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Portfolio Optimization with Many Assets: The Importance of Short-Selling

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  • Levy, Moshe
  • Ritov, Yaacov
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    Abstract

    We investigate the properties of mean-variance efficient portfolios when the number of assets is large. We show analytically and empirically that the proportion of assets held short converges to 50% as the number of assets grows, and the investment proportions are extreme, with several assets held in large positions. The cost of the no-shortselling constraint increases dramatically with the number of assets. For about 100 assets the Sharpe ratio can be more than doubled with the removal of this constraint. These results have profound implications for the theoretical validity of the CAPM, and for policy regarding short-selling limitations.

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    Bibliographic Info

    Paper provided by Anderson Graduate School of Management, UCLA in its series University of California at Los Angeles, Anderson Graduate School of Management with number qt41x4t67m.

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    Date of creation: 01 May 2001
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    Handle: RePEc:cdl:anderf:qt41x4t67m

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    Keywords: portfolio optimization; short-selling; CAPM.;

    References

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    1. Levy, Haim, 1983. "The Capital Asset Pricing Model: Theory and Empiricism," Economic Journal, Royal Economic Society, vol. 93(369), pages 145-65, March.
    2. Rudd, Andrew, 1977. "A note on qualitative results for investment proportions," Journal of Financial Economics, Elsevier, vol. 5(2), pages 259-263, November.
    3. Sharpe, William F., 1990. "Capital Asset Prices With and Without Negative Holding," Nobel Prize in Economics documents 1990-3, Nobel Prize Committee.
    4. Longstaff, Francis A. & Santa-Clara, Pedro & Schwartz, Eduardo S., 2001. "Throwing away a billion dollars: the cost of suboptimal exercise strategies in the swaptions market," Journal of Financial Economics, Elsevier, vol. 62(1), pages 39-66, October.
    5. Merton, Robert C., 1987. "A simple model of capital market equilibrium with incomplete information," Working papers 1869-87., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    6. Ravi Jagannathan & Zhenyu Wang, 1993. "The CAPM is alive and well," Staff Report 165, Federal Reserve Bank of Minneapolis.
    7. Francis A. Longstaff, 2001. "The Relative Valuation of Caps and Swaptions: Theory and Empirical Evidence," Journal of Finance, American Finance Association, vol. 56(6), pages 2067-2109, December.
    8. Merton, Robert C., 1972. "An Analytic Derivation of the Efficient Portfolio Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 7(04), pages 1851-1872, September.
    9. G. William Schwert & Paul J. Seguin, 1991. "Heteroskedasticity in Stock Returns," NBER Working Papers 2956, National Bureau of Economic Research, Inc.
    10. William F. Sharpe, 1965. "Mutual Fund Performance," The Journal of Business, University of Chicago Press, vol. 39, pages 119.
    11. Black, Fischer, 1972. "Capital Market Equilibrium with Restricted Borrowing," The Journal of Business, University of Chicago Press, vol. 45(3), pages 444-55, July.
    12. Green, Richard C & Hollifield, Burton, 1992. " When Will Mean-Variance Efficient Portfolios Be Well Diversified?," Journal of Finance, American Finance Association, vol. 47(5), pages 1785-809, December.
    13. Alexander, Gordon J, 1993. " Short Selling and Efficient Sets," Journal of Finance, American Finance Association, vol. 48(4), pages 1497-1506, September.
    14. Roll, Richard & Ross, Stephen A., 1977. "Comments on qualitative results for investment proportions," Journal of Financial Economics, Elsevier, vol. 5(2), pages 265-268, November.
    15. Levy, Haim, 1978. "Equilibrium in an Imperfect Market: A Constraint on the Number of Securities in the Portfolio," American Economic Review, American Economic Association, vol. 68(4), pages 643-58, September.
    16. Elton, Edwin J & Gruber, Martin J, 1973. "Estimating the Dependence Structure of Share Prices-Implications for Portfolio Selection," Journal of Finance, American Finance Association, vol. 28(5), pages 1203-32, December.
    17. Pulley, Lawrence B., 1981. "A General Mean-Variance Approximation to Expected Utility for Short Holding Periods," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 16(03), pages 361-373, September.
    18. Elton, Edwin J & Gruber, Martin J & Padberg, Manfred W, 1976. "Simple Criteria for Optimal Portfolio Selection," Journal of Finance, American Finance Association, vol. 31(5), pages 1341-57, December.
    19. Levy, Haim, 1973. "The Demand for Assets Under Conditions of Risk," Journal of Finance, American Finance Association, vol. 28(1), pages 79-96, March.
    20. Kroll, Yoram & Levy, Haim & Markowitz, Harry M, 1984. " Mean-Variance versus Direct Utility Maximization," Journal of Finance, American Finance Association, vol. 39(1), pages 47-61, March.
    21. Kandel, Shmuel, 1984. " On the Exclusion of Assets from Tests of the Mean Variance Efficiency of the Market Portfolio," Journal of Finance, American Finance Association, vol. 39(1), pages 63-75, March.
    22. J. G. Kallberg & W. T. Ziemba, 1983. "Comparison of Alternative Utility Functions in Portfolio Selection Problems," Management Science, INFORMS, vol. 29(11), pages 1257-1276, November.
    23. Frost, Peter A. & Savarino, James E., 1986. "An Empirical Bayes Approach to Efficient Portfolio Selection," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(03), pages 293-305, September.
    24. Fama, Eugene F & French, Kenneth R, 1992. " The Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 47(2), pages 427-65, June.
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    Cited by:
    1. Michael McKenzie & Olan T. Henry, 2007. "The Determinnts of Short Selling in the Hong Kong Equities Market," Department of Economics - Working Papers Series 1001, The University of Melbourne.

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