Calendar rebalancing is the most rudimentary rebalancing approach. This strategy simply involves analyzing the investment holdings within the portfolio at predetermined time intervals and adjusting to the original allocation at a desired frequency. Monthly and quarterly assessments are typically preferred because weekly rebalancing would be overly expensive while a yearly approach would allow for too much intermediate portfolio drift. The ideal frequency of rebalancing must be determined based on time constraints, transaction costs, and allowable drift.
A major advantage of calendar rebalancing over formulaic rebalancing is that it is significantly less time-consuming for the investor since the latter method is a continuous process. A preferred yet slightly more intensive approach to implement involves a rebalancing schedule focused on the allowable percentage composition of an asset in a portfolio. Every asset class , or individual security, is given a target weight and a corresponding tolerance range.
When the weight of any one holding jumps outside of the allowable band, the entire portfolio is rebalanced to reflect the initial target composition.
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These two rebalancing techniques, the calendar, and corridor method, are known as constant-mix strategies because the weights of the holdings do not change. Determining the range of the corridors depends on the intrinsic characteristics of individual asset classes as different securities possess unique properties that influence the decision. Transaction costs , price volatility, and correlation with other portfolio holdings are the three most important variables in determining band sizes. Intuitively, higher transaction costs will require wider allowable ranges to minimize the impact of expensive trading costs.
High volatility, on the other hand, has the opposite impact on the optimal corridor bands; riskier securities should be confined to a narrow range in order to ensure that they are not over or underrepresented in the portfolio. Finally, securities or asset classes that are strongly correlated with other held investments can acceptably have broad ranges since their price movements parallel other assets within the portfolio. A third rebalancing approach, the constant-proportion portfolio insurance CPPI strategy, assumes that as investors' wealth increases, so does their risk tolerance.
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The basic premise of this technique stems from having a preference of maintaining a minimum safety reserve held in either cash or risk-free government bonds. When the value of the portfolio increases, more funds are invested in equities whereas a fall in portfolio worth results in a smaller position toward risky assets. Maintaining the safety reserve, whether it will be used to fund a college expense or be put as a down payment on a home, is the most important requirement for the investor. For CPPI strategies, the amount of money invested in stocks can be determined with the formula:.
The investment multiplier is 1. CPPI rebalancing must be used in tandem with rebalancing and portfolio optimization strategies as it fails to provide details on the frequency of rebalancing, and only indicates how much equity should be held within a portfolio rather than providing a holding breakdown of asset classes along with their ideal corridors. Another source of difficulty with the CPPI approach deals with the ambiguous nature of "M," which will vary among investors. Portfolio rebalancing provides protection and discipline for any investment management strategy at the retail and professional levels.
The ideal strategy will balance out the overall needs of rebalancing with the explicit costs associated with the strategy chosen. Portfolio Management. Algorithm 1 shows the process of our proposed method. Before we start our proposed method, we set a replay memory and batch size and select pairs using the cointegration test.
At each epoch, we initialized total profit to 1. In the training scheme, we set a state which has spreads within the formation window and select actions which are used as trading and stop-loss boundaries. Throughout the trading window, we executed a strategy similar to a traditional pairs-trading strategy using the action selected.
After executing the strategy, we obtain a reward based on the results of the portfolio.
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Finally, for the Q-learning process, we update the Q-networks by performing a gradient descent step. We check our experiment results based on profit, maximum drawdown, and the Sharpe ratio. Profit is commonly used as a performance measure for trading strategies. It is calculated as the sum of returns taking into consideration trading cost.
Since many trades can increase total profit, it is necessary to determine the total profit taking into consideration transaction costs depending on trading volume. In this study, we set a trading cost of 5 bp; equation 21 is almost the same as equation 19 , but it does not include absolute value, and is trading cost. Maximum drawdown represents the maximum cumulative loss from the highest to the lowest values of the portfolio during a given investment period where is the value of the portfolio and is the terminal time value.
The Sharpe ratio is an indicator of the degree of excess profits from investing in risky assets used in evaluating portfolios [ 33 ]. In equation 23 , is the expected sum of portfolio returns and is the risk-free rate; we set this value to 0 and is the standard deviation of portfolio returns. The Materials and Methods section should contain sufficient details so that all procedures can be repeated. It may be divided into headed subsections if several methods are described. The lengths of the window sizes such as the formation window and trading window are selected from the performance results with the training dataset.
From these results, we select an optimized window size and compare our proposed model with traditional pairs trading, which takes a constant set of actions with the test dataset. To find the optimum window size for the optimized pairs-trading system, we experimented with six cases. We performed the experiments based on six window sizes, and the results for each window size are calculated by averaging the top-5 results for a total of 11 pairs.
