Multivariate tail covariance for generalized skew-elliptical distributions

Doubly truncated moment risk measures for elliptical distributions

In this paper, we define doubly truncated moment (DTM), doubly truncated skewness (DTS) and kurtosis (DTK). We derive DTM formulae for elliptical family, with emphasis on normal, student-$t$, logistic, Laplace and Pearson type VII distributions. We also present explicit formulas of the DTE (doubly truncated expectation), DTV (doubly truncated variance), DTS and DTK for those distributions. As illustrative example, DTEs, DTVs, DTSs and DTKs of three industry segments' (Banks, Insurance, Financial and Credit Service) stock return in London stock exchange are discussed.

Explicit expressions for joint moments of $n$-dimensional elliptical distributions

Inspired by Stein's lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive $E[X_{1}^{2}f(\mathbf{X})]$ for any measurable function $f$ satisfying some regularity conditions. Then, by applying this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of $E[X_{1}^{2}f(\mathbf{X})]$ for multivariate Student-$t$, logistic and Laplace distributions.

A note on conditional covariance matrices for elliptical distributions

In this short note we provide an analytical formula for the conditional covariance matrices of the elliptically distributed random vectors, when the conditioning is based on the values of any linear combination of the marginal random variables. We show that one could introduce the univariate invariant depending solely on the conditioning set, which greatly simplifies the calculations. As an application, we show that one could define uniquely defined quantile-based sets on which conditional covariance matrices must be equal to each other if only the vector is multivariate normal. The similar re…

The Multivariate Mixture Dynamics Model: Shifted dynamics and correlation skew

The Multi Variate Mixture Dynamics model is a tractable, dynamical, arbitrage-free multivariate model characterized by transparency on the dependence structure, since closed form formulae for terminal correlations, average correlations and copula function are available. It also allows for complete decorrelation between assets and instantaneous variances. Each single asset is modelled according to a lognormal mixture dynamics model, and this univariate version is widely used in the industry due to its flexibility and accuracy. The same property holds for the multivariate process of all assets, …

Approximation of Some Multivariate Risk Measures for Gaussian Risks

Gaussian random vectors exhibit the loss of dimension phenomena, which relate to their joint survival tail behaviour. Besides, the fact that the components of such vectors are light-tailed complicates the approximations of various multivariate risk measures significantly. In this contribution we derive precise approximations of marginal mean excess, marginal expected shortfall and multivariate conditional tail expectation of Gaussian random vectors and highlight links with conditional limit theorems. Our study indicates that similar results hold for elliptical and Gaussian like multivariate ri…

Ordering of multivariate probability distributions with respect to extreme portfolio losses

A new notion of stochastic ordering is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of ordering conditions on the spectral measures, which allows for analytical or numerical verification in practical applications. Additional comparison criteria in terms of further stochastic orderings are derived. The application examples include worst case and best case scenarios, elliptically contoured distributions, and multivariate regularly varying models with …

On the Basel Liquidity Formula for Elliptical Distributions

A justification of the Basel liquidity formula for risk capital in the trading book is given under the assumption that market risk-factor changes form a Gaussian white noise process over 10-day time steps and changes to P&L are linear in the risk-factor changes. A generalization of the formula is derived under the more general assumption that risk-factor changes are multivariate elliptical. It is shown that the Basel formula tends to be conservative when the elliptical distributions are from the heavier-tailed generalized hyperbolic family. As a by-product of the analysis a Fourier approach to…

Stress testing and systemic risk measures using multivariate conditional probability

The multivariate conditional probability distribution models the effects of a set of variables onto the statistical properties of another set of variables. In the study of systemic risk in a financial system, the multivariate conditional probability distribution can be used for stress-testing by quantifying the propagation of losses from a set of `stressing' variables to another set of `stressed' variables. In this paper I describe how to compute such conditional probability distributions for the vast family of multivariate elliptical distributions, and in particular for the multivariate Stude…

具有多元拉普拉斯分布的资产配置的解析解

An Analytic Solution for Asset Allocation with a Multivariate Laplace Distribution

本文将椭圆对称分布的多元资产配置理论专门应用于多元拉普拉斯分布，推导出解析解，并指出其与先前文章猜想的主要差异源于问题维度项的引入和方差的重新缩放。