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
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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. (2006). Also, see Asness et al. (2012) who explains that the highest risk-adjusted return is achieved not by the market but, rather, by a portfolio that over-weights safer assets. 6 It must also be noted that portfolio frontier in Figure 1 has already assumed the absence of liquidity risk, and thus, the performance expectations indicated as A and T themselves are subject to correction. 5
This paper provides an integrated analysis combining all three aspects within a model that seeks a Pareto trade-off between risk-adjusted performance and leverage level of a portfolio under liquidity risk; that is, a study of the economic efficiency between risk of concentration and risk of leveraging in the presence of market impact. Under liquidity risk, we conclude that the Sharpe-maximizing unlevered portfolio does not correspond to a tangency portfolio of the mean-variance frontier. This is in contrast to the classical portfolio theory under no market impact, i.e., the tangency portfolio T and the ‘Security Market Line’ (SML) in Figure 1. Furthermore, we show that an optimal leveraged-portfolio that maximizes the Sharpe ratio undergoes changes to its composition (risk profile) from that of an optimal unlevered portfolio when market impact is incorporated ex-ante. That is, there is no separation in the sense of Tobin’s when determining an efficient risk-averse portfolio. In particular, this result questions the standard arguments made for RP strategies based on the premise that leveraging preserves the risk profile, which indeed is based on the assumption that liquidity costs are absent. Moreover, as target return is increased under liquidity costs, the Sharpe-maximizing leveraged-portfolio is not simply obtained by proportionately-leveraging an unlevered (efficient) portfolio; hence, proportionate-leveraging is not an optimal strategy. We show that an efficient allocation of leverage exists that maximizes the economic performance, albeit with an upper limit that depends on liquidity parameters specified for a given economic state. Liquidity parameters in our model are based on permanent and temporary impact on asset prices resulting from volume and intensity of trading, two components primarily responsible for the inability to establish or liquidate portfolio positions in a timely manner at acceptable prices. We follow the basic model of Carlin et al. (2007) and Brown et al. (2010), but extend it so that the period-ending asset returns are stochastic, normally-distributed. Thus, the resulting portfolio leverage at the end of a trading trajectory is also a random variable. In order to control leverage as a managerial input, a ‘certainty-equivalent’ leverage level is employed under a risk-averse setting. Consequently, the investor’s leverage-risk-aversion level (or risk neutrality) also becomes a necessary determinant of an efficient portfolio that trades-off risk-adjusted returns with leverage risks in the presence of liquidity costs. As the investor’s leverage risk aversion increases, we show that portfolio’s economic performance (i.e., Sharpe ratio) cannot be improved for fixed levels of target return and leverage. Given a fixed leverage level of a risk-averse investor, we claim analytically that an increase in target mean is accompanied by a reduced risk-adjusted performance. This reduction in portfolio- Sharpe can be reversed to some extent with an increase in leverage; however, it can never reach the maximum Sharpe performance attainable at a lower target mean. That is, Sharpe-Leverage frontiers are separated by portfolio target mean, and frontiers of lower targets dominate those of higher targets. This is a significant departure from the classical portfolio theory that ignores 6 market impact, where the maximum Sharpe of a lower target mean can be easily-attained even at a higher target mean provided sufficient leverage is applied. Moreover, there may not exist a feasible portfolio, given the set of underlying portfolio-assets, that can achieve a desired target return when the investor faces liquidity risks and leverage restric- tions, or high risk-aversion. That is, when the investor places a higher expectation on portfolio returns, liquidity costs can compel her to take even a higher level of portfolio leverage, possibly violating preferences on leverage. We show that there exists a minimum required leverage to attain a given portfolio target, and this minimum leverage grows at an increasing rate with target return; moreover, there exists a maximum leverage level beyond which target return cannot be increased further, concepts not germane to the classical portfolio theory. Accordingly, portfolio choice is bounded by these minimum and maximum leverage levels for a specified target return when liq- uidity impact is present. Therefore, when setting portfolio targets, the investor must consider the interactions among acceptable leverage level and its risk aversion, market impact of trading due to illiquidity, and the parameters of the underlying portfolio assets, simultaneously. This integrated view is the premise of our analytical contributions in this paper - the theory of three tales of a portfolio. Under an empirical design and analysis using ETF assets, we also confirm the validity of our ex-ante analytical results. We obtain portfolio choice boundaries and illustrate the differences that exist in portfolio frontiers under liquidity impact, relative to a scenario in which liquidity costs are ignored. It also turns out that portfolio economic performance shortfall when ignoring trading impact can be substantial, a measure of ex-ante liquidity risk. However, any ex-post statistical analysis of our findings is outside the scope of this paper.
