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The short-term mean reversion of stock price and the change in trading volume

Journal of Derivatives and Quantitative Studies: 선물연구

ISSN : 1229-988X

Article publication date: 18 June 2021

Issue publication date: 23 September 2021

This study aims to analyze the effect of change in trading volume on the short-term mean reversion of the stock price in the Korean stock market. Through the variance ratio test, this paper finds that the market shows the mean reversion pattern after 2000, but not before. This study also confirms that the mean reversion property is significantly reduced if the effect of change in trading volume is excluded from the return of a stock with a significant contemporaneous correlation between return and change in trading volume in the post-2000 market. The results appear in both the Korea Composite Stock Price Index and Korea Securities Dealers Automated Quotation. This phenomenon stems from the significance of the return response to change in trading volume per se and not the sign of the response. Additionally, the findings imply that the trading volume has a term structure because of the mean reversion of the trading volume and the return also has a partial term structure because of the contemporaneous correlation between return and change in trading volume. This conclusion suggests that considering the short-term impact of change in trading volume enables a more efficient observation of the market and avoidance of asset misallocation.

• Trading volume
• Granger causality
• Variance ratio
• Korean stock market
• Contemporaneous correlation
• Mean-reversion

Jung, W. and Kang, M. (2021), "The short-term mean reversion of stock price and the change in trading volume", Journal of Derivatives and Quantitative Studies: 선물연구 , Vol. 29 No. 3, pp. 190-214. https://doi.org/10.1108/JDQS-01-2021-0003

Emerald Publishing Limited

Copyright © 2021, Woosung Jung and Mhin Kang.

Published in Journal of Derivatives and Quantitative Studies: 선물연구. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence maybe seen at http://creativecommons.org/licences/by/4.0/legalcode

1. Introduction

If the market is efficient, stock prices fully reflect all available information and it is impossible to make economic profits by trading based on the information set ( Fama, 1970 ; Malkiel, 1989 ). Therefore, stock prices follow the martingale process [ 1 ] and stock returns are not predictable. However, various studies have been conducted on the mean reversion of stock returns, which suggests that stock returns can be predictable. These findings are inconsistent with the efficient market hypothesis. Poterba and Summers (1988) observe that the divergence between market and fundamental values will eventually be eliminated by speculative forces, causing the stock price to mean-revert. This correction of “erroneous” market movements leads to the argument that stock returns must be negatively correlated at some frequency. More recently, Nagel (2012) shows that individual stocks have a negative serial correlation at daily and even monthly frequencies. Therefore, a short-term reversal strategy that buys losers and sells winners over the prior days generates profits. The mean-reverting phenomenon is not US-specific. Earlier literature shows that mean-reverting stock index returns exist in the Korean stock market ( Lee, 2002 ; Bae, 2006 ).

However, earlier research mainly focuses on the return-generating process and trading volume level. Recent studies investigate another possible channel for the mean-reversion of stock returns: the change in trading volume. Kang and Chae (2019b) find that in addition to the relationship between the trading volume and stock return, the change in trading volume also has a separate contemporaneous correlation with stock returns (hereinafter, CCRV) in the Korean stock market. They also argue that the illiquidity premium hypothesis offers an appropriate explanation for this phenomenon. Specifically, increased liquidity simultaneously causes larger trading volume and higher prices because of lowered illiquidity premiums, thereby leading to a significant and positive CCRV. Furthermore, Kang and Chae (2019a) confirm the presence of mean reversion of the trading volume in the Korean market. These findings imply that trading volume has a term structure and through the CCRV channel, this term structure affects stock price movements.

This study sheds light on the role of trading volume change in stock price mean reversion. If the trading volume mean-reverts and has a significant contemporaneous correlation with stock returns, the return is predictable, conditional on the current trading volume. In other words, the stock return process reflects the mean reversion of the trading volume and implies that the price term structure is related to the trading volume term structure. Therefore, if we extract the return component orthogonal to the volume change effect, the component becomes more comparable to the martingale process. We use the variance ratio (VR) test to empirically investigate how CCRV-orthogonalized returns become closely comparable to the martingale process than the original returns.

Why is it important to examine how trading volume change affects the mean reversion of stock returns? The study on it has important implications for investors’ risk perceptions and asset allocations. If we do not consider the impact of volume change on stock returns, the return volatility may be misestimated according to the term length. Consequently, investors may misallocate their wealth to assets, leading to market inefficiency.

Our sample includes all stocks in the Korea Composite Stock Price Index (KOSPI) and Korea Securities Dealers Automated Quotation (KOSDAQ) from September 1987 to December 2017. First, we verify whether the volume change affects the return contemporaneously in the Korean stock market. Consistent with Kang and Chae’s (2019b) findings, the distribution of CCRV in our sample shows that most stocks have positive and significant CCRV. In total, 74% of stocks have positive CCRV and 11% of stocks have negative CCRV at the 10% significance level. Only 15% of the stocks had insignificant CCRV. This result does not change in the sub-periods and the different exchanges, indicating that the mean-reversion of trading volume affects the stock price mean reversion through the CCRV channel. Based on this implication, we hypothesize that CCRV-orthogonalized returns are more comparable to random walks.

Next, we analyze our hypothesis using the VR test following Lo and MacKinlay (1988) and Poterba and Summers (1988) . If we remove the CCRV effects on stock returns, the stock returns are closer to the martingale process, leading to the VR of CCRV-orthogonalized returns being closer to 1 than the original stock returns. Our empirical results support this prediction. The differences in the average VRs between CCRV-orthogonalized returns and original stock returns are significantly positive and increase as the test horizon becomes longer. For example, the difference in the average VRs of 10-day to 5-day between CCRV-orthogonalized returns and original returns is 0.0017 ( t -values: 4.12), but the difference in the average VRs of 100-day to 5-day increases to 0.01 ( t -values: 21.91).

