Improving Value-at-Risk Estimation for Multivariate Portfolios Using a Mixture of Probabilistic Principal Component Analyzers

This paper proposes a novel approach for estimating the Value-at-Risk (VaR) of multidimensional portfolios, based on a mixture of probabilistic principal component analyzers (mPPCA) and the Akaike information criterion. The effectiveness of the proposed approach is evaluated on historical data, accounting for various numbers of mixture components in the mPPCA method. The study is performed on 100 highly diversified and 100 weakly diversified stock portfolios of the S&P 500 index over the 2009–2023 period, using rolling windows of 350 trading days. Probabilistic principal component analysis (PPCA) can model complex dependencies among assets while capturing the “heavy” tails of return distributions. As a result, the mPPCA method surpasses conventional principal component analysis (PCA) in the accuracy of VaR estimation. In addition, by reducing dimensionality, the model is computationally much more efficient and stable than a mixture of Gaussian distributions (GMM). The paper demonstrates how portfolio return volatility and tail heaviness depend both on the optimal number of components in mPPCA and on the minimal sufficient number of principal components in PCA and PPCA needed to explain 80 % of the variance in the data. The new approach, which optimizes the number of components in mPPCA, consistently achieves higher performance than GMM, PCA, or PPCA, especially for less diversified portfolios. The paper describes methods for optimizing mPPCA training and provides an extensive historical performance evaluation (backtesting). By employing just-in-time compilation, a “warm start” for mPPCA with each new window position, and a three-step algorithm for VaR estimation, the experiments are significantly accelerated compared with the standard implementation.