Multiperiod Portfolio Optimization with Many Risky Asts and General Transaction Costs∗
Victor DeMiguel Xiaoling Mei Francisco J.Nogales
January14,2014
∗DeMiguel is from London Business School and can be contacted at avmiguel@london.edu, Mei and Nogales are from the Universidad Carlos III de Madrid and can be contacted at{xmei, FcoJavier.Nogales}@est-econ.uc3m.es.We thank comments from Nicolae Garleanu,Hong Liu, Kumar Muthuraman,Raman Uppal,and minar participants at Universidad Carlos III de Madrid, the International Conference on Continuous Optimization(Lisbon),and the2013INFORMS Annual Meeting.Mei and Nogales gratefully acknowledgefinancial support from the Spanish government through project MTM2010-16519.
Multiperiod Portfolio Optimization with Many Risky Asts and General Transaction Costs
Abstract
We analyze the properties of the optimal portfolio policy for a multiperiod mean-variance investor facing a large number of risky asts in the prence of general transaction costs such as proportional,
market impact,and quadratic transaction costs.For proportional transaction costs,we give a clod-form expression for a no-trade region,shaped as a multi-dimensional parallelogram,such that if the starting portfolio is outside the no-trade region, then the optimal policy is to trade to the boundary of the no-trade region at thefirst period and hold this portfolio thereafter.Moreover,we show how the optimal portfolio policy can be efficiently computed by solving a single quadratic program.For market impact costs, we show that the optimal portfolio policy at each period is to trade to the boundary of a state-dependent rebalancing region.Moreover,wefind that the rebalancing region shrinks along the investment horizon,and as a result the investor trades throughout the entire investment horizon.Finally,we u an empirical datat on15commodity futures to show show that the utility loss associated with ignoring transaction costs or investing myopically may be large.
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Keywords:Portfolio optimization,multiperiod utility,no-trade region,market impact. JEL Classification:G11.
1Introduction
Merton(1971)showed that an investor who wishes to maximize her utility of consumption should hold afixed proportion of her wealth on each of the risky asts,and consume at a rate proportional to her
wealth.1Merton’s minal work relies on the assumptions that the investor has constant relative risk aversion(CRRA)utility,faces an infinite horizon, can trade continuously and(crucially)costlessly.Implementing Merton’s policy,however, requires one to rebalance the portfolio weights continuously,and in practice this may result in high or even infinite transaction costs.Ever since Merton’s breakthrough,rearchers have tried to address this issue by characterizing the optimal portfolio policy in the prence of transaction costs.
Rearchers focudfirst on the ca with a single-risky ast.Magill and Constantinides (1976)consider afinite-horizon continuous-time investor subject to proportional transac-tion costs and for thefirst time conjecture that the optimal policy is characterized by a no-trade interval:if the portfolio weight on the risky-ast is inside this interval,then it is optimal not to trade,and if it is outside,then it is optimal to trade to the boundary of this interval.Constantinides(1979)demonstrates the optimality of the no-trade interval policy in afinite-horizon discrete-time tting.Constantinides(1986)considers the Mer-ton framework with a single risky ast and proportional transaction costs,and computes approximately-optimal no-trade interval policies by requiring the investor’s consumption rate to be afixed proportion of her wealth,a condition that is not satisfied in general. Davis and Norman(1990)consider the same framework,show that the optimal no-
trade interval policy exists,and propo a numerical method to compute it.Dumas and Luciano (1991)consider a continuous-time investor who maximizes utility of terminal wealth,and show how to calculate the boundaries of the no-trade interval for the limiting ca when the terminal period goes to infinity.
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The ca with multiple risky asts is less tractable,and the bulk of the existing literature relies on numerical results for the ca with only two risky asts.Akian,Menaldi,and Sulem (1996)consider a multiple risky-ast version of the framework in Davis and Norman(1990), 1Meton’s result holds for either an investor facing a constant investment opportunity t,or an investor with logarithmic utility;e also Mossin(1968),Samuelson(1969),and Merton(1969,1973).
金华景区and for the restrictive ca where the investor has power utility with relative risk aversion between zero and one2and risky-ast returns are uncorrelated,they show that there exists a unique optimal portfolio policy.They also compute numerically the no-trade region for the ca with two uncorrelated stocks.Leland(2000)considers the tracking portfolio problem subject to proportional transaction costs and capital gains tax,and propos a numerical approach to approximate the no-trade region.Muthuraman and Kumar(2006)consider an infinite-horizon continuous-time investor and propo an efficient numerical approach to compute the no-trade region.Their numerical results sho
w that the no-trade region for the ca with two risky asts is characterized by four corner points,but the four corner points are not joined by straight lines,although their numerical experiments show that a quadrilateral no-trade region does provide a very clo approximation.Lynch and Tan(2010)consider afinite-horizon discrete-time investor facing proportional andfixed transaction costs,and two risky asts with predictable returns.Using numerical dynamic programming,they show that for the ca without predictability the no-trade region is cloly approximated by a parallelogram,whereas for the ca with predictability the no-trade region is cloly approximated by a convex quadrilateral.3
Most of the aforementioned papers assume an investor with CRRA utility of consump-tion who faces borrowing constraints.The assumptions render the problem untractable analytically,and hence they generally rely on numerical analysis for the ca with two risky asts.A notable exception is the work Liu(2004)who obtains an analytically tractable framework by making veral restrictive assumptions.4Specifically,he considers an investor with constant absolute risk aversion(CARA)and access to unconstrained borrowing5,who can invest in multiple uncorrelated risky asts.For this framework,Liu shows analytically that there exists a box-shaped no-trade region.
