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A New Proposal for Computing Portfolio Value-at-Risk for Semi-nonparametric Distributions

This paper proposes a semi-nonparametric (SNP) methodology for computing portfolio value-at-risk (VaR) that is more accurate than both the traditional Gaussian-assumption-based methods implemented in the software packages used by risk analysts (RiskMetrics), and alternative heavy-tailed distributions that seem to be very rigid to incorporate jumps and asymmetries in the distribution tails (e.g. the Student’s t). The outperformance of the SNP distributions lies in the fact that Edgeworth and Gram-Charlier series represent a valid asymptotic approximation of any “regular” probability density function. In fact these expansions involve general and flexible parametric representations capable of featuring the salient empirical regularities of financial data. Furthermore these distributions can be extended to a multivariate context and may be estimated in several steps and thus we propose to estimate portfolio VaR in three steps: Firstly, estimating the conditional variance and covariance matrix of the portfolio consistently with the multivariate SNP distribution; Secondly, estimating the univariate distribution of the portfolio constrained to the portfolio variance obtained from the previous step; Thirdly, computing the corresponding quantile of the portfolio distribution by implementing straightforward recursive algorithms. We estimate the VaRs obtained with such methodology for different bivariate portfolios of stock indices and interests rates finding a clear underestimation (overestimation) of VaR measures obtained from the traditional Gaussian- (Student’s t-) based methods compared to our SNP approach.