Impossible Frontiers

Article, Preprint English OPEN
Brennan, Thomas J.; Lo, Andrew W.;
(2009)
  • Publisher: Institute for Operations Research and the Management Sciences
  • Journal: volume 56,issue 6 June,pages905-923
  • Related identifiers: doi: 10.1287/mnsc.1100.1157
  • Subject: short selling, long/short, portfolio optimization, mean-variance analysis, CAPM
    • jel: jel:G1 | jel:G12 | jel:G23 | jel:G11 | jel:G32 | jel:G14

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... View more
  • References (12)
    12 references, page 1 of 2

    3 Some Examples of Impossible Frontiers 4 3.1 The Two-Asset Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 The Three-Asset Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    4 The General Case 9 4.1 Haar Measure and Covariance Matrices . . . . . . . . . . . . . . . . . . . . . 10 4.2 Linear-Factor Models and Impossibility . . . . . . . . . . . . . . . . . . . . . 13 4.3 Additional Impossibility Results . . . . . . . . . . . . . . . . . . . . . . . . . 16

    5 The One-Factor Model 18 5.1 Characterizing Impossible Tangency Portfolios . . . . . . . . . . . . . . . . . 19 5.2 The Probability of Impossible Tangency Portfolios . . . . . . . . . . . . . . . 20 5.3 A Non-Impossible Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . 22

    6 Empirical Analysis 23 6.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2 A 100-Stock Empirical Efficient Frontier . . . . . . . . . . . . . . . . . . . . 24 6.3 More Impossible Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.4 Estimation Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    A Appendix 29 A.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A.2 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A.3 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A.4 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A.5 Proof of Corollary 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.6 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.7 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.8 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.9 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.10 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A.11 Proof of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4See, for example, Brown (1976), Bawa, Brown, and Klein (1979), Frost and Savarino (1986), Jorion

    (1986), Tu and Zhou (2004, 2007, 2008), Wang (2005), DeMiguel, Garlappi, and Uppal (2007), Garlappi,

    Jarrow, R., 1980, “Heterogeneous Expectations, Restrictions on Short Sales, and Equilibrium Asset Prices”, Journal of Finance 35, 1105-1113.

    Jeffreys, H., 1961, Theory of Probability, 3rd ed. Oxford, UK: Oxford University Press.

    Jorgenson, J. and S. Lang, 2005, Posn(R) and Eisenstein Series. Berlin: Springer-Verlag.

    Jorion, P., 1986, “Bayes-Stein Estimation For Portfolio Analysis”, Journal of Financial and Quantitative Analysis 21, 279-292.

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