THE SKEW NORMAL MULTIVARIATE RISK MEASUREMENT FRAMEWORK

In this paper, we consider a random vector X = (X_1;X_2) following a multivariate Skew Normal distribution and we provide an explicit formula for the expected value of X conditioned to the event X <= X*, with X* in (-infty,+infty)^2. Such a conditional expectation has an intuitive interpretation in the context of risk measures. Suggested Reviewers: Marcel Ausloos marcel.ausloos@uliege.be He is a leading scientist in the field. Rosella Castellano rosella.castellano@unitelma.it She has a great expertise on risk measures and financial calculus. Luca Correani lcorreani@unitus.it He is able to fully understand the research. Eduard Ceptureanu eduard_ceptureanu@yahoo.com He has a deep knowledge on the financial setting here presented. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation THE SKEW NORMAL MULTIVARIATE RISK MEASUREMENT


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 [20] and the recent contributions [16,19]). 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 distributions. 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 [5] in 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 [4] in 1999, whereas in 2001 Branco and Dey [12] 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. [15] in 2004. In the next year, Azzalini ([3], 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. [14] in 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. [13], 2012 and Lee and Mclachlan [18], 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 ([8], 2013) for univariate Skew Normal distributions and their mixtures. The conditional expectation is a highly relevant concept in financial 1 Click here to access/download;Manuscript;Paper_Bernardi_Cerqueti_Palest Click here to view linked References   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 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. ([6], 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 [2] in 2016. Bernardi et al. ([7], 2017), 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 ([8] 2013) and in ([10] 2017).
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 equation (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 equation (1) can be calculated with reference to the univariate conditional distribution of the bivariate Skew Normal, see Bernardi ([8], 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. ([10], 2017), 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 equation (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 [2], 2016 and Bernardi and Catania [9], 2015) 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  The remainder of the paper can be outlined as follows: Section 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 Section 4 concludes.

The bivariate Skew Normal distribution
We are going to carry out the above procedure to determine the tail conditional expectation when the distribution is Skew Normal (see Azzalini [3], 2005, Azzalini and Dalla Valle [5], 1996, and Gupta et al. [15], 2004, among other contributions). For notational convenience, throughout this Section we suppress the symbol (≤) when referring to the TCE as defined in equation (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. ( [15], 2004). In order to lighten the notation, call , the normalising constant of the previous equation (2). The TCE of X 1 is (see Gupta et al. [15], 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 equation (3) can be easily calculated as the cumulative distribution function of the bivariate Skew Normal distribution in equation (2), as follows 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   4 MAURO BERNARDI, ROY CERQUETI, AND ARSEN PALESTINI As concerns the numerator of equation (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 , which leads to the following integral where with 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 ;   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   THE SKEW NORMAL MULTIVARIATE RISK MEASUREMENT FRAMEWORK   5 (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 −∞ Φ(t)dt, where Φ(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 Φ. Standard mathematical analysis theory establishes that Φ(γ 1 σ 2 t 2 ) = 1 2 + 1 π +∞ n=0 (−1) n (γ 1 σ 2 t 2 ) 2n+1 n!(2n + 1) .