Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures

One of the main purposes of quantitative modeling in finance is to quantify the loss of a financial portfolio. Over the past two decades, various risk measures have been proposed for measuring the risk of financial portfolios. A risk measure is represented as a map assigning an extended real number (a measure of risk) to each random loss under an implicit assumption that the true loss probability distribution is known. However, in practice, the true probability distribution is often unknown or it is prohibitively expensive to calculate the risk using the true distribution. Thus, in applications, evaluating the risk of a random variable representing the loss of a financial position often involves two steps: estimating the probability distribution from available observations or from the sampling data of the random financial loss via, e.g., Monte Carlo method and then plugging the estimated distribution into a risk measure to quantify the financial loss. This is because the risk measures are mostly law invariant, that is, they are determined only by the probability distributions of random variables. For the loss of a financial portfolio, a measure of risk computed based on the estimated distribution is known as a plug-in estimate for the risk measure [3].

Let $X$ denote the random loss of a financial portfolio on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\rho$ be a law invariant risk measure. The plug-in estimate for $\rho(X)$ is given by $\varrho(\widehat{P})$, where $\widehat{P}$ is the empirical distribution based on available observations and $\varrho$ is a risk functional defined by

see e.g. [4, 5]. In the literature, Cont et al. [2] first study the quality of statistical estimators of the law invariant risk measures using Hampel’s classical concept of qualitative robustness [6], that is, a risk functional estimator is said to be qualitatively robust if it is insensitive to the variation of the sampling data. The research is important because perceived data (particularly empirical data) may contain some noise. Without such insensitivity, financial activities based on the risk measures may cause damage. For instance, when $\rho(X)$ is applied to allocate the risk capital for an insurance company, altering the capital allocation may be costly. According to Hampel’s theorem, Cont et al. [2] demonstrate that the qualitative robustness of a statistical estimator is equivalent to the weak continuity of the risk functional, and that value at risk (VaR) is qualitatively robust whereas conditional value at risk (CVaR) is not.

Krätschmer et al [7] argue that the use of Hampel’s classical concept of qualitative robustness may be problematic because it requires the risk measure essentially to be insensitive with respect to the tail behaviour of the random variable and the recent financial crisis shows that a faulty estimate of tail behaviour can lead to a drastic underestimation of the risk. Consequently, they propose a refined notion of qualitative robustness that applies also to tail-dependent statistical functionals and that allows us to compare statistical functionals in regards to their degree of robustness. The new concept captures the trade-off between robustness and sensitivity and can be quantified by an index of qualitative robustness. Furthermore, under the new concept, Krätschmer et al [1] analyze the qualitative robustness to the law-invariant convex risk measure on Orlicz spaces and show that CVaR and spectral risk measures are all qualitatively robust when the perturbation of probability distribution is restricted to a finer topological space. Alternative generalizations of Hampel’s theorem can be found for strong mixing data (Zähle [8, 9]) and for stochastic processes in various ways (Boente et al [10] and Strohriegl and Hable [11]). For comprehensive study of statistical robustness, we refer readers to [12, 13, 14, 15] and references therein.

In this paper, we take a step further by deriving an error bound for the plug-in estimators of law invariant risk measures in terms of the variation of data and we call the analysis quantitative because no such error bound is established in the existing qualitative robust analysis. This is achieved by adopting different metrics to measure the discrepancy of the estimators and the variation of data. Specifically, we use the Fortet-Mourier metrics as opposed to the Lévy distance in Cont et al. [2] or the weighted Kolmogorov metric in Krätschmer et al. [7] to quantify the data variation (the perturbation of the true probability distributions). Moreover, we introduce a new notion of the so-called admissible laws, which effectively restrict the scope of data variation. The new metrics enable us to establish an explicit relationship between the discrepancy of the laws of the plug-in estimators (of law invariant risk measure based on the true data and perturbed data) and the discrepancy of the associated probability distributions of the data. The research is inspired by the recent work of Guo and Xu [16] where the authors derive quantitative statistical robustness for preference robust optimization models under Kantorovich metric. The main contributions of the paper can be summarized as follows.

