Multiperiod Portfolio Optimization with General Transaction Costs

Multieriod Portfolio Otimization with General Transaction Costs Victor DeMiguel Deartment of Management Science and Oerations, London Business School, London NW1 4SA, UK, avmiguel@london.edu Xiaoling Mei Deartment of Statistics, Universidad Carlos III de Madrid, 28911-Leganés (Madrid), Sain, xmei@est-econ.uc3m.es Francisco J. Nogales Deartment of Statistics, Universidad Carlos III de Madrid, 28911-Leganés (Madrid), Sain, fcojavier.nogales@uc3m.es We analyze the roerties of the otimal ortfolio olicy for a multieriod mean-variance investor facing multile risky assets in the resence of general transaction costs such as roortional, market imact, and quadratic transaction costs. For roortional transaction costs, we find that a buy-and-hold olicy is otimal: if the starting ortfolio is outside a arallelogram-shaed no-trade region, then trade to the boundary of the no-trade region at the first eriod, and hold this ortfolio thereafter. For market imact costs, we show that the otimal ortfolio olicy at each eriod is to trade to the boundary of a state-deendent rebalancing region. Moreover, we find that the rebalancing region shrinks along the investment horizon, and as a result the investor trades throughout the entire investment horizon. Finally, we show numerically that the utility loss associated with ignoring transaction costs or investing myoically may be large. Key words: Portfolio otimization; Multieriod utility; No-trade region; Market imact 1. Introduction Markowitz (1952) showed that an investor who only cares about the mean and variance of singleeriod ortfolio returns should choose a ortfolio on the efficient frontier. Two assumtions underlying Markowitz s analysis are that the investor only cares about single-eriod returns and that she is not subject to transaction costs. In this manuscrit we consider the case where these two assumtions fail to hold, and study the otimal ortfolio olicy of a multieriod mean-variance investor subject to general transaction costs, including roortional, market imact, and quadratic transaction costs. There is an extensive literature that generalizes Markowitz s work to the multieriod case. Mossin (1968), Samuelson (1969), and Merton (1969, 1971) show how an investor should otimally choose her ortfolio in a dynamic environment in the absence of transaction costs. In ractice, however, imlementing a dynamic ortfolio olicy requires one to rebalance the ortfolio weights frequently, and this may result in high 1

2 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs transaction costs. To address this issue, researchers have tried to characterize the otimal ortfolio olicies in the resence of transaction costs. The case with a single-risky asset and roortional transaction costs is well understood. In a multieriod setting, Constantinides (1979) shows that the otimal trading olicy is characterized by a no-trade interval, such that if the risky-asset ortfolio weight is inside this interval, then it is otimal not to trade, and if the ortfolio weight is outside, then it is otimal to trade to the boundary of this interval. Later, Constantinides (1986) and Davis and Norman (1990) extend this result to a continuous-time setting with a single risky asset. The case with multile risky assets is much harder to characterize, and the existent literature is scarce. Akian et al. (1996) consider a multile risky-asset version of the continuous time model by Davis and Norman (1990), and using simulations they suggest that the otimal ortfolio olicy is characterized by a multi-dimensional no-trade region. Leland (2000) develos a relatively simle numerical rocedure to comute the no-trade region based on the existence results of Akian et al. (1996). The only aer that rovides an analytical characterization of the no-trade region for the case with multile risky assets is the one by Liu (2004), who shows that under the assumtion that asset returns are uncorrelated, the otimal ortfolio olicy is characterized by a searate no-trade interval for each risky asset. 1 None of the aforementioned aers characterizes analytically the no-trade region for the general case with multile risky assets with correlated returns and transaction costs. The reason for this is that the analysis in these aers relies on modeling the asset return distribution, and as a result they must take ortfolio growth into account, which renders the roblem intractable analytically. Recently, Garleanu and Pedersen (2012) consider a setting that relies on modeling rice changes, and thus they are able to give closedform exressions for the otimal dynamic ortfolio olicies in the resence of quadratic transaction cost. Arguably, modeling rice changes is not very different from modeling stock returns, at least for daily or higher trading frequencies, yet the former aroach renders the roblem tractable. We make three contributions. Our first contribution is to use the multieriod framework roosed by Garleanu and Pedersen (2012) to characterize the otimal ortfolio olicy for the general case with multile risky assets and roortional transaction costs. Secifically, we characterize analytically that there exists a no-trade region, shaed as a arallelogram, such that if the starting ortfolio is inside the no-trade region, then it is otimal not to trade at any eriod. If, on the other hand, the starting ortfolio is outside the no-trade region, then it is otimal to trade to the boundary of the no-trade region in the first eriod, and not to trade thereafter. Furthermore, we show that the size of the no-trade region grows with the level of roortional 1 Other aers rovide numerical solutions to multieriod investment roblems in the resence of transaction costs, see for instance Muthuraman and Kumar (2006), Lynch and Tan (2010), and Brown and Smith (2011).

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 3 transaction costs and the discount factor, and shrinks with the investment horizon and the risk-aversion arameter. Our second contribution is to study analytically the otimal ortfolio olicy in the resence of market imact costs, which arise when the investor makes large trades that distort market rices. 2 Traditionally, researchers have assumed that the market rice imact is linear on the amount traded (see Kyle (1985)), and thus that market imact costs are quadratic. Under this assumtion, Garleanu and Pedersen (2012) derive closed-form exressions for the otimal ortfolio olicy within their multieriod setting. However, Torre and Ferrari (1997), Grinold and Kahn (2000), and Almgren et al. (2005) show that the square root function is more aroriate for modeling market rice imact, thus suggesting market imact costs grow at a rate slower than quadratic. Our contribution is to extend the analysis by Garleanu and Pedersen (2012) to a general case where we are able to cature the distortions on market rice through a a ower function with an exonent between one and two. For this general formulation, we show that there exists an analytical rebalancing region for every time eriod, such that the otimal olicy at each eriod is to trade to the boundary of the corresonding rebalancing region. Moreover, we find that the rebalancing regions shrink throughout the investment horizon, which means that, unlike with roortional transaction costs, it is otimal for the investor to trade at every eriod when she faces market imact costs. Finally, our third contribution is to study numerically the utility losses associated with ignoring transaction costs and investing myoically, as well as how these utility losses deend on relevant arameters. We find that the losses associated with either ignoring transaction costs or behaving myoically can be large. Moreover, the losses from ignoring transaction costs increase in the level of transaction costs, and decrease with the investment horizon, whereas the losses from behaving myoically increase with the investment horizon and are concave unimodal on the level of transaction costs. Our work is related to Dybvig (2005), who considers a single-eriod setting with mean-variance utility and roortional transaction costs. For the case with multile risky assets, he shows that the otimal ortfolio olicy is characterized by a no-trade region shaed as a arallelogram, but the manuscrit does not rovide a detailed analytical roof. Like Dybvig (2005), we consider roortional transaction costs and mean-variance utility, but we extend the results to a multi-eriod setting, and show how the results can be rigorously roven analytically. In addition, we consider the case with market imact costs. This manuscrit is organized as follows. Section 2 describes the multieriod framework under general transaction costs. Section 3 studies the case with roortional transaction costs, Section 4 the case with market imact costs, and Section 5 the case with quadratic transaction costs. Section 6 characterizes numer- 2 This is articularly relevant for otimal execution, where institutional investors have to execute an investment decision within a fixed time interval; see Bertsimas and Lo (1998) and Engle et al. (2012)

