Portfolio Theory of Three Tales: Risk-adjusted Returns, Liquidity, and Leverage Chanaka Edirisinghe* Lally School of Management, Rensselaer Polytechnic Institute, NY, U.S.A., [email protected] Jingnan Chen Singapore University of Technology and Design, Singapore, jingnan [email protected] Jaehwan Jeong College of Business and Economics, Radford University, VA, U.S.A., [email protected] January 16, 2018 We study the Pareto-efficiency between risk-adjusted return and leverage level of a portfolio when asset prices evolve continuously under liquidity impact. We show analytically the Sharpe-maximizing unlevered portfolio is no longer a tangency portfolio, and proportionate-leveraging is not an optimal strategy under liquidity risk. As target return increases, the required minimum portfolio-leverage increases at an increasing- rate, while the Sharpe-Leverage frontiers are progressively-dominated. These results are in contrast to the classical portfolio theory that assumes no liquidity risk. We perform empirical analysis to verify our analytical findings, which shows that ignoring liquidity impact can lead to severe portfolio under-performance. Keywords: Price impact of illiquidity, portfolio leverage, risk-adjusted returns, portfolio optimization, Pareto- efficiency JEL Codes: G11, C61, C44 * Corresponding author, Tel: 518-276-3336 1 2 1. Introduction A well-known quote from Tukey (1962, p. 13) states: \Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.". In this paper, the right question is: how does the economic performance of a portfolio of risky assets change when leveraging is used as a managerial choice to enhance target- return in a market scenario where illiquidity risk is inevitable? Answer to this question can be substantially-simplified when liquidity impact is completely-ignored from the question to obtain closed-form theoretical relationships, a.k.a. the modern portfolio theory (Markowitz 1952). As this study shows, the theory of leveraged portfolios is quite different when liquidity risk is taken into account. There are two related consequences of the modern portfolio theory: first, the emergence of the CAPM, see Sharpe (1964),1 in which portfolio excess return forms a linear relationship with mar- ket's excess return. The second is Tobin's (1958) separation theorem whereby portfolio choice is separated into two stages: finding an efficient (tangency) portfolio of the risky assets where risk aversion does not play a role, and then, determining the fraction of wealth to invest in the effi- cient portfolio versus a risk-free asset based on risk aversion. We show in this paper that, when illiquidity-related market impact2 is incorporated ex-ante into portfolio analysis, the above conse- quences become more complex in nature, and the conclusions are drastically-different, providing new economic insights for the three tales of a portfolio: risk-adjusted returns, leverage, and liquidity. The right question is framed in the context of rebalancing a portfolio of multiple risky assets, and a risk-free asset, in a one-period economy where trading is started at date 0, and is completed by date 1. Under a continuous-time trading paradigm facing market illiquidities, asset prices evolve under trading to affect portfolio performance adversely, evaluated at date 1. For details on how trading mechanisms affect the price formation process, see the vast literature on market micro- structure, including the seminal work in Roll (1984), Kyle (1985), and the excellent review in Hasbrouck (2007). The impact of liquidity and leveraging on portfolio performance has been addressed in the literature, but as two separate topics of interest. Our goal is to integrate liquidity and leveraging in a unified discussion toward addressing three important issues related to the right question: first, what is the potential degradation in economic performance of a portfolio designed by ignoring liquidity 1 CAPM was also introduced independently by Treynor (1961), Lintner (1965), and Mossin (1966). 2 Illiquidity may be thought of as, hypothetically, the difficulties or costs associated with reversing a trade instanta- neously after a trade is executed, which depends on market depth, volume of the trade, bid-ask spread, intermediation, and transaction costs. In this paper, we use the two terms, market impact of trading and cost of liquidity risk, in an asset synonymously. 