For robustness, we conduct a subsample analysis to examine whether the effect changes in the different subsamples. First, we divide the entire sample into two groups as of 2000. As earlier literature notes that the 1997 currency crisis had a severe impact on the Korean economy, stock returns after the crisis behave quite differently from before and the mean reversion phenomenon of stock returns may have changed. Our subsample results also show structural changes in the stock return behavior. In the first half of the sample, all VRs are greater than 1, indicating mean aversion. This result is consistent with Bae (2006) , which finds that the KOSPI shows a mean aversion phenomenon. However, in the second half of the sample, we find that stock returns show mean reversion. Bae (2006) suggests that this change contributes to an increase in foreign and institutional investors who pursue long-term investments. In both subsamples, the VRs of CCRV-orthogonalized returns are always greater than those of the original returns, consistent with the main result.

Second, stocks in the KOSPI and KOSDAQ markets have different characteristics such as investor and industry composition. Therefore, it is necessary to conduct VR tests separately in each market. However, the evidence shows no significant difference.

Finally, we also investigate whether both the significance and sign of the CCRV variable affect the results. We find that VR increases when we eliminate the volume effect of CCRV in stocks with both positive and negative CCRV. However, securities with insignificant CCRV do not show a significant change in the VR value, even after the volume effect of the CCRV is eliminated. The result suggests that regardless of its sign, the significance of CCRV plays a role in increasing the short-term variance in stock returns.

Overall, our empirical evidence supports the hypothesis that stock returns are affected by the volume change through the CCRV channel. Therefore, if we eliminate the volume change effect on stock returns, the stock returns become closely comparable to the martingale process. This evidence suggests that the mean reversion of trading volume affects the variance of short-term and long-term returns differently. Thus, we need to consider the volume effect on short-term stock returns to avoid asset misallocation.

The remainder of this paper is organized as follows. Section 2 summarizes previous studies. Section 3 describes the data and methodology. Section 4 presents our empirical results. Section 5 presents the results of robustness tests. Section 6 concludes.

2. Literature review

Our work builds on long-lasting literature on the mean reversion of stock returns. Tversky and Kahneman (1974) argue that a representativeness heuristic inclines people to overreact to new information. Many studies show that investor sentiment causes prices to swing away from the true value in the financial market ( De Bondt and Thaler , 1985, 1987 ; Shefrin and Statman, 1985 ; De Long et al. , 1990 ; Lehmann, 1990 ). If market values diverge from fundamental values, speculative forces may eliminate the difference, leading to a negative serial correlation in stock returns ( Poterba and Summers, 1988 ). Fama and French (1988) also describe how long-horizon returns lead to predictability because of slowly decaying price components. They note that the long-term mean reversion of stock returns is consistent with two alternate explanations: overreaction by irrational investors or time-varying equilibrium expected returns in an efficient market.

Some research has challenged the evidence of stock return mean reversion based on the test methodology. Richardson and Stock (1989) show that small-sample bias correction provides evidence for the mean reversion in long-horizon returns of the NYSE index and size decile portfolios. Using US data, McQueen (1992) argues that the general least square (GLS) randomization test does not reject the random walk of returns for horizons of 1 to 10 years. Jegadeesh (1991) also provides evidence against mean reversion by noting that the mean reversion phenomenon is concentrated only in January. A similar result is also observed in the market index of the London Stock Exchange. However, other studies support the mean-reverting phenomenon with more powerful panel methods or new data. Balvers et al. (2000) use a more robust test with annual panel data from 18 countries and find the mean reversion of index returns with a reversion speed of 18 to 20% per year. Gropp (2004) uses a panel method and discovers the mean reversion for the NYSE, AMEX and NASDAQ. More recently, Mukherji (2011) shows that mean reversion persists for small company stocks in one-, four- and five-year returns. Cecchetti, Lam and Mark (1990) described how the desire for consumption smoothing leads to negative autocorrelations without overreaction.

The mean reversion evidence varies across investment horizons and whether the focus is on the index or individual stocks. Sims (1980) argues that systematic short-term variations in fundamental values should be negligible in a competitive market. Therefore, price should follow a martingale process over brief time intervals, even if stock returns include a component that varies predictably over the long horizon. Early studies focus on the long-run mean reversion of stock index returns. Using the VR test, Poterba and Summers (1988) show that the S&P composite stock index has a long-term negative correlation but a short-term positive correlation because of the transitory component in stock returns. Fama and French (1988) also show that size and industry portfolio autocorrelations are weak for daily and weekly holding periods. Temporary components account for 40% of the predictable price variation of three- to five-year returns for small firms and 25% for large firms. However, stock index returns have a positive autocorrelation in the short horizon such as weekly or monthly, especially for small stocks. Lo and MacKinlay (1988) suggest that this result is mainly attributed to small firms and is not entirely explained by time-varying risk premiums and infrequent trading. Lo and MacKinlay (1990) describe further how despite negative autocorrelation in individual stock returns, weekly portfolio returns are strongly positively auto-correlated and result from important cross-autocorrelation. Nagel (2012) shows that individual stocks still have small negative autocorrelations daily, weekly and monthly. A positive high-frequency autocorrelation of the index declines and has even been negative in recent years ( Campbell, 2017 , p. 162).

There have been few studies of mean reversion in the Korean market. Lee (2002) finds that the KOSPI’s daily returns revert to the mean by using a fractionally integrated process. Bae (2006) also shows that the KOSPI and KOSDAQ’s monthly returns followed a weak mean-reverting process after the 1997 currency crisis. Our study focuses on short-term reversion in individual stock returns and suggests this phenomenon as a possible factor.

