On Multivariate Extensions of Value-at-Risk

During the last decades, researchers joined efforts to properly compare, quantify and manage risk. Regulators edict rules for bankers and insurers to improve their risk management and to avoid crises, not always successfully as illustrated by recent events.

Traditionally, risk measures are thought of as mappings from a set of real-valued random variables to the real numbers. However, it is often insufficient to consider a single real measure to quantify risks created by business activities, especially if the latter are affected by other external risk factors. Let us consider for instance the problem of solvency capital allocation for financial institutions with multi-branch businesses confronted to risks with specific characteristics. Under Basel II and Solvency II, a bottom-up approach is used to estimate a “top-level” solvency capital. This is done by using risk aggregation techniques who may capture risk mitigation or risk diversification effects. Then this global capital amount is re-allocated to each subsidiaries or activities for internal risk management purpose (“top-down approach”). Note that the solvability of each individual branch may strongly be affected by the degree of dependence amongst all branches. As a result, the capital allocated to each branch has to be computed in a multivariate setting where both marginal effects and dependence between risks should be captured. In this respect, the “Euler approach” (e.g., see Tasche, 2008) involving vector-valued risk measures has already been tested by risk management teams of some financial institutions.

Whereas the previous risk allocation problem only involves internal risks associated with businesses in different subsidiaries, the solvability of financial institutions could also be affected by external risks whose sources cannot be controlled. These risks may also be strongly heterogeneous in nature and difficult to diversify away. One can think for instance of systemic risk or contagion effects in a strongly interconnected system of financial companies. As we experienced during the 2007-2009 crisis, the risks undertaken by some particular institutions may have significant impact on the solvability of the others. In this regard, micro-prudential regulation has been criticized because of its failure to limit the systemic risk within the system. This question has been dealt with recently by among others, Gauthier et al. (2010) and Zhou (2010) who highlights the benefit of a “macro-prudential” approach as an alternative solution to the existing “micro-prudential” one (Basel II) which does not take into account interactions between financial institutions.

In the last decade, much research has been devoted to risk measures and many extensions to multidimensional settings have been investigated. On theoretical grounds, Jouini et al. (2004) proposes a class of set-valued coherent risk measures. Ekeland et al., (2012) derive a multivariate generalization of Kusuoka’s representation for coherent risk measures. Unsurprisingly, the main difficulty regarding multivariate generalizations of risk measures is the fact that vector preorders are, in general, partial preorders. Then, what can be considered in a context of multidimensional portfolios as the analogous of a “worst case” scenario and a related “tail distribution”? This is why several definitions of quantile-based risk measures are possible in a higher dimension. For example, Massé and Theodorescu (1994) defined multivariate quantiles as half-planes and Koltchinskii (1997) provided a general treatment of multivariate quantiles as inversions of mappings. Another approach is to use geometric quantiles (see, for example, Chaouch et al., 2009). Along with the geometric quantile, the notion of depth function has been developed in recent years to characterize the quantile of multidimensional distribution functions (for further details see, for instance, Chauvigny et al., 2011). We refer to Serfling (2002) for a large review on multivariate quantiles. When it turns to generalize the Value-at-Risk measure, Embrechts and Puccetti (2006), Nappo and Spizzichino (2009), Prékopa (2012) use the notion of quantile curve which is defined as the boundary of the upper-level set of a distribution function or the lower-level set of a survival function.

In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The proposed measures are based on the Embrechts and Puccetti (2006)’s definitions of multivariate quantiles. We define the lower-orthant Value-at-Risk at risk level $\alpha$ as the conditional expectation of the underlying vector of risks $\mathbf{X}$ given that the latter stands in the $\alpha$-level set of its distribution function. Alternatively, we define the upper-orthant Value-at-Risk of $\mathbf{X}$ at level $\alpha$ as the conditional expectation of $\mathbf{X}$ given that $\mathbf{X}$ stands in the $(1-\alpha$)-level set of its survival function. Contrarily to Embrechts and Puccetti (2006)’s approach, the extensions of Value-at-Risk proposed in this paper are real-valued vectors with the same dimension as the considered portfolio of risks. This feature can be relevant from an operational point of view.

