Authors: Carol Alexander, Daniel Ledermann
Abstract: Most banks employ historical simulation for Value-at-Risk (VaR) calculations, where VaR is computed from a lower quantile of a forecast distribution for the portfolio’s profit and loss (P&L) that is constructed from a single, multivariate historical sample on the portfolio’s risk factors. The implicit assumption is that history will repeat itself for certain over the forecast horizon. Until now, the only alternative is to assume the historical sample is generated by a multivariate, parametric risk factor distribution and (except in special cases where an ana- lytic solution is available) to simulate P&L via Monte Carlo (MC). This paper introduces a methodology that encompasses historical and MC VaR as special cases, which is much faster than MC simulation and which avoids the single-sample bias of historical simulation. Ran- dom orthogonal matrix (ROM) simulation is a fast matrix-based simulation method that applies directly to an historical sample, or to a parametric distribution. Each simulation matches the first four multivariate sample moments to those of the observed sample, or of the target distribution. Stressed VaR is typically computed from an historical sample using the Duffie-Pan methodology, whereby the sample is transformed to have a stressed covari- ance matrix. ROM simulation extends this methodology to generate very large samples, which furthermore have stressed values for the first four multivariate moments values.