Adaptive online portfolio selection with transaction costs

Highlights

• The adaptive online moving average method is designed to predict returns of assets.

• The net profit maximization model with transaction cost is constructed.

• The adaptive online net profit maximization algorithm is proposed.

• Multiple numerical experiments are conducted on real benchmark data sets.

Abstract

As an application of machine learning techniques in financial fields, online portfolio selection has been attracting great attention from practitioners and researchers, which makes timely sequential decision making available when market information is constantly updated. For online portfolio selection, transaction costs incurred by changes of investment proportions on risky assets have a significant impact on the investment strategy and the return in long-term investment horizon. However, in many online portfolio selection studies, transaction costs are usually neglected in the decision making process. In this paper, we consider an adaptive online portfolio selection problem with transaction costs. We first propose an adaptive online moving average method (AOLMA) to predict the future returns of risky assets by incorporating an adaptive decaying factor into the moving average method, which improves the accuracy of return prediction. The net profit maximization model (NPM) is then constructed where transaction costs are considered in each decision making process. The adaptive online net profit maximization algorithm (AOLNPM) is designed to maximize the cumulative return by integrating AOLMA and NPM together. Numerical experiments show that AOLNPM dominates several state-of-the-art online portfolio selection algorithms in terms of various performance metrics, i.e., cumulative return, mean excess return, Sharpe ratio, Information ratio and Calmar ratio.

Introduction

Online portfolio selection is attracting increasing attention from both academical researchers and industrial practitioners with the rapid development of artificial intelligence (AI). It is different from the traditional portfolio selection theory, which was firstly proposed by Markowitz in the seminal work Portfolio Selection (Markowitz, 1952). In a traditional portfolio selection problem, it is usually assumed that the return of a risky asset is subject to a certain distribution function. Based on the distribution function, the expected value and variance of the return can be calculated to measure the expected return and risk, respectively. Then the investors allocate the capital in different assets to achieve excess investment return or avert the investment risk. See Aboussalah and Lee (2020), Balter, Mahayni, and Schweizer (2021), Brandtner, Kürsten, and Rischau (2020), Cui, Gao, Shi, and Zhu (2019), Chakrabarti (2021), Gatzert, Martin, Schmidt, Seith, and Vogl (2020), Guan and An (2019), Guo, Yu, Li, and Kar (2016), Guo, Ching, Li, Siu, and Zhang (2020), Guo and Ching (2021), Ha and Zhang (2020), Kar, Kar, Guo, Li, and Majumder (2019), Li, Qin, and Kar (2010), Li, Jiang, Guo, Ching, and Yu (2020), Li, Guo, and Yu (2015), Ling, Sun, and Wang (2020), Leal, Ponce, and Puerto (2020), Markowitz (1959), Staino and Russo (2020) and Zhang, Li, and Guo (2018) for details. In contrast, online portfolio selection does not require to estimate the distribution functions of the uncertain returns in advance. It concerns more on employing artificial intelligence techniques to predict the future returns of risky assets and making the optimal investment strategy. In reality, it is difficult to find a distribution function which can well describe the real historical return data of an asset. In addition, in modern finance, large amounts of financial data are generated in every single second and investors are required to make timely decisions in a short time such as in the case of high frequency trading. It is therefore inefficient to estimate all distribution functions of risky assets before making portfolio selection decisions. Online portfolio selection focuses on exploring the most efficient and practical computational intelligence techniques to deal with real online asset trading problems, such as the stock trading. It is a type of sequential decision making optimization problem where the investment strategy is determined at the beginning of each period. In a very short time, an investor has to make a quick decision, and new market information arrives at the end of each period. For example, the investor determines an investment strategy at the beginning of one period, and holds the strategy until the end of the period. Then the real asset price at this moment is available, which can be used to determine the optimal strategy of the next period. In the following, we first give a brief introduction of the current related work on online portfolio selection and then elaborate the main contributions of our work in this paper.