4 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs ically the utility loss associated with ignoring transaction costs, and with behaving myoically. Section 7 concludes. Aendix A contains the figures, and Aendix B contains the roofs for all results in the aer. 2. General Framework Our framework is closely related to that roosed by Garleanu and Pedersen (2012); herein G&P. Like G&P, we consider a multieriod setting, where the investor tries to maximize her discounted mean-variance utility net of transaction costs by choosing the number of shares to hold of each of the N risky assets. There are three main differences between our model and the model by G&P. First, we consider a more general class of transaction costs that includes not only quadratic transaction costs, but also roortional and market imact costs. Second, we assume rice changes in excess of the risk-free rate are indeendent and identically distributed with mean µ and covariance matrix Σ, while G&P consider the more general case in which these rice changes are redictable. Third, we consider both finite and infinite investment horizons, whereas G&P focus on the infinite horizon case. The investor s decision in our framework can be written as: T where x t IR N max {x t } T t=1 t=1 [ (1 ρ) t (x t µ γ ] 2 x t Σx t ) (1 ρ) t 1 κ Λ 1/ (x t x t 1 ), (1) contains the number of shares of each of the N risky assets held in eriod t, T is the investment horizon, ρ is the discount factor, and γ is the risk-aversion arameter. The term κ Λ 1/ (x t x t 1 ) is the transaction cost for the tth eriod, where κ IR is the transaction cost arameter, Λ IR N N is the transaction cost matrix, and s is the -norm of vector s; that is, s = N i=1 s i. The transaction cost matrix Λ is a symmetric ositive-definite matrix that catures the distortions to asset rices originated by the interaction between the multile assets. G&P argue that it can be viewed as a multi-dimensional version of Kyle s lambda, see Kyle (1985). For small trades, which do not imact market rices, the transaction cost is usually assumed to be roortional to the amount traded ( = 1), and the natural choice for the transaction cost matrix is Λ = I, where I is the identity matrix. For larger trades, which result in market imact costs, the literature suggests that the transaction costs grow as a ower function with an exonent (1, 2]. Moreover, G&P consider the case with quadratic transaction costs ( = 2) and argue that a sensible choice for the transaction cost matrix is Λ = Σ. We consider this case as well as the case with Λ = I to facilitate the comarison with the case with roortional costs. 3 Our analysis relies on the following assumtion. 3 To simlify the exosition, we focus on the case where the transaction costs associated with trading all N risky assets are symmetric. It is straightforward, however, to extend our results to the asymmetry case, where the transaction costs associated with trading the jth asset at the tth eriod is: κ j (Λ 1/ (x t x t 1)) j j.

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 5 ASSUMPTION 1. Price changes in excess of the risk-free rate are indeendently and identically distributed with mean vector µ and covariance matrix Σ. 3. Proortional Transaction Costs In this section we consider the case where transaction costs are roortional to the amount traded ( = 1). These so-called roortional transaction costs are aroriate to model the cost associated with trades that are small, and thus the transaction cost originates from the bid-ask sread and other brokerage commissions. For exosition uroses, we first study the single-eriod case, and show that for this case the otimal ortfolio olicy is analytically characterized by a no-trade region shaed as a arallelogram. 4 We then study the general multieriod case, and again show that there is a no-trade region shaed as a arallelogram. Moreover, if the starting ortfolio is inside the no-trade region, then it is otimal not to trade at any eriod. If, on the other hand, the starting ortfolio is outside the no-trade region, then it is otimal to trade to the boundary of the no-trade region in the first eriod, and not to trade thereafter. Furthermore, we study how the no-trade region deends on the level of roortional transaction costs, the correlation in asset returns, the discount factor, the investment horizon, and the risk-aversion arameter. 3.1. The Single-Period Case For the single-eriod case, the investor s decision is max x (1 ρ)(x µ γ 2 x Σx) κ x x 0 1, (2) where x 0 is the starting ortfolio. Because the term x x 0 1 is not differentiable, it is not ossible to obtain closed-form exressions for the otimal ortfolio olicy for the case with roortional transaction costs. The following roosition, however, demonstrates that the otimal trading olicy is characterized by a no-trade region shaed as a arallelogram. PROPOSITION 1. Let Assumtion 1 hold, then: 1. The investor s decision roblem (2) can be equivalently rewritten as: min (x x 0 ) Σ(x x 0 ), (3) x s.t Σ(x x κ ) (1 ρ)γ, (4) where x = Σ 1 µ/γ is the otimal ortfolio in the absence of transaction costs (the Markowitz or target ortfolio), and s = max i { s i } is the infinity norm of s. 4 Our analysis for the single-eriod case with roortional transaction costs is similar to that by Dybvig (2005), but we rovide a detailed analytical roof.

6 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 2. Constraint (4) defines a no-trade region shaed as a arallelogram centered at the target ortfolio x, such that if the starting ortfolio x 0 is inside this region, then it is otimal not to trade, and if the starting ortfolio is outside this no-trade region, then it is otimal to trade to the oint in the boundary of the no-trade region that minimizes the objective function in (3). Proosition 1 shows that it is otimal not to trade if the marginal mean-variance utility from trading on any of the N risky assets is smaller than the transaction cost arameter κ. To see this, note that inequality (4), which defines the no-trade region, can be equivalently rewritten as (1 ρ)γσ(x x ) κ. (5) Moreover, it is easy to show that the vector (1 ρ)γσ(x x ) gives the marginal mean-variance utility from trading in each of the assets, because it is the gradient (first derivative) of the static mean-variance utility (1 ρ)(x µ γ 2 x Σx) with resect to the ortfolio x. For a two-asset examle, Figure 1 deicts the arallelogram-shaed no-trade region defined by inequality (4), together with the level sets for the objective function given by (3). The otimal ortfolio olicy is to trade to the intersection between the no-trade region and the tangent level set, at which the marginal utility from trading is equal to the transaction cost arameter κ. It is easy to see that the size of no-trade region defined by Equation (4) decreases with the risk aversion arameter γ. Intuitively, the more risk averse the investor, the larger her incentives to trade and diversify her ortfolio. Also, it is clear that the size of the no-trade region increases with the roortional transaction arameter κ. This makes sense intuitively because the larger the transaction cost arameter, the less attractive to the investor is to trade in order to move closer to the target ortfolio. Moreover, the following roosition shows that there exists a finite transaction cost arameter κ such that if the transaction cost arameter κ > κ, then it is otimal not to trade. PROPOSITION 2. The no-trade region is unbounded when κ κ, where κ = φ and φ is the vector of Lagrange multiliers associated with the constraint at the unique maximizer for the following otimization roblem: 3.2. The Multieriod Case max x (1 ρ)(x µ γ 2 x Σx), (6) s.t. x x 0 = 0. (7) In this section, we show that similar to the single-eriod case, the otimal ortfolio olicy for the multieriod case is also characterized by a no-trade region shaed as a arallelogram and centered around the target

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 7 ortfolio. If the starting ortfolio at the first eriod is inside this no-trade region, then it is otimal not to trade at any eriod. If, on the other hand, the starting ortfolio at the first eriod is outside this no-trade region, then it is otimal to trade to the boundary of the no-trade region in the first eriod, and not to trade thereafter. The investor s decision for this case can be written as: { T max {x t } T t=1 t=1 [(1 ρ) t (x t µ γ 2 x t Σx t ) (1 ρ) t 1 κ x t x t 1 1 ] }. (8) The following theorem demonstrates that the otimal trading olicy is characterized by a no-trade region shaed as a arallelogram. THEOREM 1. Let Assumtion 1 hold, then: 1. It is otimal not to trade at any eriod other than the first eriod; that is, x 1 = x 2 = = x T. (9) 2. The investor s otimal ortfolio for the first eriod x 1 (and thus for all subsequent eriods) is the solution to the following constrained otimization roblem: min (x 1 x 0 ) Σ (x 1 x 0 ), (10) x 1 s.t Σ(x 1 x κ ρ ) (1 ρ)γ 1 (1 ρ). (11) T where x 0 is the starting ortfolio, and x = Σ 1 µ/γ is the otimal ortfolio in the absence of transaction costs (the Markowitz or target ortfolio). 3. Constraint (11) defines a no-trade region shaed as a arallelogram centered at the target ortfolio x, such that if the starting ortfolio x 0 is inside this region, then it is otimal not to trade at any eriod, and if the starting ortfolio is outside this no-trade region, then it is otimal to trade at the first eriod to the oint in the boundary of the no-trade region that minimizes the objective function in (10), and not to trade thereafter. Similar to the single-eriod case, Theorem 1 shows that it is otimal to trade only if the marginal discounted multieriod mean-variance utility from trading in one of the assets is larger than the transaction cost arameter κ. To see this, note that inequality (11), which gives the no-trade region, can be rewritten as (γ(1 ρ)(1 (1 ρ) T )/ρ)σ(x 1 x ) κ. (12) Moreover, because it is only otimal to trade at the first eriod, the vector inside the infinite norm on the left hand-side of inequality 12 is the gradient (first derivative) of the discounted multieriod mean variance utility T t=1 (1 ρ)t ( x t µ γ 2 x t Σx t ) with resect to the ortfolio xt.

