Portfolio Theory of Three Tales: -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 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 (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 , 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 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 (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 (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 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 ‘ Market Line’ (SML) in Figure 1. Furthermore, we show that an optimal leveraged-portfolio that maximizes the 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 10: | 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

0

0 0 Leverage level () 0 Target mean () (a) (b)

Figure 3 Illustration of portfolio mean-leverage properties under liquidity impact (When price impact is ignored,

mmax(ρ) = ∞ at all leverage levels, and ρmin(m) = 0, ∀m.)

An investor with a strict leverage limit, on the other hand, should consult Figure 3(a) to obtain the maximum possible target return for her portfolio. Note that the highest attainable target return (almost) follows the law of diminishing returns for incurring leverage risk, due to the quasi- concavity. These analytical insights are further illustrated in the numerical evaluation and empirical validation in Section 6, where these functions turn out to be (strictly) concave and convex, leading to an even more severe impact on the leverage-return relationship.

5. Risk-adjusted Returns under Leverage and Liquidity

For an investor with sufficiently large target return, leveraging is unavoidable when facing asset liquidity issues, and thus, such an investor will be faced with potential trade-off between portfolio risk-adjusted performance and portfolio leverage. An unlevered portfolio can be constructed for target mean over m0, but up to a maximum of mmax(0), since ρmin(m) = 0 for m ∈ [m0, mmax(0)]. As argued earlier, for target return in this range, resulting unlevered portfolios can turn out to be inferior in risk-adjusted return; nevertheless, portfolio Sharpe may still be improved via leverag- ing even when m ∈ [m0, mmax(0)]. On the other hand, when m > mmax(0), portfolio leveraging is inevitable since a feasible portfolio exists in this case only when leverage level ρ ≥ ρmin(m) > 0. As such, the focus here is on improving portfolio risk-adjusted returns when target m ∈ [m0, mmax(0)] where a levered portfolio may be contemplated, or when m ∈ (mmax(0), m∞) where leverage is unavoidable.

Suppose the desired target mean m ∈ (m0, m∞), in which case the required leverage level must satisfy ρ ≥ ρmin(m) for the existence of a feasible portfolio under liquidity impact. For the pair max (m, ρ), the Sharpe-maximizing portfolio is determined (for fixed risk aversion π < πs ) by:

max {Ψs(x1): E[Rs(x1)] ≥ m, Ls(x1) ≤ ρ} , (36) x1 20

where the Sharpe ratio Ψs is defined in (16). When (Γ − Ms) is n.d., noting that Ψs is pseudo- concave, E[Rs] is concave, and Ls is quasi-convex in portfolio x1, it follows that the first-order necessary (KKT) conditions of the optimization in (36) are also sufficient for global optimality. How- ever, we take an alternative path by claiming in Proposition 6 that the above Sharpe-maximizing portfolio can be obtained, equivalently, by solving the following mean-variance-leverage-liquidity

(MVLL) portfolio model, specified with m ∈ (m0, m∞) and ρ ≥ ρmin(m):

 2 (MVLL) : F (m, ρ) := min σ [Rs(x1)] : E[Rs(x1)] ≥ m, Ls(x1) ≤ ρ . (37) x1

Note that (37) is a convex minimization problem; an extensive formulation of the MVLL model

∗ is presented in Appendix H. Let an optimal portfolio of (37) be denoted by x1(m, ρ), and let its Sharpe ratio be denoted by:

∗ ∗ ζSH(m, ρ) := Ψs(x1(m, ρ)). (38)

Proposition 6. (Sharpe-Leverage-Liquidity Theorem)

Let the target mean m ∈ (m0, m∞), leverage level ρ ≥ ρmin(m), and leverage risk aversion π ∈ max ∗ > ∗ ∗ [0, πs ). Suppose (Γ − Ms) is n.d. and |x1(m, ρ) (Γ − Ms)x1(m, ρ)| < w0(m − rs) for x1(m, ρ) optimal in (37). Then:

∗ i) E[Rs(x1(m, ρ))] = m holds. ∗ ii) x1(m, ρ) also solves the Sharpe-maximizing model (36), i.e, optimal value of (36) is given by ∗ ζSH(m, ρ) in (38). ∗ iii) Sharpe ratio ζSH(m, ρ) is non-decreasing and pseudo-concave in ρ for fixed m and π. ∗ iv) For fixed pair (m, ρ), ζSH(m, ρ) is non-increasing in risk aversion π. ∗ v) For fixed (ρ, π), ζSH(m, ρ) is non-increasing and pseudo-concave in target return m.

Proof. See Appendix I. 

A few remarks are in order: for specified target mean m ∈ (m0, m∞), with the leverage level cho- ∗ sen such that ρ ≥ ρmin(m), a family of mean-variance-leverage (MVL) efficient portfolios x1(m, ρ) ∗ are generated under the MVLL model in (37). In particular, x1(m, ρmin(m)) is a more ‘risk- ∗ concentrated’ efficient portfolio with the ‘least-possible’ level of leveraging, while x1(m, ρ) for ρ >

ρmin(m) is more ‘risk-diversified’. As ρ varies in [ρmin(m), ρ∞), the Sharpe-Leverage frontier is generated for the given target mean m, where increasing leverage increases the Sharpe, but at a decreasing rate of growth, as implied by the (pseudo) concavity in Proposition 6, Part iii). That

∗ is, the most risk-concentrated portfolio x1(m, ρmin(m)) has the lowest risk-adjusted returns for the ∗ fixed target m. Increasing ρ above ρmin(m) leads to improved risk-adjusted returns in x1(m, ρ), but at the expense of increased leverage risk (beyond the liquidity impact costs considered here). 21

On the other hand, at any fixed leverage level, increasing the target mean leads to worsening the Sharpe ratio, as claimed in Proposition 6, Part v). Can sufficient leverage be applied at a higher target mean to obtain a Sharpe performance level available at a lower target mean? The answer turns out to be negative, unfortunately, in the presence of liquidity risk, as concluded in the next section.

5.1. Maximum Sharpe under leveraging for fixed target mean

For fixed m ∈ (m0, m∞), since increasing leverage leads to increasing the Sharpe, consider a portfolio that yields the maximum Sharpe under unrestricted-leveraging. This portfolio can be determined by the model: ∗∗ ∗ ζSH(m) = max ζSH(m, ρ). (39) ρ≥ρmin(m) ∗ Note from Proposition 6 that ζSH(m, ρ) is non-increasing-pseudo-concave and non-decreasing- ∗ pseudo-concave, respectively, in m and ρ. Although optimizing ζSH(m, ρ) appears complicated, we will show that the solution of the maximization in (39) is accomplished directly from (37) by setting ρ = +∞. Toward this, consider the MVLL model by dropping the leverage constraint, i.e., selecting a levered portfolio with minimum variance under liquidity impact, given the target mean m:

∗  2 F (m) := min σ [Rs(x1)] : E[Rs(x1)] ≥ m . (40) x1

∗∗ The unique optimal solution of (40) is denoted by x1 (m), where the uniqueness is due to the objective function being strictly convex.

∗∗ > ∗∗ Proposition 7. Suppose (Γ − Ms) is n.d. and |x1 (m) (Γ − Ms)x1 (m)| < w0(m − rs). The leverage-unrestricted Sharpe-maximizing portfolio of (39) for fixed target return m ∈ (m0, m∞) is ∗∗ given by the unique solution x1 (m) of (40), and

∗∗ ∗∗ m − rs ζ (m) = Ψs(x (m)) = (41) SH 1 pF ∗(m)

∗∗ holds. Moreover, ζSH(m) is non-increasing and pseudo-concave in m ∈ (m0, m∞).

Proof. See Appendix J.  ∗ Although the Sharpe ζSH(m, ρ) depends on the leverage risk aversion π, observe that the ∗∗ maximum-attainable Sharpe value ζSH(m) is independent of a specific leverage risk aversion. How- ∗∗ ever, the leverage level associated with the maximum Sharpe, ζSH(m), depends on risk aversion ∗∗ ∗∗ π. Let ρmax(m) denote the leverage level that corresponds to ζSH(m). Since x1 (m) is the unique solution of (40), it follows that:

∗∗ ρmax(m) = Ls(x1 (m)) ≥ ρmin(m). (42) 22

∗ ∗∗ For ρ > ρmax(m), clearly, ζSH(m, ρ) = ζSH(m) by the optimality in (39), i.e., increasing leverage level beyond ρmax(m) does not improve the Sharpe ratio for the specified target return m. On the other ∗∗ hand, for leverage below ρmax(m), the maximum attainable Sharpe is smaller than ζSH(m), i.e., the ∗ ∗∗ inequality ζSH(m, ρ) < ζSH(m) holds for ρ ∈ [ρmin(m), ρmax(m) ). Summarizing, when a portfolio mean m is targeted in the presence of liquidity impact, a minimum leverage level of ρmin(m) is required to form a feasible portfolio, and as ρ increases from ρmin(m) ∗∗ to ρmax(m), the portfolio Sharpe increases, and reaches a maximum of ζSH(m) at ρ = ρmax(m). Moreover, as Proposition 7 claims, as the target mean m is increased, the maximum-attainable Sharpe under leveraging declines. The fact that this decline can be strict is demonstrated in our empirical analysis in Section 6.

