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Journal ArticleDOI

Dynamic Trading with Predictable Returns and Transaction Costs

01 Dec 2013-Journal of Finance (Wiley)-Vol. 68, Iss: 6, pp 2309-2340
TL;DR: In this paper, the authors derive a closed-form optimal dynamic portfolio policy when trading is costly and security returns are predictable by signals with different mean-reversion speeds, which is characterized by two principles: (1) aim in front of the target, and (2) trade partially toward the current aim.
Abstract: We derive a closed-form optimal dynamic portfolio policy when trading is costly and security returns are predictable by signals with different mean-reversion speeds. The optimal strategy is characterized by two principles: (1) aim in front of the target, and (2) trade partially toward the current aim. Specifically, the optimal updated portfolio is a linear combination of the existing portfolio and an �aim portfolio,� which is a weighted average of the current Markowitz portfolio (the moving target) and the expected Markowitz portfolios on all future dates (where the target is moving). Intuitively, predictors with slower mean-reversion (alpha decay) get more weight in the aim portfolio. We implement the optimal strategy for commodity futures and find superior net returns relative to more naive benchmarks.

Summary (3 min read)

Introduction

  • The authors derive a closed-form optimal dynamic portfolio policy when trading is costly and security returns are predictable by signals with different mean-reversion speeds.
  • An investor often uses different return predictors, e.g., value and momentum predictors, and these have different prediction strengths and mean-reversion speeds, or, said differently, different “alphas” and “alpha decays.”.
  • An investor facing transaction costs should trade more aggressively on persistent signals than on fast mean-reverting signals: the benefits from the former accrue over longer periods, and are therefore larger.
  • Section IV solves the model with persistent transaction costs.

II. Model and Solution

  • The interpretation of these assumptions is straightforward: the investor analyzes the securities and his analysis results in forecasts of excess returns.
  • The most direct interpretation is that the investor regresses the return on security s on the factors f that could be past returns over various horizons, valuation ratios, and other return-predicting variables, and thus estimates each variable’s ability to predict returns as given by βsk (collected in the matrix B).
  • While this transaction-cost specification is chosen partly for tractability, the empirical literature generally finds transaction costs to be convex (e.g., Engle, Ferstenberg, and Russell (2008), Lillo, Farmer, and Mantegna (2003)), with some researchers actually estimating quadratic trading costs (e.g., Breen, Hodrick, and Korajczyk (2002)).
  • Then the dealer’s risk is ∆x>t Σ∆xt and the trading cost is the dealer’s compensation for risk, depending on the dealer’s risk aversion reflected by λ.
  • The authors solve the model using dynamic programming.

III. Results: Aim in Front of the Target

  • The authors next explore the properties of the optimal portfolio policy, which turns out to be intuitive and relatively simple.
  • The optimal trading rate is naturally greater if transaction costs are smaller.
  • Panel A of Figure 2 shows how the optimal first trade is derived, Panel B shows the expected second trade, and Panel C shows the entire path of expected future trades.
  • Naturally, a more persistent factor has a larger effect on future Markowitz portfolios than a factor that quickly mean reverts.

IV. Persistent Transaction Costs

  • In some cases the impact of trading on prices may have a non-negligible persistent component.
  • The first of these is the temporary transaction cost as before.
  • The last term reflects that the traded shares ∆xt are assumed to be executed at the average price distortion, Dt + 1 2 C∆xt.
  • This is because, when the price impact is persistent, the trader incurs a transaction cost based on the entire cumulative trade, and therefore is more willing to incur it early in order to start collecting the benefits of a better portfolio.

V. Theoretical Applications

  • The authors next provide a few simple and useful examples of their model.
  • (22) Example 2: Relative-value trades based on security characteristics.
  • It is natural to assume that the agent uses certain characteristics of each security to predict its returns.
  • To recover the dynamic solution in a static setting, one must change not just γ and λ, but additionally the expected returns Et(rt+1) =.
  • Today’s first signal is tomorrow’s second signal Suppose that the investor is timing a single market using each of the several past daily returns to predict the next return, also known as Example 4.

A. Data

  • Table I provides summary statistics on each contract’s average price, the standard deviation of price changes, the contract multiplier (e.g., 100 ounces per contract in the case of gold), and daily trading volume.
  • Given that rolling does not change a trader’s net exposure, it is reasonable to abstract from the transaction costs associated with rolling.

