1 Introduction

Online portfolio selection problem aims to allocate the wealth among different assets at different time periods to maximize the long-term wealth. There are two models describing the problem: the Mean-Variance model [19] and Kelly’s Capital Growth model [11]. The first model uses a weighted sum of expected return (mean) and risk (variance of the return) as a trade-off between the two objectives, and it is suitable for single-period portfolio selection; the second model sees the problem as a sequential decision problem that aims to maximize the expected return at the end of multiple time periods. Kelly’s Capital Growth model has a nature of online decision making, which is widely adopted by the studies from AI and Machine Learning researchers.

In online portfolio selection problem, each asset is associated with a price in each period. The ratio of prices between current period and last period is called price relative, which reflects the return of wealth invested on the assets after one period. The agent allocates the wealth among different assets based on their price relatives at different periods. Most portfolio selection strategies follow a two-phase scheme: price relative prediction phase and portfolio optimization phase. The first phase aims to predict the price relative at next period based on historical data; while the second phase aims to compute the optimal portfolio given the prediction of price relative.

One common methodology for this problem is Mean Reversion, which assumes the portfolios that perform poorly at current period will perform well next (and vice versa). The methods include PAMR (Passive Aggressive Mean Reversion) [17], CWMR (Confidence Weighting Mean Reversion) [15], OLMAR (OnLine Moving Average Reversion) [14] and RMR (Robust Mean Reversion) [9], which adopt the mean reversion idea in different ways. They achieve superior empirical results in experiments compared with other state-of-art methods. This proves the effectiveness of mean reversion policy.

Although PAMR, CWMR and OLMAR achieve good performances, they still face some difficulties. All existing mean reversion strategies do not fully consider the noisy data and outliers (RMR is proposed to alleviate the problem), which often leads to estimation error (see [20]). Furthermore, the assumption of single-period prediction [15, 17] also leads to estimation error, which makes the performance poor. RMR (Robust Mean Reversion) [9] and OLMAR [14] uses multi-period prediction, but the algorithm sees each period equally, which ignores the temporal heterogeneity of historical price relatives and causes inaccuracy of predictions. We utilize meta learning to exploit the benefit of multi-period prediction and the periods are assigned with weights according to their performances. Moreover, this alleviates the impact of noisy data and outliers. The results show that our strategy outperforms RMR and OLMAR.

More specifically, in order to utilize multi-period historical data, we generate multiple experts for price relative prediction following typical MAR methods. Then we adopt the meta learning method for price prediction. Each expert is assigned a weight that is updated according to their performances and a weighted aggregation is used as the final prediction. Meanwhile, we choose the typical passive-aggressive learning method for portfolio optimization. This method captures the recent portfolio performances and the objective of enhancing the wealth return at each time period.

The contributions of our work are: first, to our knowledge, we are the first to exploit the Mean Reversion strategy with meta learning in online portfolio selection problems; second, we make a better use of multiple-period history, which is robust to the outliers and noises in historical data; third, we conduct extensive experiments on real-world datasets and achieve superior results compared with other state-of-the-art approaches.

The remainder of the paper is organized as follows: the next section gives a brief introduction to the related work; Sect. 3 formally introduces the online portfolio selection problem while Sect. 4 introduces some preliminary works about mean reversion theory and some related concepts. Section 5 proposes BMAR strategy that utilizes meta learning in online portfolio selection problem. Section 6 presents the results of experiments conducted on real-world datasets and a thorough comparison with the baselines. The conclusion and the future work are presented in Sect. 7.

2 Related Work

The study of online portfolio selection problem first concentrates on some benchmark algorithms, including Buy and Hold, Best Stock and Constant Rebalanced Portfolios. The Buy and Hold strategy means the agent invests wealth with an initial portfolio and holds it to the end without changing the portfolio. The Best Stock strategy means that one puts all the wealth on the stock whose performance is best in hindsight. Constant Rebalanced Portfolios is a strategy that rebalances the wealth to a fixed portfolio in all periods. The best CRP strategy which achieves highest accumulated wealth is called BCRP. BCRP is an optimal strategy if the market is i.i.d. [4]. Successive Constantly Rebalanced Portfolios (SCRP) [5] and Online Newton Step (ONS) [1] implicitly estimate next price relative via all historical price relatives with a uniform probability. However, both Best Stock and BCRP strategies have to be computed in hindsight.

