The Skew Normal multivariate risk measurement framework

In this paper, we consider a random vector X=X1,X2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X=\left( X_1,X_2\right) $$\end{document} following a multivariate Skew Normal distribution and we provide an explicit formula for the expected value of X conditioned to the event X≤X¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \le \overline{X}$$\end{document}, with X¯∈R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{X} \in \mathbb {R}^2$$\end{document}. Such a conditional expectation has an intuitive interpretation in the context of risk measures.


Introduction
The employment of nonstandard probability distributions in financial risk theory represents a growing field of research, leading to either theoretical additions as well as relevant practical implications (see e.g., the monograph Rachev and Mittnik 2000 and the recent contributions Hitaj et al. 2018;Ortobelli Lozza et al. 2018). In this context, a relevant role is played by the Skew Normal distributions. Indeed, multivariate Skew Normal distributions can be looked upon as a generalization of the Gaussian distribu-tions. One of their key features is the introduction of an additional shape parameter governing the asymmetry of the density function. In so doing, Skew Normal distributions are able to capture several aspects of applied science, meaning that they are suitable for a wide range of application, including finance and management science. They were initially introduced by Azzalini and Dalla Valle (1996). After their seminal contribution, such distributions have been extensively studied and analyzed in a large number of papers. Just to cite a few, a list of relevant statistical applications is provided in Azzalini and Capitanio (1999), whereas Branco and Dey (2001) extend Azzalini and Dalla Valle's methods to multivariate Skew Elliptical distributions. Moreover, a further class of multivariate Skew Normal distributions is introduced by Gupta et al. (2004). In the next year, Azzalini (2005) presents a rich overview on the family of Skew Normal of distributions and on their generalizations for continuous random variables sharing the same generating mechanism.
Among the most recent contributions, it is worth mentioning at least two articles where Skew Normal distributions are used to deal with real data: an application to HIV-RNA by Ghosh et al. (2007) and a methodology for measurement error based on scale mixtures of Skew Normal distributions. There is also a further stream of studies on finite mixtures of multivariate Skew Normal distributions (e.g., Cabral et al. 2012;Lee and Mclachlan 2013).
To the best of our knowledge, Skew Normal distributions have been studied in detail in the univariate case. As far as the multivariate framework is concerned, scanty attention has been paid to an explicit formulation of the expectation of such random variables conditioned to the fact that a prefixed barrier is not ever crossed. Such a conditional expectation, known as tail conditional expectation, has been calculated in Bernardi (2013) for univariate Skew Normal distributions and their mixtures. The conditional expectation is a highly relevant concept in financial applications: namely, it is quite frequent that expectation of a random variable must be calculated based on some given conditioning event. For example, the conditioning event might be something occurring in the financial market or in the bank system, such as the distress of an important institution, possibly leading to the diffusion of distress among the remaining institutions. In Bernardi et al. (2016), the contagion risk in a financial framework is taken into account, to evaluate the systemic relevance of interconnected institutions. For this purpose, the instrument constructed by Bernardi et al. is called SCoVaR and it turns out to be an extension of CoVaR, designed by Adrian and Brunnermeier (2016). Bernardi et al. (2017c), instead makes explicit use of the the Skew Normal generating mechanism to calculate the asymptotic distribution of the Network CoVaR, a statistical procedure to test the pairwise systemic dominance of a financial institution over another one. Further insights on the relation between risk assessment and Skew Normal distributions can be found in Bernardi (2013) and Bernardi et al. (2017a).
Namely, in the absence of specific formulas for conditional expectation of random variables distributed according to the Skew Normal law, the present paper intends to fill this gap. In particular, we aim at providing an explicit formula for the bivariate Tail Conditional Expectation (TCE, hereafter), defined as where X = (X 1 , X 2 ) is a bivariate Skew Normal random variable and X = X 1 , X 2 ∈ R 2 . Clearly, X can be interpreted as a suitable benchmark, based on the chosen application.
