as alternative scenarios, models, or beliefs.
Thus, under uncertainty on the probability measure, we can intuitively think on risk measurement over the product space composed of the space of random variables and the set of possible probability measures. This phenomenon is directly related to the notion of model risk relative to uncertainty concerning the choice of a model, which is studied in detail for risk measures by Kerkhof et al. (2010), Bernard and Vanduffel (2015), Barrieu and Scandolo (2015), Danielsson et al. (2016) and Kellner and Rosch (2016). From a practical point of view, The Turner Review raises the issue of model ambiguity, often called Knightian uncertainty. There is also the problem of regulatory arbitrage, as in Wang (2016), where capital requirement determinations based on risk measures can be manipulated by using distinct models linked to some probability measure used for estimation. Because the choice of probabilities may affect the risk value for a large amount, we would like to have risk measures that are robust. Robustness intuitively means functionals do not suffer relevant variations in their value from changes on the chosen probability measure. There are distinct concepts of robustness, such the qualitative one that is considered in Cont et al. (2010) and Kratschmer et al. (2014). Kratschmer et al. (2014) investigates qualitative robustness and verifies the fragility of convex risk measures in this sense. Kiesel et al. (2016) shows that many examples of risk functionals, even convex ones, are robust in the sense they are continuous in relation to the Wasserstein distance for probability measures.
From that, our main goal is to develop risk measures that are robust to the choice of a particular probability measure. To avoid troubles selecting a determined metric from the definition of robustness, we propose and study results about risk measures that consider the whole set of candidate probabilities, instead of a specific one. Moreover, we want to establish a parallel to the classic approach where no uncertainty over the probability measure is present by having a domain that is a vector space of random variables, instead of the aforementioned product space composed of positions and probabilities. In this sense, we initially propose the concept of probability-based risk measurement, which is a collection of functionals based on a given risk measure and set of probability measures. Thus, for any of the probabilities, we obtain one value for the base risk measure. From that, we develop two robust risk measures. The first one is linked to the notion of worst-case, where the robust risk measure is the supremum of a probability-based risk measurement. Extreme choices for the set of probabilities are a singleton, where the classic case is covered and the whole set of probability measures are on some measurable space. Other possible choices are closed balls around a reference probability measure based on distance, metric, divergence, or relation, such as in Shapiro (2017), Bartl et al. (2018) and Bellini et al. (2018). One can consider probability measures with specific conditions, such as in Cornilly et al. (2017), where they coincide regarding a determined number of moments. Righi (2017b) and Sun and Ji (2017) explore dual sets for coherent risk measures (see definition below). The second formulation we propose is a weighted averaging in the form of integral or expectation, based on a probability measure on the measurable space generated by the set of candidate probabilities. As any weighting scheme, it can reflect the importance given to any probability measure. For instance, more importance can be attached to more conservative or parsimonious candidates. In the sequence, we develop results regarding theoretical, statistical, and continuity properties, as well as dual representations for our two robust risk measures from the characteristics of their probability-based measurements.
This kind of structure can be applied to other functionals beyond risk measures since the uncertainty regarding probabilities is present in any modeling framework. For instance, an approach known as sub-linear expectation is gaining space in the literature. Sun and Ji (2017) considers least squares minimization, deriving its properties and guaranteeing its solution under uncertainty of the correct probability measure. In this sense, we extrapolate our results to the concept of robust deviations. The motivation is due to the fact that, beyond the notion of a monetary loss, the concept of risk is linked to the variability of an expected result. Such concept of deviation is axiomatized for convex functionals in Rockafellar et al. (2006), Pflug (2006) and Grechuk et al. (2009). The main idea is to consider generalizations of the standard deviation and similar measures in an axiomatic fashion. See Pflug and Römisch (2007) and Rockafellar and Uryasev (2013) for a comprehensive review. Recently, Righi and Ceretta (2016), Righi (2018) and Righi and Borenstein (2018) explore advantages of a more complete analysis that considers both risk and deviation measures.