Current online portfolio studies can be classified into five types. The first one is so-called “Benchmark”. One widely applied Benchmark is Uniform Buy-and-Hold strategy, which is also called the Market strategy (Market) (Li, Hoi, 2014, Li, Hoi, Zhao, Gopalkrishnan, 2011b). In this strategy, the available capital is uniformly distributed into all the risky assets in each period. Another Benchmark is called the “Best stock” (Li & Hoi, 2014), where all the capital is invested into the best asset in the whole investment process. It is a hindsight strategy due to only one so-called the “Best” asset is selected in the whole investment horizon. Constant Rebalanced Portfolios strategy (CRP) is also a popular Benchmark (Cover, 1991, Kelly, 1956) where the allocation proportions of the risky assets are the same in all periods. There are two special CRPs: Uniform Constant Rebalanced Portfolios (UCRP) (Li & Hoi, 2015) and Best Constant Rebalanced Portfolios (BCRP) (Cover, 1991). UCRP distributes the capital to all the assets equally while BCRP is an optimal offline strategy derived by solving the maximum cumulative return problem.

The second type of online portfolio selection studies focuses on the “Follow the winner” strategy. These studies are based on the momentum principle which assumes that the risky assets performing well currently will continue achieving good performance in the next period. Cover (1991) first proposed the concept of Universal Portfolio strategy (UP), which firstly distributed the capital to several base experts and derived the corresponding returns, then the performance weighted strategy could be obtained. Helmbold, Schapire, Singer, and Warmuth (1998) proposed the Exponential Gradient method (EG) in which exponentiated gradient update was employed to calculate the investment proportions based on the past return data and strategies. Agarwal, Hazan, Kale, and Schapire (2006) employed the Online Newton Step method (ONS) to tackle online portfolio selection, where the gradient and Hessian matrix of the log function of cumulative return were computed. This method is easy to implement because it only requires to compute the gradient of the log function and the inverse of Hessian matrix once in each iteration, which can be completed in $O\left(n^2\right)$ time where $n$ is the number of risky assets. Gaivoronski and Stella (2000) tracked the BCRP method by proposing Successive Constant Rebalanced Portfolios (SCRP) and Weighted Successive Constant Rebalanced Portfolios (WSCRP). Later, Gaivoronski and Stella (2003) proposed the adaptive portfolio selection approach (APS) which was able to tackle multiple portfolio selection problems including log-optimal constant rebalanced portfolio, adaptive Markowitz portfolio and index tracking.

Contrary to the above “Follow the winner” strategy, there are some “Follow the loser” approaches built on the mean reversion principle, which claims that the risky assets performing well in the past may return to normal or perform badly in the next period. Therefore, it is encouraged to buy the current under-performing risky assets and sell the over-performing assets. Borodin, El-Yaniv, and Gogan (2004) proposed the Anti-correlation method (Anticor) based on the mean reversion principle, where the proportions were transferred from the assets performing well to assets performing badly, and the explicit amounts of transferred proportions were determined by the cross-correlation matrix of different risky assets. Li, Zhao, Hoi, and Gopalkrishnan (2012) first proposed the passive-aggressive mean reversion method (PAMR) based on a loss function following the mean reversion principle. On the one hand, current portfolio will be kept if its return is below a certain return threshold under the assumption that under-performing risky assets will perform better in the next period. On the other hand, allocation proportions of the current portfolio will be readjusted if its return is above the given threshold. Similar to PAMR, Li, Hoi, Zhao, and Gopalkrishnan (2011b) and Li, Hoi, Zhao, and Gopalkrishnan (2013) first proposed the Confidence Weighted Mean Reversion method (CWMR) by modeling the portfolio vector with Gaussian distribution and updating the distribution constantly following the mean reversion principle. The above PAMR and CWMR employed the single-period mean reversion assumption where the price of asset in the next period was estimated with the price of last period, which might achieve bad performance in some data sets. To overcome this, Li, Hoi, 2012, Li, Hoi, 2015 employed the moving average method to predict the price of next period based on multiple prices of previous periods and proposed the Online Moving Average Revision method (OLMAR). Huang, Zhou, Li, Hoi, and Zhou (2016) first proposed the Robust Median Reversion strategy (RMR) where the robust $L_1$-median estimator was adopted to exploit the revision phenomenon. The RMR runs in linear time which is easy to implement in real algorithmic trading.