When dealing with bivariate Skew Normal distributions it is worth noting that the bivariate TCE defined in Eq. (1) strongly differs from the expectation of the involved conditional random variable either from a probabilistic and a risk management perspective. From the probabilistic point of view, it is undoubtedly true that a special distribution of the skew family, namely, the Extended Skew Normal distribution, is closed under marginalization and conditionalization. Therefore an alternative formulation of the tail conditional expectation in Eq. (1) can be calculated with reference to the univariate conditional distribution of the bivariate Skew Normal, see Bernardi (2013). However, such a conditional expected value, i.e., TCE = = E X 1 ≤ X 1 | X 2 = X 2 does not involve the bivariate distribution of the random variable (X 1 , X 2 ). From the risk management perspective, instead, the employed definition of tail conditional expectation is not without consequences. Indeed, as recently noted by Bernardi et al. (2017a), the TCE = risk measure suffers the lack of the consistency property since it does not preserve the stochastic ordering induced by the bivariate distribution. This is especially true for high values of the correlation between variables. The TCE ≤ defined in Eq. (1) instead preserves the stochastic ordering of the bivariate distribution and it can be effectively used as measure of risk.
Several reasonings motivate the relevance of the provided formula for the bivariate TCE under the Skew Normal assumption. First, it generalises the concept of bivariate TCE to a non-Gaussian framework being characterised by the presence of asymmetry, while collapsing to the usual Gaussian case when the involved random variables are symmetric. Second, the TCE provides a convenient way to calculate the CoES risk measure (see, e.g., Adrian and Brunnermeier 2016; Bernardi and Catania 2018) by simply plugging-in the marginal Value-at-Risk levels. More importantly, as far as it concerns the theoretical properties, the TCE satisfies the sub-additive axiom thereby being a coherent risk measure while the corresponding quantile-based measure (the extension of the VaR, namely, the CoVaR) is not necessarily coherent, see Acerbi and Tasche (2002) for a comprehensive discussion of the axiomatic theory of risk measures. The coherence property is even more relevant when mixtures of Skew distributions are involved. The provided formula for TCE easily extends to finite mixture models by adopting the approach of Bernardi (2013) and Bernardi et al. (2017b). As the final remark, the TCE can be effectively exploited to build a network-based systemic risk measure as in Bernardi et al. (2016).
The remainder of the paper can be outlined as follows: Sect. 2 introduces the statement and the basic notation of the problem. Section 3 collects the procedure and the main results for the explicit formulation of the conditional expectation and Sect. 4 concludes.
Valle 1996; Gupta et al. 2004, among other contributions). For notational convenience, throughout this section we suppress the symbol (≤) when referring to the TCE as defined in Eq. (1). In this case, the probability density function is more complex. Namely, beginning from the standard Gaussian unimodal probability density function the expression of the probability density function in the bivariate case is where σ 1 , σ 2 , ρ have the standard meanings, whereas δ 0 and γ 1 are asymmetric parameters. For an extended explanation, see Gupta et al. (2004). In order to lighten the notation, call the normalising constant of the previous Eq.
(2). The TCE of X 1 is (see Gupta et al. 2004) is defined as: where 1 (·) denotes the indicator variable, i.e., 1 (x ∈ A) = 1 if and only if x ∈ A and zero otherwise, while P X ≤ X is the joint probability that the bivariate random variable X = (X 1 , X 2 ) falls below the predetermined threshold X = X 1 , X 2 . The denominator of Eq.
(3) can be easily calculated as the cumulative distribution function of the bivariate Skew Normal distribution in Eq.
(2), as follows As concerns the numerator of Eq.
(3), we have We can exploit a typical change of variables, i.e., t j = x j − μ j σ j , for j = 1, 2. The determinant of the related Jacobian matrix is 1 σ 1 σ 2 , which leads to the following where with dt 2 which is a function of t 1 . Hereafter, the dependence of the quantities denoted by J k on the random variables has been suppressed for notational convenience. Hence, the TCE we intend to calculate explicitly is decomposed into the sum of the two integrals J 1 and J 2 . In the next section we are going to describe the approach to compute them.