There are two main categories of algorithms: follow-the-winner approach and follow-the-loser approach. The intuition behind first approach is to track the stock with best performance in history and raise the weights in the portfolio of these stocks. Most of the Follow-the-Winner approaches aim to imitate the BCRP strategy: including the universal portfolio selection (UP) [10], Exponential Gradient (EG) [8], follow-the-leader and follow-the-regularized-leader approaches. However, the prices of assets are unstable, even a good following of winner assets can not guarantee superior performances.

Follow-the-loser approach utilizes a typical assumption of mean reversion [18], which means that the good (poor)-performing assets will perform poor (good) in the following periods. The approaches in this category include Anti-Correlation (Anticor) [2], Passive-Aggressive Mean Reversion (PAMR) [17], Confidence-weighted Mean Reversion (CWMR) [15], Online Moving Average Reversion (OLMAR) [14] and Robust Mean Reversion (RMR) [9]. CRP [4, 5] implicitly envolves follow-the-loser approach since rebalancing the wealth means to transfer the wealth from winning stocks to losing stocks in some extent.

Another important category of the portfolio selection algorithms is pattern-matching, which estimates the portfolio price based on sampled similar historical patterns. Nonparametric kernel based moving window (BK) [7] measures the similarity by kernel method. Following the same framework, Nonparametric Nearest Neighbor (BNN) [12] locates the set of price relatives via nearest neighbor methods. [16] proposed Correlation-driven Nonparametric learning (CORN), which measures the similarity via correlation.

Since the mean-reversion technique is widely adopted in financial fields, it is useful in online learning algorithms as well. Passive Aggressive Mean Reversion (PAMR) [17] and Confidence Weighted Mean Reversion (CWMR) [15] estimate next price relative as the inverse of last price relative, which is in essence the mean reversion principle. Recently, [14] proposed On-Line Moving Average Reversion (OLMAR), which predicts the next price relative using moving averages and explores the multi-period mean reversion. Robust Mean Reversion is proposed to alleviate the impact of outliers and noises existing in the data. The empirical experiments indicate that OLMAR and RMR outperforms the other state-of-the-art algorithms. However, most of the Mean Reversion algorithms have important limits: first, the mean reversion strategies require to select a fixed time window for prediction, which can not be easily determined; second, the strategies treat each time period equally in prediction, which ignores the temporal heterogeneity; third, the strategies do not have strong robustness against noises and outliers. We utilize mean reversion strategies for its good depiction of reality and further use meta learning approach to tackle the limits. A detailed comparison between our strategy and existing Mean Reversion strategies is presented in Sect. 4.

3 Problem Setting

In this section, we formally introduce the Online Portfolio Selection problem. Assume that there exist m assets in market, and time is divided into T periods. Each asset i has a closing price \(p_{t,i}\) at period t and \({p_t}\) is denoted as the closing price vector (column vector for all vectors mentioned): \({p_t}=[p_{t,1},p_{t,2}, {.}{.}{.} ,p_{t,m}]\). \(x_{t,i}\) is the price relative that captures the ratio of closing prices between two consecutive periods: \(x_{t,i}=\frac{p_{t,i}}{p_{t-1,i}}\) and \({x_t}=[x_{t,1},x_{t,2}, {.}{.}{.} ,x_{t,m}]\) is the price relative vector.

In each period, the market reveals the closing prices of assets and the investor has to assign the capital with a portfolio vector: \({b_t}=[b_{t,1},b_{t,2}, {.}{.}{.} ,b_{t,m}]\) where \(b_{t,i}\) represents the proportion of wealth assigned to asset i at time t. We follow the typical assumption that no margin/short sale is allowed, therefore \(b_{t,i}\ge 0,\forall t,i\) and \(\sum _{i=1}^mb_{t,i}=1,\forall t\). An investment means to select a portfolio \({b_t}\) from the simplex: \(\{b_{t,i}\ge 0,\sum _{i=1}^mb_{t,i}=1\}\) at period t. Usually we assume the portfolio is uniformly distributed in the beginning: \({b_{0}}=[\frac{1}{m}, {.}{.}{.} ,\frac{1}{m}]\). The sequence of portfolios from \(t_1\) to \(t_2\) is denoted as \({b_{t_1}^{t_2}}\). Denote the wealth accumulated at t as \(W_t\), w.l.g. the initial wealth is assumed to be 1: \(W_0=1\). Therefore given the selected portfolio \({b_t}\) at period t, the wealth becomes \(W_t=W_{t-1}{b_t^Tx_t}=\sum _{i=1}^mb_{t,i}x_{t,i}W_{t-1}=\prod _{\tau =1}^t{b_{\tau }^Tx_{\tau }}\) (T is transpose here).