From Tables 4 and 5 , we can find that the best performance is obtained when the formation and training windows are 30 and 15, respectively, based on the profit generated by both the OLS and TLS methods.
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When we trained our networks, we set a positive reward for taking more closed positions and fewer stop-loss and exit positions. We can find the lowest ratio of portfolio closed positions based on the number of open positions, which in the formation and trading windows are for 30 and 15 days 0. Contrary to this result, the highest ratios of the number of closed positions in the formation and trading windows are for and 60 days 0. However, the highest profits reported in the formation and trading windows are for 30 and 15 days.
This can be explained when we check the ratio of the number of stop-loss portfolios. The formation and trading window sizes are 30 and 15 days and the ratio of portfolio stop-loss position is 0. This result indicates that it is important to reduce the stop-loss position while increasing the closed position.
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In addition, we can see that the trading signals made with the TLS method are better than those made with the OLS method in all six of the discrete window sizes. The reason for this is based on the difference between the hedge ratios of the two methods.
In OLS, when one side is the reference, the relative change of the other side is estimated. Since the assumption is that there is no error component on the reference side and there is an error only on the other side, the hedge ratio varies depending on the side used as the reference. However, in TLS, hedging ratios are the same regardless of which side is used as the reference. For this reason, the experimental results confirm that the TLS method is better able to determine when to execute the pairs-trading strategy. From these results, we take the optimum window size when we verify our proposed method in the test dataset.
However, we first need to ensure that the model we proposed is well-trained. It is important to check whether our reinforcement learning algorithm is trained well. Reference [ 21 ] suggested that a steadily increasing average of Q-values is evidence that the DQN is learning well. We find that the average Q-values steadily increased, indicating that our proposed model is properly trained. In addition, we provide a positive reward when the portfolio closes and a negative reward when the portfolio reaches the stop-loss threshold or exits.
Figure 4 b shows the ratio of the number of portfolio positions as training progressed. The ratio of closed to open portfolio positions increased and the ratio of portfolios reaching their stop-loss thresholds to open portfolio positions decreased. We also find that the ratio of portfolio exits to open portfolio positions slightly increased. It is possible that the rewards given for an open portfolio position compared to those given for a closed portfolio position are relatively small.
The DQN is therefore trained to prevent portfolios from reaching their stop-loss thresholds the more important objective over exiting them. This result can also serve as a basis for judging whether the proposed model is being trained properly. From this result, we can confirm that our proposed method is more profitable than the constant pairs-trading strategies.
From PTA0 to PTA5, the trading boundary and the stop-loss boundary grew larger; the numbers of open and closed portfolios and portfolios that reached their stop-loss thresholds are reduced. In other words, there is less opportunity for profit, but the probability of loss is also reduced. It is important not only to take a lot of closed positions, but also to take the best action to open and close the portfolio.
For example, if a portfolio is opened and closed by a boundary corresponding to action 0 within the same spread and if a portfolio is opened and closed by a boundary corresponding to action 1, the corresponding profit is different. Assuming that the mean reversion is certain to occur, if we take the maximum boundary condition to open a portfolio, we will obtain a larger profit than when we take a smaller boundary condition.
We can see that the PTDQN returns are higher than the strategy with the highest return among the traditional pairs trading strategies that take the constant action. Figures 5 — 8 show the changes in trading and stop-loss boundaries and the highest profit for constant action when applying the DQN method during the training period using OLS and TLS. Figure 5 consists of the spread, trading, and stop-loss boundaries.
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We find that trading and stop-loss boundaries have different values in PTDQN, showing that it has learned to find the optimal boundary according to each spread. Figures 7 and 8 exhibit the same features we see in Figures 5 and 6. The difference between these methods lies in the spreads: different results can be obtained depending on the spreads used.
Making better spreads can therefore improve performance.
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Reference [ 34 ] suggested that an average value over multiple trials should be presented to show the reproducibility of deep reinforcement learning because there may be different results from high variances across trials and random seeds. We therefore conducted five trials with different random seeds. The profit graph of DQN represents the average profit of these trials and the filled region between the maximum and minimum profit values.
We can see that PTDQN had a higher profit than the traditional pairs-trading strategies during the training period. This means that, even with the same spread, we can see how profit will change as the boundaries are changed. In other words, finding the optimal boundary for the spread is an important factor in optimizing the profitability of pairs trading.
Tables 8 and 9 show the average performance measures of each pair tested by applying the top-5 trained models. We can see that the constant action with the highest returns for each pair is different, and the TLS method is higher in all pairs than the OLS method based on profit, as shown above.
If we add the Sharpe ratio in addition to the total profit as an objective function, we can build a more optimized pairs-trading system.