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Impossible Frontiers*

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  • Thomas J. Brennan

    (School of Law, Northwestern University, Chicago, Illinois 60611)

  • Andrew W. Lo

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142; and AlphaSimplex Group, Cambridge, Massachusetts 02142)

Abstract

A key result of the capital asset pricing model (CAPM) is that the market portfolio--the portfolio of all assets in which each asset's weight is proportional to its total market capitalization--lies on the mean-variance-efficient frontier, the set of portfolios having mean-variance characteristics that cannot be improved upon. Therefore, the CAPM cannot be consistent with efficient frontiers for which every frontier portfolio has at least one negative weight or short position. We call such efficient frontiers "impossible," and show that impossible frontiers are difficult to avoid. In particular, as the number of assets, n, grows, we prove that the probability that a generically chosen frontier is impossible tends to one at a geometric rate. In fact, for one natural class of distributions, nearly one-eighth of all assets on a frontier is expected to have negative weights for every portfolio on the frontier. We also show that the expected minimum amount of short selling across frontier portfolios grows linearly with n, and even when short sales are constrained to some finite level, an impossible frontier remains impossible. Using daily and monthly U.S. stock returns, we document the impossibility of efficient frontiers in the data.

Suggested Citation

  • Thomas J. Brennan & Andrew W. Lo, 2010. "Impossible Frontiers," Management Science, INFORMS, vol. 56(6), pages 905-923, June.
  • Handle: RePEc:inm:ormnsc:v:56:y:2010:i:6:p:905-923
    DOI: 10.1287/mnsc.1100.1157
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    1. Zhenyu Wang, 2005. "A Shrinkage Approach to Model Uncertainty and Asset Allocation," Review of Financial Studies, Society for Financial Studies, vol. 18(2), pages 673-705.
    2. Victor DeMiguel & Lorenzo Garlappi & Raman Uppal, 2009. "Optimal Versus Naive Diversification: How Inefficient is the 1-N Portfolio Strategy?," Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 1915-1953, May.
    3. Stephen A. Ross, 2013. "The Arbitrage Theory of Capital Asset Pricing," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 1, pages 11-30, World Scientific Publishing Co. Pte. Ltd..
    4. Green, Richard C, 1986. "Positively Weighted Portfolios on the Minimum-Variance Frontier," Journal of Finance, American Finance Association, vol. 41(5), pages 1051-1068, December.
    5. Tu, Jun & Zhou, Guofu, 2010. "Incorporating Economic Objectives into Bayesian Priors: Portfolio Choice under Parameter Uncertainty," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 45(4), pages 959-986, August.
    6. Roll, Richard, 1977. "A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory," Journal of Financial Economics, Elsevier, vol. 4(2), pages 129-176, March.
    7. J. Tobin, 1958. "Liquidity Preference as Behavior Towards Risk," Review of Economic Studies, Oxford University Press, vol. 25(2), pages 65-86.
    8. Dybvig, Philip H, 1984. "Short Sales Restrictions and Kinks on the Mean Variance Frontier," Journal of Finance, American Finance Association, vol. 39(1), pages 239-244, March.
    9. Best, Michael J. & Grauer, Robert R., 1990. "The efficient set mathematics when mean-variance problems are subject to general linear constraints," Journal of Economics and Business, Elsevier, vol. 42(2), pages 105-120, May.
    10. Kan, Raymond & Zhou, Guofu, 2007. "Optimal Portfolio Choice with Parameter Uncertainty," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 42(3), pages 621-656, September.
    11. Jérôme Detemple & Shashidhar Murthy, 1997. "Equilibrium Asset Prices and No-Arbitrage with Portfolio Constraints," CIRANO Working Papers 97s-12, CIRANO.
    12. Rudd, Andrew, 1977. "A note on qualitative results for investment proportions," Journal of Financial Economics, Elsevier, vol. 5(2), pages 259-263, November.
    13. Heaton, John & Lucas, Deborah J, 1996. "Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset Pricing," Journal of Political Economy, University of Chicago Press, vol. 104(3), pages 443-487, June.
    14. Jarrow, Robert A, 1980. "Heterogeneous Expectations, Restrictions on Short Sales, and Equilibrium Asset Prices," Journal of Finance, American Finance Association, vol. 35(5), pages 1105-1113, December.
    15. Lorenzo Garlappi & Raman Uppal & Tan Wang, 2007. "Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach," Review of Financial Studies, Society for Financial Studies, vol. 20(1), pages 41-81, January.
    16. Jorion, Philippe, 1986. "Bayes-Stein Estimation for Portfolio Analysis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 279-292, September.
    17. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    18. 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 1785-1809, December.
    19. Best, Michael J & Grauer, Robert R, 1991. "On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results," Review of Financial Studies, Society for Financial Studies, vol. 4(2), pages 315-342.
    20. Duffie, Darrell & Garleanu, Nicolae & Pedersen, Lasse Heje, 2002. "Securities lending, shorting, and pricing," Journal of Financial Economics, Elsevier, vol. 66(2-3), pages 307-339.
    21. Sun, Wan Gui & Wang, Chun Feng, 2006. "The mean-variance investment problem in a constrained financial market," Journal of Mathematical Economics, Elsevier, vol. 42(7-8), pages 885-895, November.
    22. Isabelle Huault & V. Perret & S. Charreire-Petit, 2007. "Management," Post-Print halshs-00337676, HAL.
    23. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    24. Tu, Jun & Zhou, Guofu, 2004. "Data-generating process uncertainty: What difference does it make in portfolio decisions?," Journal of Financial Economics, Elsevier, vol. 72(2), pages 385-421, May.
    25. Best, Michael J & Grauer, Robert R, 1985. "Capital Asset Pricing Compatible with Observed Market Value Weights," Journal of Finance, American Finance Association, vol. 40(1), pages 85-103, March.
    26. Brito, Ney O, 1978. "Portfolio Selection in an Economy with Marketability and Short Sales Restrictions," Journal of Finance, American Finance Association, vol. 33(2), pages 589-601, May.
    27. 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(3), pages 293-305, September.
    28. Pogue, G A, 1970. "An Extension of the Markowitz Portfolio Selection Model to Include Variable Transactions' Costs, Short Sales, Leverage Policies and Taxes," Journal of Finance, American Finance Association, vol. 25(5), pages 1005-1027, December.
    29. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1684, August.
    30. Michael J. Best & Robert R. Grauer, 1991. "Sensitivity Analysis for Mean-Variance Portfolio Problems," Management Science, INFORMS, vol. 37(8), pages 980-989, August.
    31. Merton, Robert C., 1972. "An Analytic Derivation of the Efficient Portfolio Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 7(4), pages 1851-1872, September.
    32. Diamond, Douglas W. & Verrecchia, Robert E., 1987. "Constraints on short-selling and asset price adjustment to private information," Journal of Financial Economics, Elsevier, vol. 18(2), pages 277-311, June.
    33. Best, Michael J. & Grauer, Robert R., 1992. "Positively Weighted Minimum-Variance Portfolios and the Structure of Asset Expected Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 27(4), pages 513-537, December.
    34. 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.
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    Cited by:

    1. Chiaki Hara & Toshiki Honda, 2016. "Mutual Fund Theorem for Ambiguity-Averse Investors and the Optimality of the Market Portfolio," KIER Working Papers 943, Kyoto University, Institute of Economic Research.
    2. Wenzelburger, Jan, 2020. "Mean-variance analysis and the Modified Market Portfolio," Journal of Economic Dynamics and Control, Elsevier, vol. 111(C).
    3. Levy, Haim & Levy, Moshe, 2014. "The benefits of differential variance-based constraints in portfolio optimization," European Journal of Operational Research, Elsevier, vol. 234(2), pages 372-381.
    4. Kim, Jang Ho & Kim, Woo Chang & Fabozzi, Frank J., 2016. "Portfolio selection with conservative short-selling," Finance Research Letters, Elsevier, vol. 18(C), pages 363-369.
    5. Levy, Moshe & Levy, Haim, 2015. "Keeping up with the Joneses and optimal diversification," Journal of Banking & Finance, Elsevier, vol. 58(C), pages 29-38.

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    Replication

    This item has been replicated by:
  • Levy, Moshe & Roll, Richard, 2015. "(Im)Possible Frontiers: A Comment," Critical Finance Review, now publishers, vol. 4(1), pages 139-148, June.
  • More about this item

    Keywords

    short selling; long/short; portfolio optimization; mean-variance analysis; CAPM;
    All these keywords.

    JEL classification:

    • G1 - Financial Economics - - General Financial Markets
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading
    • G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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    1. Impossible Frontiers (Management Science 2010) in ReplicationWiki

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