Given the notations and introduction above, the online portfolio selection problem refers to a sequential decision making problem with periods from \(t=1\) to \(t=n\). In each period, the investor has to decide the portfolio based on historical closing prices of assets and the market reveals the newest closing price of assets, which leads to the change of wealth. The investor needs to strategically design portfolios \(b_1^n\) so that the accumulated wealth \(W_n\) at time \(t=n\) is maximized.

We summarize the procedures from the introduction above and formulate the whole online portfolio selection process in Algorithm 1 as [13].

figure a

4 Preliminary

In this section, we briefly introduce the mean reversion principle and how former works exploit this principle.

4.1 Mean Reversion

In each period, the algorithm tries to estimate the price relatives of assets in price prediction phase and compute the portfolio with Passive-Aggressive or Confidence Weighted Learning given the predicted price relatives. The mean reversion principle is reflected in the first phase by assuming that the poor-performing assets will have good performances in next periods (and vice versa). Denote the estimated price relative as \({\tilde{x}}_t\) and the estimated closing price as \({\tilde{p}}_t\). PAMR and CWMR assumes that the assets with high/low price relatives will have low/high price relatives in next period: \({\tilde{x}}_{t+1}=\frac{1}{x_{t}},\forall t\), which means \(\frac{{\tilde{p}}_{t+1}}{p_{t}}=\frac{p_{t-1}}{p_{t}}\). Therefore the principle assumes that \({\tilde{p}}_{t+1}=p_{t-1}\). Although the two methods work well, they can not perform consistently on some datasets. There are two reasons that lead to this inefficiency: first, the fluctuating prices may contain noises that affect the precision of mean reversion principle; second, the single period price reversion effect may not exist widely as expected.

4.2 Online Moving Average Reversion

The OLMAR (online moving average reversion) principle is proposed to model the mean reversion principle with multiple-period historical data. Denote the time window of OLMAR as w, the closing price \(p_t\) at period t is assumed to be: \({\tilde{p}}_{t}=\frac{1}{w}\sum _{\tau =t-w}^{t-1}p_{\tau }\). The price is therefore considered as the average of prices in a time window and the price relative becomes: \({\tilde{x}}_{t}=\frac{p_t}{p_{t-1}}=\frac{1}{w}(1+\frac{1}{x_{t-1}}+ {.}{.}{.} +\frac{1}{\bigotimes _{\tau =t-w}^{t-1}x_{\tau }})\). This Moving Average Reversion strategy with time window is denoted as SMAR.

Usually, it is assumed that the price at current period is closer to the price at recent periods due to the continuity of price changes. Therefore the price can be estimated with MAR by adding a decay factor \(\alpha \): \({\tilde{p}}_t=\alpha p_{t-1}+(1-\alpha {\tilde{p}}_{t-1})\), which results with a price relative: \({\tilde{x}}_t=\alpha +(1-\alpha )\frac{{\tilde{x}}_{t-1}}{x_{t-1}}\). Therefore the decay factor frees the algorithm from choosing a time window and utilizes the prices from the whole history. This Moving Average Reversion strategy with time window is denoted as EMAR.

Given the price relative predictions, OLMAR method further utilizes the online passive aggressive learning policy for portfolio optimization, which is also adopted by PAMR. Notice that the choice of time window size and decay factor determines the performance of this method and can not be pre-defined in hindsight.