2. Liquidity-Impacted Pricing and Portfolio Metrics
Portfolio positions are modeled under a continuous-time trading environment by following Car- lin et al. (2007) in partitioning price effects based on permanent and temporary components of liquidity, which follows the early work by Kraus and Stoll (1972), Holthausen et al. (1990), and several others. Accordingly, the total trading volume in an asset during the period gives rise to a permanent price impact, while the trading intensity, i.e., speed or rate, leads to temporary liquidity shortages that cause a temporary price impact. Moreover, we will assume that market impact is symmetric on buy- and sell-sides.7
7 There is evidence that buyer initiated trades in bear markets or seller-initiated trades in bull markets face increased liquidity, implying higher permanent impact in block trades for buyers in bull markets and sellers in bear markets, see Chiyachantana et al. (2004). Extensions of this information asymmetry in the permanent price impact of block trades using S&P ETF and index futures in bull and bear markets are discussed in Frino et al. (2017). 7
Consider a one-period economy in which a set of n risky assets are traded at date 0 and the portfolio performance of the assets is observed at date 1. Initial (day 0) share position in asset
n j is x0j, j = 1, . . . , n. The portfolio is rebalanced to positions-vector x1 ∈ R , which are decisions made at date 0 and their executions are completed by date 1. The transition of positions x0 → x1 faces liquidity risks depending on the trading trajectory. If the trade-size ||x1 − x0|| is large, then a significant permanent price impact may be expected, and if the position transition occurs at a high rate, then trading may also face temporary liquidity shortages.
During the position-transitions up to time t ∈ (0, 1], asset prices will progress from p0j to ptj, j = 1, . . . , n, due to the investor’s trading, but deterministically. Permanent impact on an asset price depends on the cumulative amount traded up until t; on the other hand, temporary impact depends on the rate at which the asset is traded and its price effect is instantaneous and reversible. This single asset price model by Carlin et al. (2007), proposed for liquidating a position, was extended to a portfolio of multiple assets by Brown et al. (2010) and Chen et al. (2014), which we shall follow in this paper, augmented with a stochastic adjustment at the end of the period.
Denote the position in asset j at time t ∈ [0, 1] by xtj and let the instantaneous rate of trading
dxtj be ytj = dt along an absolutely continuous trading trajectory t → xtj. A positive rate indicates buying and a negative rate indicates selling. Note that the total trading volume until time t leading R t to a permanent price impact is xtj − x0j = 0 ysjds. Let γj and λj denote the (positive) permanent impact and temporary impact coefficients, respectively. Following Brown et al. (2010), the total impact on price due to trading at time t is:8
Itj = γj(xtj − x0j) + λjytj. (1)
In the absence of price uncertainty, the asset price model in the latter reference is:
ptj = p0j + Itj, t ∈ [0, 1], j = 1, . . . , n. (2)
2.1. Stochastic asset returns model
In order to introduce price uncertainty, we leave the liquidity-based price impact Itj as determin- istic, but allow the initial price p0j to evolve stochastically under information exogenous to the investor’s trading action, i.e., the uncertain price component is unaffected by trading. The basic idea follows that of Almgren and Chriss (2000) and Almgren (2003), who split the asset price into a market impact component due to trading and an unaffected price component. They considered a
8 The model can be further generalized with nonlinear impacts due to both permanent and temporary components k k so that Itj = γj (xtj − x0j ) + λj (ytj ) , where the fixed coefficient k ∈ (0, 1]. In such a scenario, the total liquidity impact costs grow less than quadratically in the trade quantities. Our future work will focus on such an extension of the insights developed in this paper. 8 single security price that evolves according to the discrete arithmetic random walk. Gatheral and Schied (2012) employed a geometric Brownian motion (GBM) for the unaffected asset prices. To obtain tractable analytical results, we consider a stochastic ‘jump’ to the asset price at date 1, independent of the investor’s trading during the period. That is, the unaffected asset price, denoted by Ptj, is assumed to remain unchanged in the interval [0, 1 − ), but it undergoes an instantaneous
9 uncertainty at date 1. This yields Ptj = p0j for all t ∈ [0, 1) and P1j is random. Thus, the price process of the asset with trading impact is modeled as: ptj = Ptj + Itj, t ∈ (0, 1] Ptj = p0j, t ∈ [0, 1) (3) P1j = p0j + ∆j, where ∆j is the random adjustment to the unaffected price at the end of the trading period, and
Itj is given by (1). Then, the asset price at date 1 is:
p1j = p0j + ∆j + I1j = p0j + ∆j + γj(x1j − x0j). (4)
To set the notation, consider the overall return on the asset over the period, given by
p1j − p0j ∆j + I1j ∆j γj(x1j − x0j) Rj = = = + . (5) p0j p0j p0j p0j
Hence, Rj consists of two components, first the random return realized if no trading is initiated by the investor in the asset, denoted by the unaffected return
∆j rj := , (6) p0j and second, the permanent return adjustment due to trading in the asset. Hence, the observed data on asset returns, in the absence of the investor’s trading, are generated by the distribution of rj, while the second component is based on investor’s own trading activity. We shall assume the unaffected returns to be normally distributed, i.e., r ∼ N (µ, V ), where µ = E[r] ∈ Rn is the mean vector and V = Var[r] ∈ Rn×n is the covariance matrix. We denote the diagonal matrices
P0 = diag(p01, . . . , p0n), Γ = diag(γ1, . . . , γn), and Λ = diag(λ1, . . . , λn), where γj and λj are the coefficients of market impact in (1).