3 risk, in comparison to an ex-ante optimal portfolio that accounts for price impact explicitly?, second, how is this performance shortfall further impacted due to portfolio leveraging?, and third, what economic insights can be gained for setting leverage and target return levels for a portfolio in a market scenario with significant liquidity risk? In the literature on impact of liquidity, it is established empirically that asset returns are sensitive to liquidity and that sensitivity is priced, see e.g. Pastor and Staumbaugh (2003), the survey in Amihud et al. (2005), or Rahi and Zigrand (2008) who employ a general equilibrium model to measure liquidity. On the other hand, Acharya and Pedersen (2005) derive a liquidity-adjusted CAPM to explain the empirical findings within a unified framework, but without any impact of leveraging. Our focus here is neither on creating a measure of liquidity nor about modifying an existing asset pricing model with a liquidity risk premium. The issue of increasing leverage to improve portfolio performance, without liquidity concerns, has also received some attention in the literature; Jacobs and Levy (2012, 2013) proposed to augment the mean-variance (MV) utility function with an additional term for leverage aversion since the standard MV paradigm fails to capture the risks faced by leveraged portfolios. However, their model is free of market impact costs of trading, hence a liquidity impact on leveraging is absent, and their focus is to fix the MV-optimal portfolios to have leverage levels more consistent with the real-world. Leverage risks stem from over-exposure in portfolio debt that may result in, for instance, the possibility of margin calls and forced liquidations at adverse prices, and losses beyond the capital invested due to unforeseen and exogenous events or sudden price changes, see Jacobs and Levy (2012). Historical events such as the 1998 financial crisis which witnessed the downfall of Long- Term Capital Management are reminders of leverage risks.3 In the financial crisis in 2008, Lehman Brothers filed for bankruptcy due to a combination of high leverage (as much as 31:1 in 2007), risky investments, and liquidity problems. Moreover, excessive-leverage poses a real risk not only to a portfolio but also in propagating a market crisis, see Acharya et al. (2012). Despite the above leverage-related risks, fund managers often seek increased leverage in order to achieve higher return targets. Usually, an unlevered portfolio of an acceptable risk profile is further invested in by incurring a debt capital, which amplifies any risks or imbalances that exist in the unlevered portfolio, but potentially leading to a gain in portfolio's returns. It is under this premise that the so-called risk parity (RP) asset allocation strategies have emerged.4 In the logic of RP, 3 The Russian devaluation of its currency and default on its treasury's debt led to the financial crisis in August 1998, and LTCM, which once leveraged their equity as much as 30:1, lost a staggering 44% of its equity during the month of August due to massive losses under forced liquidation of the holdings to cover margin requirements. 4 RP strategies garnered popularity in the aftermath of the recent global financial crisis. By August 2015, AUM using RP strategies was close to $500B, according to J.P. Morgan Markets Research. Early development of RP can be traced to Qian (2006), and further work is discussed in, e.g., Chaves et al. (2011), Maillard et al. (2010), Choueifaty and Coignard (2008), and Carvalho et al. (2012). 4 mean Portfolio B A ݉ Unlevered portfolio T ்݉ Portfolio T proportionately‐ Tangency leveraged portfolio ݎ ଴ Risk‐free return Portfolio Std Dev 0 ߪ் ߪ஻ ߪ஺ Figure 1 Leveraging a low-risk portfolio to improve expected return (when liquidity costs are ignored) liquidity concerns are absent: choose a low-risk unlevered portfolio that allocates more weight to lower risk assets, hence, a lower portfolio return as well (e.g., the tangency portfolio or the GMV), and then, the unlevered portfolio is levered-up (referred to as proportionate-leveraging) to gain superior portfolio returns without sacrificing its risk-profile, a process referred to as generating a leverage risk premium.5 Conceptually, the notion of leverage risk premium is rooted in Tobin's separation result, as illustrated in Figure 1: given a target return m, instead of choosing a high-risk (or risk-concentrated) portfolio A, a low-risk (or risk-diversified) tangency-portfolio T with return mT may be levered-up to obtain portfolio B (incurring leverage risk) to achieve the target return m (> mT ). In so-doing, B has lowered the variance risk relative to A, i.e., σB < σA. We note, however, that the increased levels of trading required to establish the positions in B (having the same composition as T ) encounters a higher degree of the aforementioned price-effects due to asset illiquidity, thereby making the anticipated performance of B questionable. That is, the expected leverage risk premium in B faces erosion due to illiquidities, which makes the superior gains expected in RP more-hypothetical. In fact, increased leverage is only going to exacerbate the effects of liquidity risk, leading to even greater impact on performance, particularly during severe market swings. As such, an analysis of leverage premium without the context of liquidity impact on portfolio performance is a vacuous exercise.6 5 Frazzini and Pedersen (2014) presented a plausible rationale based on \leverage risk aversion and low volatility asset returns" for the existence of a leverage risk premium; the cross-sectional equity risk anomaly was first documented by Ang et al.