We also extend the literature on the relationship between abnormal trading volumes and stock returns. One pillar of research shows that heterogeneity in investment environments causes abnormal trading volumes. As this heterogeneity is resolved by trading between investors from different backgrounds, the abnormal trading volume decreases and the price becomes stable. Therefore, abnormal trading volume per se does not affect future returns. The differences in investment environments arise because of asymmetric information among investors ( Wang, 1994 ; Llorente et al. , 2002 ; Tetlock, 2010 ), differences in opinion among traders ( Harris and Raviv, 1993 ; Kandel and Pearson, 1995 ; Garfinkel and Sokobin, 2006 ) and irrational behavior by investors ( Campbell et al. , 1993 ; Odean, 1998 ; Scheinkman and Xiong, 2003 ; Baker and Stein, 2004 ; Grinblatt and Han, 2005 ; Statman Thorley and Vorkink, 2006 ; Choi et al. , 2010 ).

Another pillar suggests that abnormal trading volumes and market frictions affect stock returns. According to the investor recognition hypothesis ( Miller, 1977 ; Mayshar, 1983 ; Merton, 1987 ), stocks can be overvalued when a market has short-selling constraints. In particular, Odean and Barber (2009) argue that salient events affecting a stock such as unpredicted news, rapid price changes or unusual trading volumes, increase investors’ buying and selling demand. However, increased selling demand is not activated because of short-selling restrictions. Therefore, these events create net buying demand and higher returns on average. Gervais et al. (2001) suggest the visibility hypothesis that a sharp increase in trading volumes would attract investors’ attention to the stock, resulting in net buying pressure and, thus, in stock price increases. They find evidentiary support for this hypothesis that future returns with high trading volumes are significantly higher than those with low trading volumes. Kaniel et al. (2012) also show consistent results for stock markets across 41 countries.

Earlier research on stock returns and trading volumes in the Korean stock market has mainly focused on the causality test between these two variables. As for the stock index, various studies find that index returns are positively correlated with both trading volumes and changes in trading volumes ( Chung, 1987 ; Kho, 1997 ; Silvapulle and Choi, 1999 ; Kim and Kim, 1996 ; Chang, 1997 ). However, Lim (2016) finds that using the TGARCH model, the change in the KOSPI trading volume is affected by the return but not vice versa.

In a study on the relationship between trading volume and individual stock returns, Lee (2009) shows that using GARCH and regression methods, large and medium-sized stocks in the KOSPI market positively correlate and have a two-way Granger causal relationship with trading volumes. Small stocks, however, do not show a simultaneous relationship but only a Granger causal relationship. Eom (2013) uses the VAR model of the KOSDAQ stock to show that both past individual stock and market returns have a positive relationship with current trading volumes. Using the TGARCH and EGARCH models, Jheon and Park (2014) show a simultaneous correlation between trading volumes and returns in the KOSDAQ market. They also show that the degree of correlation between trading volumes and previous trading volumes depends significantly on the size of the stock. Finally, Lee et al. (2015) use the variance decomposition method to represent a more significant impact of the return on trading volume than the opposite.

In a study of the relationship between the change in trading volume and returns, Jinn et al. (1994) show that changes in the trading volume positively correlate with the return. Kook and Jung (2001) find that while an increase in past trading volume is likely to reverse the current return sign, the decrease continues the return trend. Kang and Chae (2019b) note that the CCRV of most stocks in the Korean market is significantly positive and the CCRV accounts for 4.22% of the total volatility of stocks. Furthermore, they argue that the liquidity premium hypothesis supports positive CCRV in the Korean market. Our study does not focus on the relationship between trading volume and returns but sheds light on how the trading volume change affects the short-term mean reversion of stock returns through the CCRV channel.

Among the studies on the relationship between abnormal trading volumes and stock returns, certain studies investigate how trading volumes positively affect the mean reversion of stock returns. Campbell et al. (1993) argue this relationship theoretically. According to their model, noisy trading causes price movements, but those movements are reverted when absorbed by liquidity providers. Further, the model assumes that such noisy trading is followed by high trading volume, while informed trading is not. The model implicitly assumes a downward-sloping demand curve and not a perfectly elastic demand curve. This assumption allows the price to be affected by trading volume. Empirically, Conrad et al. (1994) , based on a sample of NASDAQ stocks, find that high trading activity increases with reversal profitability. Avramov et al. (2006) show that the weekly or monthly negative autocorrelation of an individual stock, measured by a profit of contrarian strategy, is stronger for firms with high trading volume and high illiquidity. However, our study focuses on the impact of change, instead of the level of trading volume, on the mean-reversion of stock returns through the CCRV channel suggested by Kang and Chae (2019b) . Mean-reversion of trading volume can affect the returns when a stock has significant CCRV. Therefore, if we eliminate volume effects from returns, the short-term mean-reversion phenomenon in stock returns can be partially mitigated. This point is an additional aspect that differentiates this study from previous ones. Furthermore, to the best of our knowledge, this study is the first to investigate the relation between trading volume and mean reversion in the Korean stock market.

3. Data and methodology

3.1 methodology.

This study mainly argues that the mean reversion of stock prices is overestimated by responding to the trading volume change. Kang and Chae (2019b) show that daily volatility caused by trading volume change in the Korean stock market, on average, accounts for about 4% of the total daily volatility. However, as Kang and Chae (2019b) have stated, the volatility caused by the volume change will disappear in the long run because the trading volume mean-reverts and the price changes resulting from the change in trading volume will be restored to the original position. Therefore, Kang and Chae (2019b) also argue that it is desirable to consider the effect of the trading volume change on the return when estimating future prices based on current information. A stock return can be decomposed into the induced return by the change in trading volume and fundamental return. Figure 1 shows the concept for this argument. Figure 1(a) represents the fundamental return path of a stock, Figure 1(b) presents the return movements induced by trading volume change and Figure 1(c) is the original return path, which is the sum of Figures 1(a) and 1(b) .

