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

Siyuan Sun

Abstract

We present a fast, nonparametric algorithm for value-at-risk (VaR) and conditional VaR estimation that remains accurate for an arbitrarily large number of underlying positions. The algorithm solves the two inherent problems in VaR estimation: historical data is not directly applicable to the future, even though all predictions necessarily rely on it; and VaR estimation is equivalent to modeling a single corner of a high-dimensional space (the corner where all bets lose simultaneously). The algorithm uses only mathematical methods that do not decrease in accuracy at high dimensions. The historical data is 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 (Chicago Board Options Exchange Volatility Index, equity indexes, government 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 lies within the range 1:0% ± 0:1%, depending on the algorithm’s input parameters. Of the portfolios tested, 68% have a rate of loss exceeding 99% VaR within 1:0% ± 0:3%, and 95% within 1:0% ±0:5%.