Improving Value-at-Risk Estimation for Multivariate Portfolios Using a Mixture of Probabilistic Principal Component Analyzers
DOI:
https://doi.org/10.14529/cmse250201Keywords:
Value at Risk, VaR, PCA, PPCA, mPPCA, backtesting, dimensionality reductionAbstract
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.References
Duffie D., Pan J. An overview of value at risk. Journal of Derivatives. 1997. Vol. 4, no. 3. P. 7–49. DOI: 10.3905/jod.1997.407971.
Ledoit O., Wolf M. Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance. 2003. Vol. 10, no. 5. P. 603–621. DOI: 10.1016/S0927-5398(03)00007-0.
Fan J., Liao Y., Liu H. An overview of the estimation of large covariance and precision matrices. The Econometrics Journal. 2016. Mar. Vol. 19, no. 1. P. C1–C32. DOI: 10.1111/ectj.12061.
Pafka S., Kondor I. Noisy covariance matrices and portfolio optimization II. Physica A: Statistical Mechanics and its Applications. 2003. Vol. 319. P. 487–494. DOI: 10.1016/S0378-4371(02)01499-1.
Pearson K. LIII. On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1901. Vol. 2, no. 11. P. 559–572. DOI: 10.1080/14786440109462720.
Kelly B., Malamud S., Pedersen L.H. Principal Portfolios. The Journal of Finance. 2023. Vol. 78, no. 1. P. 347–387. DOI: 10.1111/jofi.13199.
Mavungu M. Computation of financial risk using principal component analysis. Algorithmic Finance. 2023. Vol. 10, no. 1-2. P. 1–20. DOI: 10.3233/af-220339.
Kulikov A.V., Polozov D.S., Volkov N.V. Long-term investment optimization based on Markowitz diversification. Business Informatics. 2024. Vol. 18, no. 3. P. 56–69. DOI: 10.17323/2587-814X.2024.3.56.69.
Partovi M.H., Caputo M. Principal Portfolios: Recasting the Efficient Frontier. Economics Bulletin. 2004. Jan. Vol. 7. P. 1–10.
Markowitz H. Portfolio Selection. The Journal of Finance. 1952. Vol. 7, no. 1. P. 77–91. DOI: 10.2307/2975974.
Meucci A. Managing diversification. Risk. 2009. P. 74–79.
Lohre H., Neugebauer U., Zimmer C. Diversified Risk Parity Strategies for Equity Portfolio Selection. The Journal of Investing. 2012. Aug. Vol. 21. P. 111–128. DOI: 10.3905/joi.2012.21.3.111.
Lohre H., Opfer H., Ország G. Diversifying risk parity. The Journal of Risk. 2014. June. Vol. 16. P. 53–79. DOI: 10.21314/JOR.2014.284.
Luxenberg E., Boyd S. Portfolio construction with Gaussian mixture returns and exponential utility via convex optimization. Optimization and Engineering. 2024. Vol. 25, no. 1. P. 555–574. DOI: 10.1007/s11081-023-09814-y.
Indrė Morkūnaitė D.C., Leipus R. Evaluation of Value-at-Risk (VaR) using the Gaussian Mixture Models. Research in Statistics. 2024. Vol. 2, no. 1. P. 2346075. DOI: 10.1080/27684520.2024.2346075.
Ruan L., Yuan M., Zou H. Regularized Parameter Estimation in High-Dimensional Gaussian Mixture Models. Neural Computation. 2011. Vol. 23. P. 1605–1622. DOI: 10.1162/NECO_a_00128.
Halbe Z., Bortman M., Aladjem M. Regularized Mixture Density Estimation with an Analytical Setting of Shrinkage Intensities. IEEE Transactions on Neural Networks and Learning Systems. 2013. Vol. 24. P. 460–470. DOI: 10.1109/TNNLS.2012.2234477.
Moon T. The expectation-maximization algorithm. IEEE Signal Processing Magazine. 1996. Vol. 13, no. 6. P. 47–60. DOI: 10.1109/79.543975.