∗∗ Since ζSH(m) is non-increasing in m, as target return is decreased closer to that of the ‘unlevered optimum-Sharpe’ portfolio, i.e., m = m0 + ε (for ε > 0), portfolio Sharpe attains the highest level ∗∗ max among all levered portfolios; moreover, as ε → 0, we get ζSH(m0 + ε) → ζSH , which corresponds to 0 the unlevered portfolio x1 with target mean m0, see Proposition 4.

In other words, when the investor seeks a return m > m0 and contemplates leveraging to improve ∗ max portfolio’s economic performance, the resulting optimum Sharpe ζSH(m, ρ) can never exceed ζSH at any leverage level ρ; see the graphical illustration in Figure 4(a). Also, note that Sharpe-Leverage frontier is dominated when target return is increased. The contrasting case under the classical port- folio theory is depicted in Figure 4(b), where ignoring liquidity impact allows achieving the highest Sharpe (the slope of SML) at any target return under sufficient leverage.19 Detailed empirical analyses of these analytical results are in Section 6.

∗ , Under no liquidity impact,

Under market impact Sharpe of tangency portfolio  ∗∗  ratio

ratio Increasing Increasing Sharpe Sharpe

0 Leverage level () 0 Leverage level () (a) (b)

Figure 4 Sharpe-Leverage frontiers: Effect of target mean under liquidity impact

19 On the other hand, as m decreases from m0, portfolio lending is enabled and the Sharpe can be increased, and ∗ max infinitely-so as the portfolio is completely liquidated, see Proposition 3. That is, ζSH(m, 0) > ζSH holds for m ∈ ∗ (rs, m0), and as m → rs, ζSH(m, 0) → +∞. 23

5.2. Comparison with proportionate-leveraging

0 max Recall that the lending-free portfolio x1 has the highest Sharpe (ζSH ) among all unlevered opti- mal portfolios, delivering a target return of m0. Given a target m > m0, suppose we consider 0 proportionately-leveraging the portfolio x1, rather than following the optimal leveraging approach ∗ outlined above to determine the portfolio x1(m, ρ). For a fair comparison, leverage levels of both 0 ∗ portfolios must be the same, i.e., Ls(νx1) = Ls(x1(m, ρ)) as they both deliver the target m. 0 First, note that m ≤ m¯ (x1) must hold, for if not, proportionate-leveraging cannot attain the required target, see (25). It turns out that the required multiplier ν satisfies:

0 ν  0> 0  m = E[Rs(νx1)] = (ν − 1)x1 (Γ − Ms)x1 + w0(m0 − rs) + rs. (43) w0

0> 0 As in Proposition 2, assume that (Γ−Ms) is n.d. and w0(m0 −rs) > |x1 (Γ−Ms)x1| holds. The two h i0.5 w0(m0−rs) roots of (43), ν0 and ν1, can be shown to satisfy 1 < ν0 < ν1 < 0> 0 . Noting Proposition |x1 (Γ−Ms)x1| 0 0 0 1, Ψs(x1) > Ψs(ν0x1) > Ψs(ν1x1) hold, and thus, the highest-possible Sharpe under proportionate leveraging (for target m) is achieved when ν = ν0, where

1   "  2 # 2 1 w0(m0 − rs) 1 w0(m0 − rs) w0(m − rs) (44) ν0 := 0> 0 + 1 − 0> 0 + 1 − 0> 0 . 2 |x1 (Γ − Ms)x1| 4 |x1 (Γ − Ms)x1| |x1 (Γ − Ms)x1| Then, the corresponding Sharpe ratio is (see the proof of Proposition 1):

0> 0 0 max x1 (Γ − Ms)x1 max Ψs(ν0x ) = ζ + (ν0 − 1) < ζ . (45) 1 SH p 0> 0 SH x1 P0V P0x1 0 0 Observe that the portfolio ν0x1 is feasible in model (36) with leverage ρ0 := Ls(ν0x1), whose optimal ∗ 0 solution x1(m, ρ0), for m ∈ (m0, m¯ (x1)], must satisfy:

0 ∗ ∗ max Ψs(ν0x1) ≤ ζSH(m, ρ0)) = Ψ(x1(m, ρ0)) < ζSH . (46)

Our empirical analysis in Section 6 shows that a strict inequality in (46) is achieved, implying 0 ∗ Ψs(ν0x1) < Ψ(x1(m, ρ0)). That is, proportionate-leveraging can lead to a strictly inferior portfolio, relative to an optimal risk-adjusted performance, although it achieves the required target at the specified leverage, highlighting a fundamental departure from the standard portfolio theory.

6. Empirical Validation of Analytical Results

The analytical results of the preceding sections are empirically-illustrated using actual data. We set the asset universe to be the nine Select Sector ETFs, given by the ticker symbols XLB, XLE, XLF, XLI, XLK, XLP, XLU, XLV, and XLY, indexed j = 1, . . . , n = 9. These ETFs cover the full- breadth of the S&P 500 market index.20 These assets are chosen because they are fairly-liquid, so

20 The real-estate ETF, XLRE, of the S&P 500 index was dropped from the asset universe because the inception date of XLRE is Oct 2015, while the others were incepted in 1998. 24 that an over-emphasis of our analytical results is avoided. First, the market impact parameters are estimated using the millisecond TAQ data for the 9 ETF assets for the time period, Jan 01-31,

2015, for N = 20 days of trading. The methodology we used to estimate λj and γj is in Appendix

K, and the estimated parameter values are in Table 1. Importantly, observe that (Γ − Ms) is a negative definite matrix at any (non-negative) risk-free rate.

Table 1 Estimated parameters for the market impact model ETF (Sector) R2 γ × 10−6 p-value λ × 10−6 p-value XLB (Basic Materials) 49.05% 0.6361 0 4.3737 0 XLE (Energy) 63.81% 0.7157 0 6.5823 0 XLF (Financials) 41.31% 0.0127 0 0.3776 0 XLI (Industrial Goods) 47.53% 0.2284 0 2.8244 0 XLK (Technology) 39.38% 0.0578 0 1.2855 0 XLP (Consumer Staples) 54.71% 0.0294 0.102 3.2523 0 XLU (Utilities) 54.26% 0.1797 0 4.2074 0 XLV (Health Care) 55.26% 0.3204 0 4.8528 0 XLY (Consumer Discretionary) 44.62% 0.5599 0 4.6215 0

The parameters of the liquidity-impacted asset pricing model, ptj = p0j + γj (xtj − x0j ) + λj ytj , are estimated using the procedure in Appendix K. It is evident that γj < 2λj holds for all assets j = 1,..., 9, and thus,

(Γ − Ms) is a negative definite matrix at any risk-free rate, see Footnote 12 in Section 3.

We set the day of model execution for empirical analysis (arbitrarily) to Jan 07, 2015, for which the required (daily) asset return parameters are estimated using a trailing 60-day historical time window (up to Jan 06, 2015), including the S&P 500 index-tracking ETF, SPY. The forecast parameters are in Table 2. Optimal portfolios are constructed at the beginning of Jan 07, 2015 with an initial portfolio of x0 = 0, w0 = $1 million, and no initial liability (hence, L0 = −$1 million), using the market impact parameters in Table 1, asset return parameters in Table 2, and an annualized risk-free rate of r0 = 1% (with daily-compounding). Target portfolio mean, m, is chosen such that m ≤ m∞, see (32).

6.1. Unlevered maximum-Sharpe and maximum-allowable return

The model (26) is difficult to solve directly, and thus, we employ the alternative scheme in Appendix

L to obtain m0. This yields, under market impact, m0 = 0.2033%, associated with the Sharpe, max ζSH = 0.1850. Corresponding values when ignoring market impact (by setting λj = γj = 0, ∀j) are 0.5199% and 0.3329, respectively.