8 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs The following corollary establishes how the size of the no-trade region for the multieriod case deends on the roblem arameters. COROLLARY 1. The no-trade region for the multieriod investor satisfies the following roerties: The no-trade region exands as the roortional transaction arameter κ increases. The no-trade region exands as the discount factor arameter ρ increases. The no-trade region shrinks as the investment horizon T increases. The no-trade region shrinks as the risk-aversion arameter γ increases. Similar to the single-eriod case, we observe that the size of the no-trade region grows with the transaction cost arameter κ. This is intuitive as the larger the transaction costs, the less willing the investor is to trade in order to diversify. This is illustrated in Panel (a) of Figure 2, which deicts the no-trade regions for different values of the transaction cost arameter κ for a case with two stocks. In addition, it is easy to show following the same argument used to rove Proosition 2 that there is a κ such that the no-trade region is unbounded for κ κ. The size of the no-trade region increases with the discount factor ρ. Again, this makes sense intuitively because the larger the discount factor, the less imortant the utility for future eriods and thus the smaller the incentive to trade today. This is illustrated in Figure 2, Panel (b). The size of the no-trade region decreases with the investment horizon T. To see this intuitively, note that we have shown that the otimal olicy is to trade at the first eriod and hold this osition thereafter. Then, a multieriod investor with shorter investment horizon will be more concerned about the transaction costs incurred at the first stage, comared with the investor who has a longer investment horizon. Finally, when T, the no-trade region shrinks to the arallelogram bounded by κρ/((1 ρ)γ), which is much closer to the center x. Of course, when T = 1, the multieriod roblem reduces to the static case. This is illustrated in Figure 2, Panel (c). In addition, the no-trade region shrinks as the risk aversion arameter γ increases. Intuitively, as the investor becomes more risk averse, the otimal olicy is to move closer to the diversified (safe) osition x, desite the transaction costs associated with this. This is illustrated in Figure 2, Panel (d), which also shows that the target ortfolio changes with the risk-aversion arameter, and therefore the no-trade regions are centered at different oints for different risk-aversion arameters. The no-trade region also deends on the correlation between assets. Figure 2, Panel (e) shows the no-trade regions for different correlations. When two assets are ositively correlated, the arallelogram leans to the left, because the advantages from diversification are small, whereas with negative correlation it leans to the right. In the absence of correlations the no-trade region becomes a rectangle.

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 9 4. Market Imact Costs In this section we consider market imact costs, which arise when the investor makes large trades that distort market rices. Traditionally, researchers have assumed that the market rice imact is linear on the amount traded (see Kyle (1985)), and thus that market imact costs are quadratic. Under this assumtion, Garleanu and Pedersen (2012) derive closed-form exressions for the otimal ortfolio olicy within their multieriod setting. However, Torre and Ferrari (1997), Grinold and Kahn (2000), Almgren et al. (2005), and Gatheral (2010) show that the square root function may be more aroriate for modelling market rice imact, thus suggesting market imact costs grow at a rate slower than quadratic. Therefore in this section we consider a general case, where the transaction costs are given by the -norm with (1, 2), and where we cature the distortions on market rice through the transaction cost matrix Λ. For exosition uroses, we first study the single-eriod case. 4.1. The Single-Period Case For the single-eriod case, the investor s decision is: max x (1 ρ)(x µ γ 2 x Σx) κ Λ 1/ (x x 0 ), (13) where 1 < < 2. Problem (13) can be solved numerically, but unfortunately it is not ossible to obtain closed-form exressions for the otimal ortfolio olicy. The following roosition, however, shows that the otimal ortfolio olicy is to trade to the boundary of a rebalancing region that deends on the starting ortfolio and contains the target or Markowitz ortfolio. PROPOSITION 3. Let Assumtion 1 hold, then if the starting ortfolio x 0 is equal to the target or Markowitz ortfolio x, the otimal olicy is not to trade. Otherwise, it is otimal to trade to the boundary of the following rebalancing region: where q is such that 1 + 1 q = 1. Λ 1/ Σ(x x ) q κ Λ 1/ (x x 0 ) 1 (1 ρ)γ, (14) Comaring Proositions 1 and 3, we identify three main differences between the cases with roortional and market imact costs. First, for the case with market imact costs it is always otimal to trade (excet in the trivial case where the starting ortfolio coincides with the target or Markowitz ortfolio), whereas for the case with roortional transaction costs it may be otimal not to trade if the starting ortfolio is inside the no-trade region. Second, the rebalancing region deends on the starting ortfolio x 0, whereas the notrade region is indeendent of it. Third, the rebalancing region contains the target or Markowitz ortfolio, but it is not centered around it, whereas the no-trade region is centered around the Markowitz ortfolio.

10 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Note that, as in the case with roortional transaction costs, the size of the rebalancing region increases with the transaction cost arameter κ, and decreases with the risk-aversion arameter. Intuitively, the more risk averse the investor, the larger her incentives to trade and diversify her ortfolio. Also, the rebalancing region grows with κ because the larger the transaction cost arameter, the less attractive to the investor is to trade to move closer to the target ortfolio. The following corollary gives the rebalancing region for two imortant articular cases. First, the case where the transaction cost matrix Λ = I, which is a realistic assumtion when the amount traded is small, and thus the interaction between different assets, in terms of market imact, is small. This case also facilitates the comarison with the otimal ortfolio olicy for the case with roortional transaction costs. The second case corresonds to the transaction cost matrix Λ = Σ, in analogy with the analysis by G&P in the context of quadratic transaction costs. COROLLARY 2. For the single-eriod investor defined in (13): 1. When the transaction cost matrix is Λ = I, then the rebalancing region is Σ(x x ) q κ x x 0 1 (1 ρ)γ. (15) 2. When the transaction cost matrix is Λ = Σ, then the rebalancing region is Σ 1/q (x x ) q κ Σ 1/ (x x 0 ) 1 (1 ρ)γ. (16) Note that, in both articular cases, the Markowitz strategy x is contained in the rebalancing region. To gain intuition about the form of the rebalancing regions characterized in (15) and (16), Panel (a) in Figure 3 deicts the rebalancing region and the otimal ortfolio olicy for a two-asset examle when Λ = I, while Panel (b) deicts the corresonding rebalancing region and otimal ortfolio olicy when Λ = Σ. The figure shows that, in both cases, the rebalancing region is a convex region containing the Markowitz ortfolio. Moreover, it shows how the otimal trading strategy moves to the boundary of the rebalancing region. 4.2. The Multieriod Case The investor s decision for this case can be written as: max {x t } T t=1 T t=1 [(1 ρ) (x t t µ γ ) ] 2 x t Σx t (1 ρ) t 1 κ Λ 1/ (x t x t 1 ). (17) As in the single-eriod case, it is not ossible to rovide closed-form exressions for the otimal ortfolio olicy, but the following theorem illustrates the analytical roerties of the otimal ortfolio olicy. THEOREM 2. Let Assumtion 1 hold, then:

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 11 1. If the starting ortfolio x 0 is equal to the target or Markowitz ortfolio x, then the otimal olicy is not to trade at any eriod. 2. Otherwise it is otimal to trade at every eriod. Moreover, at the tth eriod it is otimal to trade to the boundary of the following rebalancing region: where q is such that 1 + 1 q = 1. T (1 s=t ρ)s t Λ 1/ Σ(x s x ) q κ Λ 1/ (x t x t 1 ) 1 (1 ρ)γ, (18) Theorem 2 shows that for the multieriod case with market imact costs it is otimal to trade at every eriod (excet in the trivial case where the starting ortfolio coincides with the Markowitz ortfolio). Moreover, at every eriod it is otimal to trade to the boundary of a rebalancing region that deends not only on the starting ortfolio, but also on the ortfolio for every subsequent eriod. Finally, note that the size of the rebalancing region for eriod t, assuming the ortfolios for the rest of the eriods are fixed, increases with the transaction cost arameter κ and decreases with the discount factor ρ and the risk-aversion arameter γ. The following roosition shows that the rebalancing region for eriod t contains the rebalancing region for every subsequent eriod. Moreover, the rebalancing region converges to the Markowitz ortfolio as the investment horizon grows, and thus the otimal ortfolio x T converges to the target ortfolio x in the limit when T goes to infinity. PROPOSITION 4. Let Assumtion 1 hold, then: 1. The rebalancing region for the t-th eriod contains the rebalancing region for every subsequent eriod, 2. Every rebalancing region contains the Markowitz ortfolio, 3. The rebalancing region converges to the Markowitz ortfolio in the limit when the investment horizon goes to infinity. The next corollary gives the rebalancing region for the two articular cases of transaction cost matrix we consider. COROLLARY 3. For the multieriod investor defined in (17): 1. When the transaction cost matrix is Λ = I, then the rebalancing region is T (1 s=t ρ)s t Σ(x s x ) q κ x t x t 1 1 (1 ρ)γ. (19) 2. When the transaction cost matrix is λ = Σ, then the rebalancing region is T (1 s=t ρ)s t Σ 1/q (x s x ) q κ Σ 1/ (x t x t 1 ) 1 (1 ρ)γ. (20)