4.3 Robust Mean Reversion

RMR uses L1 estimator to estimate the closing prices of assets so that the resulting price has a better robustness compared to other methods: with a window size of historical periods w, RMR estimates the closing at t by minimizing this objective: \(\sum _{\tau =t-1}^{t-w}\Vert {\tilde{p}}_{t}-p_{\tau }\Vert _2\), and the price relative is estimated as \(\frac{{\tilde{p}}_t}{p_{t-1}}\). This estimation is named as L1 estimator and is relatively more robust to noises.

4.4 Temporal Heterogeneity

The price relatives of different periods are correlated in different extents. Usually, it is assumed that the price at current period is closer to the price at recent periods due to the continuity of price changes. Meanwhile, the changes of price relatives in some time windows are more similar than others (which is the foundation of CORN (CORrelation-driven Nonparametric learning) [16]). Therefore, when predicting the price relatives in the future, the algorithm should consider how to make use of the historical time windows differently.

Given these existing algorithms exploiting Mean Reversion principles, we make a comparison between our approach (Boosting Moving Average Reversion: BMAR) and these algorithms in Table 1. The multi-period column shows whether the approaches use multiple-period historical data for prediction; the Robustness shows whether the approaches are robust to noises and outliers; the Temporal Heterogenity column shows whether the approaches can utilize the data from different periods Heterogeneously. The table shows that our strategy (BMAR) preserves all the good properties, which shows the superiority.

Table 1. Comparison between our strategy (BMAR) with existing mean reversion approaches.
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5 Proposed Strategy: Boosting Moving Average Reversion

Like most methods, we solve the problem with two phases: price relative prediction and portfolio optimization. We first generate a set of experts and each expert is a predictor of the price relative following a mean reversion policy. In each period, each expert first makes its predictions on the price relatives in next period; then we compute the cumulated losses induced by each expert from their historical predictions and the true past price relatives. With the cumulated losses, we compute the weights assigned to each expert following the boosting methods introduced later and make a final prediction. Then we use Online Passive-Aggressive learning method to compute an optimized portfolio with the final prediction of price relatives in next period. When the true price relatives are revealed, we can update the cumulated losses of each predictor. The process of BMAR strategy for online portfolio selection is illustrated in Fig. 1.

Fig. 1.
figure 1

BMAR strategy for online portfolio selection

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5.1 Boosting Moving Average Reversion for Price Relative Prediction

We generate the experts of price relative prediction with different parameters from OLMAR. Denote the set of experts as E and the SMAR expert with time window w is denoted as \(E_w\), the EMAR expert with Decaying factor \(\alpha \) as \(E_{\alpha }\).

Uniform Sampling: We generate the experts by sampling the parameters uniformly from the range: \(w\sim U(w_{min},w_{max})\) and \(\alpha \sim U(0,1)\). We generate \(M=w_{max}-w_{min}+1\) experts from MAR (\(w=w_{min},w_{min}+1, {.}{.}{.} ,w_{max}\)) and N experts from EMAR (\(\alpha =\{0.1,0.2, {.}{.}{.} ,0.9\}\) when \(N=9\)). Based on the different ways of generating experts, we denote the strategy of generating experts with time window as BMAR-1 and the strategy of generating experts by sampling \(\alpha \) as BMAR-2. With the generated experts, we use weighted aggregation of their decisions to predict the price relatives at different periods. Each expert represents an approximation of the price relative in next period with a certain time window. By utilizing the predictions of these experts with a weighted scheme, we can induce temporal heterogeneity into our approach and the details will be introduced later.

As shown in Theorem 1 later, the regret of our strategy is closely related to the number of experts we generate. We will present the influence of expert numbers and sampling methods on the performances in the experiment.

We assume a weighted sum of the experts as the estimator of the price relative:

$$\begin{aligned} BMAR-1:{\tilde{x}}(t)=\frac{\sum _{i=1}^{i=N}\theta _{i,t-1}{\tilde{x}}(t,w_i)}{\sum _{i=1}^{i=N}\theta _{i,t-1}}; \; BMAR-2:{\tilde{x}}(t)=\frac{\sum _{j=1}^{j=M}\theta _{j,t-1}{\tilde{x}}(t,\alpha _j)}{\sum _{j=1}^{j=M}\theta _{j,t-1}} \end{aligned}$$
(1)

where \({\tilde{x}}(t,w)\) and \({\tilde{x}}(t,\alpha )\) are the predicted price relatives of expert \(E_w\) and \(E_{\alpha }\); \(\theta _{w,t-1}\) and \(\theta _{\alpha ,t-1}\) are the weights assigned to these experts given their performances until period \(t-1\).

