Dynamic Trading with Predictable Returns and Transaction Costs
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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Frequently Asked Questions (12)
Q2. What is the key to the out-performance of the dynamic strategy?
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.
Q3. What is the central relevance of the aim portfolio in the presence of transaction costs?
the central relevance of signal persistence in the presence of transaction costs is one of the distinguishing features of their analysis.
Q4. What is the optimal strategy for trading?
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.
Q5. What is the way to analyze the optimal portfolio?
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.
Q6. What is the strategy for a trader?
Their strategy mimics a trader who is continuously “floating” limit orders close to the midquote — a strategy that is used in practice.
Q7. What is the way to track the optimal portfolio?
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.
Q8. How do the authors calibrate the scalar for each commodity?
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.
Q9. How does Greenwood (2005) find evidence that market impact spills over to other securities?
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.
Q10. What is the way to explain the effect of persistent price-impact costs?
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.
Q11. What is the way to combine different return predictors?
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.
Q12. What is the way to predict the future of a security?
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).