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Portfolio Optimization, Short Sales, and Leverage Aversion


Copyright 2006, CFA Institute. Reproduced and republished from Financial Analysts Journal with permission from CFA Institute. All rights reserved.

Copyright 2014, Institutional Investor Journals. Reproduced and republished from Journal of Portfolio Management with permission. All rights reserved.

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

Traditional Optimization Is Not Optimal for Leverage-Averse Investors
by Bruce I. Jacobs, and Kenneth N. Levy, Journal of Portfolio Management, Winter 2014

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. 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 problem when it comes to optimization. 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.

Another issue that arises with regard to portfolios with short positions is the leverage involved. Leverage, whether in long-short portfolios or in portfolios with leveraged long positions, introduces risks that are distinct from the risk captured by a volatility measure. These include the possibility of losses beyond the capital invested and the potential for margin calls, which may necessitate forced selling, perhaps at adverse prices. These risks are not reflected in traditional mean-variance analysis, which considers only volatility risk and can lead to portfolios with very high leverage levels.

In “Traditional Optimization Is Not Optimal for Leverage-Averse Investors,” we propose that leverage aversion be included as an explicit term, along with volatility aversion, in the optimization of leveraged portfolios; this results in a mean-variance-leverage optimization model. Using enhanced active long-short equity portfolios as an example, we demonstrate that the mean-variance-leverage model shows that optimal portfolios will have modest levels of leverage (130-30 for instance) for realistic levels of leverage aversion. Mean-variance-leverage optimization selects the portfolio offering the greatest utility for a leverage-averse investor, and allows the investor to trade off expected return, volatility risk, and leverage risk.

Leverage Aversion - A Third Dimension in Portfolio Theory and Practice
Pictured (l-r) Bruce Jacobs, Ken Levy
Keynote Speakers: Bruce Jacobs and Ken Levy
Jacobs Levy Equity Management Center for
Quantitative Financial Research
Forum on Quantitative Finance
October 23, 2013
New York, NY
Bruce: “We've seen that there have been many catastrophes caused by excessive leverage, and that excessive leverage can give rise to systemic risk, market disruptions and economic crises.”  
Ken: “Just as investors are willing to sacrifice some return in order to reduce volatility risk, investors are willing to sacrifice some return in order to reduce leverage risk.”

Key Articles:

· “Traditional Optimization Is Not Optimal for Leverage-Averse Investors,” by Bruce I. Jacobs and Kenneth N. Levy, Journal of Portfolio Management, Winter 2014. article
For an investor who seeks to mitigate the unique risks of leverage, mean-variance optimization provides little guidance as to where to set a leverage constraint and cannot identify the leveraged portfolio offering the highest utility. An alternative approach—the mean-variance-leverage optimization model—allows the leverage-averse investor to determine the optimal level of leverage, and thus the highest utility portfolio, by balancing the portfolio’s expected return against the portfolio’s volatility risk and its leverage risk.

· “A Comparison of the Mean-Variance-Leverage Optimization Model and the Markowitz General Mean-Variance Portfolio Selection Model,” by Bruce I. Jacobs and Kenneth N. Levy, Journal of Portfolio Management, Fall 2013. article
The mean-variance-leverage (MVL) optimization model tackles an issue not dealt with by the mean-variance optimization inherent in the general mean-variance portfolio selection model (GPSM) — that is, the impact on investor utility of the risks that are unique to using leverage. Relying on leverage constraints with a conventional GPSM, as is commonly done today, is unlikely to lead to the portfolio offering a leverage-averse investor the highest utility. But investors can use the MVL model to find optimal portfolios that balance expected return, volatility risk, and leverage risk. The MVL model has intuitive appeal and offers straightforward implementation for portfolio selection. In contrast, practical use of a broader application of GPSM, as suggested by Markowitz in a 2013 Journal of Portfolio Management article, is dependent on successful future development of a stochastic margin-call model.

· “Leverage Aversion, Efficient Frontiers, and the Efficient Region,” by Bruce I. Jacobs and Kenneth N. Levy, Journal of Portfolio Management, Spring 2013. (1) article
We propose that portfolio theory and mean-variance optimization be augmented to incorporate investor aversion to leverage and suggest a specification for leverage aversion that captures the unique risks of leverage. We introduce mean-variance-leverage efficient frontiers, which show the tradeoffs between expected return, volatility, and leverage. We also develop the mean-variance-leverage efficient region, which illustrates that leverage aversion can have a large impact on an investor’s portfolio choice.

· “Introducing Leverage Aversion into Portfolio Theory and Practice,” by Bruce I. Jacobs and Kenneth N. Levy, Journal of Portfolio Management, Winter 2013. article
To the extent that leverage increases a portfolio’s volatility, conventional mean-variance optimization recognizes some of the risk associated with leverage. But it is silent on other risks that are unique to using leverage, including the possibility of margin calls, which can force borrowers to liquidate securities at adverse prices; losses exceeding the capital invested; and bankruptcy. We suggest replacing the risk-aversion term in conventional mean-variance analysis with two terms—the traditional risk-aversion term, renamed as volatility-aversion, and a leverage-aversion term. Recognizing leverage aversion in portfolio selection produces optimal portfolios with less leverage than portfolios produced by conventional mean-variance analysis. Less leveraged portfolios may be beneficial not only for leverage-averse investors, but also for the global economy.

· “Leverage Aversion and Portfolio Optimality,” by Bruce I. Jacobs, Kenneth N. Levy, Financial Analysts Journal, September/October 2012. (1)(2) article
A leveraged portfolio may be subject to margin calls and forced liquidations at adverse prices; it can also sustain losses beyond the capital invested. These sources of risk are different and distinct from the risks captured by traditional mean-variance optimization. We thus propose that optimization of leveraged portfolios include an explicit measure of leverage aversion in addition to the standard risk (volatility) aversion. Using enhanced active long-short portfolios as an example, we show that adding a leverage aversion term to the investor’s utility function generally results in portfolios with relatively modest levels of leverage. Explicit recognition of leverage aversion by investors might curtail some of the outsized levels of leverage and consequent market disruptions that have been experienced in recent years.

· “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.

· “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.(3) 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.(4) 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 Article:

· “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.
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 condition usually holds in practice.

Other Research Categories:

Security Selection

Plan Architecture and Portfolio Engineering

Long-Short Investing

Market Simulation

Market Crises


(1)Featured in “Pair Sees MPT Flaw Over Risks of Leverage,” by Barry B. Burr, Pensions & Investments, February 4, 2013.
(2)Featured in “Borrowing Against Yourself,” by Jason Zweig, The Wall Street Journal, September 22, 2012.
(3)Winner of a Bernstein Fabozzi/Jacobs Levy Award for Outstanding Article from The Journal of Portfolio Management.
(4)Presented at the Society of Quantitative Analysts (SQA) Seminar on "Quantitative Approaches to Market Neutral Investing," November 1997. Reprinted in Modern Portfolio Management: Active Long/Short 130/30 Equity Strategies, by Martin L. Leibowitz, Simon Emrich, and Anthony Bova, John Wiley & Sons, Hoboken, NJ, 2009.

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