We contribute to the literature since, to the best of our knowledge, this is the first paper to expose a general approach to construct robust risk measures and that explores the main features present in the literature theory, especially dual representations. Works such those of Bartl et al. (2018), Bellini et al. (2018) and Guo and Xu (2018) focus on a particular risk measure, instead of a general framework. In Pflug et al. (2012) and Guo and Xu (2018), optimization of risk measures is explored in the presence of uncertainty over probability, but the issues on risk measures theory we explore are not concerned. Shapiro (2017) and Sun and Ji (2017) develops stochastic optimization under sub-linear expectations originated from the existence of multiple probabilities by using the notion of risk measures, instead of developing robust ones. The same for Cont (2006) but regarding derivatives pricing. The stream of Cont et al. (2010), Kiesel et al. (2016), and Cornilly et al. (2017) is to consider risk functionals over a domain of distribution functions. In this case, where Law Invariance is assumed, the uncertainty is regarding the estimation of such distributions, rather than on the probability measure. Moreover, they do not develop the same issues in our framework, where statistical properties are not always assumed. Frittelli and Maggis (2018) address model risk by what they call Value and Risk measures considering functionals over a triplet composed by the initial price of an asset, a space of random variables, and a set of pricing probability measures. However, their focus is distinct from ours since they are not concentrated on robust risk measures. Laeven and Stadje (2013) and Jokhadze and Schmidt (2018) explore a concept similar to that of probability-based risk measurement and study some issues on the theory of risk measures, such as financial properties and acceptance sets. However, they do not develop in detail other points, such as dual representations.
Another relevant contribution is that we do not need a reference probability measure in our framework that is much more general and flexible. In fact, we do not impose any restrictive assumption over the set of probability measures or the base risk measure. We also do not impose any specific metric or distance on the set of probabilities. Moreover, with exception of Jokhadze and Schmidt (2018) and Müller and Righi (2018), studies from the literature are centered on the worst-case approach as the construction of robust risk measures. We also develop results for the weighted averaging. This is in consonance to the stream of forecasting literature that indicates in favor of a model combination to keep some features of each candidate probability. It is also possible to consider a space of random variables that has a measure that considers the uncertainty, such as in Maggis et al. (2018). However, this is outside the scope of our study. The same is for issues regarding multivariate or dynamic risk measures. These points are left for future research.
The remainder of this paper is structured as follows: in section 2, we present preliminaries regarding notation and definitions on our proposed framework; in section 3, we expose the main results of our study, which are focused on theoretical, statistical, and continuity properties, as well as dual representations for our proposed robust risk measures; in section 4, we adapt the framework for deviation measures by exploring results in the same fashion as those for risk measures.
Consider the real valued random result of any asset ( is a gain, is a loss) that is defined in a measurable space endowed with the point-wise partial order . All equalities and inequalities are in this respect. We define , , and as the indicator function for an event . We say that a pair of random variables is co-monotone if . We work on the space of (equivalent classes of) bounded random variables. This space is defined by the point-wise supremum norm . We have that is the cone of non-negative elements . Let be the set of all probability measures on . We have that is a non-empty set. We assume that is atom-less for any .
We consider the probability space , where is a sigma-algebra and are the Borel sets of . Moreover, , and are, respectively, the expected value, the probability function and its inverse for under . We denote by the set of all distribution functions. We have that is related to the mass probability . Note that for we have . We denote by convergence in the norm, while means point-wise convergence in . Furthermore, let be the set of probability measures on that are absolutely continuous in relation to with Radon-Nikodym derivatives . We use the convention that .
We now define risk measures in the presence of uncertainty on the chosen probability by a collection of functionals on .
A functional is a risk measure. Its probability-based risk measurement is a family of risk measures . Moreover, we assume that the functional on , defined as , is -measurable .
