Risks of large portfolios

Fan, Jianqing and Liao, Yuan and Shi, Xiaofeng (2013): Risks of large portfolios.

Abstract

Estimating and assessing the risk of a large portfolio is an important topic in financial econometrics and risk management. The risk is often estimated by a substitution of a good estimator of the volatility matrix. However, the accuracy of such a risk estimator for large portfolios is largely unknown, and a simple inequality in the previous literature gives an infeasible upper bound for the estimation error. In addition, numerical studies illustrate that this upper bound is very crude. In this paper, we propose factor-based risk estimators under a large amount of assets, and introduce a high-confidence level upper bound (H-CLUB) to assess the accuracy of the risk estimation. The H-CLUB is constructed based on three different estimates of the volatility matrix: sample covariance, approximate factor model with known factors, and unknown factors (POET, Fan, Liao and Mincheva, 2013). For the first time in the literature, we derive the limiting distribution of the estimated risks in high dimensionality. Our numerical results demonstrate that the proposed upper bounds significantly outperform the traditional crude bounds, and provide insightful assessment of the estimation of the portfolio risks. In addition, our simulated results quantify the relative error in the risk estimation, which is usually negligible using 3-month daily data. Finally, the proposed methods are applied to an empirical study.

• Item Type: MPRA Paper
• Language: English
• Keywords: High dimensionality; approximate factor model; unknown factors; principal components; sparse matrix; thresholding; risk management; volatility
• Subjects: G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C38 - Classification Methods ; Cluster Analysis ; Principal Components ; Factor Models; G - Financial Economics > G3 - Corporate Finance and Governance > G32 - Financing Policy ; Financial Risk and Risk Management ; Capital and Ownership Structure ; Value of Firms ; Goodwill; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

\bibitem{a} \textsc{Antoine, B.} (2011) Portfolio Selection with estimation risk: a test-based approach. \textit{Journal of Financial Econometrics}. \textbf{10}, 164-197.

\bibitem{A} \textsc{Antoniadis, A.} and \textsc{Fan, J.} (2001). Regularized wavelet approximations. \textit{ J. Amer. Statist. Assoc.} \textbf{96}, 939-967.

\bibitem{Bai} \textsc{Bai, J.} (2003). Inferential theory for factor models of large dimensions. \textit{Econometrica}. \textbf{71} 135-171.

\bibitem{Bab} \textsc{Bai, J.} and \textsc{Ng, S.}(2002). Determining the number of factors in approximate factor models. \textit{Econometrica}. \textbf{70} 191-221.

\bibitem{Bab} \textsc{Bannouh, K., Martens, M., Oomen, R.} and \textsc{Dijk, D.} (2012). Realized mixed-frequency factor models for vast dimensional covariance estimation \textit{Manuscript}.

\bibitem{BLa} \textsc{Bianchi, D.} and \textsc{Carvalho, C.} (2011). Risk assessment in large portfolios: why imposing the wrong constraints hurts. \textit{manuscript.}

\bibitem{BLa} \textsc{Bickel, P.} and \textsc{Levina, E.} (2008). Covariance regularization by thresholding. \textit{Ann. Statist.} \textbf{36} 2577-2604.

\bibitem{BLa} \textsc{Brodie, J., Daubechies, I., Mol, C., Giannone, D.} and \textsc{Loris, I.} (2009). Sparse and stable Markowitz portfolios. \textit{PNAS} \textbf{106}, 12267-12272.

\bibitem{Cl} \textsc{Cai, T.} and \textsc{Liu, W.} (2011). Adaptive thresholding for sparse covariance matrix estimation.\textit{ J. Amer. Statist. Assoc.} \textbf{106}, 672-684.

\bibitem{C} \textsc{Chamberlain, G.} and \textsc{Rothschild, M.} (1983). Arbitrage, factor structure and mean-variance analyssi in large asset markets. \textit{Econometrica}. \textbf{51} 1305-1324.

\bibitem{C} \textsc{Chang, C.} and \textsc{Tsay, R.} (2010). Estimation of covariance matrix via the sparse Cholesky factor with lasso. \textit{Journal of Statistical Planning and Inference}. \textbf{140} 3858-3873.

