Working Paper: NBER ID: w0996
Authors: Gary Chamberlain; Michael Rothschild
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.
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JEL Codes: No JEL codes provided
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| Absence of arbitrage opportunities (G19) | Linear functionals determining mean and cost of a portfolio are continuous (G40) |
| Covariance matrix of asset returns has k unbounded eigenvalues (C10) | Unique approximate factor structure exists (C38) |
| Unique approximate factor structure exists (C38) | Mean returns are approximately linear functions of factor loadings (C51) |
| Existence of riskless investment opportunities (G11) | Mean-variance efficiency frontier is a straight line in mean-standard deviation space (C46) |