The maximum-allowable portfolio mean under market impact is computed using (32) as m∞ = max ∞ 0.2705%, which yields πs = 0.1742%, see (34). The liability Ls(x1 ) associated with m∞, as a percent of the initial wealth, is 169.61%. Moreover, the maximum-possible return for an unlevered portfolio is mmax(0) = 0.2367%, see (29). That is, to obtain a portfolio (daily) return in excess of 0.2367% (but no more than 0.2705%), leverage must be applied with ρ ≥ ρmin(m), see (35). In 25

Table 2 Forecasts of asset return and market parameters for Jan 07, 2015 Parameter XLB XLE XLF XLI XLK XLP XLU XLV XLY SPY Mean 0.0057% -0.1818% 0.0842% 0.1130% 0.0520% 0.0935% 0.1585% 0.1233% 0.1109% 0.0630% StDev 1.212% 1.867% 0.898% 1.025% 0.949% 0.707% 0.991% 1.040% 0.891% 0.845% Asset correlations XLB 1.000 0.821 0.700 0.806 0.720 0.493 0.363 0.561 0.690 0.852 XLE 0.821 1.000 0.499 0.679 0.483 0.193 0.214 0.332 0.495 0.663 XLF 0.700 0.499 1.000 0.816 0.835 0.723 0.519 0.717 0.792 0.918 XLI 0.806 0.679 0.816 1.000 0.812 0.527 0.362 0.626 0.773 0.902 XLK 0.720 0.483 0.835 0.812 1.000 0.664 0.407 0.660 0.823 0.916 XLP 0.493 0.193 0.723 0.527 0.664 1.000 0.621 0.609 0.662 0.724 XLU 0.363 0.214 0.519 0.362 0.407 0.621 1.000 0.457 0.386 0.521 XLV 0.561 0.332 0.717 0.626 0.660 0.609 0.457 1.000 0.630 0.785 XLY 0.690 0.495 0.792 0.773 0.823 0.662 0.386 0.630 1.000 0.873 Asset return (daily) parameters for 07 January 2015 are estimated using a trailing 60-day historical time window (up to Jan 06, 2015). Actual (realized) returns on Jan 07, 2015, respectively in the order of the nine assets, are 1.1279%, 0.2130%, 1.0487%, 0.7522%, 0.8468%, 1.7101%, 0.9810%, 2.3512%, and 1.5814%, and that for the market index SPY is 1.2461%. Hence, the estimation data period is not particularly an accurate depiction of the market movement on the day of trade execution.

contrast, for a target mean in [m0, mmax(0)] ≡ [0.2033%, 0.2367%], there exist portfolios that are borrowing- and lending-free.

6.2. Sharpe-Leverage sensitivity

The effect of specifying a target mean m ≤ m∞ for a specified leverage level ρ ≥ ρmin(m) is studied under leverage risk-neutrality, i.e., π = 0. The portfolios that correspond to a range of target means are reported in Table 3, where for each specified m, the MVLL model is solved with ρ = ρmin(m). Thus, the risk-adjusted returns in Table 3 are computed for the least-possible leverage for each target mean, hence, more risk-concentrated portfolios. Observe that as m increases, Sharpe decreases, and a positive leverage is required to achieve

max the target only when m > mmax(0). At m = m0, ζSH = 0.1850 is the highest available Sharpe for a portfolio with no Cash. When target mean is strictly less than m0, the optimal portfolio holds Cash max ∗ (i.e., portfolio lending) and the Sharpe ratio is strictly greater than ζSH , e.g., ζSH(0.1850%, 0) = max 0.1959 > ζSH , agreeing with the conclusions in Sections 4 and 5.1. Also, referring to Proposition ∗ > ∗ 6, the last row in Table 3 indicates that the condition |x1(m, ρ) (Γ − Ms)x1(m, ρ)| < w0(m − rs) is satisfied for optimal portfolios at any target mean. The fact that risk-adjusted portfolio performance for a desired target mean can be improved under increased leverage is depicted in Figure 5(a). For a fixed target mean, while increasing leverage increases the risk-adjusted return, notice that it can never reach the maximum Sharpe ratio that can be attained at a lower target mean. For example, the Sharpe-Leverage frontier for target mean of 0.227% lies completely below that for target mean of 0.214%. When target mean is increased under fixed leverage, the Sharpe declines, as claimed in Proposition 6; the loss in Sharpe 26

Table 3 Optimal portfolio weights under leverage risk-neutrality (Jan 07, 2015)

m0 mmax(0) m∞ Portfolio (daily) mean target (m) Asset 0.2033% 0.2367% 0.2705% 0.1850% 0.200% 0.2143% 0.2269% 0.2395% 0.2521% Optimal portfolio (dollar) weights at beginning of period, relative to initial wealth XLB -2.09% -9.27% 0.85% -1.70% -1.98% -3.31% -5.39% -9.13% -6.75% XLE -53.78% -100.49% -85.37% -46.98% -52.27% -62.85% -76.72% -101.66% -97.62% XLF 20.00% 32.80% 63.27% 15.28% 19.10% 22.67% 26.53% 34.63% 41.30% XLI 39.60% 40.34% 61.57% 37.33% 39.20% 40.24% 40.61% 41.17% 45.93% XLK -2.38% 6.60% 32.02% -5.67% -3.00% -0.64% 2.05% 8.08% 13.67% XLP 13.33% 18.37% 32.70% 10.97% 12.97% 13.56% 14.81% 19.34% 22.47% XLU 26.55% 31.36% 42.12% 24.31% 26.17% 27.25% 28.57% 32.08% 34.41% XLV 25.46% 40.23% 60.18% 21.28% 24.64% 28.54% 33.01% 41.66% 45.93% XLY 33.23% 39.86% 62.30% 29.94% 32.65% 34.45% 36.39% 41.10% 46.04% Cash 0.00% 0.00% -169.62% 15.23% 2.50% 0.00% 0.00% -7.25% -45.39% Portfolio leverage level and risk-adjusted returns Leverage, ρ = ρmin 0 0 1.7081 0 0 0 0 0.0745 0.4550 ∗ Sharpe, ζSH(m, ρ) 0.1850 0.1330 0.1120 0.1959 0.1871 0.1764 0.1610 0.1319 0.1342 ∗> ∗ |x1 (Γ−Ms)x1 | 0.3985 0.7948 0.9890 0.3478 0.3877 0.4529 0.5552 0.8169 0.8186 w0(m−rs)

Portfolios are obtained for the least-possible level of leverage for each target mean. When m ≤ mmax(0), ρmin(m) = 0, hence the portfolio is un-leveraged; for m > mmax(0), portfolios are leveraged at ρ = ρmin(m) > 0. Note that the resulting Sharpe ratio strictly decereases as target mean increases. Also, as the last row indicates all optimal portfolios ∗ > ∗ satisfy the condition |x1(m, ρ) (Γ − Ms)x1(m, ρ)| < w0(m − rs) as required in Proposition 6. However, we do not have an analytical proof that this would be the case in general. It is evident that as m increases, the slack in this inequality becomes smaller, and as m → m∞, the condition is satisfied almost as an equality. associated with an increase in target return is quite pronounced relative to that in the case of ignoring market impact, see Figure 5(b).

∗∗ Figure 5(a) also confirms that the Sharpe reaches its maximum ζSH(m) when the leverage level reaches ρmax(m), as claimed in Proposition 7, and leveraging beyond ρmax(m) does not improve the Sharpe. In contrast, when assuming no liquidity impact, the maximum Sharpe of a lower target mean can be attained by a higher target mean, provided sufficient leverage is applied, see Figure 5(b). The latter observation is consistent with the standard portfolio theory that ignores market impact, and the maximum Sharpe identified in this case is the ‘slope’ of the SML.

0.19 0.1850 0.334

0.18 ∗∗ 0.332 0.117, 0.1781 0.17 ∗∗ 0.330 0.265, 0.1696 ratio ratio

0.16 ∗∗ 0.328 0.445, 0.1598 Sharpe 0.15 Sharpe 0.326 ∗∗ 0.697, 0.1483 0.14 0.324 Under Market Impact Ignoring Market Impact 0.13 0.322 0 0.2 0.4 0.6 0.8 1 0min 0.2 0.4min 0.6 0.8 1 =0.075 =0.455 Leverage level () Leverage level ()

m=0.203% m=0.214% m=0.227% m=0.240% m=0.252% m=0.520% m=0.630% m=0.756% m=0.882% m=1.009% (a) (b)

Figure 5 Effect of leverage level and target mean on Sharpe ratio under leverage risk-neutrality For fixed target, under liquidity impact, while risk-adjusted returns improve with increased leverage, they can never reach maximum Sharpe attained at a lower target mean. When market impact is ignored, such a phenomenon does not exist. 27

The minimum leverage required for portfolio feasibility, i.e., ρmin, and the minimum leverage

∗∗ required to achieve the maximum-possible Sharpe (ζSH), i.e., ρmax, are depicted in Figure 6(a) for the case of leverage risk-neutrality.21 Observe that when target mean increases, these minimum leverage values monotonically increase, while the gap between them monotonically decreases, and they converge to each other as m → m∞.The increasing-convexity of ρmin as m increases in Figure 6(a) is steeper than the predicted quasi-convexity in Proposition 5, implying that an increasing target mean has a significantly-worsening adverse effect on portfolio leverage. The leverage ρmax appears increasing-convex in target return, but that growth is always smaller than that of ρmin.

∗∗ Figure 6(b) shows that the maximum-possible Sharpe ζSH(m), attained with leveraging to at least ρmax(m), is strictly decreasing in target return, and the rate of decline becomes steeper at a higher target mean, which in this case is even worse than the pseudo-concave decrease predicted in

Proposition 7. To exemplify, consider the case when target mean is set at m = mmax(0) = 0.2367%. Then, the best Sharpe ratio available for an unlevered portfolio is 0.1330, see Table 3. However, for this target mean, portfolio Sharpe can be improved to a maximum of 0.1622, see Figure 6(b), by applying the maximum leverage at ρmax = 40.33%, as indicated in Figure 6(a).

