Finance Letters, 2003, 1 (3), 90-96 Measuring Value at Risk under the Conditional Edgeworth-Sargan Distribution Javier Perote a,∗ and Esther B. Del Brío b a Universidad Rey Juan Carlos, Spain b Universidad de Salamanca, Spain Abstract This paper introduces the Edgeworth-Sargan distribution on measuring Value-at-Risk of portfolios. The flexible parametric representation of this density makes it capable of improving the density fits (especially at the tails) and permits a straightforward method of percentile computation. Moreover, the time varying variance-covariance structure of the portfolio can be also estimated consistently to the Edgeworth-Sargan hypothesis since this density admits a straightforward multivariate representation. Estimates of portfolio’s VaR evidence the underestimation of VaR measures under the normality assumption and also the more accurate VaR measures under the Edgeworth-Sargan distribution. Keywords: Value at Risk, multivariate densities, Edgeworth-Sargan density, GARCH models. JEL classification code: G10, G11, C13. 1. INTRODUCTION The search of accurate Value-at-Risk (hereafter VaR) measures has been approached from many perspectives aimed to account for the behaviour of the assets, particularly in the tails of their return distribution. Different distributions have previously been used for this purpose, such as the Student’s t (Vlaar, 2000; Lucas, 2000; Huschens and Kim, 1999), Stable Paretian distributions (Mittnik et al., 2002; Khindanova et al., 2000), the Hyperbolic distribution (Bauer, 2000), or mixtures of normal distributions (Venkataraman, 1997). This paper focuses on measuring VaR under the assumption of the Edgeworth-Sargan distribution – hereafter ES – as the data generating process. The rationale under such a distribution (Sargan, 1976; 1980) lies in its ability to account for heavier tails than the normal, as well as for possible asymmetries as a result of the consideration of a general and flexible parameterisation. This flexibility is due to the density representation based on Edgeworth or Gram-Charlier expansions (Nishiyama and Robinson, 2000; Jondeau and Rockinger, 2001, respectively). From an empirical perspective, this distribution has been shown capable of accounting for salient empirical regularities of the histogram for most high frequency financial variables. This result was shown by Mauleon and Perote (2000) who tested the ES performance compared to other fitted densities like the Student’s t. Despite the Edgeworth-Sargan ability to represent the behaviour of most high frequency financial variables, this distribution is not yet widespread in the financial literature and, as far as we know, this is its first application to measuring VaR. Therefore, we propose a methodological approach to calculate the portfolio VaR by assuming that each portfolio’s asset is conditionally ES distributed. Another interesting property of such distribution is that linear transformations of ES variables are also ES distributed, thus the linear portfolios of ES variables are also ES distributed. Moreover, variance and covariance matrices can be estimated consistently under such hypothesis since multivariate ES densities can also be provided. Finally, the traditional ARIMA or GARCH models can be used to model the conditional mean and variance. Actually, their simplest versions –AR(1) and GARCH(1,1) – are used in this article. ISSN 1740-6242 © Global EcoFinance TM All Rights Reserved 90