B. Predicting Returns and Other Parameter Estimates

  • The authors use the characteristic-based model described in Example 2 in Section II, where each commodity characteristic is its own past return at various horizons.
  • The authors estimate the variance-covariance matrix Σ using daily price changes over the full sample, shrinking the correlations 50% towards zero.
  • Finally, to choose the transaction-cost matrix Λ, the authors make use of price-impact estimates from the literature.
  • Alternatively, this more conservative analysis can be interpreted as the trading strategy of a larger investor (i.e., the authors could have equivalently reduced the absolute risk aversion γ).

C. Dynamic Portfolio Selection with Trading Costs

  • The authors consider three different trading strategies: the optimal trading strategy given by Equation (27) (“optimal”), the optimal trading strategy in the absence of transaction costs (“Markowitz”), and a number of trading strategies based on a static (i.e., one-period) transaction-cost optimization as in Equation (29) (“static optimization”).
  • Table II reports the performance of each strategy as measured by, respectively, its Gross Sharpe Ratio and its Net Sharpe Ratio (i.e., its Sharpe ratio after accounting for transaction costs).
  • Panel A reports these numbers using their base-case transaction-cost estimate (discussed above), while Panel B uses their high transaction-cost estimate.
  • The dynamic strategy overcomes this problem by trading somewhat fast, but trading mainly according to the more persistent signals.
  • The authors see that the optimal portfolio is a much smoother version of the Markowitz strategy, thus reducing trading costs while at the same time capturing most of the excess return.

D. Response to New Information

  • It is instructive to trace the response to a shock to the return predictors, namely to εi,st in Equation (32).
  • Figure 5 shows the responses to shocks to each return-predicting factor, [Figure 5] namely the five-day factor, the one-year factor, and the five-year factor.
  • The optimal strategy trades much more slowly and never accumulates nearly as large a position.
  • Interestingly, since the optimal position also trades more slowly out of the position as the alpha decays, the lines cross as the optimal strategy eventually has a larger position than the Markowitz strategy.
  • The second panel shows the response to the one-year factor.

VII. Conclusion

  • This paper provides a highly tractable framework for studying optimal trading strategies in the presence of several return predictors, risk and correlation considerations, as well as transaction costs.
  • The authors derive an explicit closed-form solution for the optimal trading policy, which gives rise to several intuitive results.
  • Instead, it is optimal to take a smoother and more conservative portfolio that moves in the direction of the aim portfolio while limiting turnover.
  • Such dynamic trade-offs are at the heart of the decisions of “arbitrageurs” that help make markets efficient as per the efficient market hypothesis.
  • Net of trading costs their strategy performs significantly better, since it incurs far lower trading costs while still capturing much of the return predictability and diversification benefits.

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NBER WORKING PAPER SERIES
DYNAMIC TRADING WITH PREDICTABLE RETURNS AND TRANSACTION
COSTS
Nicolae B. Garleanu
Lasse H. Pedersen
Working Paper 15205
http://www.nber.org/papers/w15205
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
August 2009
We are grateful for helpful comments from Kerry Back, Darrell Due, Pierre Collin-Dufresne, Andrea
Frazzini, Esben Hedegaard, Hong Liu (discussant), Anthony Lynch, Ananth Madhavan (discussant),
Andrei Shleifer, and Humbert Suarez, as well as from seminar participants at Stanford Graduate School
of Business, University of California at Berkeley, Columbia University, NASDAQ OMX Economic
Advisory Board Seminar, University of Tokyo, New York University, the University of Copenhagen,
Rice University, University of Michigan Ross School, Yale University SOM, the Bank of Canada,
and the Journal of Investment Management Conference. Lasse Pedersen is affiliated with AQR Capital
Management, a global asset management firm that may apply some of the principles discussed in this
research in some of its investment products. The views expressed herein are those of the author(s)
and do not necessarily reflect the views of the National Bureau of Economic Research.
NBER working papers are circulated for discussion and comment purposes. They have not been peer-
reviewed or been subject to the review by the NBER Board of Directors that accompanies official
NBER publications.
© 2009 by Nicolae B. Garleanu and Lasse H. Pedersen. All rights reserved. Short sections of text,
not to exceed two paragraphs, may be quoted without explicit permission provided that full credit,
including © notice, is given to the source.