∗∗ Nevertheless, the leverage level of ρmax, resulting in the maximum-possible Sharpe of ζSH, may not be acceptable to an investor, say with a fixed leverage risk aversion π. In fact, the investor is not at liberty to choose the target mean mˆ and the acceptable leverage level ρˆ independently of each other in the presence of liquidity risk - indeed, the choice must satisfy the inequalities:m ˆ ≤ mmax(ˆρ) and ρmin(m ˆ ) ≤ ρˆ ≤ ρmax(m ˆ ). This range of choices is illustrated by the ‘shaded region’ in Figure

6(a) for m > m0 when leverage is applied. Then, for the pair (m, ˆ ρˆ) acceptable in this sense, the

∗ portfolio with the highest risk-adjusted performance ζSH(m, ˆ ρˆ) is determined by the MVLL model in (37). In doing so, the investor has optimally-traded-off the loss (from a potential maximum) in the risk-adjusted return for the amount of leverage (risk) mitigated, i.e., the loss in Sharpe of

∗∗ ∗ [ζSH(m ˆ ) − ζSH(m, ˆ ρˆ)] for the gain in leverage of [ρmax(m ˆ ) − ρˆ]. To illustrate, let the desired targetm ˆ = 0.24%, which yields the maximum-possible Sharpe as

∗∗ ζSH(0.24%) = 0.1598, which is associated with a leverage level of ρmax = 44.47%. However, suppose the investor’s maximum-acceptable leverage level is onlyρ ˆ= 20%. Then, the optimal Sharpe ratio

∗ available to the investor can be computed using the MVLL model in (37) to be ζSH(0.24%, 20%) = 0.1521, which represents a decline of roughly 5% in risk-adjusted performance for a reduction of about 50% in leverage level under liquidity impact.

21 When Γ − Ms is n.d., not only (29) is a convex program, (35) also can be solved via a sequence of convex programs. We omit the details here, and use standard convex (quadratic) programming software, i.e., IBMTMCplex V12.7.1. 28

1.8 0.22 1.6 1.7081 0.20 1.4 Sharpe

1.2 0.18

∗∗ 1.0 0.1622 0.16

0.8 possible Under Market Impact

Leverage Under Market Impact 0.6 0.14 40.33% 0.4

0.2705% 0.12 Maximum

0.2 0.0 0.10 0.18% 0.20% 0.22% 0.24% 0.26% 0.28% 0.18% 0.20% 0.22% 0.24% 0.26% 0.28% 0 0 Target mean (m) Target mean (m)

∗∗ Maximum Leverage (a) (b)

Figure 6 Minimum and maximum leverages as target mean varies under leverage risk-neutrality

Leverage (ρmax) required for maximum-possible Sharpe appears to be convex and increasing in m. Its difference with the minimum leverage (ρmin) required for portfolio feasibility, i.e., [ρmax(m) − ρmin(m)], is strictly decreasing in m and the difference approaches zero at the highest mean, m∞.

6.3. Sharpe-Mean frontiers and proportionate-leveraging

Sharpe and target mean efficient frontiers for the case of leverage risk-neutrality are in Figure 7, where each frontier is plotted for target mean in the interval m0 = 0.2033% to mmax(ρ) for ρ = 0,

10%, and 30%. Note that mmax(0) = 0.2367%, mmax(0.1) = 0.2405%, and mmax(0.3) = 0.2474%. As predicted by Proposition 6, Sharpe decreases as mean increases; however, the drop appears to be even worse than the predicted quasi-concavity to a more steep concave decline at higher target means. Leveraging helps reversing some of the losses in Sharpe, albeit at increased leverage risk. In contrast, when liquidity impact is ignored, the drop in Sharpe performance is somewhat subdued at higher target means due to the pseudo-concave shape, which clearly over-estimates the actual Sharpe.

To highlight the latter over-estimation, consider a portfoliox ˜1, deemed ex-ante optimal by ignoring market impact. If the performance ofx ˜1 is evaluated ex-post under market impact, its

“actual-Sharpe” is obtained as Ψs(˜x1) in (16), and plotted in Figure 8(a). While portfolio volatil- ity ofx ˜1 remains unchanged, liquidity impact severely undermines the performance, leading to negative Sharpe contrary to the positive ex-ante performance predicted in Figure 5(b). Therefore, incorporating liquidity impact is of paramount importance when determining an optimal portfolio to obtain superior risk-adjusted performance. When leveraging under liquidity impact, an optimal allocation of that liability is required to achieve the best risk-adjusted portfolio performance, instead of an allocation based on proportionate-leveraging, see Section 5.2. Under proportionate-leveraging of the Sharpe-

0 maximizing unlevered portfolio x1 (found in the first column in Table 3), the required leveraging parameter ν0 is computed by (44) when the target m is set above m0. Figure 8(b) plots the Sharpe 0 ∗ of the portfolio ν0x1, which is strictly inferior to the optimally-leveraged portfolio x1(m, ρ(ν0)) 29

0 ∗ at identical leverage levels. Therefore, the inequality Ψs(ν0x1) < Ψ(x1(m, ρ(ν0))) is strict in (46), particularly for larger target means.

0.19 0.334

0.332 0.18 0.330 0.17 0.328

0.16 0.326 Sharpe Sharpe 0.324 0.15 0.322 Under Market Impact Ignoring Market Impact 0.14 0.320

0.13 0.318 0.20% 0.21% 0.22% 0.23% 0.24% 0.25% 0.5% 0.6% 0.7% 0.8% 0.9% 1.0% 1.1% 1.2% 1.3% Target mean, m Target mean, m

ρ=0 ρ=0.1 ρ=0.3 =0 =0.1 =0.3 (a) (b)

Figure 7 Effect of target mean on optimal risk-adjusted performance under leverage risk neutrality At any fixed leverage, optimal-Sharpe declines at increased target mean at a more-steep concave rate under liquidity impact, in contrast to a more-subdued pseudo-concave drop when market impact is ignored. Increased leverage can recover some losses in risk-adjusted performance.

0.19 0.0 Actual Sharpe is computed by applying market impact for ‐0.2 an optimal portfolio computed by ignoring market impact. 0.18

ratio ‐0.4 0.17

‐0.6 0.16 ଴

Sharpe Portfolio ݔଵ is constructed under

Sharpe market impact, which is proportionately ‐0.8 0.15 levered to get the target mean under Actual market impact. ‐1.0 0.14

‐1.2 0.13 0 0.2 0.4 0.6 0.8 1 0.20% 0.21% 0.22% 0.23% 0.24% 0.25%  Leverage level ( ) Target mean, m

m=0.520% m=0.630% m=0.756% m=0.882% m=1.009% ଴ Optimal Sharpe Levered ߥ଴ݔଵ (a) (b)

Figure 8 Actual Sharpe of liquidity-ignored portfolios and proportionately-levered portfolios In (a), ex-ante performance of liquidity-ignored portfolios in Figure 5(b) significantly worsens when evaluated under market impact costs; moreover, leveraging does not help appreciably at any target. In (b), under market impact, 0 proportionate-leveraging of the optimal unlevered portfolio x1 leads to strictly-worse performance when compared with an optimal allocation of the liability, especially when target return increases.

6.4. MV frontier under leverage and liquidity

For fixed leverage ρ, Mean-Variance-Leverage (MVL) frontiers exist only for target mean satisfying m ≤ mmax(ρ), see Figure 9(a); and as ρ increases, the range of the MVL frontier increases since mmax(ρ) increases, see Figure 9(b). Note that up to the target threshold m0 = 0.2033%, MVL frontiers are identical regardless of the leverage level; moreover, the investor cannot achieve targets above mmax(0) = 0.2367% unless leverage is incurred. It is evident that as target mean is increased, 30

0.27% 0.275% 0.75 0.26% 0.270% 0.25% 0.265%

0 Mean 0.24% 0.260% Mean 0.23% 0.255% Target 0.22% 0.250%

0.21% Portfolio Under Market Impact 0.245% Under Market Impact 0.20% 0.240% Maximum 0.19% 0.235% 0.18% 0.230% 0.80% 1.00% 1.20% 1.40% 1.60% 1.80% 2.00% 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 Portfolio Standard Deviation Leverage level ()

ρ=0 ρ=0.25 ρ=0.5 ρ=0.75 (a) (b)

Figure 9 Mean-Variance-Leverage (MVL) efficient frontiers under leverage risk neutrality

In (a), at fixed leverage ρ, the MVL frontier exists only up to a target mean of mmax(ρ), which increases in ρ, see (b). For m ≤ m0, all MVL frontiers coincide. As target mean is increased, for better standard deviation risk (and improved risk-adjusted returns), portfolio leverage must be increased. for better standard deviation risk (hence, higher Sharpe), one has to increase the leverage risk. Such Sharpe-leverage trade-off was already presented in Figure 5. On the other hand, as leverage risk aversion increases, the minimum leverage required for fea- sibility, as well as the minimum leverage required to achieve the maximum-possible Sharpe, also increase (at a progressively faster rate) with target mean - sensitivity of ρmin and ρmax as π increases max in the range [0, πs = 0.1742%) is plotted in Figures 10(a) and 10(b). Increasing risk aversion leads to even more-significant rates of increases in required leverages as target mean increases. As such, an investor with a higher leverage risk aversion seeking to minimize the level of leverage exposure would have to lower the desired target mean return substantially when faced with liquidity impact of trading.