9 The uncertainty model is such that accumulated exogenous (fundamental or economic) information on the asset, as well as action of other traders, from date 0 to 1 are instantaneously revealed at date 1. In this sense, the unaffected price model here is similar to that used in the standard mean-variance model. A multivariate correlated (arithmetic) Brownian motion that evolves during the period is considered for the unaffected component in our continuing research. 9
2.2. Portfolio returns model
Recall that the positions-vector x1 is created through a continuous trading trajectory t → xtj, as R 1 specified by the trading rate ytj, where x1j −x0j = 0 ytjdt. Therefore, as prices evolve according to
(3), financing of the trading strategy t → xtj may require borrowing or lending cash, which would > make the net portfolio return at date 1 different from R x1. Assume cash (as a non-random asset) can be borrowed or lent at the continuously-compounded risk-free rate, r0(> 0) per trading period. Let K(y) be the net cash generated by the trading strategy. Then,
Z 1 > r0(1−t) K(y) = − pt yte dt. (7) 0 Observe that K(y) allows the possibility of borrowing exogenous cash, if required to leverage, in the process of constructing the portfolio x1 along a chosen trading trajectory.
Denote the initial (cash) liability at day 0 of the portfolio by L0. A positive or negative L0 indicates an initial debt level or surplus cash position, respectively. The total dollar return at day 1 is composed of the non-random net cash (generated or required), plus the random asset value, and the initial net asset value. Denoting the portfolio rate of return random variable for the period by R(x1), > r0 K(y) + [p0 + ∆ + Γ(x1 − x0)] x1 − e L0 R(x1) = > − 1, (8) p0 x0 − L0 where ∆ = P0r and assuming that the initial liability (or surplus cash) is discounted at the risk- free rate. Note that it is impossible to evaluate the portfolio return in (8) and its distributional parameters without specifying a trading strategy. We shall be concerned only with the case of static trading strategies in this paper. A static strategy is one determined in advance of trading. Such an example is a constant trading rate, e.g. Brown et al. (2010) and Chen et al. (2014), which is a volume-weighted average price (VWAP) approach. A dynamic strategy is one in which the trade size depends on the stock price during execution of the order, such as in the case of a Delta-hedging strategy. For the special case of a single stock liquidation problem, Almgren and Chriss (2000) showed the optimality of static strategy under arithmetic Brownian motion, while Gatheral and Schied (2012) obtained a dynamic strategy as optimal under GBM. In particular, if the price process has no random term or the random component is independent of the current stock price, then a statically-optimal strategy will be dynamically-optimal for the problem of a single asset pure liquidation problem (where terminal equity value is maximized). Motivated by this, we employ the static constant-rate trading strategy although we cannot prove its optimality for our portfolio problem. We shall use the subscript ‘s’ in all relevant parameters and variables when there is explicit dependence of their values in the trading strategy. 10
x1j −x0j Under the constant-rate trading strategy, yj ≡ ytj = 1 = x1j − x0j. Following (7), the net cash ‘generated’ is: Z 1 Z 1 > r0(1−t) r0 > > > −r0t Ks(x1) = −pt yte dt = −e [p0 y + y Λy + ty Γy]e dt 0 0 > > > > = −x1 Msx1 + 2(Msx0) x1 − x0 Msx0 − κsp0 (x1 − x0), (9) where we have defined:
1 1 er0 − 1 r0 r0 Ms := (e − 1) Λ + 2 (e − r0 − 1) Γ and κs := . (10) r0 r0 r0
Ms is a positive definite matrix if Γ and Λ are positive definite, i.e., λj, γj > 0. Note that when the risk-free rate r0 → 0, we have Ms → (Λ + 0.5Γ) and κs → 1. Defineκ ˆs = κs − 1. Then, the portfolio expected return is obtained from (8) as
> r0 > Ks + [p0 + ∆ + Γ(x1 − x0)] x1 − e L0 − (p0 x0 − L0) E[Rs(x1)] = E > p0 x0 − L0 1 > > φs = x1 (Γ − Ms)x1 + (2Msx0 − Γx0 − κˆsp0 + P0µ) x1 + − 1 (11) w0 w0 by defining the initial portfolio net wealth at date 0:
> w0 := p0 x0 − L0 (> 0) (12) and the trading strategy dependent constant:
> > r0 φs := κsp0 x0 − x0 Msx0 − e L0. (13)
The variance of the portfolio return is 1 1 Var[R (x )] = Var ∆>x = x>P V P x , (14) s 1 2 1 2 1 0 0 1 (w0) (w0)
1 p > and its standard deviation is denoted by σ[Rs(x1)] := x P0V P0x1, which is independent of w0 1 the trading strategy.