12 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs To gain intuition about the shae of the rebalancing regions characterized in (19) and (20), Panel (a) in Figure 4 shows the otimal ortfolio olicy and the rebalancing regions for a two-asset examle with an investment horizon T = 3 when Λ = I, whereas Panel (b) deicts the corresonding otimal ortfolio olicy and rebalancing regions when Λ = Σ. The figure shows, in both cases, how the rebalancing region for each eriod contains the rebalancing region for subsequent eriods. Moreover, every rebalancing region contains, but is not centered at, the Markowitz ortfolio x. In articular, for each stage, any trade is to the boundary of the rebalancing region and the rebalancing is towards the Markowitz strategy x. Note also that the figure shows that, for the case with Λ = I, it is otimal for the investor to buy the second asset in the first eriod, and then sell it in the second eriod. This may aear subotimal from the oint of view of market imact costs, but it turns out to be otimal when the investor considers the trade off between multieriod mean-variance utility and market imact costs. Finally, we study numerically the imact of the market imact cost growth rate on the otimal ortfolio olicy. Figure 5 shows the rebalancing regions and trading trajectories for investors with different transaction growth rates = 1, 1.25, 1.5, 1.75, 2. When the transaction cost matrix Λ = I, Panel (a) shows how the rebalancing region deends on. In articular, for = 1 we recover the case with roortional transaction costs, and hence the rebalancing region becomes a arallelogram-shaed no-trade region. For = 2, the rebalancing region becomes an ellise. And for values of between 1 and 2, the rebalancing regions become suerellises but not centered at the target ortfolio x. On the other hand, Panel (b) in Figure 5 shows how the trading trajectories deend on for a articular investment horizon of T = 10 days. We observe that, as grows, the trading trajectories become less curved and the investor converges towards the target ortfolio at a slower rate. We have also considered the case Λ = Σ, but the insights are similar and thus we do not include the figure. 5. Quadratic Transaction Costs We now consider the case with quadratic transaction costs. The investor s decision is: max {x t } T t=1 T t=1 [ (1 ρ) t (x t µ γ ] 2 x t Σx t ) (1 ρ) t 1 κ Λ 1/2 (x t x t 1 ) 2 2. (21) As exlained in Section 2, our general framework differs from that considered by G&P in three resects: (i) G&P assume rice changes are redictable, whereas we assume rice changes are iid, (ii) G&P consider an infinite horizon, whereas we allow for a finite investment horizon, and (iii) we consider a general transaction costs matrix Λ whereas G&P focus on the articular case Λ = Σ. The next theorem rovides an extension of the results in G&P to obtain an exlicit characterization of the otimal ortfolio olicy.

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 13 THEOREM 3. Let Assumtion 1 hold, then: 1. The otimal ortfolio x t, x t+1,..., x t+t 1 satisfies the following equations: x t = A 1 x + A 2 x t 1 + A 3 x t+1, for t = 1, 2,..., T 1 (22) x t = B 1 x + B 2 x t 1, for t = T. (23) where A 1 = (1 ρ)γ [(1 ρ)γσ + 2κΛ + 2(1 ρ)κλ] 1 Σ, A 2 = 2κ [(1 ρ)γσ + 2κΛ + 2(1 ρ)κλ] 1 Λ, A 3 = 2(1 ρ)κ [(1 ρ)γσ + 2κΛ + 2(1 ρ)κλ] 1 Λ, with A 1 + A 2 + A 3 = I, and B 1 = (1 ρ)γ[(1 ρ)γσ + 2κΛ] 1 Σ, B 2 = 2κ[(1 ρ)γσ + 2κΛ] 1 Λ, with B 1 + B 2 = I. 2. The otimal ortfolio converges to the Markowitz ortfolio as the investment horizon T goes to infinity. Theorem 3 shows that the otimal ortfolio for each stage is a combination of the Markowitz strategy (the target ortfolio), the revious eriod ortfolio, and the next eriod ortfolio. The next corollary shows the secific otimal ortfolios for two articular cases of transaction cost matrix. Like G&P, we consider the case where the transaction costs matrix is roortional to the covariance matrix, that is Λ = Σ. In addition, we also consider the case where the transaction costs matrix is roortional to the identity matrix; that is Λ = I. COROLLARY 4. For a multieriod investor with objective function (21): 1. When the transaction cost matrix is Λ = I, then the otimal trading strategy satisfies x t = A 1 x + A 2 x t 1 + A 3 x t+1, for t = 1, 2,..., T 1 (24) x t = B 1 x + B 2 x t 1, for t = T (25) where A 1 = (1 ρ)γ [(1 ρ)γσ + 2κI + 2(1 ρ)κi] 1 Σ, A 2 = 2κ [(1 ρ)γσ + 2κI + 2(1 ρ)κi] 1, A 3 = 2(1 ρ)κ [(1 ρ)γσ + 2κI + 2(1 ρ)κi] 1,

14 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs with A 1 + A 2 + A 3 = I, and B 1 = (1 ρ)γ[(1 ρ)γσ + 2κI] 1 Σ, B 2 = 2κ[(1 ρ)γσ + 2κI] 1. with B 1 + B 2 = I. 2. When the transaction cost matrix is Λ = Σ, then the otimal trading strategy satisfies x t = α 1 x + α 2 x t 1 + α 3 x t+1, for t = 1, 2,..., T 1 (26) x t = β 1 x + β 2 x t 1, for t = T. (27) where α 1 = (1 ρ)γ/((1 ρ)γ + 2κ + 2(1 ρ)κ), α 2 = 2κ/((1 ρ)γ + 2κ + 2(1 ρ)κ), α 3 = 2(1 ρ)κ/((1 ρ)γ + 2κ + 2(1 ρ)κ) with α 1 + α 2 + α 3 = 1, and β 1 = (1 ρ)γ/((1 ρ)γ + 2κ) β 2 = 2κ/((1 ρ)γ + 2κ) with β 1 + β 2 = 1. 3. When the transaction cost matrix is Λ = Σ, then the otimal ortfolios for eriods t = 1, 2,..., T lay on a straight line. Corollary 4 shows that, when Λ = Σ, the solution becomes simler and easier to interret than when Λ = I. Note that when Λ = Σ, matrices A and B in Theorem 3 become scalars α and β, resectively, and hence the otimal ortfolio at eriod t can be exressed as a linear combination of the Markowitz ortfolio, the revious eriod ortfolio and the next eriod ortfolio. For this reason, it is intuitive to observe that the otimal trading strategies for all the eriods must lay on a straight line. To conclude this section, Figure 6 rovides a comarison of the otimal ortfolio olicy for the case with quadratic transaction costs (when Λ = Σ), with those for the cases with roortional and market imact costs (when Λ = I), for a multieriod investor with T = 4. We have also considered other transaction cost matrices, but the insights are similar. The figure confirms that, for the case with quadratic transaction costs, the otimal ortfolio olicy is to trade at every eriod along a straight line that converges to the Markowitz ortfolio. It can also be areciated that the investor trades more aggressively at the first eriods comared to the final eriods. For the case with roortional transaction costs, it is otimal to trade to the boundary of the no-trade region shaed as a arallelogram in the first eriod and not to trade thereafter. Finally, for the case with market imact costs, the investor trades at every eriod to the boundary of the corresonding rebalancing region. The resulting trajectory is not a straight line. In fact, as in the examle discussed in Section 4, it is otimal for the investor to buy the second asset in the first eriod and then sell it in the second eriod; that is, the otimal ortfolio olicy is inefficient in terms of market imact costs, but it is otimal in terms of the tradeoff between market imact costs and discounted utility.

