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

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... ) 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. Active investors and asset managers -such as hedge funds, mutual funds, and proprietary 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 predictors, 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