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Arbitrage and Mean-Variance Analysis on Large Asset Markets


  • Gary Chamberlain
  • Michael Rothschild


We examine the implications of arbitrage in a market with many assets. The absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous; hence there exist unique portfolios that represent these functionals. The mean variance efficient set is a cone generated by these portfolios. Ross [16, 18J showed that if there is a factor structure, then the distance between the vector or mean returns and the space spanned by the factor loadings is bounded as the number of assets increases. We show that if the covariance matrix of asset returns has only K unbounded eigenvalues, then the corresponding K eigenvectors converge and play the role of factor loadings in Ross' result. Hence only a principal components analysis is needed to test the arbitrage pricing theory. Our eigenvalue conditional can hold even though conventional measures of the approximation error in a K factor model are unbounded. We also resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available.

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  • Gary Chamberlain & Michael Rothschild, 1981. "Arbitrage and Mean-Variance Analysis on Large Asset Markets," NBER Technical Working Papers 0015, National Bureau of Economic Research, Inc.
  • Handle: RePEc:nbr:nberte:0015 Note: ME

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    References listed on IDEAS

    1. Amemiya, Takeshi, 1973. "Regression Analysis when the Dependent Variable is Truncated Normal," Econometrica, Econometric Society, vol. 41(6), pages 997-1016, November.
    2. White, Halbert, 1980. "Nonlinear Regression on Cross-Section Data," Econometrica, Econometric Society, vol. 48(3), pages 721-746, April.
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    Cited by:

    1. Korajczyk, Robert A, 1996. "A Measure of Stock Market Integration for Developed and Emerging Markets," World Bank Economic Review, World Bank Group, vol. 10(2), pages 267-289, May.
    2. Merton, Robert, 1990. "Capital market theory and the pricing of financial securities," Handbook of Monetary Economics,in: B. M. Friedman & F. H. Hahn (ed.), Handbook of Monetary Economics, edition 1, volume 1, chapter 11, pages 497-581 Elsevier.
    3. Bruce N. Lehmann & David M. Modest, 2003. "Diversification and the Optimal Construction of Basis Portfolios," NBER Working Papers 9461, National Bureau of Economic Research, Inc.
    4. Bruce N. Lehmann, 1992. "Empirical Testing of Asset Pricing Models," NBER Working Papers 4043, National Bureau of Economic Research, Inc.

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