2.3. Liquidation-based risk-free rate and Sharpe ratio
In order to evaluate the Sharpe ratio of a risky-investment strategy, the return on investment in the risk-free asset under total portfolio liquidation must be determined, herein termed the effective risk-free rate for the investor. Given the day 0 (initial) portfolio net value w0(> 0) of risky and risk-free investments, and under a constant-rate liquidation strategy, the liquidated portfolio has x1 = 0 at date 1. Then, referring to (11), the portfolio rate of return upon liquidation is:
φs rs := E[Rs(0)] = − 1. (15) w0 11
10 where φs is in (13). We shall assume that φs > 0 for the initial portfolio x0. Consequently, portfolio return of a risky investment x1 is compared with rs. For better risk-adjusted performance, portfolio Sharpe ratio,
> > E[Rs(x1)] − rs x1 (Γ − Ms)x1 + (2Msx0 − Γx0 − κˆsp0 + P0µ) x1 Ψs(x1) := = , (16) σ[R (x )] p > s 1 x1 P0V P0x1 must be increased. In doing so, x1 may be allowed to incur portfolio debt; however, the level of leverage in portfolio x1 must be ‘acceptable’ to the investor.
3. Model of portfolio leverage
Noting that portfolio (cash) liability at date 0 is L0 (which indicates an initial surplus cash position > if negative), the net asset position is p0 x0 − L0. Define leverage ratio by ‘total liability divided by > total assets’. Thus, the leverage ratio at date 0 is ρ0 := max{L0, 0}/(p0 x0 − L0). To determine the leverage ratio at date 1, the period-ending liability at date 1 is:
r0 Ls(x1) = e L0 − Ks(x1)
> > = x1 Msx1 + (κsp0 − 2Msx0) x1 − φs. (17)
Since Ms is p.d., Ls is convex in x1. If Ls > 0, it is a liability, and Ls < 0 indicates an excess ending cash position in the portfolio.11 The period-ending net asset position at date 1 is
> As(x1) = p1 x1 − Ls(x1)
> > = x1 (Γ − Ms)x1 + [∆ + (2Ms − Γ)x0 − κˆsp0] x1 + φs. (18)
While the liability level Ls is non-random, the asset position As is random (and normally- distributed since ∆ is normally-distributed). Hence, the leverage ratio at date 1, i.e.,
+ + LR(x1, ∆) := Ls (x1)/As(x1) where Ls (x1) := max{Ls(x1), 0}, (19) is random. A random variable cannot be controlled in managing leverage; instead, a ‘certainty equivalent’ (CE) form of LR(x1, ∆) is employed. We assume the investor is risk-averse to portfolio leverage. Then, noting that As is normally-distributed, LR(x1, ∆) is replaced by the CE function:
+ Ls (x1) 0 ≤ Ls(x1) := , (20) E[As(x1)] − π Var[As(x1)]
10 If φs > w0 > 0 holds, then liquidating the initial portfolio yields rs > 0. 11 While x1 has long/short positions, Ls(x1) in (17) does not treat short positions themselves as liabilities. If short P positions are included in defining leverage, total liability becomes Ls + j p1j max{0, −x1j }, which is a random variable since p1 is random. Statistical difficulties stemming from that are addressed in our continuing research under a chance-constraint approach to leverage control. 12 provided the ‘price of risk’ in leverage, herein referred to as the investor’s leverage-risk-aversion level π (≥ 0), is sufficiently small such that there exists a feasible portfolio with the denominator of (20) being positive. If π is sufficiently-large, indicating an extremely leverage-risk-averse investor,
E[As]−π V ar[As] ≤ 0 holds for all portfolios (except for the liquidating portfolio, x1 = 0), and thus, the investor must set Ls(x1) ≤ 0 to prohibit portfolio debt. As π decreases, incurring portfolio debt becomes increasingly less-restrictive, and when π = 0, the investor is leverage-risk-neutral. Noting (18), the expected asset value is:
> > E[As(x1)] = x1 (Γ − Ms)x1 + [(2Ms − Γ)x0 − κˆsp0 + P0µ] x1 + φs (21)
= w0 [E[Rs(x1)] + 1] and the variance is:
> > Var[As(x1)] = Var ∆ x1 = x1 P0V P0x1. (22)
Given an investor-prescribed allowable maximum leverage level ρ (≥ 0), and a risk aversion coef- ficient π ≥ 0, portfolio leverage level is then controlled by the constraint:
0 ≤ Ls(x1) ≤ ρ. (23)
When the liquidity impact parameter matrix (Γ − Ms) is negative-definite (n.d.), it can be shown that Ls(x1) is a quasi-convex function, see Appendix A. This implies, then, that the set of portfolios satisfying (23) is a convex set, which is an essential property in the development of our analytical results. To interpret the latter negative definiteness, note that (11) yields,
> > w0 (E[Rs(x1)] − rs) = x1 (Γ − Ms)x1 + [P0µ − κˆsp0 + (2Ms − Γ)x0] x1, (24) which is concave in the asset positions in this case, that is, portfolio excess mean return obeys the economic law of diminishing returns under liquidity impact. Therefore, economically-speaking, the required negative definiteness may be expected to hold across most asset-sets. In the empirical evidence provided in this paper, price impact parameter estimates reported in Table 1 verify that
12 Γ − Ms is n.d., as we shall assume throughout the paper.