Denote the loss of expert \(E_w\) by period t as l(w, t) and the loss of weighted expert (BMAR) by period t as l(t). The cumulated losses of expert \(E_w\) and weighted expert by period T are denoted as \(L(w,T)=\sum _{t=1}^{t=T}l(w,t)\) and \(L(T)=\sum _{t=1}^{t=T}l(t)\) respectively. The difference between the two losses is seen as the regret of weighted expert with respect to expert \(E_w\): \(R(w,T)=L(T)-L(w,T)\).

We introduce the weights assigned to the expert, i.e. exponential weights:

$$\begin{aligned} \theta _{i,t-1}=\frac{e^{-\eta R_{i,t-1}}}{\sum _{j=1}^Ne^{-\eta R_{j,t-1}}} \end{aligned}$$
(2)

where \(\eta \) is a nonnegative parameter. Notice that \(R(w,T)=L(T)-L(w,T)\), the exponential weights make the predictions simpler:

$$\begin{aligned} \theta _{i,t-1}=\frac{e^{-\eta L_{i,t-1}}}{\sum _{j=1}^Ne^{-\eta L_{j,t-1}}} \end{aligned}$$
(3)

It has been proved that this expert learning procedure guarantees a proper upper bound of regret in prediction, as shown in the theorems from [3]:

Theorem 1

Assume that the loss function is convex in its first argument and takes values from [0, 1], then the regret of exponentially weighted average predictor satisfies (N is the number of experts, n is the number of periods and \(\eta \) is the parameter in exponential weights):

$$\begin{aligned} {\hat{L}}_n-\min _{i=1, {.}{.}{.} ,N}L_{i,n}\le \frac{ln N}{\eta }+\frac{n\eta }{2} \end{aligned}$$
(4)

The details of the proofs can be found in [3] and we omit the details here. Given these theorems, we can design loss functions that satisfy the requirements:

$$\begin{aligned} l({\hat{p}},y)=\frac{1}{N\varepsilon }\times \Vert {\hat{p}}-y\Vert _2^2 \end{aligned}$$
(5)

where \(\varepsilon \) is the constant that rescales \(l({\hat{p}},y)\) into [0, 1]. And it is easy to verify that the function is convex, which satisfies the requirement of the theorems. Notice that \(\varepsilon \) actually works as coefficients of \(l({\hat{p}},y)\) with \(\eta \), we can simply tune the value of \(\eta \) to adjust the performances, therefore when using this loss function, we do not explicitly set the value of \(\varepsilon \).

Remarks on Robustness: Notice that we do not explicitly model robustness in our prediction, however the utilization of multiple experts involves robustness: if the outliers and noises causes degradation of the experts’ prediction accuracy, the weights assigned to these affected experts are lowered, which prevents the final predictions suffering from the noises and outliers.

5.2 Portfolio Optimization

Given the predicted price relatives shown in former section, we utilize the passive aggressive learning procedure to solve an optimal portfolio. The basic idea of passive aggressive learning is to keep the portfolio the same if the predefined requirement is satisfied, otherwise the portfolio is computed to satisfy the requirement with a minimal change. More specifically, we formulate the optimization problem as follows:

(6)

where \(\epsilon \) is the threshold for the return at each period. Usually \(\epsilon \) is a constant greater than 1 to ensure the return under predicted price relative is increasing. The optimum to this problem is the portfolio assigned for period t. Notice that if we keep the portfolio same with that in last period and the return under predicted price relatives still exceeds the required value, we will keep the portfolios unchanged; otherwise, we will try to minimize the change between current portfolio and that in last period as long as the return can exceed the requirement. Since this optimization problem is convex, we can derive the portfolio in a closed form. The solution without considering the nonnegativity constraint is presented in the following proposition:

$$\begin{aligned} b_{t+1}=b_t-\alpha _{t+1}({\hat{x}}_{t+1}-{\bar{x}}_{t+1}\cdot {\mathbf {1}}) \end{aligned}$$
(7)

where \({\bar{x}}_{t+1}=\frac{1}{d}({\mathbf {1}}\dot{\hat{x}}_{t+1})\) denotes the average predicted price relative and \(\alpha _{t+1}\) is the Lagrangian multiplier calculated as,

$$\begin{aligned} \alpha _{t+1}=\min \{0,\frac{{\hat{x}}_{t+1}b_t-\epsilon }{\Vert {\hat{x}}_{t+1}-{\bar{x}}_{t+1}\cdot {\mathbf {1}}\Vert ^2}\} \end{aligned}$$
(8)

In order to ensure that the portfolio is non-negative, we project the above portfolio into the simplex domain as [14].