Our choice for the space of bounded random variables is considered in the classic approach, as in Delbaen (2002) and Föllmer and Schied (2002). Extensions of risk measures from to larger spaces, such as of integrable random variables under by keeping theoretical properties are conducted on Theorem 2.2 of Filipović and Svindland (2012) and Theorem 4.3 of Koch-Medina and Munari (2014) under some conditions. Moreover, under this framework, we avoid considering functionals on the product space , keeping the pattern of classic approach. Note that the choice for has no warmth because it is not affected by the chosen probability measure . The assumption regarding measurable property is to avoid indefiniteness of integrals.
We now provide some examples of probability-based risk measures. We intend to use such examples to illustrate the distinct steps of our analysis. We choose those risk measures because they frequently appear in literature:
Expected Loss (EL): This is a trivial risk measure defined conform . It is a very parsimonious case, indicating the expected value (mean) of a loss.
Value at Risk (VaR): This is the most prominent example of risk measure. It is defined as , and represents some quantile of interest for the loss. VaR is a leading risk measure in both academics and industry, recognized by its simplicity.
Expected Shortfall (ES): This risk measure is proposed by, among others under distinct names, Acerbi and Tasche (2002) and is defined conform , . It represents the expected value of a loss, given it is beyond the -quantile of interest.
Entropic risk measure (ENT): This functional is proposed by Föllmer and Schied (2002) and is defined as . It is based on the exponential utility function and has relevant financial properties.
Maximum loss (ML): This is an extreme risk measure defined as . Such a risk measure does not depend on the choice for , but it leads to an adverse and restrictive situation.
From these examples, it is clear the role played by the chosen probability measure. We now present a formal definition of robustness.
A risk measure is robust if , , , exists such that implies in , where is a metric.
Typical choices for the metric are, for instance, the well-known Prohorov, Lévy, Kantorovich, Kolgomorov or Wasserstein.
Note that such definition of robustness is dependent on the chosen metric. To avoid such choices, we propose functionals that consider the whole rather than some specific . The first formulation we consider is related to the worst-case approach, which is a protective oriented framework. In this case, one chooses the supremum of probability-based risk measurement. We now define it.
Let be a probability-based risk measurement. Its worst-case risk measure is a functional defined as:
This risk measure is robust by definition. Of course, choices for affect the value of . More specifically, if , then . If the set of probability measures is finite, i.e., , then the supremum is attained. For practical applications, this is typically the case where a limited number of estimation models are considered. Nonetheless, the robustness regarding a specific probability measure is preserved. For instance, let be a collection of sets . If we would like to have robustness regarding the choice of we could have the worst-case functional defined as
where, , and we would have our worst-case risk measure again but over a larger set of probabilities.
The second formulation is linked to the concept of model combination by a weighted averaging of over . This reasoning is due to the fact that a combination considers aspects of the whole without leading to an extreme conservative risk measurement. We now expose a formal definition.
Let be a probability-based risk measurement. Its weighted risk measure is a functional defined as:
This risk measure also is robust by definition. Since is -measurable, the integral is well defined. We have that , hence this risk measure is not so aggressive as the worst case. Moreover, one could choose as a capacity (non-additive probability measure). However, we restrict our analysis to the case of a probability measure since it covers most practical situations of model risk. One can argue about uncertainty regarding the choice of . This is a source of potential discussion, making the worst-case approach less subjective than the weighted averaging. Nonetheless, note that for probability measures on we have that , thus is somehow robust in relation to the choice of . Nonetheless, when is finite we have that . Furthermore, attains any value in the interval . For completeness sake, we now provide a formal result for this claim.
Let be a probability-based risk measurement and be defined conform (2). Then, may assume, , any value in the interval conform the choice for .
Let be some desired value for . If there exists such that , then we get the result by choosing , i.e., the Dirac measure on . Otherwise, exist such that . In this case there is, by interpolation, a positive scalar with such that . Thus, we get the result by choosing , which is a probability measure on . This completes the proof. ∎
By solving the linear system in the proof of the last proposition, we get . This is a canonical solution to interpolation. Moreover, may attain both the infimum and supremum, which are limits of the interval, for some given in the case is discrete.
