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On the use of the Moore–Penrose generalized inverse in the portfolio optimization problem

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  • Lee, Miyoung
  • Kim, Daehwan

Abstract

When the number of assets (N) exceeds the number of time periods (T), the sample covariance matrix is singular, and the portfolio optimization problem cannot be solved via traditional mean-variance algebra. In such a case, the Moore–Penrose (MP) generalized inverse becomes handy: In this paper, we critically examine the MP solution of the portfolio optimization problem. Our findings include: i) the MP solution leads to a portfolio of “pseudo-riskfree composite assets”; ii) it is orthogonal to principal components, iii) most importantly, it is poorly diversified. We illustrate our findings using equity market data.

Suggested Citation

  • Lee, Miyoung & Kim, Daehwan, 2017. "On the use of the Moore–Penrose generalized inverse in the portfolio optimization problem," Finance Research Letters, Elsevier, vol. 22(C), pages 259-267.
    • Handle: RePEc:eee:finlet:v:22:y:2017:i:c:p:259-267
    DOI: 10.1016/j.frl.2016.12.017
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    1. William F. Sharpe, 1963. "A Simplified Model for Portfolio Analysis," Management Science, INFORMS, vol. 9(2), pages 277-293, January.
    2. Roll, Richard & Ross, Stephen A, 1980. "An Empirical Investigation of the Arbitrage Pricing Theory," Journal of Finance, American Finance Association, vol. 35(5), pages 1073-1103, December.
    3. Ryan, Peter J. & Lefoll, Jean, 1981. "A Comment on Mean-Variance Portfolio Selection with Either a Singular or a Non-Singular Variance-Covariance Matrix," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 16(3), pages 389-395, September.
    4. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    5. Chamberlain, Gary & Rothschild, Michael, 1983. "Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets," Econometrica, Econometric Society, vol. 51(5), pages 1281-1304, September.
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    Cited by:

    1. Munish Kansal & Manpreet Kaur & Litika Rani & Lorentz Jäntschi, 2023. "A Cubic Class of Iterative Procedures for Finding the Generalized Inverses," Mathematics, MDPI, vol. 11(13), pages 1-18, July.
    2. Andrew Grant & Oh Kang Kwon & Steve Satchell, 2024. "Properties of risk aversion estimated from portfolio weights," Journal of Asset Management, Palgrave Macmillan, vol. 25(5), pages 427-444, September.

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    More about this item

    Keywords

    Portfolio optimization with singular covariance matrix; Moore–Penrose generalized inverse; Minimum norm; Principal component; Diversification;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C38 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Classification Methdos; Cluster Analysis; Principal Components; Factor Analysis

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