If we assume that the fundamental return path follows the Wiener process, for t < T, the relation between the short-term return volatility σ f ( t ) 2 and the long-term return volatility σ f ( T ) 2 can be expressed as σ f ( t ) 2 / t = σ f ( T ) 2 / T . In contrast, the relation between long- and short-term volatility observed in a separate CCRV induced return path, which has negative autocorrelation caused by mean-reversion of trading volume and CCRV, can be expressed as σ CCRV ( t ) 2 / t > σ CCRV ( T ) 2 / T . Therefore, the long-term original return volatility σ ( T ) 2 / T = σ f ( T ) 2 / T + σ CCRV ( T ) 2 / T is less than σ ( t ) 2 / t = σ f ( t ) 2 / t + σ CCRV ( t ) 2 / t and σ ( T ) 2 / T < σ ( t ) 2 / t holds. As short-term volatility is more affected by CCRV-induced components having negative autocorrelation than long-term volatility, a short-term structural mean-reversion in price is observed. To confirm this, we conduct the following process.

First, we extract the effect of change in trading volume from the original return path to compare the mean reversion phenomenon in the original return with that in the CCRV-orthogonalized return. We construct CCRV-orthogonalized time-series returns for each stock by using the following regression specification: (1) r i , t = a i + β i Δ V i , t + ε i , t , where Δ V i , t is defined as the change in log volume turnover of stock i at time t. We use ε i,t s as CCRV-orthogonalized stock returns. Significant β i implies that the return of stock i is affected by the trading volume change.

Next, we compare the mean reversion of original returns with that of CCRV-orthogonalized returns using the VR test following Lo and MacKinlay (1988) and Poterba and Summers (1988) [ 2 ]. The test is based on the fact that the stock variance should be proportional to the return horizon if the returns follow a random walk. We examine the variance of returns at different horizons relative to the variance over the base interval. For daily returns, the VR statistics are, therefore: (2) V R l ( k ) = v a r ( r t k ) k / v a r ( r t l ) l   , where r t τ = ∑ i = 0 τ − 1 r t − i , t is the test interval and l is the base interval. When conducting the VR test for the CCRV-orthogonalized returns, we substitute r t with ε t .

If daily returns do not have serial autocorrelations through time, the VR l ( k ) statistics should converge to 1. However, if daily returns are autocorrelative, VR l ( k ) varies depending on the sign of autocorrelation and the length of the test interval. If a positive (negative) autocorrelation exists in the daily returns, VR l ( k ) is greater (less) than one and increases (decreases) as k increases.

We set the base interval, l , as five days [ 3 ] and the test interval, k , to 1, 10, 20, 30, 50 and 100 days to calculate the VR statistics for short-, mid- and long-term test intervals. We compute two VR statistics for each test interval using the original and CCRV-orthogonalized returns and compare these statistics. If both VR values are less than 1, but the VR value from CCRV-orthogonalized returns is greater than that from the original stock returns, the mean reversion phenomenon of stock returns is weakened after excluding the volume effects. We suggest that CCRV-orthogonalized returns are more comparable to random walks.

Our sample covers all common stocks listed on the KOSPI and KOSDAQ from September 1, 1978 to December 31, 2015. We exclude ETNs, ETFs, REITs, SPACs, KDRs, preferred stocks and common stocks with less than one year of stock returns. We obtain stock information from FnDataGuide.

4. Empirical analysis

4.1 distribution of contemporaneous correlation between return and change in trading volume.

We eventually show that the stock price’s mean reversion phenomenon in the CCRV-orthogonalized returns becomes weaker than that in the original returns. However, before conducting this discussion, we first need to verify whether the volume change affects the return contemporaneously. Although Kang and Chae (2019b) show the Korean stock market’s result, it is applied to each stock each year. As our analysis is performed without year-by-year classification, it is essential to reaffirm the volume effect on the price for each level. Therefore, we use equation (1) as a regression specification to estimate β i , which refers to the sensitivity of returns to volume changes. Figure 2 shows the histogram of the t -values of the estimated β i and Panel A in Table 1 shows the overall distribution of the estimated β i .

Figure 2 shows that stocks with significantly positive CCRV (located on the right side of t -values 1.645 on the x-axis) are the majority, while stocks with significantly negative CCRV (located on the left side of t -values −1.645 on the x-axis) are the minority. Panel A in Table 1 represents the supporting result that the positive CCRV stocks account for 74% of the total, while the negative CCRV stocks account only for 11%. These findings are consistent with those of Kang and Chae (2019b) . This panel also describes the results for the sub-period before and after 2000 and for the KOSPI and KOSDAQ markets after 2000. These sub-sample results are not different from those in the entire sample, except that the proportion of negative CCRV stocks decreases to approximately 7% before 2000. The fact that more than 80% of stocks in our sample have significant CCRV indicates that the returns of most stocks are affected by trading volume changes.

In addition, we analyze the difference in characteristics of the significant and insignificant CCRV stocks. First, we classify each stock’s size, Be/Me ratio and return volatility into 10 groups each year, assigning one to the smallest group and 10 to the largest group by sequentially increasing them. Additionally, stocks belonging to KOSPI are assigned 0 and stocks belonging to KOSDAQ are assigned 1 to distinguish the exchanges. Panel B in Table 1 shows the statistics for the group numbers calculated through this process. We then apply the daily Carhart model ( Carhart, 1997 ) to calculate the adjusted- R 2 and alpha of each stock. Panel B also reports the statistics for these results. Significant CCRV stocks have a larger size and less volatility than insignificant CCRV stocks. Further, adjusted- R 2 and alpha of Significant CCRV stocks are also significantly higher. However, the difference in size is only 1.24 and considering the dispersion of standard deviation within each group, it may be unreasonable to consider size as a major determinant of the significance of CCRV [ 4 ]. This evidence is also applicable for adjusted- R 2 and alpha.