8.00 8.00

7.00 7.00

6.00 6.00

5.00 5.00 Under Market Impact 4.00 Under Market Impact 4.00 3.00 3.00

2.00 2.00

1.00 0 1.00

0.00 0.00 0.180% 0.200% 0.220% 0.240% 0.260% 0.280% 0.180% 0.200% 0.220% 0.240% 0.260% 0.280% Target mean (m) Target mean (m)

=0 =0.0005 =0.001 =0.0015 =0 =0.0005 =0.001 =0.0015 (a) (b)

Figure 10 Sensitivity of leverage thresholds on target mean and leverage risk aversion In (a), as risk aversion increases, the required minimum leverages to construct a feasible portfolio increases sharpely. In (b), the same is true for the required leverage to obtain a portfolio with the maximum-possible Sharpe. As the target increases, the rate of increases in these leverage thresholds also increase significantly. Hence, highly risk averse investors must set their target returns sufficiently small. 31

7. Concluding Remarks

This paper presents an analysis of the optimal trade-off between portfolio risk-adjusted perfor- mance and leverage level in the presence of liquidity risk. The direct consideration of liquidity impact costs significantly alters our understanding of the interplay among risk, return, and lever- age. The Sharpe-maximizing portfolio under market impact, in the absence of leveraging, is no longer the tangency portfolio of the MV frontier. Furthermore, (proportionately) leveraging of a given portfolio, which preserves the portfolio composition, is easily dominated, in the sense of risk-adjusted performance, by an optimally-constructed portfolio with identical levels of leverage and target mean. Market impact causes significant deterioration in portfolio performance as target return is increased, although some of the adverse effects can be countered with increased lever- age risk. In this sense, the trade-off that exists between leverage risk and risk-adjusted return is important in the presence of market impact of trading. Even with fairly-liquid ETF assets considered in our empirical analysis, portfolio performance can be significantly and adversely affected if liquidity costs are ignored. For further insight, a larger sample of various equities may be employed, and (out-of-sample) portfolio excess returns, relative to the classical MV model, can be analyzed. For instance, such excess returns may be risk-adjusted using a standard asset pricing model in the search for a potential liquidity-based leverage risk premium. These topics are outside the scope of the current paper, and they will be addressed in our future research.

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APPENDICES “Portfolio Theory of Three Tales: Risk-adjusted Returns, Liquidity, and Leverage”

Appendix A: Leverage function Ls(x1) is quasi-convex

Claim 1. Suppose the liquidity impact parameter matrix (Γ − Ms) is negative definite, and the investor’s leverage risk-aversion, π, is sufficiently-small such that the set of portfolios X := n {x1 ∈ < : E[As(x1)] − π Var[As(x1)] > 0} is non-empty, i.e., X =6 ∅. Then, the leverage function

Ls(x1) in (20) is quasi-convex on X.

+ + Proof. Since Ls (x1) = max{Ls(x1), 0}, Ls is nonnegative, and convex in x1 (because Ls(x1) is convex). Moreover, the denominator of (20) is positive and concave on the set of portfolios given in the proposition. Then, the result follows directly from Mangasarian (1970). 

Appendix B: Proof of Proposition 1

Writing the Sharpe ratio in (16) as Ψs(x1) = f1(x1)/f2(x1), under the negative definiteness, f1 is concave. Denote the vector of the square-roots of the eigen values of the positive definite covariance matrix V by e, and let the diagonal matrix E = diag(e1, . . . , en). Writing V using > its eigen value-decomposition, there exists a matrix Q such that V = QEEQ . Then, f2(x1) = p > > > x1 P0QEEQ P0x1 = ||EQ P0x1||2, and thus is convex due to the property of subadditivity of the norm. Since f1 is concave, differentiable (and nonnegative) and f2 is convex, differentiable

(and positive) on the open set X given in the proposition, it follows that the ratio function Ψs is pseudo-concave; see Mangasarian (1970). To show the second part, for ν > 0, (16) yields

> x1 (Γ − Ms)x1 Ψs(νx1) = Ψs(x1) + (ν − 1) . (47) f2(x1)

Thus, Ψs(νx1) is linear in ν with a negative slope under the negative definiteness. Moreover, for

ν > 1, Ψs(νx1) < Ψs(x1), and for 0 < ν < 1, Ψs(νx1) > Ψs(x1). 

Appendix C: Proof of Proposition 2

For a given portfolio x1 with expected return m (> rs), under proportionate leveraged positions by multiplier ν > 0,

1  2 > >  g(ν) = E[Rs(νx1)] = ν x1 (Γ − Ms)x1 + ν(2Msx0 − Γx0 − κˆsp0 + P0µ) x1 + rs w0 ν  >  = (ν − 1)x1 (Γ − Ms)x1 + w0(m − rs) + rs, (48) w0 and thus, g(ν) is concave in ν with g(1) = m. For g(ν) < m, we must have

 >  2  >  f(ν) ≡ x1 (Γ − Ms)x1 ν + w0(m − rs) − x1 (Γ − Ms)x1 ν − w0(m − rs) < 0, 35

implying that ν < 1 or ν > νmax must hold, noting νmax > 1 under the stated-condition that m−rs > 1 > |x (Γ − Ms)x1|. Moreover, for ν ∈ (1, νmax), g(ν) > m follows, proving part (i). For part (ii), due w0 1 to the concavity of g, the maximum of f occurs when

0 > f (ν) = (2ν − 1)x1 (Γ − Ms)x1 + w0(m − rs) = 0, which yields ν =ν ¯ as the solution. For ν ∈ (1, ν¯], g is increasing, while it is decreasing for ν >

ν¯. Substitutingν ¯ in (48), the maximum portfolio mean under proportionate leveraging of x1 is obtained.

1 > On the other hand, if m − rs ≤ |x (Γ − Ms)x1|, then νmax ≤ 1 holds and g(ν) < m for ν > 1. This w0 1 completes the proof. 

Appendix D: Proof of Proposition 3

n Define the set X := {x1 ∈ < : Ls(x1) ≤ 0}. Since Ls is convex, X is a convex set. Moreover,

Ls(0) = −φs < 0, i.e., 0 ∈ int(X), the interior of X. Let portfoliox ¯1 satisfy Ls(¯x1) = 0. Then, for all

ν ∈ [0, 1), we have νx¯1 ∈ int(X) . Since (Γ − Ms) is negative definite, due to Proposition 1, Ψs(νx¯1) is decreasing in ν over the interval (0, 1), which implies that the supremum of Ψs is attained at x1 = 0.

To show the second part of the proposition, let l1, l2 ∈ (−φs, 0] such that l1 > l2. Suppose an optimal portfolio of ζSH(l1) is denoted byx ˆ1. Consider proportionate-leveraging ofx ˆ1 with factor

ν(> 0) such that the portfolio νxˆ1 satisfies Ls(νxˆ1) = l2 - it is only necessary to show that the only n nonnegative solution satisfies ν < 1. Considering (17), and denoting c = κsp0 − 2Msx0 ∈ R ,

2 > > Ls(νxˆ1) = (ν) xˆ1 Msxˆ1 + νc xˆ1 − φs,

and thus, ν satisfying Ls(νxˆ1) = l2 are solutions of the quadratic equation:

> 2 > f(ν) := (ˆx1 Msxˆ1)(ν) + (c xˆ1)ν − (l2 + φs) = 0,

where l2 + φs > 0 and f is convex since Ms is p.d. It is straightforward to show that the only positive solution of f(ν) = 0 is  q  1 > > 2 > νˆ = > −c xˆ1 + (c xˆ1) + 4ˆx1 Msxˆ1(l2 + φs) > 0. 2ˆx1 Msxˆ1

> 2 > > > 2 To show thatν ˆ < 1, it is only necessary to claim (c xˆ1) + 4ˆx1 Msxˆ1(l2 + φs) < (c xˆ1 + 2ˆx1 Msxˆ1) . > > This strict inequality holds sincex ˆ1 Msxˆ1 +c xˆ1 −φs = l1 and l1 > l2. Hence, the portfolio x1 =ν ˆxˆ1 satisfies Ls(x1) = l2, implying that ζSH(l2) ≥ Ψs(ˆνxˆ1). But, sinceν ˆ < 1, it follows from Proposition

1 that Ψs(ˆνxˆ1) > Ψs(ˆx1) = ζSH(l1). Therefore, ζSH(l2) > ζSH(l1) for l2 < l1, completing the proof.  36