车库英文2Janeˇc ek and Shreve(2004)show that relative risk aversion parameters between one and zero lead
to intolerably risky behavior.
3Brown and Smith(2011)also consider the ca with proportional transaction costs and return pre-dictability.Specifically,they propo veral heuristic trading strategies for afinite-horizon discrete-time investor facing proportional transaction costs and multiple asts with predictable returns,and u upper bounds bad on duality theory to evaluate the optimality of the propod heuristics.
4Another important exception is Muthuraman and Zha(2008)who u a simulation-bad numerical optimization to approximate the optimal portfolio policy of a continuous-time investor who maximizes her long-term expected growth rate for cas with up to ven risky asts.Also,in their early paper Magill and Constantinides(1976)conjecture the existence of a box-shaped no-trade region for the ca where the portfolio weights are small.
5He does impo constraints to preclude arbitrage portfolio policies.
Recently,Garleanu and Pedern(2013),herein G&P,consider a more tractable frame-work that allows them to provide clod-form expressions for the optimal portfolio policy in the prence of quadratic transaction costs.Their investor maximizes the prent value of the mean-variance utility of her wealth changes at multiple time periods,she has access to unconstrained borrowing,and she fac
es multiple risky asts with predictable price changes. Several features of this framework make it tractable.First,the focus on utility of wealth changes(rather than consumption)plus the access to unconstrained borrowing imply that there is no need to track the investor’s total wealth evolution,and instead it is sufficient to track wealth change at each period.Second,the focus on price changes(rather than returns)implies that there is no need to track the risky-ast price evolution,and instead it is sufficient to account for price changes.Finally,the aforementioned features,combined with the u of mean-variance utility and quadratic transaction costs places the problem in the category of linear quadratic control problems,which are tractable.肺气肿会传染吗
In this paper,we u the path-breaking formulation of G&P to study analytically the optimal portfolio policies for general transaction costs.Our portfolio lection framework is both more general and more specific than that considered by G&P.It is more general becau we consider a broader class of transaction costs that includes not only quadratic transaction costs,but also the less tractable proportional and market impact costs.It is more specific becau,consistent with most of the literature on proportional transaction costs,we consider the ca with constant investment opportunity t,whereas G&P’s work focus on the impact of predictability.
We make three contributions.Ourfirst contribution is to characterize analytically the optimal portfolio
policy for the ca with many risky asts and proportional transaction costs.Specifically,we provide a clod-form expression for a no-trade region,shaped as a multi-dimensional parallelogram,such that if the starting portfolio is inside the no-trade region,then it is optimal not to trade at any period.If,on the other hand,the starting portfolio is outside the no-trade region,then it is optimal to trade to the boundary of the no-trade region in thefirst period,and not to trade thereafter.Moreover,we show how the optimal portfolio policy can be computed by solving a quadratic program—a class of optimization problems that can be efficiently solved for cas with up to thousands of risky
asts.Finally,we u the clod-form expressions of the no-trade region to show how its size grows with the level of proportional transaction costs and the discount factor,and shrinks with the investment horizon and the risk-aversion parameter.
Our cond contribution is to study analytically the optimal portfolio policy in the pres-ence of market impact costs,which ari when the investor makes large trades that distort market prices.6Traditionally,rearchers have assumed that the market price impact is linear on the amount traded(e Kyle(1985)),and thus that market impact costs are quadratic.Under this assumption,Garleanu and Pedern(2013)derive clod-form ex-pressions for the optimal portfolio policy within their multiperiod tting.However,Torre and Ferrari(1997),Grinold and Kahn(2000),and
Almgren,Thum,Hauptmann,and Li (2005)show that the square root function is more appropriate for modeling market price impact,thus suggesting market impact costs grow at a rate slower than quadratic.Our contribution is to extend the analysis by G&P to a general ca where we are able to cap-ture the distortions on market price through a a power function with an exponent between one and two.For this general formulation,we show analytically that there exists a state-dependent rebalancing region for every time period,such that the optimal policy at each period is to trade to the boundary of the corresponding rebalancing region.Moreover,we find that the rebalancing regions shrink throughout the investment horizon,which means that,unlike with proportional transaction costs,it is optimal for the investor to trade at every period when she faces market impact costs.
Finally,our third contribution is to u an empirical datat with the prices of15 commodity futures to evaluate the utility loss associated with ignoring transaction costs and investing myopically,as well as identifying how the utility loss depend on relevant parameters.Wefind that the loss associated with either ignoring transaction costs or behaving myopically can be large.Moreover,the loss from ignoring transaction costs increa in the level of transaction costs,and decrea with the investment horizon,whereas the loss from behaving myopically increa with the investment horizon and are unimodal on the level of transaction costs.
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6This is particularly relevant for optimal execution,where institutional investors have to execute an investment decision within afixed time interval;e Bertsimas and Lo(1998)and Engle,Ferstenberg,and Rusll(2012)