4.2 Contemporaneous correlation between return and change in trading volume effect on mean reversion of stock return

4.2.1 determination of the base interval for variance ratio test..

As we have seen in Section 4.1, our sample confirms that most shares are affected by a change in trading volume. For stocks with significant CCRV, β i Δ V i , t in equation (1) operates as a determinant factor for the stock return to explain the discrepancy between the original return, r i , t and the CCRV-orthogonalized return, ε i , t . Therefore, in this section, we demonstrate that using the VR test, these stocks have significant mean reversion differences between CCRV-orthogonalized and original returns.

Before we proceed with the VR test, it is necessary to select the base interval in equation (2) . As described in Section 3.1, stock returns and variances are measured at frequencies of 1, 5, 10, 20, 30, 50 and 100 days. In measuring variance, we exclude data with fewer than 20 consecutive observations. The left side of Table 2 shows the statistics of the estimated variances and the right side displays the return variance scaled by the interval for comparison between the results of each interval.

Panel A shows the entire sample result. The average variances scaled by each interval peak at 0.00205 for the test interval of one day and gradually decreases to 0.00158 as the test interval increases to 100 days. Panel B represents the result of significant CCRV stocks. The average return variance/interval is 0.00207 for the one-day interval, similar to that in Panel A, but peaks at 0.00224 for the five-day interval. Then, it gradually declines to 0.00169 for the 100-day return. This difference is caused by approximately 400 stocks with insignificant CCRV, excluded from the sample in Panel B. This result is consistent with the prediction in Figure 1 that greater variance will be observed in stocks significantly affected by trading volume changes. Return variance/interval of stocks with significant CCRV is maximized at the five-day test interval because, as demonstrated by Kang and Chae (2019a) , the abnormal trading volume is halved in two to three days; thus, most of the variance enhanced by the trading volume change disappears in five days. Based on these results, we determined five days as the base interval, l , in equation (2) .

Additionally, Panel C provides the result for CCRV-orthogonalized return. Comparing the results in Panel B and Panel C, we confirm that CCRV-orthogonalization decreases the return variance. This result may be expected because the Panel B and Panel C samples are composed of significant CCRV stocks. However, it contains more information. As shown in the return variance/interval column, the proportion of decrement for each interval is different. The proportion decreases gradually as the interval increases. This evidence implies that the CCRV effect is relatively stronger on short-term prices.

These results have important implications for option pricing. One of the most important requirements for option pricing and hedging is the volatility estimation of the forward price. Options are often traded through the over-the-counter (OTC) market for hedging the existing portfolios, stock grants and product development for customers. For options that are not traded in such a market, implied volatility cannot be used for volatility estimation, and therefore, it relies heavily on historical volatility. As shown in Panel C of Table 2 , if the long-term forward price volatility is estimated based on the short-term historical return, the long-term volatility of significant CCRV stock can be over-estimated. In this case, because of the convexity of the options price, a small error in volatility estimation may cause a large error in the options price [ 5 ]. In addition, volatility misestimation also affects Greeks, causes errors in the hedging ratio, increases book management costs and may also affect risk management. As described above, misestimation of volatility has important implications in dealing with derivatives. Therefore, the forward prices’ volatility from historical returns must be estimated carefully.

4.2.2 Variance ratio test.

In this section, we compare the mean reversion in original stock returns with that in CCRV-orthogonalized returns. Table 3 presents the results of the VR test for each test interval. First, column (a) shows that the average of VR (1) in the whole sample is 1.0669, which is significantly greater than 1. This result suggests that the variance of the five-day return is less than five times that of the unit-day return because of a negative autocorrelation between one-day returns within five days.

On the other hand, VR (10) to VR (100) are significantly smaller than 1 and monotonically decrease as the test interval increases. This evidence indicates a negative autocorrelation between the unit-day returns, even for longer than 10 days. The evidence suggests that, overall, the mean reversion of stock returns exists across all test intervals.

Unlike the results for the whole sample, the original returns of significant CCRV stocks have different results for VR (1). The mean of VR (1) is 0.9548, which is significantly less than 1. This evidence shows that positive autocorrelations exist between one-day returns within a base interval of five days. On the other hand, for the test intervals of more than 10 days, all VR values are significantly less than 1 and monotonically decrease as the test interval increases, consistent with results for the whole sample. The original return of significant CCRV stocks shows momentum for a short period of fewer than five days and reversal for a more extended period.

CCRV-orthogonalized returns show similar results in most cases to the original returns of stocks with significant CCRV. All test values of VR (1) through VR (100) are significantly less than 1. In other words, even CCRV-orthogonalized returns present momentum for shorter periods of less than five days and reversal for longer periods.

Despite similar VR values between the original return and CCRV-orthogonalized returns, we should pay attention to the change in VR values. The last column (d) indicates that the VRs of the CCRV-orthogonalized return in all test intervals except VR (1) is significantly greater than those of the original return. If we consider the average half-lives of abnormal trading volumes in the Korean stock market, five days may be insufficient for the trading volume shock to disappear. As a result, the abnormal volume-induced returns do not disappear in five days. This hypothesis is consistent with the evidence that there is no difference in VR (1) between the original and CCRV-orthogonalized returns. Meanwhile, VR (10) to VR (100) in column (d) indicate that mean reversion of CCRV-orthogonalized returns is weaker for the test intervals of more than 10 days. These findings suggest that the market can be more efficient if we eliminate the short-term price impact of trading volume changes [ 6 ].

