
Portfolio Optimization, Short Sales, and Leverage Aversion
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, longshort is not a twoportfolio strategy, in which a portfolio of longs is somehow combined with a separately optimized portfolio of shorts. Rather, it is a oneportfolio 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 longshort portfolio not constrained by benchmark weights. The ensuing benefits are described in “LongShort Management: An Integrated Approach.” This article, along with “On the Optimality of LongShort Strategies,” describes the conditions under which a dollar or betaneutral 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 LongShort Portfolios.” Our research indicates that the same algorithms used for optimizing longonly 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 longshort 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 meanvariance analysis, which considers only volatility risk and can lead to portfolios with very high leverage levels. In “Traditional Optimization Is Not Optimal for LeverageAverse 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 meanvarianceleverage optimization model. Using enhanced active longshort equity portfolios as an example, we demonstrate that the meanvarianceleverage model shows that optimal portfolios will have modest levels of leverage (13030 for instance) for realistic levels of leverage aversion. Meanvarianceleverage optimization selects the portfolio offering the greatest utility for a leverageaverse investor, and allows the investor to trade off expected return, volatility risk, and leverage risk.
Key Articles: · “Traditional Optimization Is Not Optimal for LeverageAverse 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, meanvariance 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 meanvarianceleverage optimization model—allows the leverageaverse 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 MeanVarianceLeverage Optimization Model and the Markowitz General MeanVariance Portfolio Selection Model,” by Bruce I. Jacobs and Kenneth N. Levy, Journal of Portfolio Management, Fall 2013. article The meanvarianceleverage (MVL) optimization model tackles an issue not dealt with by the meanvariance optimization inherent in the general meanvariance 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 leverageaverse 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 margincall 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 meanvariance 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 meanvarianceleverage efficient frontiers, which show the tradeoffs between expected return, volatility, and leverage. We also develop the meanvarianceleverage 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 meanvariance 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 riskaversion term in conventional meanvariance analysis with two terms—the traditional riskaversion term, renamed as volatilityaversion, and a leverageaversion term. Recognizing leverage aversion in portfolio selection produces optimal portfolios with less leverage than portfolios produced by conventional meanvariance analysis. Less leveraged portfolios may be beneficial not only for leverageaverse 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 meanvariance 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 longshort 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 LongShort 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 longshort portfolios using fast algorithms that were originally designed with longonly 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 factorandscenario 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 longshort portfolio optimization problem. We introduce the concept of "trimability" for longshort portfolios, and show that the same fast algorithms that were designed for longonly portfolios can be used, virtually unchanged, for longshort portfolio optimization, provided the portfolio is "trimable." This trimability condition usually holds in practice.
·
“LongShort 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
·
“On the Optimality of LongShort Strategies,” by Bruce I. Jacobs,
Kenneth N. Levy, and David Starer, Financial Analysts Journal, March/April
1998.(4)
article Other Research Categories: Plan Architecture and Portfolio Engineering ___________________________________________
(1)Featured in “Pair Sees MPT Flaw Over Risks of Leverage,” by Barry B. Burr, Pensions & Investments, February 4, 2013. 