Tipping M.E., Bishop C.M. Probabilistic Principal Component Analysis. Journal of the Royal Statistical Society Series B. 1999. Vol. 61, no. 3. P. 611–622. DOI: 10.1111/1467-9868.00196.
Tipping M.E., Bishop C.M. Mixtures of Probabilistic Principal Component Analyzers. Neural Computation. 1999. Feb. Vol. 11, no. 2. P. 443–482. DOI: 10.1162/089976699300016728.
Lyu Y., Zhou L., Cong Y., et al. Multirate mixture probability principal component analysis for process monitoring in multimode processes. IEEE Transactions on Automation Science and Engineering. 2023. Vol. 21, no. 2. P. 2027–2038. DOI: 10.1109/TASE.2023.3253285.
Tra V., Amayri M., Bouguila N. Unsupervised fault detection for building air handling unit systems using deep variational mixture of principal component analyzers. IEEE Transactions on Automation Science and Engineering. 2024. Vol. 21, no. 4. P. 6787–6803. DOI: 10.1109/tase.2023.3331347.
Zhang J., Chen M., Hong X. Nonlinear process monitoring using a mixture of probabilistic PCA with clusterings. Neurocomputing. 2021. Vol. 458. P. 319–326. DOI: 10.1016/j.neucom.2021.06.039.
Zhao B., Xiao X., Zhang W., et al. Self-Paced Probabilistic Principal Component Analysis for Data with Outliers. ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). 2020. P. 3737–3741. DOI: 10.1109/icassp40776.2020.9054487.
Chao Han S.L., House L. Covariance-Guided Mixture Probabilistic Principal Component Analysis (C-MPPCA). Journal of Computational and Graphical Statistics. 2015. Vol. 24, no. 1. P. 66–83. DOI: 10.1080/10618600.2014.891460.
Tzagkarakis G., Caicedo-Llano J., Dionysopoulos T. Exploiting market integration for pure alpha investments via probabilistic principal factors analysis. Journal of Mathematical Finance. 2013. Vol. 3, no. 1. P. 192–200. DOI: 10.4236/jmf.2013.31A018.
Begušic S., Kostanjcar Z. Cluster-Specific Latent Factor Estimation in High-Dimensional Financial Time Series. IEEE Access. 2020. Vol. 8. P. 164365–164379. DOI: 10.1109/ACCESS.2020.3021898.
Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control. 1974. Dec. Vol. 19, no. 6. P. 716–723. DOI: 10.1109/TAC.1974.1100705.
Brutti Righi M., Sergio Ceretta P. On the existence of an optimal estimation window for risk measures. Applied Finance Letters. 2015. Nov. Vol. 4, no. 12. P. 28–32. DOI: 10.24135/afl.v4i1and2.30.
Volkov N.V. Estimating VaR of diversified portfolios using PCA and PPCA dimensionality reduction methods. Proceedings of MIPT. 2025. Vol. 17, no. 1. P. 127–141. (in Russian).
Kupiec P.H. Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives. 1995. Vol. 3, no. 2. P. 73–84. DOI: 10.3905/jod.1995.407942.
Lam S.K., Pitrou A., Seibert S. Numba: A LLVM-based Python JIT Compiler. Proceedings of the SecondWorkshop on the LLVM Compiler Infrastructure in HPC, LLVM 2015, Austin, TX, USA, November 15, 2015. ACM, 2015. P. 1–6. DOI: 10.1145/2833157.2833162.
Ypma T.J. Historical development of the Newton–Raphson method. SIAM Review. 1995. Vol. 37, no. 4. P. 531–551. DOI: https://doi.org/10.1137/1037125.
Volkov N. MPPCA for VaR Estimation: Source Code Repository. 2025. URL: https://gitlab.com/n.volkovsky/mppca-for-var-estimation (accessed: 20.03.2025).
MacQueen J. Some methods for classification and analysis of multivariate observations. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1, Statistics. Vol. 5. University of California Press. 1967. P. 281–298.
Lloyd S. Least squares quantization in PCM. IEEE Transactions on Information Theory. 1982. Vol. 28, no. 2. P. 129–137. DOI: 10.1109/TIT.1982.1056489.