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 15 6. Numerical Analysis In this section, we study numerically the utility loss associated with ignoring transaction costs and investing myoically, as well as how these utility losses deend on the transaction cost arameter, the rice-change correlation, the investment horizon, and the risk-aversion arameter. We first consider the case with roortional transaction costs, and then study how the monotonicity roerties of the utility losses change when transaction costs are quadratic. We have also considered the case with market imact costs ( = 1.5), but the insights are similar to those from the case with quadratic transaction costs and thus we do not reort the results to conserve sace. For each tye of transaction cost (roortional or quadratic), we consider three different ortfolio olicies. First, we consider the target ortfolio olicy, which consists of trading to the target or Markowitz ortfolio in the first eriod and not trading thereafter. This is the otimal ortfolio olicy for an investor in the absence of transaction costs. Second, the static ortfolio olicy, which consists of trading at each eriod to the solution to the single-eriod roblem subject to transaction costs. This is the otimal ortfolio olicy for a myoic investor who takes into account transaction costs. Third, we consider the multieriod ortfolio olicy, which is the otimal ortfolio olicy for a multieriod investor who takes into account transaction costs. Finally, we evaluate the utility of each of the three ortfolio olicies using the aroriate multieriod framework; that is, when considering roortional transaction costs, we evaluate the investor s utility from each ortfolio with the objective function in equation (8); and when considering quadratic transaction costs, we evaluate the investor s utility using the objective function (21). 6.1. Proortional Transaction Costs 6.1.1. Base Case. We consider a base case with roortional transaction costs of 30 basis oints, riskaversion arameter γ = 10 5, which corresonds to a relative risk aversion of one for an investor managing M = 10 5 dollars, annual discount factor ρ = 5% 5, and an investment horizon of T = 22 days (one month). We consider four risky assets (N = 4), with starting rice of one dollar, asset rice change correlations of 0.2, and we assume the starting ortfolio is equally-weighted across the four risky assets with a total number of M = 10 5 shares. We randomly draw the annual average rice changes from a uniform distribution with suort [0.1, 0.25], and the annual rice change volatilities from a uniform distribution with suort [0.1, 0.4]. 5 Garleanu and Pedersen (2012) consider, for a larger investor facing quadratic transaction costs, an absolute risk aversion γ = 10 9, which corresonds to an investor managing M = 10 9 dollars, and a discount factor ρ = 1 ex( 0.02/260) which corresonds with an annual discount of 2%.

16 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs For our base case, we observe that the utility loss associated with investing myoically (that is, the difference between the utility of the multieriod ortfolio olicy and the static ortfolio olicy) is 70.01%. The utility loss associated with ignoring transaction costs altogether (that is, the difference between the utility of the multieriod ortfolio olicy and the target ortfolio olicy) is 13.90%. Hence we find that the loss associated with either ignoring transaction costs or behaving myoically can be substantial. The following subsection confirms this is also true when we change relevant model arameters. 6.1.2. Comarative statics. We study numerically how the utility losses associated with ignoring transaction costs (i.e., with the static ortfolio), and investing myoically (i.e., with the target ortfolio) deend on the transaction cost arameter, the rice-change correlation, the investment horizon, and the risk-aversion arameter. Panel (a) in Figure 7 deicts the utility loss associated with the target and static ortfolios for values of the roortional transaction cost arameter κ ranging from 0 basis oint to 120 basis oints. As exected, the utility loss associated with ignoring transaction costs is zero in the absence of transaction costs and increases monotonically with transaction costs. Moreover, for large transaction costs arameters, the utility loss associated with ignoring transaction costs grows linearly with κ and can be very large. The utility losses associated with behaving myoically are concave unimodal in the transaction cost arameter, being zero for the case with zero transaction costs (because both the single-eriod and multieriod ortfolio olicies coincide with the target or Markowitz ortfolio), and for the case with large transaction costs (because both the single-eriod and multieriod ortfolio olicies result in little or no trading). The utility loss of behaving myoically reaches a maximum of 80% for a level of transaction costs of around 5 basis oints Panel (b) in Figure 7 deicts the utility loss associated with the target and static ortfolio olicies for values of rice change correlation ranging from 0.3 to 0.4. The grah shows that while the utility loss associated with the static ortfolio is monotonically decreasing in correlation, the utility loss of the target ortfolio increases with the correlation. The exlanation for this stems from the fact that the static ortfolio is less diversified than the multieriod ortfolio (because it assigns a higher weight to transaction costs), whereas the target ortfolio is more diversified than the multieriod ortfolio (because it ignores transaction costs). Hence, because the benefits from diversification are greater for smaller correlations, the utility loss from the static ortfolio decreases with correlation, and the utility loss of the target ortfolio increases for large correlations. Panel (c) in Figure 7 deicts the utility loss associated with investing myoically and ignoring transaction costs for investment horizons ranging from T = 5 (one week) to T = 22 (one month). Not surrisingly, the utility loss associated with behaving myoically grows with the investment horizon. Also, the utility

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 17 loss associated with ignoring transaction costs is very large for short-term investors, and decreases monotonically with the investment horizon. The reason for this is that the size of the no-trade region for the multieriod ortfolio olicy decreases monotonically with the investment horizon, and thus the target and multieriod olicies become similar for long investment horizons. This makes sense intuitively: by adoting the Markowitz ortfolio, a multieriod investor incurs transaction cost losses at the first eriod, but makes mean-variance utility gains for the rest of the investment horizon. Hence, when the investment horizon is long, the transaction losses are negligible comared with the utility gains. Finally, we find that the utility losses associated with investing myoically and ignoring transaction costs do not deend on the risk-aversion arameter, because the utility for these three ortfolio olicies decrease at the same relative rate as γ increases. 6.2. Quadratic Transaction Costs In this section we study whether and how the resence of quadratic transaction costs (as oosed to roortional transaction costs) affects the utility losses of the static and target ortfolios. 6.2.1. The Base Case. We consider the same arameters as in the base case with roortional transaction costs, lus we assume the transaction cost matrix Λ = Σ, and the transaction cost arameter κ = 1.5 10 4. Similar to the case with roortional transaction costs, we find that the losses associated with either ignoring transaction costs or behaving myoically are substantial. For instance, for the base case with find that the utility loss associated with investing myoically is 30.81%, whereas the utility loss associated with ignoring transaction costs is 116.20%. Moreover, we find that the utility losses associated with the target ortfolio are relatively larger, comared to those of the static ortfolio, for the case with quadratic transaction costs. The exlanation for this is that the target ortfolio requires large trades in the first eriod, which are enalized heavily in the context of quadratic transaction costs. The static ortfolio, on the other hand, results in smaller trades over successive eriods and this will result in overall smaller quadratic transaction costs. 6.2.2. Comarative Statics. Panel (a) in Figure 8 deicts the utility loss associated with investing myoically and ignoring transaction costs for values of the quadratic transaction cost arameter κ ranging from 5 10 5 to 5 10 4. Our findings are very similar to those for the case with roortional transaction costs. Not surrisingly, the utility losses associated with ignoring transaction costs are small for small transaction costs and grow monotonically as the transaction cost arameter grows. Also, the utility loss associated with the static ortfolio olicy is concave unimodal in the level of transaction costs. The intuition behind these results is similar to that rovided for the case with roortional transaction costs.