First, we introduce the notion of admissible laws induced by a probability metric, which is a class of probability distributions whose discrepancy with the law of the Dirac measure at $0$ is finite. The admissibility effectively restricts the scope of data perturbation. Using the notion, we compare the admissibility under $\phi$-topology and the Fortet-Mourier metric.

Second, we propose to use the Fortet-Mourier metric to quantify the variation of the probability measure. The metric enables us to establish an explicit relationship between the discrepancy of the laws of the plug-in estimators of law invariant risk measure based on the true data and perturbed data by noise and the change of the true underlying probability measures when the risk functional is Lipschitz continuous on a class of admissible laws. We find that the risk functionals associated with the general moment-type convex risk measures are Lipschitz continuous.

Third, we introduce the concept of Lipschitz continuity for a general statistical functional on a class of admissible laws induced by the Fortet-Morier metric and find that for the Lipschitz continuous risk measure, the parameter of the Fortet-Mourier metric allows us to compare the tail-dependent risk measures with regard to their degree of robustness, i.e., the index of statistical robustness.

Fourth, we apply the new approach to examine the quantitative statistical robustness of a range of well known risk measures, including CVaR, optimized certainty equivalent, shortfall risk measure and conclude that under mild conditions, they are all quantitatively robust, and the indexes of quantitative robustness to them are also calculated.

The rest of the paper is organized as follows. In Section 2, we set up the background of the problem for research. In Section 3, we introduce the concept of Fortet-Mourier metric and admissible laws. In section 4, we establish the quantitative statistical robustness theory and compare with the qualitative statistical robustness theory. In section 5, we apply our theory to risk measures and give some examples. Some technical details are given in the appendix.

In this section, we discuss the background of statistical robustness in the context of law invariant risk measures. We begin by a brief review of law invariant risk measures and its estimation, and then move to explain the issues when the data may contain noise.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be an atomless probability space, where $\Omega$ is a sample space with sigma algebra $\mathcal{F}$ and $\mathbb{P}$ is a probability measure. Let $X:(\Omega,\mathcal{F},\mathbb{P})\rightarrow{\rm I\!R}$ be a financial loss and $F_{X}(x):=\mathbb{P}(X\leq x)$ be the law or the probability distribution of $X$. For $p\geq 1$, let $\mathscr{L}^{p}(\Omega,{\cal F},\mathbb{P})$ ($\mathscr{L}^{p}$ for short) denote the space of random variables mapping from $(\Omega,{\cal F},\mathbb{P})$ to ${\rm I\!R}$ with finite $p$-th order moments. We say that a map $\rho:\mathscr{L}^{1}\to\overline{{\rm I\!R}}:={\rm I\!R}\cup\{+\infty\}$ is a convex risk measure¹¹1We note that the canonical model space for law invariant convex risk measure is $\mathscr{L}^{1}$ [17]. [18] if it satisfies the following properties:

1. (i)

   Monotonicity: $\rho(X)\leq\rho(Y)$ for $X,Y\in\mathscr{L}^{1}$ with $X\leq Y$ $\mathbb{P}$-almost surely;

2. (ii)

   Translation invariance: $\rho(X+c)=\rho(X)+c$ for $X\in\mathscr{L}^{1}$ and $c\in{\rm I\!R}$;

3. (iii)

   Convexity: $\rho(\lambda X+(1-\lambda)Y)\leq\lambda\rho(X)+(1-\lambda)\rho(Y)$ for $X,Y\in\mathscr{L}^{1}$ and $\lambda\in[0,1]$.

Moreover, if $\rho$ satisfies positive homogeneity, i.e., for any $\alpha\geq 0$, $\rho(\alpha X)=\alpha\rho(X)$, then $\rho$ is a coherent risk measure, see [19, 18] for the original definitions of these concepts. A risk measure $\rho$ is said to be law invariant if $\rho(X)=\rho(Y)$ for $X$ and $Y$ having the same law. We refer readers to Föllmer and Weber [20] for a recent overview of risk measures.

As discussed in [2, 7], it is a widely-accepted procedure to estimate the risk of a financial loss by means of a Monte Carlo method or from a set of available observations. Such a procedure is particularly sensible when $\rho$ is law invariant. The following proposition states that the law invariance of a risk measure $\rho$ is equivalent to the existence of a risk functional $\varrho$ in (1.1).