12 1 Also, observe from (10) that Msj increases with the risk-free rate, and at r0 = 0, Msj = λj + 2 γj . Hence, Γ − Ms 1 is n.d. for any risk-free rate provided 2 γj < λj holds for all assets, i.e., the permanent impact coefficient is less than twice the temporary impact coefficient. Such an assumption was made in Brown et al. (2010) for each asset in the portfolio, while a similar assumption is also made in Almgren and Chriss (2000). The estimated parameters reported in Table 1 for ETF assets confirm that the preceding condition is satisfied for each asset, and thus, Γ − Ms is n.d. 13
3.1. Pitfalls in proportionate leveraging
Improving portfolio’s risk-adjusted returns is of fundamental importance in portfolio management. A standard method of improving a given (unlevered) portfolio is to apply proportionate leveraging, as presented below. When ignoring market impact of trading, such practice is an optimal portfolio strategy under leveraging. However, as we will show in the sequel, this is not an optimal strategy for allocating portfolio debt when trading encounters market impact.
Given a portfolio x1, we say x1 is proportionately-levered to a portfoliox ˜1 with factor (or multi- plier) ν ifx ˜1j = νx1j, ∀j. When there is no market impact, i.e., Γ = 0 = Λ (so, Ms = 0), the Sharpe ratio Ψs in (16) is ‘positively homogeneous’ (of degree 0) in asset positions. This implies that portfolio Sharpe is invariant to proportionate increases in portfolio positions, as is well-known. In contrast, the situation is very different under liquidity impact:
Proposition 1. Suppose the matrix (Γ − Ms) is n.d. Then, the Sharpe ratio Ψs in (16) is pseudo- concave in x1 ∈ X := {x1 : E[Rs(x1)] > rs}. For a given scalar ν, Ψs(νx1) is decreasing in ν > 0.
Moreover, Ψs(νx1) < Ψs(x1) if ν > 1 and Ψs(νx1) > Ψs(x1) if 0 < ν < 1.
Proof. See Appendix B. Thus, it follows that proportionate leveraging (with ν > 1) of a portfolio under liquidity impact leads to strictly worsening the risk-adjusted returns (whereas trimming positions proportionately lead to Sharpe increases). The above conclusion does not mean that portfolio expected returns will not increase under leveraging, but the increase in mean relative to the increase in standard deviation risk progressively-worsens. However, if the extent of leveraging becomes excessive, even the portfolio mean improvement may get hampered, as shown next.
Proposition 2. Suppose the matrix (Γ − Ms) is n.d. and portfolio x1 satisfies E[Rs(x1)] = m > rs 1 > and m − rs > |x (Γ − Ms)x1|. For a proportionate-levered portfolio x˜1 = νx1: w0 1
i) E[Rs(˜x1)] > m if ν ∈ (1, νmax(x1)); moreover, E[Rs(˜x1)] < m if ν < 1 or ν > νmax(x1), where
w0(m−rs) νmax(x1) := > . |x1 (Γ−Ms)x1| ii) E[Rs(˜x1)] is increasing in ν ∈ (1, ν¯(x1)], and E[Rs(˜x1)] is decreasing for ν > ν¯(x1), where w0(m−rs) ν¯(x1) := 0.5 1 + > < νmax(x1). |x1 (Γ−Ms)x1|
iii) The maximum portfolio mean under proportionate leveraging of x1 is obtained at ν =ν ¯(x1), and is given by
2 (¯ν) > m¯ (x1) := E[Rs(¯νx1)] = |x1 (Γ − Ms)x1| + rs (> m). (25) w0
Proof. See Appendix C. 14
First, noting the case of ν < 1 in Proposition 2(i), and combining with that in Proposition 1, we conclude that proportionate-deleveraging, while improving the portfolio’s Sharpe ratio, leads to worsening the portfolio’s expected returns.
