Testing Portfolio Efficiency When the Zero-Beta Rate Is Unknown: A Note

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. A FUNDAMENTAL PROBLEM IN empirical finance is that of determining whether a given portfolio is on the mean-variance efficient frontier. From a normative perspective, a solution to this problem has relevance for any investor wishing to maximize expected portfolio return for a given level of risk. Since a number of equilibrium models, including the well-known Capital Asset Pricing Model (CAPM), yield predictions that particular portfolios are efficient, the problem is also important from a positive economic perspective. Fama [3], Roll [9], and Ross [10] observe that the efficiency of a portfolio is equivalent to the existence of a positive linear relation between the expected returns of the component securities and their betas computad relative to the portfolio. If borrowing and lending at a known riskless rate, r, is assumed, then the "zero-beta" intercept in the linear relation must equal r, and the econometric analysis is greatly simplified. Gibbons, Ross, and Shanken [5] show that a transformation of the likelihood ratio test (LRT) statistic has an exact F distribution in this case, assuming joint normality of returns. The analysis is more complicated when the zero-beta intercept is unknown and must be treated as an additional parameter in the econometric specification. Shanken [11] introduces a multivariate "cross-sectional regression test" (CSRT) of efficiency for this case and derives an approximate sampling distribution for the test. An upper bound on the exact distribution functions of the CSRT and the LRT is also obtained.1 Under certain circumstances, this bound enables the researcher to infer, without appeal to asymptotic approximations, that the null hypothesis of efficiency cannot be rejected on the basis of the given statistics. In this paper, we present a complementary small-sample result. A lower bound on the distribution function of the LRT is derived which, in some cases, permits the researcher to infer that the null hypothesis should be rejected. An application involving the CRSP equal-weighted stock index illustrates the usefulness of the result.