18 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Panel (b) in Figure 8 deicts the utility loss associated with investing myoically and ignoring transaction costs for values of rice change correlation ranging from -0.3 to 0.4. Similar to the case with roortional transaction costs, the utility loss associated with the static ortfolio is monotonically decreasing with correlation. The reason for this is again that the static ortfolio is less diversified than the multieriod ortfolio, and this results in larger losses when correlations are small because the otential gains from diversification are larger. Unlike in the case with roortional transaction costs, however, the utility loss of the target ortfolio is monotonically decreasing with correlation. This is surrising as the target ortfolio is more diversified than the multieriod ortfolio and thus its losses should be relatively smaller when correlations are low. We find, however, that because the target ortfolio requires large trades in the first eriod and thus it incurs very high quadratic transaction costs, the benefits of diversification are swamed by the quadratic transaction costs. Consequently, the utility loss of the target ortfolio is larger for small (negative) correlations, because in this case the target ortfolio takes extreme long and short ositions that result in very large trades in the first eriod, and thus very large quadratic transaction costs. Panel (c) in Figure 8 deicts the utility loss associated with investing myoically and ignoring transaction costs for values investment horizon T ranging from 5 days to 22 days. Our findings are similar to those for the case with roortional transaction costs. The target ortfolio losses are monotonically decreasing with the investment horizon as these two ortfolios become more similar for longer investment horizons, where the overall imortance of transaction costs is smaller. The static ortfolio losses are monotonically increasing with the investment horizon because the larger the investment horizon the faster the multieriod ortfolio converges to the target, whereas the rate at which the static ortfolio converges to the target does not change with the investment horizon. Finally, we find that the utility loss associated with investing myoically and ignoring transaction costs is monotonically decreasing for values of the risk-aversion level γ ranging from 5 10 6 to 5 10 5. The exlanation for this is that when the risk-aversion arameter is large, the mean-variance utility is relatively more imortant comared to the quadratic transaction costs, and thus the target and static ortfolios are more similar to the multieriod ortfolio. This is in contrast to the case with roortional transaction costs, where the losses did not deend on the risk-aversion arameter. The reason for this difference is that with quadratic transaction costs, the mean-variance utility term and the transaction cost term are both quadratic, and thus the risk-aversion arameter does have an imact on the overall utility loss. Finally, we have reeated our analysis for a case with market imact costs ( = 1.5) and we find that the monotonicity roerties of the utility losses are roughly in the middle of those for the case with roortional transaction costs ( = 1) and quadratic transaction costs ( = 2), and thus we do not reort the results to conserve sace.

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 19 7. Conclusions We study the otimal ortfolio olicy for a multieriod mean-variance investor facing multile risky assets subject to roortional, market imact, or quadratic transaction costs. We demonstrate analytically that, in the resence of roortional transaction costs, the otimal strategy for the multieriod investor is to trade in the first eriod to the boundary of a no-trade region shaed as a arallelogram, and not to trade thereafter. For the case with market imact costs, the otimal ortfolio olicy is to trade to the boundary of a statedeendent rebalancing region. In addition, the rebalancing region converges to the Markowitz ortfolio as the investment horizon grows large. We contribute to the literature by characterizing the no-trade region for a multieriod investor facing roortional transaction costs, and studying the analytical roerties of the otimal trading strategy for the model with market imact costs. Finally, we also show numerically that the utility losses associated with ignoring transaction costs or investing myoically may be large, and study how they deend on the relevant arameters. Acknowledgement We thank comments from seminar articiants at Universidad Carlos III de Madrid. Mei and Nogales gratefully acknowledge financial suort from the Sanish government through roject MTM2010-16519.

20 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Aendix A: Figures Figure 1 No-trade region and level sets. This figure lots the no-trade region and the level sets for a single-eriod investor facing roortional transaction costs with κ = 0.005, discount factor ρ = 0.5, risk-aversion arameter γ = 10, rice-change mean µ = [0.02; 0.01], rice-change volatilities [0.02; 0.01] and rice-change correlation 0.2.

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 21 Figure 2 No-trade region: comarative statics. This figure shows how the no-trade region for a multieriod investor subject to roortional transaction costs deends on relevant arameters. In the base case, we consider the roortional transaction cost arameter κ = 0.005, discount factor ρ = 0.5, riskaversion arameter γ = 10, rice-change means µ = [0.02; 0.01], rice-change volatilities [0.02; 0.01] and rice-change correlation 0.2. (a) No-trade regions for different κ (b) No-trade regions for different ρ (c) No-trade regions for different T (d) No-trade regions for different γ (e) No-trade regions for different correlations

22 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Figure 3 Rebalancing region for single-eriod investor This figure deicts the rebalancing region for a single-eriod investor subject to market imact costs. Panel (a) deicts the rebalancing region when Λ = I, with = 1.5, risk-aversion arameter γ = 1, transaction cost arameter κ = 0.0015, rice-change correlation 0.2, and annual discount factor ρ = 0.02. Panel (b) deicts the rebalancing region when Λ = Σ, with κ = 0.15. In both cases, we consider the rice-change means [0.08; 0.12] and rice-change volatilities [0.037; 0.146]. (a) Rebalancing region when Λ = I (b) Rebalancing region when Λ = Σ Figure 4 Rebalancing region for multieriod investor This figure deicts the rebalancing region for a multieriod investor subject to market imact costs, and investment horizon T = 3. Panel (a) deicts the rebalancing region when Λ = I, with = 1.5, risk-aversion arameter γ = 1, transaction cost arameter κ = 0.0015, rice-change correlation 0.2, and annual discount factor ρ = 0.02. Panel (b) deicts the rebalancing region when Λ = Σ, with κ = 0.09. In both cases, we consider the rice-change means [0.08; 0.12] and rice-change volatilities [0.037; 0.146]. (a) Rebalancing region when Λ = I, T = 3 (b) Rebalancing region when Λ = Σ, T = 3

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 23 Figure 5 Rebalancing regions and trading trajectories for different exonents. This figure deicts the rebalancing regions and trading trajectories for the market imact costs model deending on exonent. Panel (a) deicts the rebalancing regions for the single-eriod ortfolio, with transaction cost arameter κ = 0.008, and annual discount factor ρ = 0.09. Panel (b) deicts the multieriod otimal trading trajectories for the multieriod ortfolio when T = 10, with transaction cost arameter κ = 0.15, and annual discount factor ρ = 0.05. In both cases, we consider transaction costs matrix Λ = I, the risk-aversion arameter γ = 1, correlation 0.4, rice-change means [0.1; 0.1] and rice-change volatilities [0.2; 0.2]. (a) Rebalancing regions deending on exonent. (b) Trading trajectories deending on exonent. Figure 6 Trading trajectories for different transaction costs. This figure deicts the multieriod trading trajectories for a multieriod investor facing different transaction costs and an investment horizon T = 4. We consider a risk-aversion arameter γ = 1, rice-change correlation 0.2, annual discount factor ρ = 0.05. In the roortional transaction cost case, the transaction costs arameter κ = 0.008. In the market imact cost case, κ = 0.002 and Λ = I. In the quadratic transaction cost case, κ = 2 and Λ = Σ. For all cases, we consider the rice-change means [0.08; 0.12] and rice-change volatilities [0.037; 0.146].

24 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Figure 7 Utility losses with roortional transaction costs. This figure deicts the utility loss deending on relevant model arameters. Panel (a) deicts the utility loss as a function of the roortional transaction costs arameter κ. Panel (b) deicts the utility loss as a function of the rice-change correlation. Panel (c) deicts the utility loss as a function of the investment horizon T. In the base case, we consider number of assets N = 4, roortional transaction costs arameter κ = 0.0030, risk-aversion arameter γ = 1e 5, annual discount factor ρ = 0.05, rice-change correlations equal to 0.2, and investment horizon T = 22. The annual rice-change means are randomly generated from interval [0.1, 0.25] and the rice-change volatilities are generated from interval [0.1, 0.4]. (a) Utility losses deending on κ. (b) Utility losses deending on correlation. (c) Utility losses deending on investment horizon T.