The fourth type of online portfolio selection studies focuses on pattern matching based approaches, where both the “Follow the winner” and “Follow the loser” strategies are considered simultaneously. There are usually two steps in pattern matching based approaches. The first step is sample selection intended for selecting the historical price patterns which are similar to the latest price pattern. Here each pattern represents a historical price sequence of a given window size. The selected historical price patterns are used to estimate the return vector of the whole portfolio in the next period. The second step is to construct the portfolio optimization model based on the selected price patterns. Györfi, Lugosi, and Udina (2006) employed the nonparametric kernel-based sample selection method to search for similar price patterns by comparing the Euclidean distance of different patterns, and constructed a log-optimal portfolio based on the capital growth theory. Later, Györfi, Udina, and Walk (2008) employed the nonparametric nearest neighbor-based sample selection and proposed the nonparametric nearest neighbor log-optimal investment strategy. Li, Hoi, and Gopalkrishnan (2011a) employed the correlation-driven nonparametric sample selection method by using the correlation coefficient of different patterns, and proposed the Correlation-driven Nonparametric learning algorithm (CORN).

The fifth type of online portfolio selection studies is the meta-learning approach. In this approach, multiple base experts are defined where each expert is equipped with different strategies and outputs one portfolio. Then all the output portfolios are combined together into a final portfolio. Vovk and Watkins (1998) employed the Aggregating Algorithm (AA) to tackle online portfolio selection problem, and generalized the worst-case bound of UP algorithm. Das and Banerjee (2011) studied different meta algorithms such as Online Gradient Update (OGU) and Online Newton Update (ONU), and demonstrated the effectiveness of them in online portfolio selection. Hazan and Seshadhri (2009) first proposed the Follow the Leading History algorithm (FLH) where the set of base experts was updated constantly and each base expert made predictions of future prices starting from a different time point in history. More meta-learning algorithms can be found in Akcoglu, Drineas, Kao, 2002, Akcoglu, Drineas, Kao, 2004; Vovk (1990).

This paper is an extended work of the OLMAR in Li and Hoi (2015) and Li, Hoi, Zhao, Gopalkrishnan, 2011b, Li, Hoi, Zhao, Gopalkrishnan, 2013. Although the OLMAR achieves better performance over many online portfolio selection algorithms, there are still some limitations which should be overcome carefully. Firstly, all the risky assets share the same decaying factor in OLMAR, which is set as a constant in all the investment periods. However, in practice, the price movement of each asset is subject to its own distribution function. It might be better to employ different decaying factors for different assets. In addition, the decaying factor of the risky asset should be adjusted constantly with the change of financial market. Secondly, in the decision making process of OLMAR, transaction costs are not considered for convenience of computation such that the solution can be derived by using a linear update formula. However, in its numerical experiments, the transaction costs are counted in computing the cumulative return and risk-related ratios based on the previous solution, which might cause considerable transaction costs in real trading activities and reduce total investment profit. Therefore, we propose the adaptive online net profit maximization algorithm (AOLNPM) by considering the adjusting decaying factors of different assets and transaction costs simultaneously. By combining the adaptive online moving average method (AOLMA) and the net profit maximization model (NPM), our AOLNPM algorithm contributes to the literature in four-fold.

• A novelly designed decaying factor is introduced, by which we gain predicted returns with better relative errors.

• Transaction costs are considered by maximizing the net profit of the whole portfolio in each period.

• The change of variables is employed to transform the nonlinear NPM into an equivalent linear programming problem, which is easier to solve and implement.

• The AOLNPM algorithm outperforms traditional online portfolio selection algorithms in multiple numerical experiments with different benchmark data sets.

The remainder of the paper is structured as follows. Section 2 gives a brief introduction of the online portfolio selection problem. Section 3 introduces the adaptive online moving average method and adaptive online net profit maximization algorithm. Section 4 provides the numerical experiments to demonstrate the effectiveness and practicability of our proposed algorithm. Finally, Section 5 gives the conclusions and discusses future research issues.