Interfaces with Other DisciplinesSimulation methods for robust risk assessment and the distorted mix approach
Introduction
Capital requirements are an instrument to limit the downside risk of financial companies. They constitute an important part of banking and insurance regulation, for example, in the context of Basel III, Solvency II, and the Swiss Solvency test. Their purpose is to provide a buffer to protect policy holders, customers, and creditors. Within complex financial networks, capital requirements also mitigate systemic risks.
The quantitative assessment of the downside risk of financial portfolios is a fundamental, but arduous task. The difficulty of estimating downside risk stems from the fact that extreme events are rare; in addition, portfolio downside risk is largely governed by the tail dependence of positions which can hardly be estimated from data and is typically unknown. Tail dependence is a major source of model uncertainty when assessing the downside risk.
In practice, when extracting information from data, various statistical tools are applied for fitting both the marginals and the copulas – either (semi-)parametrically or empirically. The selection of a copula is frequently made upon mathematical convenience; typical examples include Archimedean copulas, meta-elliptical copulas, extreme value copulas, or the empirical copula, see e.g. McNeil, Frey, & Embrechts (2015). The statistical analysis and verification is based on the available data and is center-focused due to limited observations from tail events. This approach is necessarily associated with substantial uncertainty. The induced model risk thus affects the computation of monetary risk measures, the mathematical basis of capital requirements. These functionals are highly sensitive to tail events by their nature – leading to substantial misspecification errors of unknown size.
In this paper, we suggest a novel approach to deal with this problem. We focus on the downside risk of portfolios. Realistically, we assume that the marginal distributions of individual positions and their copula in the central area can be estimated sufficiently well. We suppose, however, that a satisfactory estimation of the dependence structure in the tail area is infeasible. Instead, we assume that practitioners who deal with the estimation problem share viewpoints on a collection of copulas that potentially capture extremal dependence. However, practitioners are uncertain about the appropriate choice among the available candidates.
The family of copulas that describes tail dependence translates into a family of joint distributions of all positions and thus a collection of portfolio distributions. To combine the ingredients to joint distributions, we take a particularly elegant approach: The Distorted Mix (DM) method developed by Li, Yuen, & Yang (2014) constructs a family of joint distributions from the marginal distributions, the copula in the central area and several candidate tail copulas. A DM copula is capable of handling the dependence in the center and in the tail separately. We use the DM method as the starting point for a construction of a convex family of copulas and a corresponding set of joint distributions.1
Once a family of joint distributions of the positions is given, downside risk in the face of uncertainty can be computed employing a classical worst case approach. To quantify downside risk, we focus on robust average value at risk (AV@R). The risk measure AV@R is the basis for the computation of capital requirements in both the Swiss solvency test and Basel III. As revealed by the axiomatic theory of risk measures, AV@R has many desirable properties such as coherence and sensitivity to tail events, see Föllmer & Schied (2004). In addition, AV@R is m-concave on the level of distributions, see Bellini & Bignozzi (2015), and admits the application of well-known optimization techniques as described in Rockafellar & Uryasev (2000) and Rockafellar & Uryasev (2002).
To be more specific, we consider a -dimensional random vector with given marginals and a copula in a set of distorted mix copulas . Considering aggregate losses for some measurable function , we study the worst case riskwhere signifies AV@R at some fixed level. The exact problem will be described in Section 2.2.
Our model setup leads to a continuous stochastic optimization problem to which we apply a combination of stochastic approximation and sample average approximation. We explain how these techniques may be used to reduce the dimension of the mixture space of copulas. We discuss the solution technique in detail and illustrate its applicability in several examples.
The main contributions of the paper are:
- 1.
For a given family of copulas modeling tail dependence, we describe a DM framework that conveniently allows worst case risk assessment.
- 2.
We provide an efficient algorithm that numerically computes the worst case risk and identifies worst case copulas in a lower-dimensional mixture space.
- 3.
We successfully apply our framework in selected case studies. The considered examples are financial markets and cyber risk.
The paper is structured as follows. Section 2 explains the DM approach to model uncertainty and formulates the optimization problem associated to the computation of robust AV@R. In Section 3, we develop an optimization solver combining stochastic approximation (i.e., the projected stochastic gradient method) and sample average approximation: stochastic approximation identifies candidate copulas and a good approximation of the worst-case risk; in many cases, risk is insensitive to certain directions in the mixture space of copulas, enabling us to use sample average approximation to identify worst-case solutions in lower dimensions. Section 4 discusses two applications of our framework, namely to financial markets and cyber risk. Section 5 concludes with a discussion of potential future research directions.
The concept of model uncertainty or robustness is a challenging topic in practice that has also been intensively discussed in the academic literature. Risk refers to situations in which possible scenarios and their associated probabilities are known; uncertainty in contrast describes circumstances in which the probabilities of events are not known, but are characterized by a nontrivial set of probability measures. This paper considers model uncertainty of this type.
The underlying key assumption concerns the structure of the considered probability measures. We assume that marginal distributions and the dependence structure of typical events is known; uncertainty arises from tail dependence. Risk is measured on the basis of a worst-case approach, a suitable algorithm is suggested and implemented in case studies. These structural assumptions on the class of probability measures are motivated by typical considerations in practice and distinguish our analysis from previous approaches in the literature.