Also, we observe that leveraging a portfolio x1 to improve its mean return under market impact follows the ‘law of diminishing returns’, whilst decreasing its Sharpe; moreover, if the leveraging is
1 > excessive, mean return can even decline over that of x1, provided m − rs > |x (Γ − Ms)x1|. To w0 1 interpret the latter condition, using the case of zero initial risky asset positions and zero risk-free
> > rate, (24) yields w0(m − rs) = −|x1 (Γ − Ms)x1| + P0µ x1 when Γ − Ms is n.d. Thus, the condition 1 > > > that m − rs > |x (Γ − Ms)x1| implies P0µ x1 > 2|x (Γ − Ms)x1| must hold. Intuitively, this w0 1 1 means that the unaffected mean return, see (6), should be at least twice the (guaranteed) loss in return due to trading under liquidity risk, i.e., market impact parameters must be sufficiently small in order to consider leveraging a portfolio. However, we cannot offer a rigorous proof that the condition holds generally for a given non-efficient portfolio. Nevertheless, for the efficient portfolios selected under liquidity impact in this paper, our empirical analyses show that this condition is always satisfied, see Table 3.
To illustrate graphically, consider a portfolio x1 with expected return m (> rs), which is pro- portionately leveraged to yield expected return, g(ν) where ν is the leveraging multiplier. Since
(Γ − Ms) is n.d., g(ν) is concave in ν and g(1) = m. Under market impact, however, it follows that
0 13 g (1) < m − rs, see Figure 2. On the other hand, if liquidity impact is ignored in leveraging, the expected portfolio return increases as ν increases at a constant rate of m − rs, while the Sharpe ratio remains a constant.
Standard MV leveraging of Standard MV leveraging of ratio under NO market impact under NO market impact: Slope Return, Sharpe
Ψ Leveraging of under market Leveraging of under market impact and trade dynamics impact and trade dynamics Expected Case of 1 0: | Γ | Fixed portfolio 1 Leverage multiplier ( ) 1 ̅ Leverage multiplier ( )
(a) (b)
Figure 2 Effect of market impact when a given portfolio x1 undergoes proportionate-leveraging
13 1 > It can be shown that if the condition m − rs > |x (Γ − Ms)x1| is violated for a portfolio x1 with mean return m, w0 1 0 then g (1) ≤ 0 holds, implying that proportionate-leveraging of x1 does not increase portfolio return under liquidity impact. 15
Although proportionate-leveraging of x1 preserves its risk profile, this practice carries two related pitfalls: first, the maximum achievable target mean is limited bym ¯ (x1), see (25), and second, portfolio Sharpe will decline from that of x1 at all targets up tom ¯ (x1), see Proposition 1. Therefore, in order to improve portfolio Sharpe under leveraging, a new risk profile (or composition) must be devised following a path different from proportionate leveraging. This is the concept of optimal leveraging under liquidity impact as advocated in this paper. To determine an optimally-leveraged portfolio, net cash generated from portfolio trading in (9) must be allowed to be negative, to be funded by risk-free borrowing. In order to avoid excessive leverage, portfolio borrowings must be controlled via the leverage constraint for a specified input- pair (π, ρ), in addition to a portfolio target return requirement, if any, in the process of maximizing portfolio Sharpe in (16). Then, the trade-off between Sharpe ratio and portfolio leverage provides important insights in determining efficient portfolios under liquidity impact.
4. Unlevered Optimal-Sharpe and Leverage Impact
First, consider the case when portfolio leverage is not allowed, i.e., ρ = 0, and thus, Ls(x1) = 0 or Ls(x1) ≤ 0. Would the portfolio that maximizes Sharpe ratio in this case allow ‘lending’, i.e.,
Ls(x1) < 0? It turns out that a non-levered portfolio that allows lending achieves a maximum Sharpe when it (asymptotically) becomes the liquidating portfolio:
Suppose (Γ − M ) is n.d. Then, sup {Ψ (x ): L (x ) ≤ 0} is attained when Proposition 3. s x1 s 1 s 1 x1 → 0, i.e., the liquidating portfolio with investment only in the risk-free asset. Furthermore, define: ζ (l) := sup {Ψ (x ): L (x ) = l}, for scalar l, Then, the maximum Sharpe ratio ζ (l) SH x1 s 1 s 1 SH is non-increasing for l ∈ (−φs, 0].
Proof. See Appendix D.