3 Main results
In this section, we explore results regarding relevant topics for risk measures, such as theoretical, statistical, and continuity properties, as well as dual representations. Thus, we expose results related to such issues for both and .
3.1 Theoretical properties
A key aspect in risk measures theory is the role of theoretical properties, also known as axioms that such functionals may or not fulfill. Such properties affect posterior results in almost any subject in the theme. We now formally define such concepts.
A risk measure may possess the following properties:
Monotonicity: if , then .
Translation Invariance: .
Positive Homogeneity: .
Co-monotonic Additivity: with co-monotone.
We have that is called monetary if it fulfills Monotonicity and Translation Invariance, convex if it is monetary and respects Convexity, coherent if it is convex and fulfills Positive Homogeneity, and co-monotone if it attends Co-monotonic Additivity. In this paper, we are working with normalized risk measures in the sense of .
For a review on details regarding interpretation of such properties, we recommend the mentioned books on the classic theory. Convexity and Positive Homogeneity imply Sub-additivity, which is defined as . Co-monotonic Additivity means absence of diversification and induces Positive Homogeneity. Normalization is easily obtained through a translation, beyond it the fact it is implied from Positive Homogeneity. Other properties appear in literature, such as cash sub-additivity: , quasi-convexity: , and surplus-invariance under monotonicity: . However, we choose to focus on those needed for our main analysis. We have that VaR is a monetary risk measure, ENT is a convex risk measure, while EL, ES, EVaR, and ML are coherent risk measures. Moreover, we have that EL, VaR, and ES are co-monotone.
We now link the theoretical properties of our robust risk measures to those of the family that generates them. The first proposition is regarding , while the second is a similar result for the weighted risk measure . We will keep this ordering throughout the paper.
Let be a probability based risk measurement and defined as in (1). If posses, , any property among Monotonicity, Translation Invariance, Convexity and Positive Homogeneity, then fulfills the same property. Moreover, if is co-monotone , then is sub-additive with co-monotone.
The results are based on properties of the supremum. For Monotonicity, let . Since , we get
Regarding Translation Invariance, we get for
About Convexity, since , we have the following:
For Positive Homogeneity, we have that
Concerning Co-monotonic Additivity, let be a co-monotonic pair. Then we obtain
This concludes the proof. ∎
Under some conditions on the set , such as being tight, i.e., there is some compact set satisfying , and exists and such that , Bartl et al. (2018) provide situations where the supremum in definition of is attained. In this case, the worst-case risk measure is co-monotone when is for any . We do not pursue such goal in this paper and choose to keep results in a more general fashion.
The results follow from properties of the integral. Regarding Monotonicity, let . Since , we get
For Translation Invariance, we have that for any
Concerning Convexity, since , we have
For Positive Homogeneity, we have that
For Co-monotonic Additivity, let be a co-monotonic pair. Then we obtain
This concludes the proof. ∎
The concept of risk measures is attached to a primitive notion of acceptance set that represents those with no positive risk. In this sense, (monetary) risk measures represent an amount of capital that should be added to the initial position to make it acceptable. We now formally define this kind of structure.
Let be a risk measure. We have that the acceptance set of is defined as .
We now expose a result that links the properties of an acceptance set from those of the risk measure that induces it.
Theorem 3.6 (Proposition 4.6 of Föllmer and Schied (2016)).
Let be the acceptance set defined by . Then:
If fulfills Monotonicity, then , and implies in ;
If fulfills Translation Invariance, then ;
If is a monetary risk measure, then is non-empty, closed with respect to the supremum norm, and ;
If attends Convexity, then is a convex set;
If fulfills Positive Homogeneity, then is a cone.
If is a coherent risk measure, then its acceptance is a convex cone that contains , and has no intersection with the complement of . Moreover, if for co-monotone , then . On the other hand, for a non-empty , the risk measure associated with this set is . It is also possible to establish the properties of from those of , see Proposition 4.7 of Föllmer and Schied (2016). However, for monetary risk measures in , which is always the case in this paper, it is well known that .