Several properties have been derived. In particular, we show that the lower-orthant Value-at-Risk and the upper-orthant Value-at-Risk both satisfy the positive homogeneity and the translation invariance property. We compare the components of these vector-valued measures with the univariate VaR of marginals. We prove that the lower-orthant Value-at-Risk (resp. upper-orthant Value-at-Risk) turns to be more conservative (resp. less conservative) than the vector composed of univariate VaR. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. In particular, we show that, for Archimedean families of copulas, the lower-orthant Value-at-Risk and the upper-orthant Value-at-Risk are both increasing with respect to the risk level whereas their behavior is different with respect to the degree of dependence. In particular, an increase of the dependence amongts risks tends to lower the lower-orthant Value-at-Risk whereas it tends to widen the upper-orthant Value-at-Risk. In addition, these two measures may be useful for some applications where risks are heterogeneous in nature. Indeed, contrary to many existing approaches, no arbitrary real-valued aggregate transformation is involved (sum, min, max,$\ldots$).

The paper is organized as follows. In Section 1, we introduce some notations, tools and technical assumptions. In Section 2, we propose two multivariate extensions of the Value-at-Risk measure. We study the properties of our multivariate VaR in terms of Artzner et al. (1999)’s invariance properties of risk measures (see Section 2.1). Illustrations in some Archimedean copula cases are presented in Section 2.2. We also compare the components of these multivariate risk measures with the associated univariate Value-at-Risk (see Section 2.3). The behavior of our ${\rm VaR}$ with respect to a change in marginal distributions, a change in dependence structure and a change in risk level $\alpha$ is discussed respectively in Sections 2.4, 2.5 and 2.6. In the conclusion, we discuss open problems and possible directions for future work.

In this section, we first introduce some notation and tools which will be used later on.

From the usual definition in the univariate setting, the Value-at-Risk is the minimal amount of the loss which accumulates a probability $\alpha$ to the left tail and $1-\alpha$ to the right tail. Then, if $F_{X}$ denotes the cumulative distribution function associated with risk $X$ and $\overline{F}_{X}$ its associated survival function, then

$${\rm VaR}_{\alpha}(X):=\inf\left\{x\in\mathbb{R}:F_{X}(x)\geq\alpha\right\}$$

and equivalently,

$${\rm VaR}_{\alpha}(X):=\inf\left\{x\in\mathbb{R}:\overline{F}_{X}(x)\leq 1-\alpha\right\}.$$

Consequently, the classical univariate VaR can be viewed as the boundary of the set $\left\{x\in\mathbb{R}:\right.$ $\left.F_{X}(x)\geq\alpha\right\}$ or, similarly, the boundary of the set $\left\{x\in\mathbb{R}:\overline{F}_{X}(x)\leq 1-\alpha\right\}$.

This idea can be easily extended in higher dimension, keeping in mind that the two previous sets are different in general as soon as $d\geq 2$. We propose a multivariate generalization of Value-at-Risk for a portfolio of $d$ dependent risks. As a starting point, we consider Definition 17 in Embrechts and Puccetti (2006). They suggest to define the multivariate lower-orthant Value-at-Risk at probability level $\alpha$, for a increasing function $\underline{G}$ : $\mathbb{R}^{d}\rightarrow[0,1]$, as the boundary of its $\alpha$–upper-level set, i.e., $\partial\{\textbf{x}\in\mathbb{R}^{d}:\underline{G}(\textbf{x})\geq\alpha\}$ and analogously, the multivariate upper-orthant Value-at-Risk, for a decreasing function $\overline{G}$ : $\mathbb{R}^{d}\rightarrow[0,1]$, as the boundary of its $(1-\alpha)$–lower-level set, i.e., $\partial\{\textbf{x}\in\mathbb{R}^{d}:\overline{G}(\textbf{x})\leq 1-\alpha\}$.

