Value-at-Risk and Expected Shortfall for Linear Portfolios with elliptically distributed RisK Factors

In this paper, we generalize the parametric Delta-VaR method from portfolios with normally distributed risk factors to portfolios with elliptically distributed ones. We treat both expected shortfall and the Value-at-Risk of such portfolios. Special attention is given to the particular case of a multivariate t-distribution.Delta Elliptic VaR, Delta Elliptic ES, Delta Student VaR, Delta Student ES

VaR and ES for Linear Portfolios with mixture of Generalized Laplace Distributed Risk Factors

RiskMetrics Delta-Normal VaR, Delta-GLD-VaR, Delta-MGLD, Delta-GLD ES, Delta-MGLD, Hedge Funds Risk.

VaR and ES for linear Portfolis with mixture of elliptically distributed Risk Factors.

The particular subject of this paper, is to give an explicit formulas that will permit to obtain the linear VaR or Linear ES, when the joint risk factors of the Linear portfolios, changes with mixture of t-Student distributions. Note that, since one shortcoming of the multivariate t- distribution is that all the marginal distributions must have the same degrees of freedom, which implies that all risk factors have equally heavy tails, the mixture of t-Student will be view as a serious alternatives, to a simple t-Student-distribution. Therefore, the methodology proposes by this paper seem to be interesting to controlled thicker tails than the standard Student distribution.sadefo-kamdem

VaR and ES for linear Portfolios with mixture of elliptically distributed Risk Factors.

The particular subject of this paper, is to give an explicit formulas that will permit to obtain the linear VaR or Linear ES, when the joint risk factors of the Linear portfolios, changes with mixture of t-Student distributions. Note that, since one shortcoming of the multivariate t- distribution is that all the marginal distributions must have the same degrees of freedom, which implies that all risk factors have equally heavy tails, the mixture of t-Student will be view as a serious alternatives, to a simple t-Student-distribution. Therefore, the methodology proposes by this paper seem to be interesting to controlled thicker tails than the standard Student distribution.Delta mixture Elliptic VaR, Delta mixture Student VaR, Delta mixture Elliptic ES, Delta mixture Student ES, VaR Models.

Measuring the risk of a nonlinear portfolio with fat tailed risk factors through probability conserving transformation

This paper presents a new heuristic for fast approximation of VaR (Value-at-Risk) and CVaR (conditional Value-at-Risk) for financial portfolios, where the net worth of a portfolio is a non-linear function of possibly non-Gaussian risk factors. The proposed method is based on mapping non-normal marginal distributions into normal distributions via a probability conserving transformation and then using a quadratic, i.e. Delta–Gamma, approximation for the portfolio value. The method is very general and can deal with a wide range of marginal distributions of risk factors, including non-parametric distributions. Its computational load is comparable with the Delta–Gamma–Normal method based on Fourier inversion. However, unlike the Delta–Gamma–Normal method, the proposed heuristic preserves the tail behaviour of the individual risk factors, which may be seen as a significant advantage. We demonstrate the utility of the new method with comprehensive numerical experiments on simulated as well as real financial data

VAR FOR QUADRATIC PORTFOLIO'S WITH GENERALIZED LAPLACE DISTRIBUTED RETURNS

This paper is concerned with the e±cient analytical computation of Value-at-Risk (VaR) for portfolios of assets depending quadratically on a large number of joint risk factors that follows a multivariate Generalized Laplace Distribution. Our approach is designed to supplement the usual Monte-Carlo techniques, by providing an asymptotic formula for the quadratic portfolio's cumulative distribution function, together with explicit error-estimates. The application of these methods is demonstrated using some financial applications examples.

Multivariate Elliptical Truncated Moments

In this study, we derived analytic expressions for the elliptical truncated moment generating function (MGF), the zeroth, first, and second-order moments of quadratic forms of the multivariate normal, Student’s t, and generalised hyperbolic distributions. The resulting formulae were tested in a numerical application to calculate an analytic expression of the expected shortfall of quadratic portfolios with the benefit that moment based sensitivity measures can be derived from the analytic expression. The convergence rate of the analytic expression is fast – one iteration – for small closed integration domains, and slower for open integration domains when compared to the Monte Carlo integration method. The analytic formulae provide a theoretical framework for calculations in robust estimation, robust regression, outlier detection, design of experiments, and stochastic extensions of deterministic elliptical curves results

Measuring the risk of financial portfolios with nonlinear instruments and non-Gaussian risk factors

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The focus of my research has been computationally efficient means of computing measures of risk for portfolios of nonlinear financial instruments when the risk factors might be possibly non-Gaussian. In particular, the measures of risk chosen have been Value-at-Risk (VaR) and conditional Value-at-Risk (CVaR). I have studied the problem of computation of risk in two types of financial portfolios with nonlinear instruments which depend on possibly non-Gaussian risk factors: 1. Portfolios of European stock options when the stock return distribution may not be Gaussian; 2. Portfolios of sovereign bonds (which are nonlinear in the underlying risk factor, i.e. the short rate) when the risk factor may or may not be Gaussian. Addressing both these problems need a wide array of mathematical tools both from the field of applied statistics (Delta-Gamma-Normal models, characteristic function inversion, probability conserving transformation) and systems theory (Vasicek stochastic differential equation model, Kalman filter). A new heuristic is proposed for addressing the first problem, while an empirical study is presented to support the use of filter-based models for addressing the second problem. In addition to presenting a discussion of these underlying mathematical tools, the dissertation also presents comprehensive numerical experiments in both cases, with simulated as well as rea