When in the context of probability-based risk measurement, we have a family of acceptance sets . Moreover, it is possible to obtain some financial intuition from acceptance sets.
We now provide specific cases of acceptance sets for the risk measures of Example 2.2:
EL: In this case we get the positions with non-negative expectation under , conform .
MSD: The acceptable positions are based on performance criteria, under , similar to a Sharpe ratio, conform , where .
VaR: For this risk measure, the acceptance set represents positions with a probability of loss, under , no greater than , i.e., .
ES: In this case, we obtain positions with non-negative expectation, under , of their “restriction” to values below the quantile of interest, conform .
EVaR: The acceptable positions are based on performance criteria, under , similar to a Omega ratio, i.e., .
ENT: This risk measure is linked to the concept of exponential utility, and its acceptance set is composed by positions with bounded expectation, under , of an exponential composition conform .
ML: This is a restrictive risk measure, where acceptable positions are those with null probability of loss, conform . Note that is invariant to the choice of , what reflects the fact that this risk measure is not affected by such choice.
Based on the results from Theorem 3.6 with Propositions 3.2 and 3.4, it is straightforward that properties of the acceptance sets and are obtained from those robust risk measures that generate them. Moreover, it is possible to construct from the collection of . We now expose this result in a formal way.
Let be a probability-based risk measurement and be defined as in (1). Then .
From the definition of acceptance set we have:
This concludes the proof. ∎
3.2 Statistical properties
Beyond theoretical properties, statistical features are also relevant for the risk management step. Thus, in this section, we address such properties. An important property in the classical theory of risk measures is Law Invariance, which assures the value of a risk measure is determined by its distribution. This is a significant feature regarding practical matters where data-based models are considered. However, due to the nature of our proposed risk measures and , it is necessary to work with stronger versions of this property that take into account a family of probability measures. We now define such concepts.
Let be a probability-based risk measurement. Then, we have the following properties:
-Law Invariance: if , then ;
-Law Invariance: is -law invariant ;
Cross Law Invariance: if , then , .
Note that Cross Law Invariance is stronger than their or even based counterparts, see Laeven and Stadje (2013) for instance. Of course, if , then both and when -Law Invariance is present. Furthermore, it is straightforward to verify that Cross Law Invariance of implies that if and , then . Under -Law Invariance, monetary and convex risk measures respects stochastic dominance of, respectively, first and second order, see the results in Bäuerle and Müller (2006). We have that cross-law invariance is respected in all cases exposed in Example 2.2 since we have that . Thus, all such risk measures are directly dependent of the quantile function and, consequently, of the cumulative distribution.
In this framework, we have that risk measurement can be understood as a functional over distribution functions conform , where . This conception is proposed by Cont et al. (2010) and is called risk functional. Note that equality in distribution induces an equivalence relation over ; thus, it can be represented as a union of all its (disjoint) equivalence classes. Moreover, it is well-known that has the same distribution of under for any (see Lemma A.25 in Föllmer and Schied (2016), for instance), where is uniformly distributed over , i.e., . Thus, we have, with some abuse of notation, that . Discussions of properties regarding risk functionals of this kind are present in Acciaio and Svindland (2013), Frittelli et al. (2014) and Frittelli and Maggis (2018).
Another highlighted statistical property in risk measures theory is Elicitability, which is directly linked to risk forecasting since the score functions work as criteria for comparing competing models. See Ziegel (2016) and the references therein for more details. If a risk measure has this attribute, it can be recovered from minimization of some score function. We formally define it now.
A map is called scoring function if it has the following properties:
if, and only if, ;
is increasing for and decreasing for , for any ;
is continuous in , for any .
A risk measure is elicitable on if exists a scoring function such that:
In the Definition 3.11, we have that expectation in (3) is indeed over . Since , the expectation term is equivalent to and well defined. Furthermore, if a risk measure is elicitable on , then it has convex levels sets, i.e., implies in . See Delbaen et al. (2016) for details.
Elicitability can be restrictive since, depending on what theoretical properties (axioms) are demanded, we end up with only one example of risk measure that satisfies the requisites. We now expose such results.