Therefore, as claimed, increasing portfolio lending improves the Sharpe ratio. As l → −φs, achieved with the liquidating portfolio, portfolio Sharpe is the highest. As liability (l) approaches zero, the portfolio lending disappears, while the portfolio remains unlevered, in which case the max Sharpe ratio is the worst among all unlevered portfolios, denoted by ζSH ≡ ζSH(0). In this case, 0 the portfolio is invested only in the risky assets, denoted x1, and it is given by:
max ζSH = sup {Ψs(x1): Ls(x1) = 0} . (26) x1 For the classical theory by ignoring liquidity risk, Λ = 0 = Γ must be set in (26) so that trade execution is instantaneous. In that case, the ‘tangency’ portfolio of the MV efficient frontier of the risky assets determines the maximum Sharpe ratio, see portfolio ‘T ’ in Figure 1. Then, the value of (26) is the slope dE[Rs] evaluated at the optimal portfolio (with no-lending and no-borrowing), dσ[Rs] vis-´a-visSML. In a significant departure, incorporating liquidity impact leads to a very different counterpart result: 16
0 Proposition 4. Denote an optimal solution of (26) by x1. Then, the maximum Sharpe ratio under no-leveraging is:
max dE[Rs] 0 dLs ζSH = − θ0σ[Rs(x1)] , (27) dσ[Rs] 0 dσ[Rs] 0 x1 x1 where the scalar: 0> 0 x1 (Γ − Ms)x1 θ0 = 0> 0 0 . (28) w0(x1 Msx1 + φs)σ[Rs(x1)]
If (Γ − Ms) is n.d., then θ0 < 0 and the optimal Sharpe ratio strictly decreases if sufficiently small portfolio debt is applied.
Proof. See Appendix E. Note that the first term in (27) is the slope of the tangent to the mean-variance frontier under
0 market impact at the optimal portfolio x1 which is completely invested in risk assets with no borrowing. Hence, the Sharpe-maximizing portfolio is not a ‘tangency-portfolio’ of the MV-frontier 0 0 max dLs(x1) at x1. Since θ0 is negative, the optimal Sharpe ζSH is greater or less depending on if 0 dσ[Rs(x1)] is positive or negative, respectively.14 Consequently, a simple result comparable to SML does not
0 exist in this case. In the ‘immediate vicinity’ of the Sharpe-maximizing unlevered portfolio x1, the equation (27) represents the security market relationship (SMR) under liquidity impact. As an
0 investor consuming the portfolio x1 contemplates leverage (to improve portfolio return), she would allocate the liability Ls optimally such that (27) is satisfied in order to obtain a new portfolio with 0 the best-Sharpe possible, instead of proportionately-leveraging the portfolio x1. Indeed, SMR is the SML in the special case when Λ = 0 = Γ.15
It is noteworthy that θ0 < 0 implies the sensitivity result that the maximum possible Sharpe ratio under sufficiently small portfolio debt will strictly decrease from the “no-leverage Sharpe”
max 0 max of ζSH . That is, given the lending- and debt-free portfolio x1 with maximum Sharpe of ζSH , any leveraging under liquidity risk results in strictly-decreasing portfolio’s risk-adjusted returns. This conclusion is fundamentally different from the corresponding result when ignoring market impact, where the Sharpe ratio is known to remain invariant under risk-free borrowing, i.e., when leveraging the tangency portfolio. Under liquidity impact, in contrast, increased portfolio variance risk under leveraging is not sufficiently offset by portfolio mean return increases, due to losses from market impact. Summarizing:
2M (x0 −x )+κ p 0 14 dLs = (w )2σ[R (x0)] Pn jj 1j 0j s 0j , where Vˆ = P V P . Thus, in general, the sign of dLs(x1) dσ[R ] 0 s 1 j=1 ˆ (j) 0 0 0 dσ[R (x0)] s V x1 s 1 0 cannot be predicted. However, if p0j is sufficiently large relative to 2Mjj (x1j − x0j ), and the optimal portfolio is 0 dLs(x1) positively correlated with each asset, then 0 is positive. dσ[Rs(x )] 1 15 max dE[Rs] In this case when price impact is zero, (28) yields θ0 = 0, and thus, ζSH = follows from (27), which is dσ[Rs] 0 x1 the slope of the tangency portfolio as known in the standard portfolio theory. 17
max 0 Sharpe ratio of ζSH under liquidity risk is achieved when target return is m0 ≡ E[Rs(x1)], 0 corresponding to the unlevered and lending-free portfolio x1; any leveraging (proportionate or max otherwise) of this portfolio strictly decreases the risk-adjusted returns to below ζSH . It was already claimed in Proposition 1 that proportionate-leveraging of a portfolio leads to worsening Sharpe. What is concluded in the preceding paragraph is that a marginal increase of debt that is optimally-allocated in an otherwise unlevered Sharpe-maximized portfolio also leads to worsening Sharpe. In the sequel, we shall show such an optimal allocation of debt is still preferable in the sense of economic performance (as measured by Sharpe ratio), relative to proportionate- leveraging.
