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Impossible Frontiers

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  • Thomas J. Brennan
  • Andrew W. Lo

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 derive conditions on asset-return means, variances, and covariances that yield impossible frontiers. With the exception of the two-asset case, we show that impossible frontiers are difficult to avoid. Moreover, 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 shortselling across frontier portfolios grows linearly with n, and even when shortsales 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, 2008. "Impossible Frontiers," NBER Working Papers 14525, National Bureau of Economic Research, Inc.
  • Handle: RePEc:nbr:nberwo:14525
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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. 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.
    3. 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.
    4. Levy, Moshe & Levy, Haim, 2015. "Keeping up with the Joneses and optimal diversification," Journal of Banking & Finance, Elsevier, vol. 58(C), pages 29-38.

    More about this item

    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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