Let be an elicitable -law invariant monetary risk measure. Then:
If is convex, then , known as Shortfall Risk, where , where is some increasing and convex function, and is contained in the range of , with ;
If is coherent, then , with ;
If is comonotonic coherent, then , with ;
If is comonotonic but not convex, then , with .
The entropic risk measure is contemplated on item (i) of Theorem 3.12 by choosing and . Moreover, we have that ES, MSD, and ML are not, at least directly, elicitable risk measures.
Similarly to the concept of Law Invariance, Elicitablity is not directly applied to our robust risk measures and since they are not dependent on a fixed probably measure. Nonetheless, based on the elicitability property for the base , we can have a characterization of and . We now provide such results.
Let be a probability based risk measurement, a convex set, and defined as in (1). If is elicitable underscore function , then we have that:
Let . We have that is a compact set. Moreover, we have from Definition 3.11 that for and when . Thus, the minimization of is not altered if we replace by . Then, we get that:
The last step in this deduction is due to the minimax theorem, which is valid since both and are convex, is compact, plus both and possess the necessary continuity properties. This concludes the proof. ∎
Under this framework, we have that can be obtained as the minimizer of worst-case expectation for the score function when the base is elicitable. Such fact is in consonance with the reasoning behind this risk measure. The technical condition regarding convexity for is necessary for the minimax theorem used in the proof.
Let be a probability-based risk measurement and defined as in (2). If is elicitable underscore function , then we have that:
where is defined as .
From the hypotheses we get the following:
We have used the interchange between minimum and integral, see Theorem 14.60 in Rockafellar and Wets (2009), for instance. This concludes the proof. ∎
The probability measure is the barycenter of and possess an intuitive composition as an averaging of those . Note that we do not necessarily have that . If exists a reference dominant measure such that , then . We have that preserves the usual properties of an expectation such as monotonicity and linearity. Thus, as a byproduct, it can be an alternative to the sub-linear expectation .
3.3 Continuity properties
Continuity properties are important because risk measures are basically functionals that require these properties to ensure certain mathematical results, such as existence of derivatives or optimal values. In this sense, we now define continuity properties that appear in the literature of risk measures.
A risk measure is said to be:
Continuous: if implies in ;
Uniformly continuous: if exists some such that implies in ;
Lipschitz continuous: if exists some constant such that ;
Continuous from above: if , and implies in ;
Continuous from below: if , and implies in ;
Fatou continuous: if and implies in ;
Lebesgue continuous: if and implies in .
Lipschitz continuity implies the uniform one is stronger than the regular one. Monetary risk measures are Lipschitz continuous in . The more general definition for Fatou and Lebesgue continuities is applied to bounded sequences, which is the case for ; thus, we drop it. For convex risk measures, continuity from below implies its counterpart from above, which is equivalent to Fatou continuity in this case, and, consequently, the Lebesgue one. See the chapter 4 of Föllmer and Schied (2016) for details on such facts. Moreover, it is known from Theorem 2.1 of Jouini et al. (2006) and Proposition 1.1 of Svindland (2010) that -law invariant convex risk measures are automatically Fatou continuous in atom-less probability spaces. All risk measures from Example 2.2 are Lipschitz and Fatou continuous. From Monotonicity, they are also continuous from above. Moreover, EL, VaR, ES, EVaR, and ENT are continuous from below and, consequently, in the Lebesgue sense.
We now present a result that links the continuity properties of our robust risk measures to those of the family that generates them. In fact, we do not make any imposition regarding theoretical properties, and these results can be applied to any functional under the assumptions of Definition 2.1. This is useful in the context of robust finance.
Let be a probability based risk measurement and defined as in (1). Then:
If posses, , any property among regular, uniform, Lipschitz and Fatou continuity, then also does.
If fulfills, , continuity from above, below or Lebesgue, then is Fatou continuous for decreasing, increasing or any sequences, respectively.
For item (i), let such that . Then, for given we have that for any exists such that . Let . Thus, for we get that