4.1. Target return and portfolio leveraging
0 The Sharpe-maximizing portfolio x1 attains a target return of m0 without any leveraging. Can the return be improved beyond m0 without incurring any leverage? The answer is in the affirmative, but there exists an upper limit on target return; moreover, the Sharpe declines from the no-leverage max ζSH . It also turns out that when portfolio leverage is applied, a further increase in target mean can still be obtained, along with a possible improvement in Sharpe.
Consider the problem of determining the maximum target return, mmax(ρ), under a prescribed leverage level, ρ:
mmax(ρ) = max {E[Rs(x1)] : Ls(x1) ≤ ρ} . (29) x1 0 The following remarks are noteworthy: first, m0 ≤ mmax(0) holds since the portfolio x1 that solves
(26) is feasible in (29) for ρ = 0. However, if m0 < mmax(0), then the Sharpe of an unlevered portfolio 0 max xˆ1 that solves (29) for ρ = 0 would be inferior to that of the unlevered portfolio x1, i.e., ζSH >
Ψs(ˆx1). Second, for an investor specifying leverage parameters (π, ρ), along with a desired target return m > mmax(ρ), there does not exist any feasible portfolio, i.e., the investor has to decrease the required return to form a levered portfolio. Third, mmax(ρ) is finite when Γ − Ms is n.d., because
E[Rs(x1)] is a (strict) concave function and (29) is a solvable convex program. Finally, it also follows from (29) that as leverage becomes infinite, the resulting target mean m∞ ≡ mmax(∞) is still finite.
In contrast, when market impact is ignored as in the classical portfolio theory, mmax(ρ) = +∞ 16 holds for any leverage level, whereas the highest possible return m∞ under liquidity impact is associated with the portfolio:
∞ x1 := arg max {E[Rs(x1)] : Ls(x1) ≤ ∞} = arg max {E[Rs(x1)]} (30) x1 x1 1 = [(2Msj − γj)x0j + (µj − κˆs)p0j] , ∀j = 1, . . . , n, (31) 2|γj − Msj|
16 A portfolio having any target mean can be constructed with or without incurring leverage by taking long/short positions appropriately given the absence of liquidity costs. 18 see Appendix F for details. Moreover,
n 2 1 X [(2Msj − γj)x0j + (µj − κˆs)p0j] m∞ = + rs, (32) 4w0 |γj − Msj| j=1 and the leverage level corresponding to the maximum return is denoted by:
∞ ρ∞ := Ls(x1 ) ≥ 0. (33)
∞ Noting that x1 in (31) is independent of π, define the upper limit on risk aversion by:
∞ max E[As(x1 )] πs := ∞ . (34) Var[As(x1 )]
max 17 Then, for π < πs , leverage function Ls is well-defined.
Next, consider the problem of determining the minimum leverage level, ρmin (≥ 0), needed before a feasible portfolio can be constructed for a prescribed target return m (< m∞):
ρmin(m) := min {Ls(x1): E[Rs(x1)] ≥ m} . (35) x1
18 The following properties hold for ρmin(m) and mmax(ρ):
∞ max Proposition 5. Suppose (Γ − Ms) is n.d. and Ls(x1 ) > 0. Then, for fixed π ∈ [0, πs ):
i) ρmin(m) = 0 holds for m ∈ [rs, mmax(0)]. Moreover, ρmin(m) is positive, non-decreasing, and
quasi-convex in m for m > mmax(0).
ii) mmax(ρ) is non-decreasing and quasi-concave in ρ ≥ 0.
iii) mmax(ρ∞) = m∞ and ρmin(m∞) = ρ∞, where m∞ and ρ∞ are given by, respectively, (32) and
(33). Moreover, for ρ > ρ∞, mmax(ρ) = m∞. ∞ Conversely, if Ls(x1 ) ≤ 0, then ρmin(m) = 0 holds for m ∈ [rs, m∞].
Proof. See Appendix G.
When portfolio debt is not allowed, target mean cannot exceed mmax(0), i.e., an unlevered portfolio can be constructed so long as the target mean is no more than mmax(0), see the graphical illustration in Figure 3(a). Observe from Figure 3(b) that increasing target mean for a portfolio (say, to m) requires an accompanying (minimum) increase in portfolio leverage to ρmin(m), resulting in a more-concentrated risk-profile; further increases of leverage beyond ρmin(m) yield better-diversified risk profiles at the expense of additional leverage risk. The (quasi) convex increase in ρmin(m) in m under liquidity risk may particularly be unsettling for an investor with higher return targets.
17 n max This is because the set {x1 ∈ < : E[As(x1)] − π Var[As(x1)] > 0} is non-empty for π < πs . 18 (29) and (35) admit the equivalent (dual) representation: ρmin(m) := min {ρ : mmax(ρ) ≥ m} and mmax(ρ) := max {m : ρmin(m) ≤ ρ} . This yields: mmax (ρmin(m)) = m. 19