Appendix E: Proof of Proposition 4

0 An optimal solution x1 of (26) satisfies the first order necessary KKT conditions, see Bazaraa et al. (2006), because the ‘constraint qualification’ is satisfied noting that every KKT point is regular. For the Lagrangian:

F(x1, λ) = Ψs(x1) − θLs(x1), (49)

0 denoting the optimal Lagrange multiplier by θ0 ∈ <, at the optimal solution for x1 = x1, we have:

0 0 0 0 0 ) ∇E[Rs(x1)] − Ψs(x1)∇σ[Rs(x1)] − θ0σ[Rs(x1)]∇Ls(x1) = 0 (50) 0 Ls(x1) = 0, which yields

 0 0   0 0  0 ∂E[Rs(x1)].∂σ[Rs(x1)] 0 ∂Ls(x1).∂σ[Rs(x1)] Ψ(x1) = − θ0σ[Rs(x1)] , j = 1, . . . , n. (51) ∂xj ∂xj ∂xj ∂xj

Summing (51) over j = 1, . . . , n, and then, dividing both sides by n,

n  0 0  n  0 0  0 1 X ∂E[Rs(x1)].∂σ[Rs(x1)] 0 1 X ∂Ls(x1).∂σ[Rs(x1)] Ψ(x1) = − θ0σ[Rs(x1)] n ∂xj ∂xj n ∂xj ∂xj j=1 j=1 0 0 dE[Rs(x1)] 0 dLs(x1) = 0 − θ0σ[Rs(x1)] 0 , (52) dσ[Rs(x1)] dσ[Rs(x1)]

0 as required in (27). Next, taking the inner product of the first equation in (50) with x1:

0> 0 0 0> 0 0 0> 0 x1 ∇E[Rs(x1)] − Ψs(x1) x1 ∇σ[Rs(x1)] − θ0σ[Rs(x1)]x1 ∇Ls(x1) = 0. (53)

0 22 0 0 Let the optimal expected return, E[Rs(x1)] = m0 (> rs). ∇Ls(x1) = 2Msx1 + (κsp0 − 2Msx0), and 0> 0 0> 0 0 0> 0 1 0> 0 thus, x ∇Ls(x ) = x Msx +φs since Ls(x ) = 0. Similarly, x ∇ [Rs(x )] = x (Γ−Ms)x + 1 1 1 1 1 1 E 1 w0 1 1 0> 0 0 (m0 − rs), and x1 ∇σ[Rs(x1)] = σ[Rs(x1)]. Therefore,

1 0> 0 0 0 0 0> 0 x1 (Γ − Ms)x1 + (m0 − rs) − Ψs(x1) σ[Rs(x1)] − θ0σ[Rs(x1)](x1 Msx1 + φs) = 0. (54) w0

0 0 Noting (m0 − rs) − Ψs(x1) σ[Rs(x1)] = 0, the expression for θ0 in (28) is obtained. When Γ − Ms is n.d., the optimal Lagrange multiplier θ0 is strictly negative. Observe that θ0 = [dΨs(x1)/dε] 0 x1 where ε is a small positive increase in liability, as given by Ls(x1) = ε > 0. Therefore, for a small increase portfolio debt, the Sharpe ratio declines. This completes the proof. 

22 Liquidating the initial portfolio x0 yields portfolio return rs, see (15). 37

Appendix F: Maximum allowable target mean under market impact

Claim 2. Suppose (Γ − Ms) is n.d. Then, the maximum expected return in (30) is obtained by the unique leverage-unrestricted portfolio:

∞ 1 x1j = [(2Msj − γj)x0j + (µj − κˆs)p0j] , ∀j = 1, . . . , n, (55) 2|γj − Msj| and the maximum portfolio expected return under market impact is:

n 2 1 X [(2Msj − γj)x0j + (µj − κˆs)p0j] m∞ := + rs. (56) 4w0 |γj − Msj| j=1

Proof. Given Γ − Ms is n.d., the problem maxx1 E[Rs(x1)] is concave maximization where

1  > >  E[Rs(x1)] = x1 (Γ − Ms)x1 + (2Msx0 − Γx0 − κˆsp0 + P0µ) x1 + rs, (57) w0 see (11). The first-order conditions are necessary and sufficient for optimality, i.e.,

∂E[Rs(x1)]/∂x1j = 0, ∀j, which yields

2(Γ − Ms)x1 + (2Ms − Γ)x0 − κˆsp0 + P0µ = 0

∞ 1 (58) ⇒ x1j = [(2Msj − γj)x0j + (µj − κˆs)p0j] , ∀j. 2|γj − Msj|

2 ∞ 1 Pn [(2Msj −γj )x0j +(µj −κˆs)p0j ] Substituting x in (57) gives m∞ = + rs. 1 4w0 j=1 |γj −Msj |  ∞ ∞ Note that the portfolio x1 may be a leveraged portfolio if the liability level Ls(x1 ) > 0. Defining the constants:

aj := κsp0j − 2Msjx0j, bj := (1 + µj)p0j − γjx0j, and cj := −aj + bj, (59) it follows that n X cj L (x∞) = 0.25 [M (a + b ) − 2a γ ] − φ . (60) s 1 2 sj j j j j s |γj − Msj| j=1 ∞ The sign of Ls(x1 ) depends on the asset mean returns, initial positions, and initial prices, in addition to the liquidity impact parameters. When initial risky asset positions are zero and

∞ the risk-free rate is zero, i.e., x0 = 0 and κs = 1, the portfolio liability simplifies to: Ls(x1 ) = h i Pn 2 µj Msj 2 ∞ 0.25 − 2 µj(p0j) − φs. A sufficiently small µ implies Ls(x ) > 0. j=1 |γj −Msj | |γj −Msj | 1 Appendix G: Proof of Proposition 5

Part (i): For ρ = 0, let an optimal solution of (29) bex ˜1, and thus, Ls(˜x1) = 0 and E[Rs(˜x1)] = ∞ mmax(0) < m∞ since Ls(x1 ) > 0. Then,x ˜1 is also feasible in (35) for m ≤ mmax(0), with Ls(˜x1) = 0, and thus, ρmin(m) ≤ 0. Therefore, ρmin(m) = 0 for m ≤ mmax(0), i.e., an unlevered portfolio is always feasible when m ≤ mmax(0). 38

When m > mmax(0), there does not exist a portfolio x1 with Ls(x1) ≤ 0, for if not, E[Rs(x1)] >

E[Rs(˜x1)], violating the optimality ofx ˜1 for (29) when ρ = 0. Therefore, for m > mmax(0), every feasible portfolio of (35) satisfies Ls(x1) > 0, implying ρmin(m) > 0. Moreover, as m increases the feasible set in (35) becomes more restrictive, and thus, ρmin(m) is non-decreasing.

To show the quasi-convexity, for the parametric value function in (35), for arbitrary m1 and 1 2 m2 (m1 =6 m2), let the optimal solutions of (35) be denoted, respectively, by x1 and x1. For some

α ∈ [0, 1], by concavity of E[Rs(x1)], we have:

1 2 1 2 E[Rs(αx1 + (1 − α)x1)] ≥ αE[Rs(x1)] + (1 − α)E[Rs(x1)] ≥ αm1 + (1 − α)m2.

α 1 2 α α Denoting mα := αm1 + (1 − α)m2 and x1 := αx1 + (1 − α)x1, we have ρmin(mα) ≤ Ls(x1 ) since x1 is feasible in (35) for m = mα. Due to Claim 1, Ls(x1) is quasi-convex. Therefore,

α 1 2 ρmin(mα) ≤ Ls(x1 ) ≤ max{Ls(x1), Ls(x1)} = max{ρmin(m1), ρmin(m2)}, which implies that ρmin(m) is quasi-convex.

Part (ii): From (29), it is clear that mmax(ρ) is non-decreasing because the feasible region cannot shrink as ρ increases. To show the quasi-concavity, consider arbitrary ρ1 and ρ2 such that ρ2 > ρ1 ≥

0. By the non-decreasing property, mmax(ρ1) ≤ mmax(ρ2). Moreover, for some α ∈ [0, 1], denote ρα :=

αρ1 + (1 − α)ρ2, and thus, ρα ≥ ρ1. This implies, mmax(ρα) ≥ mmax(ρ1) = min{mmax(ρ1), mmax(ρ2)}.

That is, mmax(ρ) is quasi-concave. ∞ Part (iii): At m = m∞, portfolio x1 is the unique feasible in (35) with portfolio leverage level ∞ given in (33), the uniqueness due to x1 being the unconstrained maximum of the strict concave ∞ function, E[Rs(x1)]. Therefore, ρmin(m∞) = ρ∞. On the other hand, when ρ ≥ ρ∞, x1 is feasible in

(29), and thus, mmax(ρ∞) ≥ m∞. But, m∞ is the ‘unconstrained-maximum’ portfolio return, and thus, we must have mmax(ρ∞) = m∞. ∞ ∞ Finally, if Ls(x1 ) ≤ 0, x1 is feasible in (35) for m ≤ m∞, which yields, ρmin(m) ≤ ρmin(m∞) = 0, and thus, ρmin(m) = 0 must hold for m ∈ [rs, m∞]. This completes the proof. 