Our methodology parallels the one chosen by Glasserman & Xu (2014) in the sense that both papers study the worst-case in a given class of models. However, Glasserman & Xu (2014) consider probability measures that are in some neighborhood of a given benchmark model in terms of relative entropy or another divergence measure. Similarly, Hu & Hong (2013) and Breuer & Csiszár (2016) study ambiguity sets defined by relative entropy. While divergence measures are not proper metrics, optimal transport costs such as Wasserstein distances are metrics under regularity conditions on the cost function; Blanchet & Murthy (2019) investigate model uncertainty in such a framework. Bartl, Drapeau, & Tangpi (2020) use a similar approach and study robust optimized certainty equivalent risk measures in the context of optimal transport costs.
Another, complementary perspective on robustness comes from statistics, as suggested by Hampel (1971). By Hampel’s famous theorem, the classical notion of robustness can be characterized by continuity properties of functionals with respect to the weak topology. Distribution-based convex risk measures such as average value of risk are not continuous in this sense and thus not Hampel-robust, see Cont, Deguest, & Scandolo (2010) and Kou, Peng, & Heyde (2013). A refined notion of Hampel robustness that corresponds to finer topologies on sets of probability measures is suggested in the seminal paper Krätschmer, Schied, & Zähle (2014) and applied to risk measures; the proposal in Krätschmer et al. (2014) lifts the corresponding issues of robust statistics to a higher level that permits a comprehensive analysis.
In the current paper, we focus on worst-case AV@R in a multi-factor model. We are interested in the worst-case risk if the dependence structure is uncertain. This is closely related to papers that derive bounds in the face of partial information about dependence, cf. Embrechts, Puccetti, & Rüschendorf (2013), Bernard, Jiang, & Wang (2014), Bernard & Vanduffel (2015), Rüschendorf (2017), Puccetti, Rüschendorf, Small, & Vanduffel (2017), Li, Shao, Wang, & Yang (2018), Embrechts, Liu, & Wang (2018), Weber (2018), and Hamm, Knispel, & Weber (2020). In contrast to these contribution, we propose an algorithmic DM approach that is based on candidate copulas; this setting is very flexible in terms of the marginal distributions and copulas that are considered. Our framework is suitable for capturing uncertainty about the dependence in specific regions of a joint distribution. A simpler setting of mixture distributions is studied in Zhu & Fukushima (2009) and Kakouris & Rustem (2014).
Our algorithm builds on sampling-based stochastic optimization techniques. Applications of stochastic approximation and stochastic average approximation to the evaluation of risk measures were investigated by Rockafellar & Uryasev (2000), Rockafellar & Uryasev (2002), Dunkel & Weber (2007), Bardou, Frikha, & Pagès (2009), Dunkel & Weber (2010), Meng, Sun, & Goh (2010), Sun, Xu, & Wang (2014), and Bardou, Frikha, & Pagès (2016). The contributions discuss different risk measures including AV@R and utility-based shortfall risk, efficient estimation including variance reduction, portfolio optimization and hedging but do not concentrate on model uncertainty. Without a specific focus on risk measures, the relevant simulation techniques are also discussed in Kushner & Yin (2003), Shapiro (2003), Fu (2006), Bhatnagar, Prasad, & Prashanth (2013), and Kim, Pasupathy, & Henderson (2015). Ghosh & Lam (2019) study worst-case approximations for performance measures in the face of uncertainty, based on stochastic approximation; their analysis does, however, not specifically consider uncertainty about dependence in different regions of multivariate distributions as captured by the DM approach in this paper.
Section snippets
Distorted mix copula
Letting be an atomless probability space, we consider the family of random variables . The task consists in computing the risk of an aggregate loss random variable for a risk measure . A finite distribution-based monetary risk measure is a functional with the following three properties:
- •
Monotonicity:
- •
Cash-invariance:
- •
Distribution-invariance:
We consider a specific factor structure of aggregate
Optimization solver
In this section, we develop an algorithm solving problem (6) that builds on two classical approaches: Stochastic Approximation (SA) and Sample Average Approximation (SAA). While SA is an iterative optimization algorithm that is based on noisy observations, SAA first estimates the whole objective function and transforms the optimization into a deterministic problem. We combine both approaches.
The standard stochastic gradient algorithm of SA quickly approximates the worst-case risk, but the
Applications
Our method is flexible and can be used in multiple application domains. For the purpose of illustrating its applicability, we consider two case studies. The first example in Section 4.1 is based on a substantial amount of financial market data and allows a good calibration of copulas. Model risk can thereby be reduced.5 For cyber risk, the second example discussed in Section 4.2, only few observations are
Conclusion
Uncertainty requires suitable techniques for risk assessment. In this paper, we combined stochastic approximation and stochastic average approximation to develop an efficient algorithm to compute the worst case average value at risk in the face of tail uncertainty. Dependence was modelled by the distorted mix method that flexibly assigns different copulas to different regions of multivariate distributions. The method is computationally efficient and allows at the same time to identify copulas
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