Note that the generalizations of Value-at-Risk by Embrechts and Puccetti (2006) (see also Nappo and Spizzichino, 2009; Tibiletti, 1993) are represented by an infinite number of points (an hyperspace of dimension $d-1$, under some regularly conditions on the functions $\underline{G}$ and $\overline{G}$). This choice can be unsuitable when we face real risk management problems. Then, we propose more parsimonious and synthetic versions of the Embrechts and Puccetti (2006)’s measures. In particular in our propositions, instead of considering the whole hyperspace $\partial\{\textbf{x}:\underline{G}(\textbf{x})\geq\alpha\}$ (or $\partial\{\textbf{x}:\overline{G}(\textbf{x})\leq 1-\alpha\}$) we only focus on the particular point in $\mathbb{R}_{+}^{d}$ that matches the conditional expectation of X given that X stands in this set. This means that our measures are real-valued vectors with the same dimension as the considered portfolio of risks.

In addition, to be consistent with the univariate definition of $\rm{VaR}$, we choose $\underline{G}$ (resp. $\overline{G}$) as the $d-$dimensional loss distribution function $F$ (resp. the survival distribution function $\overline{F}$) of the risk portfolio. This allows to capture information coming both from the marginal distributions and from the multivariate dependence structure, without using an arbitrary real-valued aggregate transformation (for more details see Introduction).

In analogy with the Embrechts and Puccetti’s notation we will denote $\underline{{\rm VaR}}$ our multivariate lower-orthant Value-at-Risk and $\overline{{\rm VaR}}$ the upper-orthant one.

In the following, we will consider non-negative absolutely-continuous random vector⁴⁴4We restrict ourselves to $\mathbb{R}^{d}_{+}$ because, in our applications, components of $d-$dimensional vectors correspond to random losses and are then valued in $\mathbb{R}_{+}$. ${\textbf{X}=(X_{1},\ldots,X_{d})}$ (with respect to Lebesgue measure $\lambda$ on $\mathbb{R}^{d}$) with partially increasing multivariate distribution function⁵⁵5A function $F(x_{1},\ldots,x_{d})$ is partially increasing on $\mathbb{R}^{d}_{+}\setminus(0,\ldots,0)$ if the functions of one variable $g(\cdot)=F(x_{1},\ldots,x_{j-1},\cdot,x_{j+1},\ldots,x_{d})$ are increasing. About properties of partially increasing multivariate distribution functions we refer the interested reader to Rossi (1973), Tibiletti (1991). $F$ and such that ${\operatorname{\mathbb{E}}(X_{i})<\infty},$ for $i=1,\ldots,d$. These conditions will be called regularity conditions.

However, extensions of our results in the case of multivariate distribution function on the entire space $\mathbb{R}^{d}$ or in the presence of plateau in the graph of $F$ are possible. Starting from these considerations, we introduce here a multivariate generalization of the VaR measure.

Note that, under the regularity conditions, $\partial\underline{L}(\alpha)=\{\textbf{x}\in\mathbb{R}^{d}_{+}:F(\textbf{x})=\alpha\}$ has Lebesgue-measure zero in $\mathbb{R}^{d}_{+}$ (e.g., see Property 3 in Tibiletti, 1990). Then we make sense of Definition 2.1 using the limit procedure in Feller (1966), Section 3.2:

$$\operatorname{\mathbb{E}}[\,X_{i}\,|\,F(\textbf{X})=\alpha\,]=\lim_{h\rightarrow 0}\,\operatorname{\mathbb{E}}[\,X_{i}\,|\,\alpha<F(\textbf{X})\leq\alpha+h\,]\\ =\lim_{h\rightarrow 0}\,\frac{\int_{Q_{X_{i}}(\alpha)}^{\infty}x\left(\int_{\alpha}^{\alpha+h}f_{(X_{i},F(\textbf{X}))}(x,y)\,{\mathrm{d}}y\right){\mathrm{d}}x}{\int_{\alpha}^{\alpha+h}f_{F(\textbf{X})}(y)\,{\mathrm{d}}y},$$

(1)

for $i=1,\ldots,d$.