Appendix H: Mean-variance portfolio model under leverage and liquidity impact (MVLL) A compact extensive formulation of the MVLL model in (37) is given below. 1 F (m, ρ) = min x>P V P x 2 1 0 0 1 x1 (w0) > > s.t. x1 (Γ − Ms)x1 + [(2Ms − Γ)x0 − κˆsp0 + P0µ] x1 ≥ w0(m − rs) h Lev − ρ x>(Γ − M )x + [(2M − Γ)x + P µ − κˆ p ]> x 1 s 1 s 0 0 s 0 1 (61) >  − π x1 P0V P0x1 ≤ ρφs

 > >  − Lev + x1 Msx1 − (2Msx0 − κsp0) x1 ≤ φs Lev ≥ 0. 39

Appendix I: Proof of Proposition 6

dE[Rs(νx1)] 1 > ∗ Part i): From (48), we have |ν=1 = x (Γ − Ms)x1 + (m − rs). In particular, at x ≡ dν w0 1 1 ∗ ∗> ∗ x1(m, ρ), due to the condition in the proposition that |x1 (Γ − Ms)x1| < w0(m − rs), it follows that ∗ dE[Rs(νx1)] ∗ ∗ ∗ dν |ν=1 > 0. Therefore, E[Rs(x1)] can be strictly decreased such that E[Rs(νx1)] < E[Rs(x1)] for large enough ν < 1.

∗ Assume by contradiction that E[Rs(x1)] > m. Then, there must exist a sufficiently small δ > 0 ∗ ∗ such that E[Rs(νx1))] ≥ m for ν ∈ (1 − δ, 1). To check the feasibility of νx1 for some ν ∈ (0, 1) in ∗ ∗ ∗ (37), we consider the two possibilities for the optimal solution x1: either Ls(x1) ≤ 0 or Ls(x1) > 0. ∗ + ∗ ∗ First, suppose Ls(x1) ≤ 0. Then, Ls (x1) = 0 and (37) is feasible with Ls(x1) = 0 where ∗ ∗ ∗ E[As(x1)] − πV ar[As(x1)] ≥ 0. At νx1, it can be shown for ν ∈ (0, 1) that

∗ ∗ E[As(νx1)] − πV ar[As(νx1)]

∗ ∗ ∗> ∗ = ν(E[As(x1)] − πV ar[As(x1)]) + ν(ν − 1)x1 (Γ − Ms − πP0V P0)x1 + (1 − ν)φs ≥ 0,

since Γ − Ms is negative definite and φs > 0. Moreover, for ν ∈ (0, 1),

∗ ∗ ∗> ∗ Ls(νx1) = νLs(x1) + ν(ν − 1)x1 Msx1 + (ν − 1)φs ≤ 0, (62)

+ ∗ ∗ since Ms is positive definite, which yields Ls (νx1) = 0. Hence, Ls(νx1) = 0 for ν ∈ (0, 1). Combining ∗ the above results, therefore, it follows that the constraints of (37) are satisfied for the solution νx1 for any ν ∈ (1 − δ, 1).

∗ On the other hand, suppose Ls(x1) > 0. According to (62),

∗ dLs(νx1) ∗> ∗ ∗ = x1 Msx1 + Ls(x1) + φs > 0. dν ν=1

∗ Thus, there exists a sufficiently small  > 0 such that for ν ∈ (1 − , 1) we have Ls(νx1) ≥ 0. To ∗ check if Ls(νx1) ≤ ρ still holds,

+ ∗ ∗ ∗ Ls (νx1) − ρ(E[As(νx1)] − πV ar[As(νx1)])

+ ∗ ∗ ∗ ∗> ∗ = ν(Ls (x1) − ρ(E[As(x1)] − πV ar[As(x1)])) + ν(ν − 1)x1 [Ms − ρ(Γ − Ms − P0V P0)]x1

+ (ν − 1)(1 + ρ)φs ≤ 0,

¯ ¯ ∗ for any ν ∈ (1 − , 1). That is, defining δ := min{δ, }, there exists ν ∈ (1 − δ, 1) such that νx1 is a 2 ∗ 2 2 ∗ 2 ∗ feasible solution in (37) with an objective value σ [Rs(νx1)] = ν σ [Rs(x1)] < σ [Rs(x1)] = F (m, ρ), ∗ ∗ contradicting to the optimality of x1. Therefore, E[Rs(x1(m, ρ))] = m.

Part ii): Suppose, by contradiction, that in an optimal solutionx ˜1 of (36), E[Rs(˜x1)] > m. Following a proof analogous to that of Part i) above, it follows then that there must exist sufficiently 40

small δ > 0 such that νx˜1 is feasible in (36) for any ν ∈ (1 − δ, 1). This implies due to Proposition

1 that Ψs(νx˜1) > Ψs(˜x1) when Γ − Ms is n.d. The latter strict inequality violates the optimality of x˜1 in (36). Therefore, we must have E[Rs(˜x1)] = m. ∗ Now, since x1 optimal in (37) satisfies the constraints in (36),

m − rs m − rs Ψs(˜x1) = ≥ ∗ σ[Rs(˜x1)] σ[Rs(x1)]

2 ∗ 2 holds, which yields F (m, ρ) ≡ σ [Rs(x1)] ≥ σ [Rs(˜x1)]. Sincex ˜1 is also feasible in (37), F (m, ρ) ≤ 2 2 ∗ 2 ∗ σ [Rs(˜x1)]. Combining the two inequalities, σ [Rs(x1)] = σ [Rs(˜x1)], and thus Ψs(˜x1) = Ψs(x1), which completes the proof. Part iii): It follows from Part ii) of the proposition:

∗ ∗ ζSH(m, ρ) = Ψs(x1(m, ρ)) = max {Ψs(x1): E[Rs(x1)] ≥ m, Ls(x1) ≤ ρ} . (63) x1

∗ As ρ increases, the feasible region of (63) enlarges (cannot ‘shrink’), and thus, ζSH(m, ρ) cannot decrease. Moreover, when Γ − Ms is n.d., the optimization in (63) is a parametric maximization of a pseudo-concave function on a convex feasible set that is parametrized by ρ via the quasi-convex

∗ function Ls(x1). Therefore, ζSH(m, ρ) is pseudo-concave in ρ. Part iv):

As π increases, the denominator in the definition of Ls(x1) decreases, and thus, Ls(x1) increases. Therefore, the feasible region of (63) shrinks (cannot enlarge), hence the Sharpe ratio cannot increase. Part v): Increasing m in (36) ‘shrinks’ the feasible region and thus, the Sharpe ratio is non- increasing in m. To show the required pseudo-concavity in m, note that the optimal objective of (36) is given by ζ∗ (m, ρ) = √m−rs due to Part ii) of the proposition. Appealing to the proof of SH F (m,ρ) Proposition 1 in Appendix B, it can be shown in an analogous manner that pF (m, ρ) is convex

m−rs in m. Since m − rs is linear in m, √ is pseudo-concave in m, see Mangasarian (1970). F (m,ρ) This completes the proofs of the parts. 

Appendix J: Proof of Proposition 7

dE[Rs(νx1)] 1 > ∗∗ ∗∗ From (48), we have |ν=1 = x (Γ − Ms)x1 + (m − rs). In particular, at x ≡ x (m), dν w0 1 1 1 ∗∗> ∗∗ due to the condition in the proposition that |x1 (Γ − Ms)x1 | < w0(m − rs), it follows that ∗∗ dE[Rs(νx1 )] ∗∗ ∗∗ dν |ν=1 > 0. Therefore, E[Rs(x1 )] can be strictly decreased such that E[Rs(νx1 )] < ∗∗ E[Rs(x1 )] for large enough ν < 1. ∗ Assume by contradiction that E[Rs(x1)] > m. Then, there must exist a sufficiently small δ > 0 ∗∗ ∗∗ such that E[Rs(νx1 )] ≥ m for ν ∈ (1 − δ, 1). That is, νx1 is feasible in (40) for ν ∈ (1 − δ, 1), and 41

2 ∗∗ 2 2 ∗∗ 2 ∗∗ ∗ it has an objective value σ [Rs(νx1 )] = ν σ [Rs(x1 )] < σ [Rs(x1 )] = F (m), contradicting the ∗∗ ∗∗ optimality of x1 . Therefore, E[Rs(x1 (m))] = m.

