Trimability and Fast Optimization of Long–Short Portfolios

Abstract

Optimization of long–short portfolios through the use of fast algorithms takes advantage of models of covariance to simplify the equations that determine optimality. Fast algorithms exist for widely applied factor and scenario analysis for long-only portfolios. To allow their use in factor and scenario analysis for long–short portfolios, the concept of "trimability" is introduced. The conclusion is that the same fast algorithms that were designed for long-only portfolios can be used, virtually unchanged, for long–short portfolio optimization—provided the portfolio is trimable, which usually holds in practice.

We discuss fast optimization algorithms for long–short portfolios, including market-neutral equity portfolios that have zero market exposure and enhanced active equity portfolios that have a full market exposure, such as 120–20 portfolios (with 120 percent of the capital long and 20 percent short). Fast algorithms that greatly enhance the speed and ease of portfolio optimization were designed for long-only portfolios. Whether fast algorithms can be used to optimize long–short portfolios has not been apparent, because such portfolios may violate assumptions used in formulating the long-only portfolio optimization problem. We describe a sufficient condition under which a portfolio optimization algorithm designed for long-only portfolios will find the correct long–short portfolio, even if the algorithm's use would violate certain assumptions made in the formulation of the long-only problem. This condition, the "trimability condition," is widely satisfied in practice.

Fast portfolio optimization algorithms for long-only portfolios achieve their speed by taking advantage of models that define new fictitious securities whose magnitudes are linearly related to the magnitudes of the real securities in such a way that the covariance matrix of the securities' returns becomes diagonal, or almost so, and idiosyncratic terms are uncorrelated. Under these conditions, portfolio optimization amounts to the solution of sparse, well-structured sets of equations and can be performed rapidly.

This approach may not be applicable to long–short portfolios, however, because of the possibility that such portfolios may hold long and short positions in the same security. In that case, the idiosyncratic terms will not be uncorrelated. Yet, despite this violation of the assumption of no correlation of the idiosyncratic terms, we find that fast algorithms are still applicable to long–short portfoliosif the trimability condition holds. This condition essentially requires that if a portfolio with short and long positions in the same stock is feasible, then it is also feasible to net the long and short positions while keeping the holdings of all other risky securities the same and not reducing the expected return of the portfolio.

We present two cases of particular interest. In the first case, if one uses a factor or scenario model of covariance and the trimability condition is satisfied, then existing fast algorithms for long-only portfolios will find the correct long–short portfolio. In the second case, if one uses a historical covariance model (where the number of securities greatly exceeds the number of observations), then, again, existing fast algorithms for long-only portfolios will produce the correct long–short portfolio and, in this case, will do so whether or not the trimability condition holds.

We also discuss the incorporation of practical and regulatory constraints into the optimization of long–short portfolios. Examples include budget constraints, the U.S. Federal Reserve Board's Regulation T margin requirements, upper bounds on long or short positions in individual or groups of assets, and the requirement that the difference between the sum of long positions and the sum of short positions be close to an investor-chosen value (e.g., 0 for market-neutral portfolios or 1 for enhanced active 120–20 portfolios). To our knowledge, all such constraints—whether imposed by regulators, brokers, or investors themselves—are expressible as linear equalities or weak inequalities. Therefore, they can be incorporated into the general portfolio selection model.