The result is well-known, see for instance Delage et al. [21] for random variables defined in $\mathscr{L}^{\infty}$. The usefulness of the representation is that it naturally captures the law invariance and allows one to define any law invariant risk measure directly over the space of probability measures $\mathscr{P}({\rm I\!R})$ induced by random variables in $\mathscr{L}^{\infty}$ (also known as probability distributions), see Fritelli et al. [22]. Dentcheva and Ruszczyński [23] take it further to define a class of law invariant risk measures in the space of quantile functions directly. In a more recent development, Haskell et al. [24] extend the research to a broad class of multi-attribute choice functions defined over the space of survival functions. Let $P:=\mathbb{P}\circ X^{-1}$ be the push-forward probability measure on ${\rm I\!R}$ induced by $X$. Since $\mathbb{P}(X\leq x)$ coincides with $P((-\infty,x])$ ($P(x)$ for short), we also call $P$ the distribution or the law of $X$ interchangeably throughout the paper. Consequently, we can write (2.1) as (1.1).

In this paper, we are not concerned with the definition of risk measures over the space of probability distributions or the space of quantile functions, rather we concentrate on the stability of statistical estimators of law invariant risk measures. The risk functional $\varrho(P)$ with the law $P=\mathbb{P}\circ X^{-1}$ can be used in a natural way to construct an estimator for the risk $\rho(X)$ of $X\in\mathscr{L}^{1}$. All one needs to do is to take an estimate $P_{N}$ of $P$ based on the available observations of $X$ and then to plug this estimator into the risk functional $\varrho$ to obtain the desired estimator of $\rho(X)$, i.e.,

where in this paper, $P_{N}$ can be seen as the empirical distribution of an independent and identically distributed (i.i.d., for short) sequence $\xi^{1},\xi^{2},\ldots,\xi^{N}$ of historical observations or Monte Carlo simulations, i.e.,

Here and later on $\mathbf{1}_{A}$ denotes the indicator function of event $A$. Indeed, $P_{N}$ can be a fairly general estimates, for instance, $P_{N}$ can be a smoothed empirical distribution based on uncensored data or empirical distribution based on censored data, see, e.g., [3] or empirical distribution based on identically distributed dependent data, see, e.g., [9].

We can see that $\widehat{\varrho}_{N}$ is a mapping from ${\rm I\!R}^{N}$ to ${\rm I\!R}$. Figure 1 illustrates the relationship between the risk functionals, their estimators and the spaces associated.

In practice, the samples obtained from empirical data may contain noise. In that case, we might regard the samples as generated by a perturbed random variable $Y$ with law $Q$, that is, $Q=\mathbb{P}\circ Y^{-1}$. Let $\tilde{\xi}^{1},\cdots,\tilde{\xi}^{N}$ be i.i.d samples from $Y$. Then the practical empirical distribution function for estimating the law of $X$ is

and the practical estimator is $\tilde{\varrho}_{N}=\widehat{\varrho}_{N}(\tilde{\xi}^{1},\cdots,\tilde{\xi}^{N}):=\varrho(Q_{N})$ with perceived empirical data whereas $\widehat{\varrho}_{N}$ is a statistical estimator with noise being detached. Since we are unable to obtain the latter, we tend to use the former as a statistical estimator of $\rho(X)$ and this works only if the two estimators are sufficiently close.

To quantify the closeness, we may look into the discrepancy between the laws of the two estimators under some metric $\mathsf{d\kern-0.70007ptl}$, i.e.,

