A Krylov Subspace Approach to Large Portfolio Optimization

34 Pages Posted: 18 Dec 2011

See all articles by Isabelle G. Bajeux-Besnainou

Isabelle G. Bajeux-Besnainou

McGill University - Desautels Faculty of Management

Wachi Bandara

Pluribus Labs

Efstathia Bura

Vienna University of Technology - Institute for Mathematical Methods in Economics

Date Written: November 17, 2011

Abstract

With a large number of securities (N) and fewer observations (T), deriving the global minimum variance portfolio requires the inversion of the singular sample covariance matrix of security returns. We introduce the Break-Down Free Generalized Minimum RESidual (BFGMRES), a Krylov subspaces method, as a fully automated approach for deriving the minimum variance portfolio. BFGMRES is a numerical algorithm that provides solutions to singular linear systems without requiring ex-ante assumptions on the covariance structure. Moreover, it is robust to illiquidity and potentially faulty data. US and international stock data are used to demonstrate the relative robustness of BFGMRES to illiquidity when compared to the “shrinkage to market” methodology developed by Ledoit and Wolf (2003). The two methods have similar performance as assessed by the Sharpe ratios and standard deviations for filtered data. In a simulation study, we show that BFGMRES is more robust than shrinkage to market in the presence of data irregularities. Indeed, when there is an illiquid stock shrinkage to market allocates almost 100% of the portfolio weights to this stock, whereas BFGMRES does not. In further simulations, we also show that when there is no illiquidity, BFGMRES exhibits superior performance than shrinkage to market when the number of stocks is high and the sample covariance matrix is highly singular.

Keywords: Krylov subspaces, singular systems, sample covariance matrix, global minimum variance portfolio

JEL Classification: C02, G11

Suggested Citation

Bajeux-Besnainou, Isabelle G. and Bandara, Wachindra and Bura, Efstathia, A Krylov Subspace Approach to Large Portfolio Optimization (November 17, 2011). Available at SSRN: https://ssrn.com/abstract=1974031 or http://dx.doi.org/10.2139/ssrn.1974031

Isabelle G. Bajeux-Besnainou

McGill University - Desautels Faculty of Management ( email )

1001 Sherbrooke St. West
Montreal, Quebec H3A1G5 H3A 2M1
Canada

Wachindra Bandara (Contact Author)

Pluribus Labs ( email )

1331 7th St.
Berkeley, CA 94710
United States

Efstathia Bura

Vienna University of Technology - Institute for Mathematical Methods in Economics ( email )

Argentinierstr. 8
Vienna, 1040
Austria

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