In addition to the above analysis, we conduct VR tests for each sub-period by dividing the entire sample period since 2000 into two. Bae (2006) demonstrates that the KOSPI shows momentum before the financial crisis but changes to mean reversion subsequently. The KOSDAQ index also presents mean reversion after the financial crisis. A sub-period analysis is necessary to investigate whether these results persist, even in short-term returns of individual stocks, not the index. Moreover, with a sub-period analysis, we examine how volatility in the original and CCRV-orthogonalized returns of stocks with signature CCRV varies over time.

Panel A of Table 4 shows the sub-period result from 1987 to 1999, which is quite different from Table 3 . All values of VR (10) to VR (100) are significantly greater than 1 for both the entire sample and significant CCRV stocks. This result implies that stock prices do not show a short-term mean-reversion pattern [ 7 ]. Meanwhile, Panel B of Table 4 shows the sub-period result after 2000, similar to Table 3 . Among them, in the results of the whole sample, the values of VR (10) to VR (100) are less than 1, which is consistent with Bae (2006) , who argues that stock prices revert to the mean after the 1997 currency crisis. Likewise, in both the original and CCRV-orthogonalized return of significant CCRV stocks, short-term mean-reversion patterns are observed at intervals greater than 10.

The last columns (d) of Panels A and B in Table 4 show that the difference in VR values is significantly positive for the test interval of greater than 10 days, consistent with the results in Table 3 . However, this should be interpreted carefully. The methodology of this study, which examines the short-term effects of trading volumes on returns, cannot be applied because we do not observe the short-term mean reversion phenomenon of stock returns before 2000. Therefore, the previous argument that the stock market becomes more efficient when we eliminate short-term impact in stock returns because of trading volume changes should apply only to the market after 2000.

Next, we divide the whole sample into KOSPI and KOSDAQ samples and proceed with the above analysis. The KOSPI and KOSDAQ markets may show different market efficiencies because of different investors and industry compositions ( Lee et al. , 2006 ; Park et al. , 2007 ). Such differences in market efficiency may cause differences in the mean reversion patterns between the two markets. We conduct the analysis only on the post-2000 sample where the mean reversion phenomenon of the stock price exists. Table 5 describes the results.

First, results between the KOSPI and KOSDAQ markets do not contrast significantly for the whole sample. In both markets, VR (1) is significantly greater than 1. VR (10) to VR (100) are significantly less than 1 and monotonically decrease as the test interval increases. This evidence shows a short-term mean reversion in the KOSPI and KOSDAQ markets after 2000.

However, we observe discrepant results from the original return in the significant CCRV stock sample. In both markets, VR (10) to VR (100) are smaller than 1 and decrease as the test interval increases. However, while VR (1) in the KOSPI market is not significantly different from 1 (mean: 1.0194, standard error: 0.0143), VR (1) in the KOSDAQ market is significantly smaller than 1 (mean = 0.9745, standard error = 0.0097). The results suggest that while short-term momentum is observed in the KOSDAQ market even for less than five days, this is not the case in the KOSPI market. These results are similar to the results of the CCRV-orthogonalized returns.

In addition, the results in the different columns show that the VR (10) to VR (100) of CCRV-orthogonalized returns are significantly greater than those of original returns in both the KOSPI and KOSDAQ markets. Therefore, our argument that the market can be observed more efficiently when the CCRV effect is eliminated does not depend on the market.

4.3 Sign of contemporaneous correlation between return and change in trading volume and mean reversion of stock return

In the previous sections, we examine our hypothesis that CCRV affects stock mean reversion and, eventually, market efficiency. However, as we have seen in Table 1 , CCRV is categorized into three groups based on its significance and sign: a significantly positive CCRV, a significantly negative CCRV and an insignificant CCRV. Thus, the overall response of the return to trading volume changes will be different for each group, as shown in Figure 3 .

The price moves in the same direction as the volume change in the significantly positive CCRV stocks and moves in the opposite direction in the significantly negative CCRV stocks. In the insignificant CCRV stocks, the price is uncorrelated with the volume change. As the mean reversion of trading volume is related to price movements only in stocks with significant CCRV, the VR of the CCRV-orthogonalized return is significantly larger than that of the original return in both the significantly positive and significantly negative CCRV stock samples but do not show significant differences in the insignificant CCRV stock sample. To demonstrate this argument empirically, we divide the post-2000 sample into three groups based on the sign of CCRV and investigate the mean reversion phenomenon in each subsample [ 8 ].

Table 6 presents the results for each subsample. First, in the positive and negative CCRV groups, the VR difference between the original and CCRV-orthogonalized returns is significantly positive in all test intervals of greater than 10 days. However, the difference in the insignificant CCRV group is small (0.0001) and the t -values are also not significant. This evidence is consistent with our expectation that the impact of CCRV on return mean reversion depends on the significance of CCRV, not on the sign.

On the other hand, in the positive CCRV group, VR (1) is significantly less than 1, at 0.9376, indicating short-term momentum in returns within five days, while in the negative CCRV group, VR (1) is 1.0714, indicating short-term mean reversion. However, this difference may result from each stock’s characteristics rather than the CCRV’s sign because CCRV is the mechanical response of returns to volume changes. According to Kang and Chae (2019b) , the sign of CCRV depends on stock characteristics such as size and liquidity. This evidence suggests that the difference in VR (1) between stocks with positive and negative CCRV is caused by latent stock characteristics rather than the sign of CCRV per se [ 9 ].

Table 7 shows the subsample results of applying the analysis of Table 6 to the KOSPI and KOSDAQ markets. In most cases, the results do not differ between markets and are similar to those in Table 6 . VR (10) to VR (100) in both the positive and negative CCRV groups are significantly larger than before when the CCRV effect is removed from stock returns, but there is no significant change in the insignificant CCRV group. Although the differences of VR (30) and VR (50) in the insignificant CCRV group in the KOSDAQ market are significant ( t = 1.69 for VR(30) and t = 2.65 for VR(50)), the difference is 0.0005, which is not economically meaningful. Based on the empirical results, we can conclude that the significance of CCRV per se , rather than its sign, induces the impact of trading volume change on return and, thus, we can observe the market more efficiently by considering the CCRV.