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 25 Figure 8 Utility losses with quadratic transaction costs. This figure deicts the utility loss as a function of relevant model arameters. Panel (a) deicts the utility loss as a function of the quadratic transaction cost arameter λ. Panel (b) deicts the utility loss as a function of the rice-change correlation. Panel (c) deicts the utility loss as a function of the investment horizon arameter T. In the base case, we consider number of assets N = 4, quadratic transaction costs arameter κ = 1.5e 4, risk-aversion arameter γ = 1e 5, annual discount factor ρ = 0.05, rice-change correlations equal to 0.2, and investment horizon T = 22. The annual rice-change means are randomly generated from interval [0.1, 0.25] and rice-change volatilities are generated from interval [0.1, 0.4]. (a) Utility losses deending on κ (b) Utility losses deending on correlation. (c) Utility losses deending on investment horizon T

26 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Aendix B: Proof of Proosition 1 Proofs of all results Define the subdifferential Ω of κ x x 0 1 as s Ω = { u u (x x 0 ) = κ x x 0 1, u κ }, (28) where s denotes a subgradient of κ x x 0 1. If we write κ x x 0 1 = max s κ s (x x 0 ), then objective function (2) can be sequentially rewritten as: max x = max x (1 ρ)(x µ γ 2 x Σx) κ x x 0 1 min s κ (1 ρ)(x µ γ 2 x Σx) s (x x 0 ) = min max (1 s κ x ρ)(x µ γ 2 x Σx) s (x x 0 ). (29) Diferentiating the inner objective function in (29) with resect to x gives and hence x = 1 γ Σ 1 (µ 1 1 ρ min s κ min s κ 0 = (1 ρ)(µ γσx) s, (30) s) for s Ω. Plugging x into (29), [ 1 γ Σ 1 (µ 1 ] 1 ρ s) µ γ [ 1 2 γ Σ 1 (µ 1 [ 1 s γ Σ 1 (µ 1 ] 1 ρ s) x 0 ] [ 1 1 ρ s) Σ γ Σ 1 (µ 1 1 ρ s) 1 2γ (µ 1 1 ρ s) Σ 1 (µ 1 1 ρ s) + s x 0. (31) Note that from (30), we can attain s = (1 ρ)(µ γσx) = (1 ρ)[γσ(x x)] as well as s κ, where x = 1 γ Σ 1 µ denoting Markowitz strategy. Plugging s back into (31), we conclude that roblem (31) is equivalent to the following γ min x 2 x Σx γx Σx 0. (32) s.t. (1 ρ)γσ(x x ) κ. (33) If we now add a constant term γ 2 x 0 Σx 0 to the objective function (32), it follows immediately that min x min x min x γ 2 x Σx γx Σx 0 γ 2 x Σx γx Σx 0 + γ 2 x 0 Σx 0 (x x 0 ) Σ(x x 0 ). (34) Rearranging terms in constraint (33), we conclude the objective function (2) is equivalent to min (x x 0 ) Σ(x x 0 ), (35) x s.t. κ Σ(x x ) (1 ρ)γ. (36) To show that constraint (4) defines a no-trade region, note that when the initial osition x 0 satisfies constraint (4), then x = x 0 minimizes the objective function and it is feasible with resect to the constraint. On the other hand, when ]

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 27 x 0 is not in the region defined by the constraint, the otimal x must be the oint on the boundary of the feasible region that minimizes the objective. Finally, note that constraint Σ(x x ) κ is equivalent to γ e (1 ρ)γ (1 ρ)κ Σ(x x ) γ e, hence we conclude that the no-trade region defined by constraint (4) is a arallelogram centered (1 ρ)κ around x. Proof of Proosition 2 Consider the following otimization roblem min x (1 ρ)(x µ γ 2 x Σx) (37) s.t. x x 0 = 0 (38) Noting that the otimal solution to the above otimization is x = x 0. Define φ as the Lagrange multiliers corresonding to constraint (38) and further the l 1 enalty function for otimization roblem (37)-(38) as φ(x, κ) = (1 ρ)(x µ γ 2 x Σx) + κ x x 0 1, (39) which is also equivalent to objective (2). Let κ = φ, when κ > κ, the solution to roblem (37)-(38) x = x 0 is also the minimizer of enalty function (39) (see Theorem 17.3 of Nocedal and Wright (2006)). We conclude that when κ > κ, the no-trade region will become unbounded. Proof of Theorem 1 Part 1. Define subdifferential Ω t of κ x t x t 1 1 as s t Ω t = { u t u t (x t x t 1 ) = κ x t x t 1 1, u t κ }, (40) where s t denotes a subgradient of κ x t x t 1 1, t = 1, 2,, T. If we write κ x t x t 1 1 = max st κ s T t (x t x t 1 ), objective function (8) can be sequentially rewritten as max {x t } T t=1 = max T [(1 ρ) (x t t µ γ ) ] 2 x t Σx t (1 ρ) t 1 κ x t x t 1 1 t=1 min T {x t } T s t=1 t κ t=1 = min max T s t κ {x t } T t=1 t=1 [(1 ρ) (x t t µ γ ) ] 2 x t Σx t (1 ρ) t 1 s t (x t x t 1 ) Differentiating the inner objective function of (41) with resect to x t gives hence x t = 1 γ Σ 1 (µ + s t+1 ) [(1 ρ) (x t t µ γ ) ] 2 x t Σx t (1 ρ) t 1 s t (x t x t 1 ). (41) 0 = (1 ρ)(µ γσx t ) s t + (1 ρ)s t+1. (42) 1 (1 ρ)γ Σ 1 s t, for s t Ω t, s t+1 Ω t+1 (43) Denote x t as the otimal solution for stage x t, then there exists s t and s t+1 so that x t = 1 γ Σ 1 (µ + s t+1) 1 (1 ρ)γ Σ 1 s t, t (44)

28 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs If we let s t = 1 (1 ρ)t t+2 ρ s T, for t = 1, 2,, T 1 and s T = (1 ρ)(µ γσx T ) and rewrite x t as x t = 1 1 γ Σ 1 (µ + s t+1) (1 ρ)γ Σ 1 s t = 1 γ Σ 1 (µ + s r+1) 1 (1 ρ)γ Σ 1 s r = x r, t, r (45) where s t κ. By this means, we find the values of s t such that x t = x r for all t r. we conclude that x 1 = x 2 = = x T satisfies the otimality conditions. Part 2. Rewrite objective function (8) into the following by combining the fact that x 1 = x 2 = = x T, { T [(1 ρ) (x t t µ γ ) ] } 2 x t Σx t (1 ρ) t 1 κ x t x t 1 1 max {x t } T t=1 = max x 1 { T t=1 t=1 [(1 ρ) t (x 1 µ γ 2 x 1 Σx 1 )] κ x 1 x 0 1 } = max x 1 (1 ρ) (1 ρ) T +1 ρ Define subgradient of κ x 1 x 0 1 as s and the subdifferential Ω as Substitute (1 ρ) in Theorem 1 with to (x 1 µ γ 2 x 1 Σx 1 ) κ x 1 x 0 1. (46) s Ω = { u u (x 1 x 0 ) = κ x 1 x 0 1, u κ }. (47) (1 ρ) (1 ρ)t +1 ρ, it follows immediately that objective function (46) is equivalent s.t min x 1 (x 1 x 0 ) Σ (x 1 x 0 ), (48) κ ρ Σ(x 1 x ) (1 ρ)γ 1 (1 ρ). T (49) Part 3. Note that constraint (11) is equivalent to κ (1 ρ)γ ρ 1 (1 ρ) T e Σ(x 1 x ) κ (1 ρ)γ ρ 1 (1 ρ) T e, which is a arallelogram centered at x. To show that constraint (11) defines a no-trade region, note that when the starting ortfolio x 0 satisfies constraint (11), then x 1 = x 0 minimizes the objective function (10) and is feasible with resect to the constraint. On the other hand, when x 0 is not inside the region defined by (11), the otimal solution x 1 must be the oint on the boundary of the feasible region that minimizes the objective. We then conclude that constraint (11) defines a no-trade region. Proof of Proosition 3 Differentiating objective function (13) with resect to x t gives (1 ρ)(µ γσx) κλ 1/ Λ 1/ (x x 0 ) 1 sign(λ 1/ (x x 0 )) = 0, (50) where a 1 denotes the absolute value to the ower of 1 for each comonent: a 1 = ( a 1 1, a 2 1,, a N 1 ),