Dynamic Trading with Predictable Returns and Transaction Costs
Nicolae B. Garleanu and Lasse H. Pedersen
NBER Working Paper No. 15205
August 2009
JEL No. G11,G12
ABSTRACT
We derive a closed-form optimal dynamic portfolio policy when trading is costly and security returns
are predictable by signals with different mean-reversion speeds. The optimal strategy is characterized
by two principles: 1) aim in front of the target and 2) trade partially towards the current aim. Specifically,
the optimal updated portfolio is a linear combination of the existing portfolio and an "aim portfolio,"
which is a weighted average of the current Markowitz portfolio (the moving target) and the expected
Markowitz portfolios on all future dates (where the target is moving). Intuitively, predictors with slower
mean reversion (alpha decay) get more weight in the aim portfolio. We implement the optimal strategy
for commodity futures and find superior net returns relative to more naive benchmarks.
Nicolae B. Garleanu
Haas School of Business
F628
University of California, Berkeley
Berkeley, CA 94720
and NBER
garleanu@haas.berkeley.edu
Lasse H. Pedersen
Copenhagen Business School
Solbjerg Plads 3, A5
DK-2000 Frederiksberg
DENMARK
and NYU
and also NBER
lpederse@stern.nyu.edu

Active investors and asset managers such as hedge funds, mutual funds, and propri-
etary traders try to predict security returns and trade to profit from their predictions.
Such dynamic trading often entails significant turnover and transaction costs. Hence, any
active investor must constantly weigh the expected benefit of trading against its costs and
risks. An investor often uses different return predictors, e.g., value and momentum pre-
dictors, and these have different prediction strengths and mean-reversion speeds, or, said
differently, different “alphas” and “alpha decays.” The alpha decay is important because it
determines how long the investor can enjoy high expected returns and, therefore, affects the
trade-off between returns and transactions costs. For instance, while a momentum signal
may predict that the IBM stock return will be high over the next month, a value signal
might predict that Cisco will perform well over the next year.
This paper addresses how the optimal trading strategy depends on securities’ current
expected returns, the evolution of expected returns in the future, their risks and correlations,
and their transaction costs. We present a closed-form solution for the optimal dynamic
portfolio strategy, giving rise to two principles: 1) aim in front of the target and 2) trade
partially towards the current aim.
To see the intuition for these portfolio principles, note that the investor would like to
keep his portfolio close to the optimal portfolio in the absence of transaction costs, which
we call the “Markowitz portfolio.” The Markowitz portfolio is a moving target, since the
return-predicting factors change over time. Due to transaction costs, it is obviously not
optimal to trade all the way to the target all the time. Hence, transaction costs make it
optimal to slow down trading and, interestingly, to modify the aim, thus not to trade directly
towards the current Markowitz portfolio. Indeed, the optimal strategy is to trade towards
an “aim portfolio,” which is a weighted average of the current Markowitz portfolio (the
moving target) and the expected Markowitz portfolios on all future dates (where the target
is moving).
While new to finance, these portfolio principles have close analogues in other fields such as
the guidance of missiles towards moving targets, shooting, and sports. For example, related
2

dynamic programming principles are used to guide missiles to an enemy airplane in so-called
“lead homing” systems. Similarly, hunters are reminded to “lead the duck” when aiming
their weapon.
1
The most famous example from the sports world is perhaps the following
quote from the “great one”:
A great hockey player skates to where the puck is going to be, not where it is.
Wayne Gretzky
Another way to state our portfolio principle is that the best new portfolio is a combination
of 1) the current portfolio (to reduce turnover), 2) the Markowitz portfolio (to partly get
the best current risk-return trade-off), and 3) the expected optimal portfolio in the future
(a dynamic effect).
Figure 1 illustrates this natural trading rule. The solid line illustrates the expected path [Figure 1]
of the Markowitz portfolio, starting with large positions in both security 1 and security 2,
and gradually converging towards its long-term mean (e.g., the market portfolio). The aim
portfolio is a weighted-average of the current and future Markowitz portfolios so it lies in
the “convex hull” of the solid line or, equivalently, between the current Markowitz portfolio
and the expected aim portfolio next period. The optimal new position is achieved by trading
partially towards this aim portfolio.
In this example, the curvature of the solid line means that the Markowitz position in
security 1 decays more slowly as the predictor that currently “likes” security 1 is more
persistent. Therefore, the aim portfolio has a larger position in security 1, and, consequently,
the optimal trade buys more shares in security 1 than it would otherwise. We show that it
is in fact a more general principle that predictors with slower mean reversion (alpha decay)
get more weight in the aim portfolio. An investor facing transaction costs should trade more
aggressively on persistent signals than on fast mean-reverting signals: the benefits from the
former accrue over longer periods, and are therefore larger.
The key role played by each return predictor’s mean reversion is an important implication
of our model. It arises because transaction costs imply that the investor cannot easily change
his portfolio and, therefore, must consider his optimal portfolio both now and in the future.
3