5.3 Transaction Costs

In this section, we will introduce the transaction cost, which is an important factor in practical scenarios. In practice, each transaction of wealth from one asset to another is charged with transaction fees. The transaction cost is imposed by markets, and a portfolios behavior cannot change the properties of transaction costs, such as commission rates or tax rates. Usually we assume the transaction fee follows a proportional model, which means rebalancing a portfolio incurs transaction costs on every buy and sell operation, based upon a transaction cost rate of \(\gamma \in (0,1)\). Therefore the transaction cost for a rebalancing from \({\hat{b}}_{t-1}\) to \(b_t\) is computed as:

$$\begin{aligned} \frac{\gamma }{2}\times \sum _{i=1}^m|b_{t,i}-{\hat{b}}_{t-1,i}| \end{aligned}$$
(9)

Therefore the cumulated wealth after n periods becomes:

$$\begin{aligned} W_n^{\gamma }=W_0\prod _{t=1}^n[(b_t\cdot x_t)\times (1-\frac{\gamma }{2}\times \sum _{i=1}^m|b_{t,i}-{\hat{b}}_{t-1,i}|)] \end{aligned}$$
(10)

Notice that the main intuition of Passive-Aggressive portfolio optimization is to keep the portfolio unchanged unless the requirement can not be satisfied. This avoids unnecessary rebalancing of wealth among assets and saves the transaction costs induced.

6 Experiment

We conduct extensive experiments on several real-world datasets to evaluate the performances of our strategy and make comparisons with state-of-the-art approaches.

6.1 Experiment Setting

In our experiment, we use the real-world datasets that are frequently used in related works. There are four datasets that contain price relatives of assets from US and Global markets. The time frames of these datasets range from decades to years, which reflect the performances of both long-term and short-term portfolio selections. The details of the datasets are listed in Table 2. In the experiment, we use the metrics that are adopted in the literatures for evaluation: i.e. the total wealth achieved at the final period.

Table 2. Statistics of the real-world datasets for experiment.
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6.2 Comparison Approaches

We select the state-of-art algorithms (most of them have been introduced in the related works) for comparison, including those Benchmark algorithms (Market, Best-Stock and BCRP), follow-the-winner algorithms (UP, EG, ONS), pattern-matching algorithms (\(B^k\), \(B^{NN}\), CORN, Anticor) and all the variants of mean reversion algorithms (PAMR, CWMR, OLMAR, RMR). For all the algorithms above, we choose the parameters with best performances as reported in related works. Notice that we select some algorithms that use information in hindsight for comparison (which are strong baseline algorithms). For the default setting of BMAR, we set \(W=8\) and \(N=9\) for BMAR-1 and BMAR-2. The other parameters are chosen as: \(\eta =1,\varepsilon =5\) for all datasets.

Table 3. Cumulative Wealth on four datasets.
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Table 4. Significance Test of the Performances on experimental datasets. MER means Mean Excess Return; All the statistics are preferred to be higher; p-value is expected to be low.
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6.3 Performance Evaluation

We present the cumulative wealth of our strategy and the comparative approaches in Table 3. As shown in the table, our strategy achieves the best performance on all datasets and outperforms other comparative algorithms significantly on long-term portfolio selection problems, i.e. on NYSE(O) and NYSE(N). Notice that the parameters fit for each dataset can be different, we also list the best performances of algorithms for comparison. The results show that the (including best or conventional) performances achieved by BMAR are better than other comparison algorithms. We also conduct significance test (following [6]) on the performances and the results are listed in Table 4. The significance tests shown above indicate that the performance of our strategy is significantly better on all datasets, which is not the consequence of luck. Notice that our approach outperforms other baselines significantly, especially on datasets NYSE(O)and NYSE(N). The reason is that the NYSE datasets contain relatively long time periods, and the wealth accumulated has a “Matthew Effect” (the accumulated wealth will be increased with time passing by, the longer it goes, the more wealth will be accumulated).

