Portfolio Optimization Including Short Positions

As we researched the idea of using short positions in conjunction with long positions in a portfolio framework, we soon realized the real benefits of this approach emerge only if one employs a single "integrated optimization" that considers long positions and short positions simultaneously. In this framework, long-short is not a two-portfolio strategy, in which a portfolio of longs is somehow combined with a separately optimized portfolio of shorts. Rather, it is a one-portfolio strategy in which the long and short positions are determined jointly within an optimization that takes into account the expected returns of the individual securities, the standard deviation of those returns, and the correlations between them, as well as the investor's tolerance for risk.

Only with an integrated optimization is a long-short portfolio not constrained by benchmark weights. Rather than having to move away from or toward benchmark weights in order to pursue return or control risk, the investor can allocate capital without regard to the securities' weights in the underlying benchmark, as offsetting long and short positions can be used to control risk. The investor does not have to hold securities that have no expected excess return nor does the investor have to restrict the portfolio's holdings of securities with especially good expected returns merely in order to ensure that the portfolio's return does not stray too far from an underlying benchmark's return. The ensuing benefits are described in "Long-Short Management: An Integrated Approach." This article, along with "On the Optimality of Long-Short Strategies," describes the conditions under which a dollar- or beta-neutral portfolio is optimal.

Portfolios with both long and short positions, however, present a technical problem when it comes to optimization. The optimization problem in general is tractable because one can take certain shortcuts. Some models in wide use for long-only portfolios—for example, factor models and scenario models—allow the investor to apply "fast" algorithms that greatly simplify the optimization problem. It is not readily apparent that such models are applicable when portfolios hold short as well as long positions.

We examined this problem closely, most recently in "Trimability and Fast Optimization of Long-Short Portfolios." Our research indicates that the same algorithms used for optimizing long-only portfolios can be used, unchanged, for portfolios that contain short positions—provided a certain condition holds. This condition, which we term "trimability," usually holds in practice.


· “Trimability and Fast Optimization of Long-Short Portfolios,” by Bruce I. Jacobs, Kenneth N. Levy, and Harry M. Markowitz, Financial Analysts Journal, March/April 2006. article
This paper discusses the optimization of long-short portfolios using fast algorithms that were originally designed with long-only portfolios in mind. Fast algorithms that take advantage of various models of covariance gain speed by greatly simplifying the equations. Fast algorithms currently exist for factor, scenario, or mixed factor-and-scenario models of covariance, but they generally apply only to portfolios of long positions. It is desirable to be able to apply factor and scenario models to the long-short portfolio optimization problem. We introduce the concept of "trimability" for long-short portfolios, and show 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." This trimability condition usually holds in practice.

· “Portfolio Optimization with Factors, Scenarios, and Realistic Short Positions,” by Bruce I. Jacobs, Kenneth N. Levy, and Harry M. Markowitz, Operations Research, July/August 2005. article
This paper presents fast algorithms for calculating mean-variance efficient frontiers when the investor can sell securities short as well as buy long, and when a factor and/or scenario model of covariance is assumed. Currently, fast algorithms for factor, scenario, or mixed factor and scenario models exist, but (except for a special case of the results reported here) apply only to portfolios of long positions. Factor and scenario models are used widely in applied portfolio analysis, and short sales have been used increasingly as part of large institutional portfolios. Generally, the critical line algorithm (CLA) traces out mean-variance efficient sets when the investor's choice is subject to any system of linear equality or inequality constraints. Versions of CLA that take advantage of factor and/or scenario models of covariance gain speed by greatly simplifying the equations for segments of the efficient set. These same algorithms can be used, unchanged, for the long-short portfolio selection problem provided a certain condition on the constraint set holds. This conditional usually holds in practice.

· “Long-Short Portfolio Management: An Integrated Approach,” by Bruce I. Jacobs, Kenneth N. Levy, and David Starer, The Journal of Portfolio Management, Winter 1999; and abstracted in The CFA Digest, Fall 1999.(1) article
With the freedom to sell short, an investor can benefit from stocks with negative expected returns as well as from those with positive expected returns. The benefits of combining short positions with long positions in a portfolio context, however, depend critically on the way the portfolio is constructed. Only an integrated optimization that considers the expected returns, risks, and correlations of all securities simultaneously can maximize the investor's ability to trade off risk and return for the best possible performance. This holds true whether or not the long-short portfolio is managed relative to an underlying asset class benchmark. Despite the incremental costs associated with shorting, a long-short portfolio, with its enhanced flexibility, can be expected to perform better than a long-only portfolio based on the same set of insights.

· “On the Optimality of Long-Short Strategies,” by Bruce I. Jacobs, Kenneth N. Levy, and David Starer, Financial Analysts Journal, March/April 1998.(2) article
This article considers the optimality of portfolios not subject to short-selling constraints and derives conditions that a universe of securities must satisfy for an optimal active portfolio to be dollar neutral or beta neutral. Following the common practice of constraining long-short portfolios to have zero net holdings or zero betas is generally suboptimal. Only under specific unlikely conditions will such constrained portfolios optimize an investor's utility function. The article derives precise formulas for optimally equitizing an active long-short portfolio using exposure to a benchmark security. The relative sizes of the active and benchmark exposures depend on the investor's desired residual risk relative to the residual risk of a typical portfolio and on the expected risk-adjusted excess return of a minimum-variance active portfolio. Optimal portfolios demand the use of integrated optimizations.


Other Research Categories:

Security Selection

Plan Architecture and Portfolio Engineering

Long-Short Investing

Market Simulation

Market Crisis

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(1)The Journal of Portfolio Management Bernstein Fabozzi/Jacobs Levy Award, Outstanding Article winner.
(2)Presented at the Society of Quantitative Analysts (SQA) Seminar on "Quantitative Approaches to Market Neutral Investing," November 1997.
 

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