In contrast, absent transaction costs, the investor can re-optimize at no cost and needs to
consider only the current investment opportunities without regard to alpha decay.
Our specification of transaction costs is sufficiently rich to allow for both purely transitory
and persistent costs. Persistent transaction costs means that trading leads to a market
impact and this effect on prices persists for a while. Indeed, since we focus on market-
impact costs, it may be more realistic to consider such persistent effects, especially over
short time periods. We show that, with persistent transaction costs, the optimal strategy
remains to trade partially towards an aim portfolio and to aim in front of the target, though
the precise trading strategy is different and more involved.
Finally, we illustrate our results empirically in the context of commodity futures markets.
We use returns over the past five days, 12 months, and five years to predict returns. The
five-day signal is quickly mean reverting (fast alpha decay), the 12-month signal mean reverts
more slowly, whereas the five-year signal is the most persistent. We calculate the optimal
dynamic trading strategy taking transaction costs into account and compare its performance
to the optimal portfolio ignoring transaction costs and to a class of strategies that perform
static (one-period) transaction-cost optimization. Our optimal portfolio performs the best
net of transaction costs among all the strategies that we consider. Its net Sharpe ratio is
about 20% better than that of the best strategy among all the static strategies. Our strategy’s
superior performance is achieved by trading at an optimal speed and by trading towards an
aim portfolio that is optimally tilted towards the more persistent return predictors.
We also study the impulse-response of the security positions following a shock to return
predictors. While the no-transaction-cost position immediately jumps up and mean reverts
with the speed of the alpha decay, the optimal position increases more slowly to minimize
trading costs and, depending on the alpha decay speed, may eventually become larger than
the no-transaction-cost position, as the optimal position is reduced more slowly.
The paper is organized as follows. Section I describes how our paper contributes to
the portfolio-selection literature that started with Markowitz (1952). We provide a closed-
form solution for a model with multiple correlated securities and multiple return predictors
4

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TL;DR: In this paper, the authors consider the execution of portfolio transactions with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact, and they explicitly construct the efficient frontier in the space of time-dependent liquidation strategies, which have minimum expected cost for a given level of uncertainty.
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Frequently Asked Questions (12)
Q1. What have the authors contributed in "Nber working paper series dynamic trading with predictable returns and transaction costs" ?

The authors are grateful for helpful comments from Kerry Back, Darrell Due, Pierre Collin-Dufresne, Andrea Frazzini, Esben Hedegaard, Hong Liu ( discussant ), Anthony Lynch, Ananth Madhavan ( discussant ), Andrei Shleifer, and Humbert Suarez, as well as from seminar participants at Stanford Graduate School of Business, University of California at Berkeley, Columbia University, NASDAQ OMX Economic Advisory Board Seminar, University of Tokyo, New York University, the University of Copenhagen, Rice University, University of Michigan Ross School, Yale University SOM, the Bank of Canada, and the Journal of Investment Management Conference. Lasse Pedersen is affiliated with AQR Capital Management, a global asset management firm that may apply some of the principles discussed in this research in some of its investment products. The views expressed herein are those of the author ( s ) and do not necessarily reflect the views of the National Bureau of Economic Research. 

The key to the out-performance is that the dynamic strategy gives less weight to the five-day signal because of its fast alpha decay. 

the central relevance of signal persistence in the presence of transaction costs is one of the distinguishing features of their analysis. 

The optimal strategy is to chase a moving target, adjusting the aim for alpha decay and trading patiently by always edging partially towards the aim. 

The authors see that the optimal portfolio is a much smoother version of the Markowitz strategy, thus reducing trading costs while at the same time capturing most of the excess return. 

Their strategy mimics a trader who is continuously “floating” limit orders close to the midquote — a strategy that is used in practice. 

The optimal portfolio tracks an “aim portfolio,” which is analogous to the optimal portfolio in the absence of trading costs in its trade-off between risk and return, but different since more persistent return predictors are weighted more heavily relative to return predictors with faster alpha decay. 

Since Greenwood (2005) provides the only estimate of these transaction-cost spill-overs in the literature using the assumption Λ = λΣ and since real-world transaction costs likely depend on variance as well as turnover, the authors stick to this specification and calibrate λ as the median across the estimates for each commodity. 

Greenwood (2005) finds evidence that market impact in one security spills over to other securities using the specification Λ = λΣ, where the authors recall that Σ is the variance-covariance matrix. 

if the frequency of trading is large relative to the the resiliency of prices, then the investor will be affected by persistent price-impact costs. 

Their framework constitutes a powerful tool to optimally combine various return predictors taking into account their evolution over time, decay rate, and correlation, and trading off their benefits against risks and transaction costs. 

For instance, one can imagine that each security is associated with a value characteristic (e.g., its own book-to-market) and a momentum characteristic (its own past return).