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Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets

46 Pages Posted: 15 Mar 2004  

Gary Chamberlain

Harvard University - Department of Economics; National Bureau of Economic Research (NBER)

Michael Rothschild

Princeton University; National Bureau of Economic Research (NBER)

Date Written: October 1982

Abstract

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. These portfolios span the mean-variance efficient set. We resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available. Ross 112, 141 showed that if there is a factor structure, then the mean returns are approximately linear functions of factor loadings. We define an approximate factor structure and show that this weaker restriction is sufficient for Ross' result. If the covariance matrix of the asset returns has only K unbounded eigenvalues, then there is an approximate factor structure and it is unique. The corresponding K eigenvectors converge and play the role of factor loadings. Hence only a principal component analysis is needed in empirical work.

Suggested Citation

Chamberlain, Gary and Rothschild, Michael, Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets (October 1982). NBER Working Paper No. w0996. Available at SSRN: https://ssrn.com/abstract=303488

Gary Chamberlain (Contact Author)

National Bureau of Economic Research (NBER)

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Harvard University - Department of Economics ( email )

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Michael Rothschild

Princeton University ( email )

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National Bureau of Economic Research (NBER) ( email )

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