where $P^{\otimes N}$ and $Q^{\otimes N}$ denote the probability measures on measurable space $\left({\rm I\!R}^{N},{\cal B}({\rm I\!R})^{\otimes N}\right)$ with marginals $P$ and $Q$ on each $({\rm I\!R},{\cal B}({\rm I\!R}))$ respectively, $\mathcal{B}({\rm I\!R})$ denotes the corresponding Borel sigma algebra of ${\rm I\!R}$. Since neither $P$ nor $Q$ is known, we want the discrepancy to be uniformly small for all $P$ and $Q$ over a subset of admissible laws on $\mathscr{P}({\rm I\!R})$ so long as $Q$ is sufficiently close to $P$ under some metric $\mathsf{d\kern-0.70007ptl}^{\prime}$. The uniformity may be interpreted as robustness. Qualitative robustness refers to the case that the relationship between $\mathsf{d\kern-0.70007ptl}\left(P^{\otimes N}\circ\widehat{\varrho}_{N}^{-1},Q^{\otimes N}\circ\tilde{\varrho}_{N}^{-1}\right)$ and $\mathsf{d\kern-0.70007ptl}^{\prime}(P,Q)$ is implicit whereas quantitative robustness refers to the case that the relationship is explicit, i.e., a function of the latter can be used to bound the former, and this is what we aim to achieve in this paper because qualitative robustness have been well investigated, for instance, in [2, 1, 7].

There are two essential elements in investigating both the qualitative and quantitative statistical robustness of a risk functional: One is the specific choice of probability metrics but not just the topologies generated by them, see, e.g., [12, 2, 7], to quantify the change of the law $P$ and to estimate the discrepancy between the laws of two estimators, i.e., (2.5); the other is the determination of the subset $\mathscr{M}$ of admissible laws in $\mathscr{P}({\rm I\!R})$ (see, e.g., [7, 9]), containing all empirical distributions: $\mathscr{M}_{1,\mathrm{emp}}\subset\mathscr{M}$, to restrict the perturbation of the law $P$. For instance, the subset $\mathscr{M}$ may be specified via some generalized moment conditions, which are interesting in econometric or financial applications.

To introduce these two essential elements thoroughly, some preliminary notions and results in probability theory and statistics such as $\phi$-weak topology are required. We first give a sketch of them to prepare our discussions in the follow-up sections. Let $\phi:{\rm I\!R}\to[0,\infty)$ be a continuous function and ${\cal M}_{1}^{\phi}:=\left\{P^{\prime}\in\mathscr{P}({\rm I\!R}):\int_{{\rm I\!R}}\phi(t)P^{\prime}(dt)<\infty\right\}$. In the particular case when $\phi(\cdot):=|\cdot|^{p}$ and $p$ is a positive number, write ${\cal M}_{1}^{p}$ for ${\cal M}_{1}^{|\cdot|^{p}}$. Note that ${\cal M}_{1}^{\phi}$ defines a subset of probability measures in $\mathscr{P}({\rm I\!R})$ which satisfies the generalized moment condition of $\phi$. From the definition, we can see that ${\cal M}_{1}^{p_{2}}\subset{\cal M}_{1}^{p_{1}}$ for any positive numbers $p_{1},p_{2}$ with $p_{1}<p_{2}$ due to Hölder inequality.

Clearly, $\phi$-weak topology is finer than the weak topology, and the two topologies coincide if and only if $\phi$ is bounded. It is well known (see [7, Lemma 3.4]) that $\phi$-weak convergence is equivalent to weak convergence, denoted by ${P_{l}}\xrightarrow[]{w}P$, together with $\int_{{\rm I\!R}}\phi(t)P_{l}(dt)\to\int_{{\rm I\!R}}\phi(t)P(dt)$. Moreover, it follows by [7, 1] that the $\phi$-weak topology on ${\cal M}_{1}^{\phi}$ is generated by the metric $\mathsf{d\kern-0.70007ptl}_{\phi}:{\cal M}_{1}^{\phi}\times{\cal M}_{1}^{\phi}\to{\rm I\!R}$ defined by

for $P,Q\in{\cal M}_{1}^{\phi}$, where $\mathsf{d\kern-0.70007ptl}_{\mathrm{Prok}}:\mathscr{P}({\rm I\!R})\times\mathscr{P}({\rm I\!R})\to{\rm I\!R}_{+}$ is the Prokhorov metric defined by

where $A^{\epsilon}:=A+B_{\epsilon}(0)$ denotes the Minkowski sum of $A$ and the open ball centred at $0$ on ${\rm I\!R}$ and $\mathcal{B}({\rm I\!R})$ is the corresponding Borel sigma algebra on ${\rm I\!R}$. We note that the Prokhorov metric metrized the weak topology on ${\rm I\!R}$ see, e.g., [25].