Suppose an optimal solution of the maximization in (39), for fixed m, isx ˜1(m) at some opti- malρ ˜(m). Due to Proposition 6-Part i), E[Rs(˜x1(m))] = m, and thus,x ˜1(m) is feasible in (40),

2 ∗∗ 2 m−rs ∗∗ ∗∗ implying σ [Rs(x (m))] ≤ σ [Rs(˜x1(m))]. Therefore, √ = Ψs(x (m)) ≤ ζ (m). Conversely, 1 F ∗(m) 1 SH ∗∗ ∗∗ ∗∗ for x1 (m) optimal in (40), Ls(x1 (m)) ≥ ρmin(m) must hold due to Proposition 5. That is, (x1 (m) ∗∗ ∗∗ ∗∗ ∗ ∗∗ is feasible for the maximization in (39) with ρ := Ls(x1 (m)). Therefore, ζSH(m) ≥ ζSH(m, ρ ) = ∗∗ ∗∗ ∗∗ Ψs(x1 (m)). Hence, we must have: Ψs(x1 (m)) = ζSH(m), proving the first assertion. To prove the second assertion on the maximum-attainable Sharpe, note ζ∗∗(m) = √m−rs as SH F ∗(m) proved above. Appealing to the proof of Proposition 1 in Appendix B, pF ∗(m) is convex in m.

∗∗ Then, following a proof similar to that in Proposition 6-Part v), it follows that ζSH(m) is pseudo- concave in m. To claim that ζ∗∗(m) is non-increasing in m, we will show that √m−rs is non-increasing in m. SH F ∗(m)

First, observe that (40) is well-defined for m ≥ rs. Then, by Taylor’s first order condition for the

p ∗ convex function F (m), considering the the two points m and rs satisfying m > m0 > rs:

dpF ∗(m) ∗0 p ∗ p ∗ p ∗ F (m) F (rs) ≥ F (m) + (rs − m) = F (m) − 0.5 (m − rs), (64) dm pF ∗(m)

∗0 ∗ ∗ where F (m) is the first derivative of F (m). Notice that F (rs) = 0 since rs is achieved under

p ∗ portfolio liquidation, i.e., x1 = 0, hence zero variance. Then, multiplying (64) by F (m) > 0,

0 p ∗ p ∗ ∗ ∗ 0 = F (rs) F (m) ≥ F (m) − 0.5F (m)(m − rs). (65)

Since ζ∗∗(m) = √m−rs , the first derivative satisfies: SH F ∗(m)

0 dζ∗∗(m) F ∗(m) − 0.5(m − r )F ∗ (m) SH = s ≤ 0 (66) dm F ∗(m)pF ∗(m)

∗ ∗∗ due to the inequality in (65) and since F (m) > 0. Thus, ζSH(m) is non-increasing in m. This completes the proofs of the parts. 

Appendix K: Market Impact Parameter Estimation Market impact parameters are estimated using TAQ (Millisecond Trade and Quote) data from NYSE. The trade (execution) transactions data for a given day (for given stock or ETF) are aggregated into five-minute intervals during regular trading time.23 In the present estimation, the

23 Shorter intervals are likely to contain more white noise that can adversely affect the trade impact estimation. If the interval is too long, then market impact can be diluted due to over-aggregation. Consequently, a 5-min trading interval is chosen as a compromise. 42 day of week effects, as well as any attention to exogenous events affecting the market liquidity, are ignored. Then, the total of 390 minutes per day (from 09:30-16:00 hrs) is divided into T = 78 five-minute intervals, for each trading day in the sample of N number of days. In what follows, we describe the estimation model and obtain the market impact parameters.

K.1. Trade classification To determine net trading volume in an interval, each trade execution needs to be classified as a ‘buy’ or a ‘sell’, i.e., whether the transaction is initiated by a buyer or seller. When the TAQ data is used, since the trades are not classified in the data, it becomes necessary to infer the trade direction. There are several rules used in the literature, for example, tick test, quote set, or the Lee-Ready algorithm, see Lee and Ready (1991). Also, see Ellis et al. (2000) who compare the accuracy of these rules using a proprietary data set from NASDAQ. However, with only TAQ data being used in this paper (and with no bid-ask data), the only classification method we can use is the tick rule to assign trade direction. As explained in Asquith et al. (2009), “tick test classifies a trade as a buy (sell) if the trade price is higher (lower) than the previous trade. If the current and previous trade prices are the same, the trade is classified by the next previous trade. Because the test depends on data from previous trades, transactions at the beginning of the day are not classified.” In our implementation, two consecutive trades are deemed the ‘same’ only if the price difference is no more than $0.005.

For a given day d, at some time t, the price of an asset j is denoted by pt,d(j). The discussion here proceeds with a generic asset, and thus, the dependence on j is suppressed for the convenience of exposition. Hence, the open price for the day is p0,d. Having classified each trade by the above tick rule, for each five-minute interval (denoted t) in the day d, the following trade statistics are calculated:

1. Total trade volume: vt,d (shares) 24 2. Net trade volume:v ˆt,d 25 3. Dollar trading volume: st,d o c 4. Open and close prices (in the 5-min interval): pt,d and pt,d 5. Given an interval (t, d), let there be n distinct trades.The price change of two consecutive

trades is denoted δpi, if greater than $0.005, otherwise, 0. The sum of absolute values of price Pn changes in the 5-min interval: ∆p(t, d) = i=1 |δpi|

24 The net volume of a stock is a consolidated total of the positive and negative movements of the security over the period, i.e., up-tick volume minus its down-tick volume in the 5-min interval. A positive (negative) net volume indicates greater upward (downward) movement associated with net ‘buying’ (‘selling’) in the security over the five minutes. 25 The dollar volume is computed by multiplying each trade size by the execution price of the trade, and summing up over all trades in the 5-min interval. 43

K.2. Estimation under trade dynamics Pt The total net volume traded in a day until time t is τ=1 vˆτ,d. This leads to the permanent price c change by the end of period t given by (pt,d − p0,d), the price differential at the end (close) of the time period and the price at the beginning of the day. On the other hand, the temporary impact is due to trading at higher rates causing a temporary lack of liquidity to absorb the required trading rate. Given a 5-min time period, the temporary price effect is measured by the product of the level

L L of illiquidity in the period, It,d, and the (effective) trade price against illiquidity, that is, It,d × vw pt,d , where volume-weighted average price (VWAP) is used for the effective price. We employ the liquidity measure developed by Amihud (2002), which is one of the most widely used in the finance literature. Lou and Shu (2016) report that over one hundred papers published in the top three finance journals use the Amihud (2002) measure in their empirical analysis. Thus, adopting from the above literature, and defining L ∆p(t, d) It,d = (67) st,d as the measure of illiquidity during a 5-min interval, the temporary effect is

L vw ∆p(t, d) st,d ∆p(t, d) It,d × pt,d = × = . (68) st,d vt,d vt,d Combining the permanent and temporary price effects, we estimate the model:

t c X (pt,d − p0,d) + ∆p(t, d) = γ vˆτ,d + λvt,d + εt,d, t = 1, . . . T, d = 1,...N. (69) τ=1

Appendix L: Evaluation of Lending-free Unlevered Portfolio

Since the non-convex model (26) is difficult to solve directly to obtain m0, we employ the MVLL model in (37) with ρ = 0 instead. The MVLL model is executed in an iterative search over m,

∗ ∗ starting from rs, to obtain an optimal solution x1(m, 0) such that the liability variable Ls(x1(m, 0)) is zero. That is, ∗ m0 = min {m : |Ls(x1(m, 0))| < ε} , (70)

−5 where ε = w0 × 10 is specified as a numerical tolerance. Figure 11 depicts the evaluation in (70) for the cases of market impact as well as ignoring market impact.

References Amihud, Y. (2002). Illiquidity and stock returns: Cross-section and time series effects. Journal of Financial Markets 5 31–56.

Asquith, P., Oman, R., and Safaya, C. (2009). Short sales and trade classification algorithms. Journal of Financial Markets 13 157–173. 44

$3,000 $2,500

$2,500 $2,000 2015 2015

$2,000 07, 07,

$1,500 Jan Jan ‐ ‐ $1,500 $1,000 $1,000 Assuming No Market Impact Under Market Impact $500 ABS(Liability) ABS(Liability) $500 m0= 0.5199% m0= 0.2033% $0 $0 0.202% 0.203% 0.204% 0.205% 0.206% 0.207% 0.208% 0.5185% 0.5190% 0.5195% 0.5200% 0.5205% 0.5210% 0.5215% 0.5220% Target mean (m) Target mean (m)

Figure 11 Sharpe maximizing return (m0) under no-leverage

Bazaraa, M.S., Sherali, H.D., and Shetty, C.M. (2006). Nonlinear Programming, Theory and Applications John Wiley & Sons, New York, 3rd edition.

Ellis, K., Michaely, B., and O’Hara, M. (2000). The accuracy of trade classification rules: evidence from Nasdaq. Journal of Financial and Quantitative Analysis 35 529–551.

Lee, C. and Ready, M. (1991). Inferring trade direction from intraday data. Journal of Finance 46 733–746.

Lou, X. and Shu, T. (2016). Price Impact or Trading Volume: Why is the Amihud (2002) Illiquidity Measure Priced?. Available at SSRN: http://ssrn.com/abstract=2291942.

Mangasarian, O.L. (1970). Convexity, Pseudo-convexity, and Quasi-convexity of Composite Functions. Cahiers du Centre d’Etudes de Recherche Operationelle 12 114–122.