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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 (1992, 1993, 1995), based on market capitalization and value. In contrast to Fama and French (1992, 1993, 1995), we find that their restricted IOS is significantly different from the unrestricted IOS with the implication that the Fama-French tests are misspecified. (Investment Opportunity Set; Long/Short Portfolios; Spanning; Intersection)
1. Introduction and Results Summary
Self-financing portfolios arise extensively in the theory and practice of finance. A self-financing portfolio is a set of long and short investments such that the net portfolio investment is zero. This implies that the sum of a self-financing portfolio's investment weights is zero.1 In practice, self-financing portfolios are combined with standard portfolios. The standard portfolio, with weights summing to one and to which the self-financing portfolio is added, is also known as the host, core, or benchmark portfolio. Applications include arbitraging, asset swapping, hedging, information tilting, currency and asset overlaying, market-neutral investing, market timing, rebalancing, and reallocating.2 Despite the widespread applica- tion of self-financing portfolios, surprisingly little has been published regarding the analytics, test statistics, and geometry of the self-financing portfolio investment opportunity set (SFIOS). In contrast, the proper- ties of standard portfolio investment opportunity sets (IOS) are well known from the originating work of Markowitz (1952), and the subsequent work of Sharpe (1964), Merton (1972), and Roll (1977), in particular. There does not seem to exist a specific analysis of the investment opportunity set mathematics and the geometry of self-financing portfolios. However, some literature is closely related. Roll (1992) analyzes benchmark tracking and Brennan (1993) investigates asset...