Moreover, column (a) of Panel A in Table 7 shows that the VR (1) for the KOSPI market is 0.9893 in the positive CCRV group and 1.2105 in the negative CCRV group. As shown in Table 5 , the VR (1) of the KOSPI is 1.0194, which is not significant and is larger than 1. Now, we can confirm that the result is because of differences in the VR (1) between the two different CCRV groups. These results are similar to VR (1) for the KOSDAQ market. In Table 5 , the VR (1) for the KOSDAQ market is 0.9745, but in Table 7 , it is 0.9476 in the positive CCRV group and 1.1336 in the negative CCRV group. In addition, VR (1) increases significantly in the positive CCRV group when the volume effect is removed from stock returns and decreases significantly to close to 1 in the negative CCRV group. This evidence also suggests that removing the volume effect from stock returns enables a more efficient observation of the market [ 10 ].

5. Robustness test

5.1 controlling the effects of the previous trading volume level.

Previous studies focus on the relationship between trading volume level and return and the effect of the current trading volume on the following return. In the context of those studies, Kang and Chae (2019b) argue that the CCRV is a phenomenon that acts in addition to the existing relationship between the trading volume level and returns.

Following Kang and Chae (2019b) , we assume that the effect of CCRV on stock price mean-reversion is an additional phenomenon overlapping the relation between trading volume level and return. Therefore, the results of this study must hold when the effect of trading volume level on stock return is removed. To confirm this, we must control the lead-lag effect observed in the time series of trading volumes and returns. In this regard, Lee (2002) , Lee (2009) and Lee et al. (2015) verify the Granger causality for the Korean stock market, but the results of daily data at the individual stock level are yet to be reported. Therefore, we check whether the Granger causality appears in the daily trading volume and return in the Korean market as a prerequisite for the main analysis. Panel A in Table 8 below shows the results of a Granger causality test for the Korean market from 2001 to 2015 by using the five lagged daily returns and trading volumes from t −1 to t −5.

As shown in Panel A of Table 8 , in the Korean stock market, 84.36% of the stocks’ returns Granger-cause the trading volumes, 48.87% of the stocks’ trading volumes Granger-cause the returns and 44.81% of the stocks shows a bidirectional Granger causality.

Further, a VR test is conducted to verify if VR appears closer to one in the CCRV-orthogonalized return when the influence of previous trading volumes and returns on the current return is removed. The test is conducted on the variance observed in the two-type residuals of the following two-stage regression equations. The following regression equation (3a) controls the five lagged returns and trading volumes applied to the Granger causality test. In addition, in the regression equation (3b) , the residual of the regression equation (3a) , ϵ 1, i , t , is the dependent variable and the change in trading volume is the explanatory variable: (3a) r i , t = α 1 , i + ∑ j = 1 m γ i , j r t − j + ∑ j = 1 m β i , j T V t − j + ϵ 1 , i , t , (3b) ϵ i , t = α 2 , i + β i ′ Δ T V t + ϵ 2 , i , t ,

Panel B in Table 8 shows the VR test results for the two residuals ϵ 1, i , t and ϵ 2, i , t calculated in the above process. As shown in column (a), in ϵ 1, i , t , where the previous return and trading volume level are controlled, VR(1) is 1.0415 and in VR(10)∼VR(100), it is significantly smaller than 1. This evidence shows the mean-reversion pattern that appears. The result of ϵ 2, i , t is similar, wherein VR(1) is 1.0365 and VR(10) ∼ VR(100) shows a mean-reversion pattern that is significantly smaller than 1, between 0.9630 and 0.9756. In addition, the difference between the two residuals in VR(10) to VR(100) is 0.0045 to 0.0119, which increases as the period increases and all t -values appear at statistically significant levels. The difference is similar to the difference observed in the Korean market from 2000 to 2015 in panel B of Table 4 , which is 0.0019 to 0.0097. Therefore, even when the Granger causality between the return and the trading volume level in the Korean market is controlled, the VR of the CCRV-orthogonalized return appears closer to one. Further, this result supports our main argument that the mean reversion of stock prices is overestimated by responding to the trading volume change.

5.2 Testing for efficiency by using the approximate entropy

We use the approximate entropy methodology (hereinafter, ApEn ) proposed by Pincus (1991) to supplement the results from the previous test.

ApEn is a measure that quantifies and represents the complexity, unpredictability and irregularity observed in time series and is similar to the verification contents of the market efficiency hypothesis of Fama (1970) . Pincus and Kalman (2004) , Kim et al. (2005) , Oh et al. (2007) , Bhaduri (2014) and Pele et al. (2017) tests the market efficiency by applying this methodology to the asset market. The method of calculating ApEn is as follows: A p E n ( S N , m , r ) = ∑ i = 1 N − m + 1 ln ⁡ [ C m i ( r ) ] ( N − m + 1 ) − ∑ i = 1 N − m ln ⁡ [ C m + 1 i ( r ) ] ( N − m ) C m i ( r ) = B i ( N − m + 1 ) ,     C m + 1 i ( r ) = B i ( N − m ) B i = θ ( r − d [ x ( u ) , x ( j ) ] (4) d [ x ( u ) , x ( j ) ] = max ⁡ k = 1 , 2 , … , m ( | S i + k − 1 − S j + k − 1 | )   , where S N is an instantaneous time-series, m is a pattern length based on embedding dimension, r is similarity based on the threshold, d is the maximum distance between x ( i ) elements and x(j) elements and B i is Heaviside function having 1 if d < r and 0 if d  ≥  r . We set m as two and r as 0.2 times the standard deviation of returns following the definition and assumption of Kim et al. (2005) .

