Beyond the lognormal

Hosking, J., Bonti, G., and Siegel, D. (2000). Risk, 13:5 (May 2000), 59-62.

Introduction. Most risk managers recognise that market returns are not lognormally distributed but exhibit so-called fat tails. This well-established phenomenon is important when measuring value-at-risk, as the quantiles in VAR are directly related to the tails of the distributions of market parameters. The more extreme a quantile required, the more sensitive is the result to the correct modelling of the tails of the distribution.

Two of the most widely used approaches to calculating VAR, however, are based on the (log-)normality assumption. They are the covariance method of JP Morgan's RiskMetrics and the Monte Carlo simulation approach. The historical simulation technique does not suffer from this drawback but can be criticised for other reasons (Jorion, 1997, pages 195-196).

As part of a research project between Deutsche Bank and IBM, we have devised a method for calculating VAR that overcomes the problems associated with the assumption of normality. Our basic approach uses Monte Carlo simulation, which has the desirable property of being able to generate large scenario sets, thereby improving the statistical properties of the VAR estimator. We generate the scenarios by modifying the classical Monte Carlo approach so as to take into account the fat-tailed distributions of market variables. The method uses new statistical techniques for identifying and estimating fat-tailed distributions, and includes a model of statistical dependence between quantities that are not normally distributed.

In this paper we describe the rationale behind our approach, give details of the mathematical framework, and report encouraging results obtained by applying the method to several real-world portfolios.

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