Conditional tail risk measures for the skewed generalised hyperbolic family

This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.     

Some results on the CTE-based capital allocation rule

Tasche [Tasche, D., 1999. Risk contributions and performance measurement. Working paper, Technische Universität München] introduces a capital allocation principle where the capital allocated to each risk unit can be expressed in terms of its contribution to the conditional tail expectation (CTE) of the aggregate risk. Panjer [Panjer, H.H., 2002. Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, University of Waterloo, Research Report 01-15] derives a closed-form expression for this allocation rule in the multivariate normal case. Landsman and Valdez [Landsman, Z., Valdez, E., 2002. Tail conditional expectations for elliptical distributions. North American Actuarial J. 7 (4)] generalize Panjer’s result to the class of multivariate elliptical distributions.In this paper we provide an alternative and simpler proof for the CTE-based allocation formula in the elliptical case. Furthermore, we derive accurate and easy computable closed-form approximations for this allocation formula for sums that involve normal and lognormal risks.     

Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalised hyperbolic distributions

This paper addresses one of the main challenges faced by insurance companies and risk management departments, namely, how to develop standardised framework for measuring risks of underlying portfolios and in particular, how to most reliably estimate loss severity distribution from historical data. This paper investigates tail conditional expectation (TCE) and tail variance premium (TVP) risk measures for the family of symmetric generalised hyperbolic (SGH) distributions. In contrast to a widely used Value-at-Risk (VaR) measure, TCE satisfies the requirement of the “coherent” risk measure taking into account the expected loss in the tail of the distribution while TVP incorporates variability in the tail, providing the most conservative estimator of risk. We examine various distributions from the class of SGH distributions, which turn out to fit well financial data returns and allow for explicit formulas for TCE and TVP risk measures. In parallel, we obtain asymptotic behaviour for TCE and TVP risk measures for large quantile levels. Furthermore, we extend our analysis to the multivariate framework, allowing multivariate distributions to model combinations of correlated risks, and demonstrate how TCE can be decomposed into individual components, representing contribution of individual risks to the aggregate portfolio risk.     

Tail Risk of Multivariate Regular Variation

Tail risk refers to the risk associated with extreme values and is often affected by extremal dependence among multivariate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. Asymptotic expressions for tail risk are established in terms of the intensity measure that characterizes multivariate regular variation. Tractable bounds for tail risk are derived in terms of the tail dependence function that describes extremal dependence. Various examples involving Archimedean copulas are presented to illustrate the results and quality of the bounds.     

Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions

The Conditional Tail Expectation (CTE), also known as the Expected Shortfall and Tail-VaR, has received much attention as a preferred risk measure in finance and insurance applications. A related risk management exercise is to allocate the amount of the CTE computed for the aggregate or portfolio risk into individual risk units, a procedure known as the CTE allocation. In this paper we derive analytic formulas of the CTE and its allocation for the class of multivariate normal mean–variance mixture (NMVM) distributions, which is known to be extremely flexible and contains many well-known special cases as its members. We also develop the closed-form expression of the conditional tail variance (CTV) for the NMVM class, an alternative risk measure proposed in the literature to supplement the CTE by capturing the tail variability of the underlying distribution. To illustrate our findings, we focus on the multivariate Generalized Hyperbolic Distribution (GHD) family which is a popular subclass of the NMVM in connection with Lévy processes and contains some common distributions for financial modelling. In addition, we also consider the multivariate slash distribution which is not a member of GHD family but still belongs to the NMVM class. Our result is an extension of the recent contribution of Ignatieva and Landsman (2015).     

Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.     

Tail dependence for elliptically contoured distributions

The relationship between the theory of elliptically contoured distributions and the concept of tail dependence is investigated. We show that bivariate elliptical distributions possess the so-called tail dependence property if the tail of their generating random variable is regularly varying, and we give a necessary condition for tail dependence which is somewhat weaker than regular variation of the latter tail. In addition, we discuss the tail dependence property for some well-known examples of elliptical distributions, such as the multivariate normal, t, logistic, and Bessel distributions.     

Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models

Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event.” We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.     

Multivariate measures of skewness for the skew-normal distribution

The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) [10], Malkovich and Afifi (1973) [9], Isogai (1982) [17], Srivastava (1984) [15], Song (2001) [14], Móri et al. (1993) [11], Balakrishnan et al. (2007) [3] and Kollo (2008) [7] measures of skewness. The exact expressions of all measures of skewness, except for Song’s, are derived for the family of skew-normal distributions, while Song’s measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions.     

A closed-form solution of the Black–Litterman model with conditional value at risk

We consider a portfolio optimization problem of the Black–Litterman type, in which we use the conditional value-at-risk (CVaR) as the risk measure and we use the multi-variate elliptical distributions, instead of the multi-variate normal distribution, to model the financial asset returns. We propose an approximation algorithm and establish the convergence results. Based on the approximation algorithm, we derive a closed-form solution of the portfolio optimization problems of the Black–Litterman type with CVaR.     

