Testing Portfolio Efficiency when the Zero‐Beta Rate is Unknown: A Note

  • Author(s): JAY SHANKEN
  • Published: Apr 30, 2012
  • Pages: 269-276
  • DOI: 10.1111/j.1540-6261.1986.tb04506.x


A lower bound on the distribution function of the likelihood ratio test of portfolio efficiency is derived. An empirical application demonstrates that the bound may sometimes be used to infer rejection of the null hypothesis without appeal to asymptotic statistical approximations. A procedure for incorporating partial information about the zero‐beta intercept, in the multivariate framework, is also developed and applied.

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