Dividing numerator and denominator in (1) by $h$, we obtain, as $h\rightarrow 0$

$$\operatorname{\mathbb{E}}[X_{i}\,|F(\textbf{X})=\alpha]=\frac{\int_{Q_{X_{i}}(\alpha)}^{\infty}x\,f_{(X_{i},F(\textbf{X}))}(x,\alpha)\,{\mathrm{d}}x}{K^{\prime}(\alpha)},$$

(2)

for $i=1,\ldots,d$, where $K^{\prime}(\alpha)=\frac{{\mathrm{d}}K(\alpha)}{{\mathrm{d}}\alpha}$ is the Kendall distribution density function. This procedure gives a rigorous sense to our $\underline{{\rm VaR}}_{\alpha}(\textbf{X})$ in Definition 2.1. Remark that the existence of $f_{(X_{i},F(\textbf{X}))}$ and $K^{\prime}$ in (2) is guaranteed by the regularity conditions (for further details, see Proposition 1 in Imlahi et al., 1999 or Proposition 4 in Chakak and Ezzerg, 2000).

In analogy with Definition 2.1, we now introduce another possible generalization of the VaR measure based on the survival distribution function.

As for $\partial\underline{L}(\alpha)$, under the regularity conditions, $\partial\overline{L}(\alpha)=\{\textbf{x}\in\mathbb{R}^{d}_{+}:\overline{F}(\textbf{x})=1-\alpha\}$ has Lebesgue-measure zero in $\mathbb{R}^{d}_{+}$ (e.g., see Property 3 in Tibiletti, 1990) and we make sense of Definition 2.2 using the limit Feller’s procedure (see Equations (1)-(2)).

From now on, we denote by $\underline{{\rm VaR}}^{1}_{\alpha}(\textbf{X})$, $\ldots$, $\underline{{\rm VaR}}^{d}_{\alpha}(\textbf{X})$ the components of the vector $\underline{{\rm VaR}}_{\alpha}(\textbf{X})$ and by $\overline{{\rm VaR}}^{1}_{\alpha}(\textbf{X})$, $\ldots$, $\overline{{\rm VaR}}^{d}_{\alpha}(\textbf{X})$ the components of the vector $\overline{{\rm VaR}}_{\alpha}(\textbf{X})$.

Note that if X is an exchangeable random vector, ${\underline{\mbox{VaR}}_{\alpha}^{i}(\textbf{X})=\underline{\mbox{VaR}}_{\alpha}^{j}(\textbf{X})}$ and ${\overline{\mbox{VaR}}_{\alpha}^{i}(\textbf{X})=\overline{\mbox{VaR}}_{\alpha}^{j}(\textbf{X})}$ for any $i,j=1,\ldots,d$. Furthermore, given a univariate random variable $X$, $\operatorname{\mathbb{E}}[X\,|F_{X}(X)=\alpha]=\operatorname{\mathbb{E}}[X\,|\overline{F}_{X}(X)=1-\alpha]=\rm{VaR}_{\alpha}(X),$ for all $\alpha$ in $(0,1)$. Hence, lower-orthant VaR and upper-orthant VaR are the same for (univariate) random variables and Definitions 2.1 and 2.2 can be viewed as natural multivariate versions of the univariate case. As remarked above, in Definitions 2.1-2.2 instead of considering the whole hyperspace $\partial\underline{L}(\alpha)$ (or $\partial\overline{L}(\alpha)$), we only focus on the particular point in $\mathbb{R}_{+}^{d}$ that matches the conditional expectation of X given that X falls in $\partial\underline{L}(\alpha)$ (or in $\partial\overline{L}(\alpha)$).