Calculation of the TCE
The complete calculation of the TCE (4) must be carried out in successive steps. Our procedure can be outlined as follows: (i) since both integrals (6) and (7) contain the same integral, which will be denoted by J 3 , we focus on the explicit calculation of J 3 ; (ii) since J 3 can be further decomposed into the sum of 2 integrals, say J 4 and J 5 , we separately consider them; we recognize that J 4 can be calculated, whereas J 5 contains an integral of the kind y k e −y 2 dy, where k is an integer number; (iii) the 2 occurrences regarding J 5 have to be treated separately: if k is odd, the integral can be computed; if k is even, J 5 contains an integral of the kind J 6 = a −∞ (t)dt, where (t) is the cumulative distribution function of the standard unimodal Gaussian distribution; (iv) J 6 can be approximated by series expansion, consequently the expression of J 3 is accomplished; (v) finally, J 1 and J 2 can be calculated by replacing the above formulation of J 3 ; in this form, some coefficients and integrals appear, but they can be determined by comparison with the original formulations of J 1 and J 2 .
As can be easily inferred, this procedure needs to be separated in several steps, due to the complications which will occur during the computation. The starting point is the integral in square brackets appearing in both (6) and (7). Call it J 3 , defined as A simple transformation leads to The integral in (9) can be approximated by using a series expansion of the Gaussian function . Standard mathematical analysis theory establishes that Plugging (10) into (9) yields: where Integrals J 4 and J 5 appearing in (11) can be treated by 2 different changes of variables. First, consider J 4 . By setting we have On the other hand, J 5 needs another change of variable. Call consequently we have In order to simplify the notation, call The integral in the right-hand side of (15) can be solved by distinguishing the cases of k even or odd.
(i) k is an odd number. This means that there exists an integer h such that k = 2h + 1. By making the change of variable t = y 2 , we obtain that y 2h+1 e −y 2 dy = 1 2 t h e −t dt.
By iterating the integration by parts and replacing the variable t with y 2 at the end of the procedure, we have: (ii) k is an even number. We can state that there exists an integer h such that k = 2h. Even in this case integration by parts is necessary. After iterating it, the involved integral becomes: since lim z→−∞ (z (z) + φ(z)) = 0. By plugging (19) into (18) we obtain: Now, the expressions (17) (for odd values of k) and (20) (for even values of k) have to be inserted into (15). By distinguishing the odd and even cases, integral J 5 turns out to be: Although the approximated expression of J 5 is particularly complex, it can be finally employed to accomplish a suitable form for J 3 . Integrals J 4 and J 5 are going to be inserted into J 3 , subsequently J 3 will be inserted into J 1 . In other words, (14) and (15) are plugged into (11), which is plugged into (6). The expression of α in (16) must be considered as well. At the end of this tedious and long process, and for a sufficiently large value of , we obtain: which reduces to where the α's are constants that can be determined by comparing the two terms of the equality in (23). By rearranging terms, (23) is equal to: . The cases α = 2n +2, 2n +2−s, 4n +3−s have been already treated above. Note that α = 2n + 2 is even whereas α = 2n + 2 − s (α = 4n + 3 − s) is even when s is even (s is odd) and odd otherwise. When α is even, the integral J 7α can be approximated as in (20). When α is odd, then the integral J 7α is as in (17). Also J 8 can be properly treated by using the series expansion of the Gaussian in (10) and the binomial expansion. Recall that We have Consider the exponential function in (28). The exponent can be rewritten as follows: (−1) n (α 3 t 1 ) 2n+1 n!(2n + 1) Now an intuitive change of variable can be made:

Conclusions and further developments
This paper is entirely devoted to the complex calculation of the TCE of a random vector which is composed of bivariate Skew Normal random variables. We outlined the whole procedure and developed most steps in an extended form. From the risk management perspective it is worth noting that the provided risk measure can be effectively used to assess the systemic risk contributions of different institutions belonging to a given market extending the SCoVaR measure introduced by Bernardi et al. (2016). Furthermore, the provided risk measure enjoys the consistency property since it preserves the stochastic ordering induced by the bivariate distribution.