A mean-variance analysis of self-financing portfolios

This paper develops the analytics and geometry of the investment opportunity set (IOS) and the test statistics for self-financing portfolios. A self-financing portfolio is a set of long and short investments such that the sum of their investment weights, or net investment, is zero. This contrasts with a standard portfolio that has investment weights summing to one. Examples of self-financing portfolios are hedges, overlays, arbitrage portfolios, swaps, and long/short portfolios. A standard portfolio plus the IOS of self-financing portfolios form a restricted IOS hyperbola with restricted efficient set constants that differ from the usual constants. The restrictions affect statistical tests of portfolio efficiency, which are developed for the self-financing restrictions. As an application, we consider the self-financing portfolios formed by Fama and French, based on market capitalization and value. In contrast to Fama and French, we find that their restricted IOS is significantly different from the unrestricted IOS with the implication that the Fama–French tests are misspecified.