Estimating VAR of diversified portfolios using PCA and PPCA dimensionality reduction methods

This paper examines the estimation of Value at Risk (VaR) for portfolios composed of a large number of assets, employing dimensionality reduction techniques such as Principal Component Analysis (PCA) and Probabilistic PCA (PPCA). The study utilizes open daily returns data of Nasdaq-listed stocks and the S&P 500 index over the period from 2005 to 2024. An optimal window size for both PCA and PPCA in VaR estimation is determined. The VaR estimates obtained through these methods are compared via backtesting to assess whether the number of VaR exceptions aligns with a binomial distribution. A comprehensive comparison is conducted for various portfolios composed of Nasdaq stocks and the S&P 500 index. For a range of portfolio collections, including both well-diversified and poorly diversified sets, the classical PCA method proves less accurate than PPCA in estimating VaR5%1. This finding is supported by statistically significant binomial tests of the number of exceptions, particularly in the case of poorly diversified portfolios. Consequently, the PPCA method demonstrates greater effectiveness and reliability in financial risk assessment compared to conventional PCA.