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 29 and sign(λ 1/ (x x 0 )) is the vector containing the sign of each comonent for Λ 1/ (x x 0 ). Given that Λ is symmetric, rearranging (50) (1 ρ)λ 1/ γσ(x x) = κ Λ 1/ (x x 0 ) 1 sign(λ 1/ (x x 0 )). (51) Kee in mind that x = x 0 can not be the otimal solution unless the initial osition x 0 satisfies x 0 = x. Otherwise, take q-norm on both sides of (51): Λ 1/ Σ(x x ) q = κ (1 ρ)γ Λ1/ (x x 0 ) 1 sign(λ 1/ (x x 0 )) q, (52) where q is the value such that 1 + 1 = 1. Note that q Λ1/ (x x 0 ) 1 sign(λ 1/ (x x 0 )) q = x x 0 1, we conclude that the otimal trading strategy satisfies It follows immediately Proosition 3. Λ 1/ Σ(x x ) q κ = Λ 1/ (x x 0 ) 1 (1 ρ)γ. (53) Proof of Theorem 2 Differentiating objective function (17) with resect to x t gives (1 ρ) t (µ γσx t ) (1 ρ) t 1 κλ 1/ Λ 1/ (x t x t 1 ) 1 sign(λ 1/ (x t x t 1 )) + (1 ρ) t κλ 1/ Λ 1/ (x t+1 x t ) 1 sign(λ 1/ (x t+1 x t )) = 0. (54) Secifically, given that Λ is symmetric, the otimality condition for stage T reduces to (1 ρ)λ 1/ (µ γσx T ) = κ Λ 1/ (x T x T 1 ) 1 sign(λ 1/ (x T x T 1 )), (55) Kee in mind that x T = x T 1 can not be the otimal solution unless x T 1 = x. Otherwise, take q-norm on both sides of (55) and rearrange terms, Iterating the otimal condition recursively gives Λ 1/ Σ(x T x ) q κ Λ 1/ (x T x T 1 ) 1 (1 ρ)γ. (56) κ x t x t 1 1 sign(x t x t 1 ) = T (1 ρ) s t+1 γλ 1/ Σ(x x s ). (57) Kee in mind that x t = x t 1 can not be the otimal solution unless that x t 1 = x. Otherwise, take q-norm on both sides, it follows straightforwardly that s=t T (1 s=t ρ)s t Λ 1/ Σ(x s x ) q κ = Λ 1/ (x t x t 1 ) 1 (1 ρ)γ. (58) We conclude that the otimal trading strategy for eriod t satisfies (58) whenever the initial osition is not x. Proof of Proosition 4 Part 1. Define g(x) = (1 ρ)λ 1/ Σ(x x ). For eriod T it holds that g(x T ) q κ γ Λ1/ (x T x T 1 ) 1.

30 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Moreover, for the following last eriod it holds g(x T 1 ) + (1 ρ)g(x T ) q κ γ Λ1/ (x T 1 x T 2 ) 1. Noting that A + B q A q B q, it follows immediately that κ γ Λ1/ (x T 1 x T 2 ) 1 g(x T 1 ) + (1 ρ)g(x T ) q g(x T 1 ) q (1 ρ) g(x T ) q g(x T 1 ) q (1 ρ) κ γ Λ1/ (x T x T 1 ) 1, where the last inequality holds because g(x T ) q κ γ Λ1/ (x T x T 1 ) 1. Rearranging terms, it imlies that g(x T 1 ) q κ γ Λ1/ (x T 1 x T 2 ) 1 + (1 ρ) κ γ Λ1/ (x T x T 1 ) 1, g(x T 1 ) q κ Λ 1/ (x T 1 x T 2 ) 1 γ + (1 ρ)κ γ Λ 1/ (x T x T 1 ) 1 Λ 1/ (x T 1 x T 2 ) 1 g(x which is a wider area than the rebalancing region defined for x T : T ) q κ. Λ 1/ (x T x T 1 ) 1 γ Similarly, g(x T 2 ) + (1 ρ)g(x T 1 ) + (1 ρ) 2 g(x T ) q κ γ Λ1/ (x T 2 x T 3 ) 1, it follows κ γ Λ1/ (x T 2 x T 3 ) 1 g(x T 2 ) + (1 ρ)g(x T 1 ) + (1 ρ) 2 g(x T ) q g(x T 2 ) + (1 ρ) g(x T 1 ) (1 ρ) 2 g(x T ) q, g(x T 2 ) + (1 ρ) g(x T 1 ) (1 ρ) 2 κ γ Λ1/ (x T x T 1 ) 1, where the last inequality holds because g(x T ) q κ γ Λ1/ (x T x T 1 ) 1. rearranging terms g(x T 2 ) + (1 ρ) g(x T 1 ) κ γ Λ1/ (x T 2 x T 3 ) 1 + (1 ρ) 2 κ γ Λ1/ (x T x T 1 ) 1, It imlies that, g(x T 2 ) + (1 ρ) g(x T 1 ) x T 2 x T 3 1 κ γ + (1 κ Λ 1/ (x T x T 1 ) 1 ρ)2, γ Λ 1/ (x T 2 x T 3 ) 1 which is a region wider than the rebalancing region defined by g(x T 1)+(1 ρ) g(x T ) κ for x Λ 1/ (x T 1 x T 2 ) 1 γ T 1. Recursively, we can deduce the rebalancing region corresonding to each eriod shrinks along t. Part 2. Note that the rebalancing region for eriod t relates with the trading strategies thereafter. Moreover, The condition x 1 = x 2 = = x T = x satisfies inequality (18), we then conclude that the rebalancing region for stage t contains Markowitz strategy x. Part 3. The otimality condition for eriod T satisfies (1 ρ)(µ γσx T ) κλ 1/ Λ 1/ (x T x T 1 ) 1 sign(λ 1/ (x T x T 1 )) = 0. (59) Let ω to be the vector such that lim T x T = ω. Taking limit on both sides of (59): (1 ρ)(µ γσω) κλ 1/ Λ 1/ (ω ω) 1 sign(λ 1/ (ω ω)) = 0, (60),

DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs 31 Noting that lim T x T = lim T x T 1 = ω, it follows (1 ρ)(µ γσω) = 0, which gives that ω = 1 γ Σ 1 µ = x. We conclude the investor will move to Markowitz strategy x in the limit case. Proof of Theorem 3 Part 1. For t = 1, 2,, T 1, differentiating objective function (21) with resect to x t gives (1 ρ)(µ γσx t ) κ(2λx t 2Λx t 1 ) κ(1 ρ)(2λx t 2Λx t+1 ) = 0, (61) rearranging terms [(1 ρ)γσ + 2κΛ + 2κ(1 ρ)λ] x t = (1 ρ)µ + 2κΛx t 1 + 2(1 ρ)κλx t+1. (62) The solution can be written exlicitly as following x t =(1 ρ)γ [(1 ρ)γσ + 2κΛ + 2κ(1 ρ)λ] 1 Σx + 2κ [(1 ρ)γσ + 2κΛ + 2κ(1 ρ)λ] 1 Λx t 1 + 2(1 ρ)κ [(1 ρ)γσ + 2κΛ + 2κ(1 ρ)λ] 1 Λx t+1. (63) Define A 1 = (1 ρ)γ [(1 ρ)γσ + 2κΛ + 2(1 ρ)κλ] 1 Σ, A 2 = 2κ [(1 ρ)γσ + 2κΛ + 2(1 ρ)κλ] 1 Λ, A 3 = 2(1 ρ)κ [(1 ρ)γσ + 2κΛ + 2(1 ρ)κλ] 1 Λ, where A 1 + A 2 + A 3 = I to conclude the result. For t = T 1, the otimality condition is (1 ρ)(µ γσx T ) κ(2λx T 2Λx T 1 ) = 0, (64) the exlicit solution is straightforwardly as following x T = (1 ρ)γ [(1 ρ)γσ + 2κΛ] 1 Σx + 2κ [(1 ρ)γσ + 2κΛ] 1 Λx T 1. (65) Define B 1 = (1 ρ)γ[(1 ρ)γσ + 2κΛ] 1 Σ, B 2 = 2κ[(1 ρ)γσ + 2κΛ] 1 Λ, where B 1 + B 2 = I, we conclude the results. Part 2. As T, taking limit on both sides of (65) we then conclude that lim T x T = x.

32 DeMiguel, Mei, and Nogales: Multieriod Portfolio Otimization with General Transaction Costs Proof of Corollary 4 Substituting Λ with Λ = I and Λ = Σ resectively we obtain the otimal trading strategy. To show that the trading trajectory of the case Λ = Σ follows a straight line, noting that x T is a linear combination of x T 1 and x, which indicates that x T, x T 1 and x are on a straight line. Contrarily, assuming that x T 2 is not on the same line, then x T 2 can not be exressed as linear combination of x T, x T 1 and x. According to equation (26) when t = T 1, it follows x T 1 = α 1 x + α 2 x T 2 + α 3 x T, (66) rearranging terms we get x T 2 = 1 α 2 x T 1 α 1 α 2 x α 3 α 2 x T. Note that 1 α 2 α 1 α 2 α 3 α 2 = 1, it is contradictory with the assumtion that x T 2 is not a linear combination of x T, x T 1 and x. By this means, we can show recursively that all the olicies corresonding to the model when Λ = Σ lay on the same straight line.

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