Dependence structure of conditional Archimedean copulas

In this article, copulas associated to multivariate conditional distributions in an Archimedean model are characterized. It is shown that this popular class of dependence structures is closed under the operation of conditioning, but that the associated conditional copula has a different analytical form in general. It is also demonstrated that the extremal copula for conditional Archimedean distributions is no longer the Fréchet upper bound, but rather a member of the Clayton family. Properties of these conditional distributions as well as conditional versions of tail dependence indices are also considered.     

Finite sample tail behavior of multivariate location estimators

A finite sample performance measure of multivariate location estimators is introduced based on “tail behavior”. The tail performance of multivariate “monotone” location estimators and the halfspace depth based “non-monotone” location estimators including the Tukey halfspace median and multivariate L-estimators is investigated. The connections among the finite sample performance measure, the finite sample breakdown point, and the halfspace depth are revealed. It turns out that estimators with high breakdown point or halfspace depth have “appealing” tail performance. The tail performance of the halfspace median is very appealing and also robust against underlying population distributions, while the tail performance of the sample mean is very sensitive to underlying population distributions. These findings provide new insights into the notions of the halfspace depth and breakdown point and identify the important role of tail behavior as a quantitative measure of robustness in the multivariate location setting.     

从SPVⅡ分布生成的新的多元偏态t分布的有关性质

随机向量的t分布属于椭球等高分布族,然而,它是对称分布.在许多诸如经济学、生理学、社会学等领域中,有时回归模型中的随机误差不再满足对称性,通常表现出高度的偏态性(skewness).于是就有了偏态椭球等高分布族.本文在已有的多元偏态t分布的基础上,着重研究它的分布性质,包括线性组合分布、边缘分布、条件分布及各阶矩.     

Limit distributions of multivariate kurtosis and moments under Watson rotational symmetric distributions

The limit distribution of Mardia's measure of sample multivariate kurtosis is derived for a wide class of multivariate distributions which includes both the family of elliptical and Watson rotational symmetric distributions. Explicit expressions are given for the higher-order moments of Watson rotational symmetric distributions. The problem of constructing approximate confidence intervals for the kurtosis parameter is also discussed.     

Asymptotic properties of conditional quantiles of the Cauchy distribution in Hilbert space

In this paper, we consider the conditional distributions that are induced by finite-dimensional projections of a σ-additive Cauchy measure defined in a real Hilbert space. In the case where the dimension of projections tends to infinity, we establish the almost sure convergence of “conforming” sequences of finite-dimensional conditional quantiles and prove the strong law of large numbers for the triangular array scheme applied to a family of conditional distributions. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Generalized symmetrized dirichlet distributions

In this paper we introduce a family of multivariate distributions, which consists of scale mixtures of symmetrized Dirichlet distributions. This family is a symmetrization of multivariate Liouville distributions and contains the well-known spherically symmetric distributions as a special case. The basic properties of this family such as stochastic representation, probability density functions, marginal and conditional distributions and components' independence are studied. A criterion of the invariance of statistics is also given.     

Stein's Lemma for elliptical random vectors

For the family of multivariate normal distribution functions, Stein's Lemma presents a useful tool for calculating covariances between functions of the component random variables. Motivated by applications to corporate finance, we prove a generalization of Stein's Lemma to the family of elliptical distributions.     

Sur les estimateurs de Bayes minimax: une méthode constructive

Bayes estimation of the mean of a multivariate normal distribution is considered under quadratic loss. We show that, when a variance mixture of normal distributions is used as a prior, superharmonicity of the square root of the marginal density provides a viable method for constructing Bayes minimax estimators. Examples illustrate the theory. In particular, we show that a scaled multivariate Student-t prior yields a proper Bayes minimax estimate.     

On the Tail Mean-Variance optimal portfolio selection

In the present paper we propose the Tail Mean-Variance (TMV) approach, based on Tail Condition Expectation (TCE) (or Expected Short Fall) and the recently introduced Tail Variance (TV) as a measure for the optimal portfolio selection. We show that, when the underlying distribution is multivariate normal, the TMV model reduces to a more complicated functional than the quadratic and represents a combination of linear, square root of quadratic and quadratic functionals. We show, however, that under general linear constraints, the solution of the optimization problem still exists and in the case where short selling is possible we provide an analytical closed form solution, which looks more “robust” than the classical MV solution. The results are extended to more general multivariate elliptical distributions of risks.     

New multivariate orderings based on conditional distributions

In this paper, we propose and study new multivariate extensions of the dispersive, right‐spread, decreasing mean residual life and new better than used in expectation univariate orders. These new orders are based on the comparison of univariate marginal distributions conditional on survival data for the rest of the components. Relationships among multivariate orders and applications to some multivariate random vectors are also provided. Copyright © 2011 John Wiley & Sons, Ltd.