6.4 Parameter Sensitivity

Notice that our strategy has several parameters: W (which is the maximum window size of experts generated from moving mean reversion) and \(\eta \) for strategy BMAR-1; \(\alpha \) and \(\eta \) for strategy BMAR-2. The threshold for portfolio optimization \(\varepsilon \) is also a key parameter. We conduct experiments on all datasets with different values of the parameters.

Impact of \({\eta }\varvec{.}\) Notice that we use \(P_2\) norm of the difference between prediction and true price relative as loss function, the value of \(\eta \) therefore has two effects: first, it scales the loss function into [0, 1]; second, it evaluates the weights assigned to each expert considering their performances. Therefore we conduct experiments with different values of \(\eta \) on the datasets to show their impact in Figs. 2 and 3. Notice that the two strategies are different according to their ways of estimating the price relatives at each period. The impact of \(\eta \) also varies with the two strategies. Generally, choosing \(\eta =1\) guarantees relative good performances on all the datasets, which is adopted in the experiments shown in Table.3.

Fig. 2.
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Parameter sensitivity of \(\eta \) with \(\varepsilon =5,W=8\) in BMAR-1

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Fig. 3.
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Parameter sensitivity of \(\eta \) with \(\varepsilon =5,N=9\) in BMAR-2

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Fig. 4.
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Parameter sensitivity of W with \(\varepsilon =5,\eta =1\) in BMAR-1

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Impact of W. BMAR-1 generates experts with different window sizes \(w\in [2,W]\), each expert estimates the asset price as the mean of prices in most recent w periods. We choose different values of W to generate experts, where each W means \(W-1\) experts with window sizes are generated. The results are shown in Fig. 4. Notice that the impacts of W are different on the datasets, this is due to the fact that the optimal window sizes for the experts to work on different datasets are also different: the optimal window size for MSCI can be relatively low compared with other datasets. We find that the strategy can achieve consistently good performances when \(W\in [8,10]\) and set it as a conventional value.

Impact of \(\varepsilon \) . The passive-aggressive portfolio optimization technique is applied in our scheme, which tends to keep the portfolios same unless they fail to reach the requirement of return from each period. Therefore, we conduct experiments with different values of \(\varepsilon \) to show the impact. The impact of \(\varepsilon \) on our strategies are similar. As shown in Fig. 5, both BMAR-1 and BMAR-2 achieve good performances on all datasets when \(\varepsilon \in [5,10]\). Similarly, we choose \(\varepsilon =5\) as a conventional setting.

6.5 Performance Under Transaction Costs

We also conduct experiments with different transaction cost ratio since it is an unavoidable issue in practice. We alter the transaction cost ratio from \(0\%\) to \(1\%\) and compute the cumulative wealth of different strategies. The results are presented in Fig. 6.

Fig. 5.
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Parameter sensitivity of \(\varepsilon \) with \(\eta =1,w=8\) in BMAR-1 and \(\eta =1,N=9\) in BMAR-2

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Fig. 6.
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Performance with different transaction cost ratios

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Judging from the results, the transaction costs has a significant impact on the wealth return. When the transaction cost ratio is greater than 0.005, the wealth achieved on most datasets is close to 0. Since our algorithms can still outperform the baselines, they have good scalability for transaction costs. Notice that the real transaction cost ratio is usually below 0.005, our algorithm can work well in practice.

7 Conclusion

In this paper, we consider the online portfolio selection problem from the perspective of mean reversion and meta learning. So far, mean reversion strategies have achieved best empirical results, however they face limits of unknown window size and ignores the temporal heterogeneity of different periods. Meanwhile they are easily affected by outliers and noises in the data. We utilize meta learning to tackle the limits and propose Boosting Moving Average Reversion (BMAR) strategies. The experiments on real-world datasets show that BMAR outperforms state-of-the-art strategies. We believe more accurate prediction of price relatives can further improve the performances and we will consider this as future works.