\bibitem{C} \textsc{Connor, G.} and \textsc{Korajczyk, R.} (1993). A Test for the number of factors in an approximate factor model. \textit{Journal of Finance}. \textbf{48}, 1263-1291.

\bibitem{C} \textsc{DeMiguel, V., Garlappi, L., Nogales, F.} and \textsc{Uppal, R.} (2009). A generalized approach to portfolio optimization: improving performance by constraining portfolio norms. \textit{Management Science}. \textbf{55}, 798-812.

\bibitem{FF} \textsc{El Karoui, N.} (2010). High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation. \textit{Annals of Statistics}. \textbf{38}, 3487-3566.

\bibitem{FF} \textsc{Fama, E.} and \textsc{French, K.} (1992). The cross-section of expected stock returns. \textit{Journal of Finance}. \textbf{47} 427-465.

\bibitem{FF} \textsc{Fama, E.} and \textsc{French, K.} (1993). Common risk factors in the returns on stocks and bonds \textit{Journal of Financial Economics}. \textbf{33} 3-56..

\bibitem{FL} \textsc{Fan, J., Fan, Y.} and \textsc{Lv, J.} (2008). High dimensional covariance matrix estimation using a factor model. \textit{J. Econometrics}.

\bibitem{FL} \textsc{Fan, J., Liao, Y.} and \textsc{Mincheva, M.} (2011). High dimensional covariance matrix estimation in approximate factor models. \textit{Ann. Statist}. {\bf 39}, 3320-3356.

\bibitem{FLM} \textsc{Fan, J., Liao, Y.} and \textsc{Mincheva, M.} (2013). Large covariance estimation by thresholding principal orthogonal complements. (with discussion) \textit{Journal of the Royal Statistical Society Series B}, to appear.

\bibitem{fz} \textsc{Fan, J., Zhang, J.,} and \textsc{Ke, Y.} (2012). Asset Allocation and Risk Assessment with Gross Exposure Constraints for Vast Portfolios. {\em Journal of the American Statistical Association}, \textbf{107}, 592-606.

\bibitem{FY} \textsc{Fan, J. and Yao, Q.} (2003). {\em Nonlinear Time Series: Nonparametric and Parametric Methods}, 576 pp. Springer-Verlag, New York.

\bibitem{fad} \textsc{Fryzlewicz, P.} (2012). High-dimensional volatility matrix estimation via wavelets and thresholding. \textit{Manuscript}. London School of Economics and Political Science

\bibitem{d} \textsc{Gandy, A.} and \textsc{Veraart, L.} (2012). The effect of estimation in high-dimensional portfolios. \textit{Mathematical Finance.} \textbf{}

\bibitem{d} \textsc{Gomez, K.} and \textsc{Gallon, S.} (2011). Comparison among high dimensional covariance matrix estimation methods. \textit{Revista Columbiana de Estadistica. } \textbf{34}, 567-588.

\bibitem{d} \textsc{Jacquier, E.} and \textsc{Polson, N.} (2010). Simulation-based-estimation in portfolio selection. \textit{Manuscript}.

\bibitem{d} \textsc{Lai, T., Xing, H.} and \textsc{Chen, Z.} (2011). Mean–variance portfolio optimization when means and covariances are unknown. \textit{Annals of Applied Statistics.} \textbf{5}, 798-823.

\bibitem{d} \textsc{Ledoit, O.} and \textsc{Wolf, M.} (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. \textit{Journal of Empirical Finance.} \textbf{10}, 603-621.

\bibitem{d} \textsc{Peligrad, M.} (1996). On the asymptotic normality of sequences of weak dependent random variables. \textit{Journal of Theoretical Probability.} \textbf{9}, 703-715.

\bibitem{d} \textsc{Pesaran, M. H.} and \textsc{Zaffaroni, P.} (2008). Optimal asset allocation with factor models for large portfolios.

\bibitem{R}\textsc{Rothman, A., Levina, E.} and \textsc{Zhu, J.} (2009). Generalized thresholding of large covariance matrices. \textit{J. Amer. Statist. Assoc.} \textbf{104} 177-186.

\bibitem{SW2}\textsc{Stock, J.} and \textsc{Watson, M.} (2002). Forecasting using principal components from a large number of predictors. \textit{ J. Amer. Statist. Assoc.} \textbf{97}, 1167-1179.