A generic nonparametric value-at-risk estimator for high dimensions

Abstract

We present in this article a non-parametric value-at-risk (VaR+CVaR) algorithm that remains accurate for an arbitrarily large number of underlying positions. The algorithm solves the two inherent problems of VaR estimation. First, past history is not directly applicable to the future, but all predictions of the future are based on the past. Second, VaR estimation is equivalent to modeling a single corner of a high-dimensional space (the corner where all bets lose simultaneously). The algorithm only uses mathematical methods that strictly do not degrade in accuracy at high-dimensions. Historical data are then directly incorporated with all high-dimensional relationships present, without manipulation. We test the algorithm with an ensemble of 500 portfolios with random positions across 49 distinct liquid futures of different expiries (VIX, equity indexes, gov. bonds, rates, energy, metals, livestock, agriculture, and softs). All VaR estimations are performed strictly blind to the future. The median portfolio rate of loss exceeding the 99% confidence daily VaR estimate is between $1.0\pm0.1$% depending on algorithm input parameters. 68% of portfolios have a rate of loss exceeding 99% VaR between $1.0\pm0.3$%, and 95% of portfolios between $1.0\pm0.5$%.