As shown in the above equation (4) , the value of ApEn decreases as similar patterns appear in the time series and its value increases as predictions become difficult. Thus, higher values imply higher efficiency.

We compare the ApEn value in each stock’s original return with that in the CCRV- orthogonalized return. The results are reported in Table 9 . The average value of ApEn is 1.5802 in the original return and 1.5883 in the CCRV-orthogonalized return. The average difference between the two values measured in each stock appears to be 0.0081 and the corresponding t -values are 12.65. The significant increase in the ApEn value observed in the CCRV-orthogonalized return indicates the increased randomness in the time series of the CCRV-orthogonalized return. Therefore, the results of the ApEn analysis are consistent with the VR test. This evidence shows that the market can be observed more efficiently if the trading volume change is considered.

6. Conclusion

This study analyzed the short-term mean reversion of stock return in the Korean market from 1987 to 2015. Mainly focusing on the effect of the change in trading volume on stock returns, we compare the mean reversion patterns in the CCRV-orthogonalized return with that in the original return using the VR test.

The empirical analysis confirms the existence of short-term mean reversion of stock price in the Korean market after 2000, but not before 2000. In addition, for stocks whose price has been affected by changes in trading volume since 2000, the VR increases from the original and is closer to 1 if the trading volume effect is excluded. These results appear in the significant CCRV stocks in both KOSPI and KOSDAQ, regardless of the sign of CCRV.

Based on the above results, we confirm that the partial term structure of the stock return is related to the term structure of the trading volume and CCRV and the stock return’s term structure also partially affects the short-term mean reversion of price. This term structure of stock prices amplifies the volatility in short-term returns over volatility in long-term returns. However, such a trading volume effect is predictable and extractable by the consideration of the CCRV channel. Consequently, we conclude that the market is observed more efficiently if we remove the effect of trading volume change on the stock price. Moreover, by considering the CCRV effect, we can avoid the misestimation of the forward price volatility that may cause mispricing and hedging errors when dealing with OTC market’s options, which usually depend on the historical data when estimating volatility.

On the subject of mean reversion of a stock price, it would be a good research topic to compare the long-term and short-term mean reversions for each stock and examine the factors that affect each phenomenon. Moreover, revealing the determinants of the mean reversion of stock prices can be a promising future work. Given our finding that the mean reversion pattern appears differently because of the sign of CCRV, other factors may play an important role in the mean reversion of a stock price. If we can reveal the causal relationship between these hidden determinants and the mean reversion, we can have a deeper understanding of market efficiency and we can better understand the characteristics of the market. Finally, the discussion on CCRV, the main variable of this study, is still in progress and more research is needed. Therefore, we hope that this study enriches the existing discussion and serves as a useful ingredient for future research.

Return path decomposition

CCRV distribution (histogram)

Return response to the change in trading volume

CCRV distribution

Sample	> 1.645	| | < 1.645	< −1.645	Total
Whole sample	1,838	386	275	2,499
(1987∼2015)	(73.55%)	(15.45%)	(11.00%)
1987∼1999	765	183	82	1,030
	(74.27%)	(17.77%)	(7.96%)
2000∼2015	1,717	355	286	2,358
	(72.82%)	(15.06%)	(12.13%)
KOSPI(2000∼)	683	137	109	929
	(73.52%)	(14.75%)	(11.73%)
KOSDAQ(2000∼)	1,034	219	177	1,430
	(72.31%)	(15.31%)	(12.38%)

	Significant CCRV			Insignificant CCRV			Difference
Variables	N.OBS	Mean	SD	N.OBS	Mean	SD	Mean	-value
Size	29,544	5.642	2.838	3,302	4.405	2.838	1.24	23.69
Be/me	29,544	5.326	2.897	3,302	5.327	2.897	0.00	−0.02
Std(rt)	29,544	5.484	2.831	3,302	5.732	2.831	−0.25	−4.32
Exchange	2,289	0.516	0.500	409	0.521	0.500	0.00	−0.18
Adj. (Carhart4)	2,289	16.41%	8.99%	409	11.62%	8.99%	4.79%	7.73
Alpha (Carhart4)	2,289	0.00%	0.13%	409	−0.05%	0.13%	0.05%	3.93

Notes: -values of the coefficients on the change in trading volume. CCRV estimation is also adopted for the sub-period samples before 2000 and after 2000 and for KOSPI and KOSDAQ. Based on the level of -values as the classification standard, each cell reports the number of observations above the parentheses and the proportion within the parentheses. Panel B reports the characteristic distribution of CCRV according to the size, be/me ratio, return standard deviation, exchange, adjusted and risk-adjusted return. For size, be/me and return standard deviation, we classified stocks into 10 groups for each year. The lowest groups are assigned values of one and the highest groups are assigned values of 10 in an increasing sequence. For stock exchanges, a stock that belongs to KOSPI is assigned zero value and a stock that belongs to KOSDAQ is assigned values of one. The values reported for size, be/me, standard deviation and exchange are for those number of groups. For adjusted and risk-adjusted return, the values are estimated based on the daily four-factor model under the regression methodology with one-day lead-lag variables

Interval		Return variance		Return variance/interval
		Average	Std.err	Average	Std.err
1	2,473	0.00205	0.00005	0.00205	0.00005
5	2,452	0.00942	0.00014	0.00188	0.00003
10	2,434	0.01853	0.00029	0.00185	0.00003
20	2,388	0.03692	0.00059	0.00185	0.00003
30	2,350	0.05422	0.00093	0.00181	0.00003
50	2,223	0.08794	0.00155	0.00176	0.00003
100	1,830	0.15779	0.00267	0.00158	0.00003
1	2,102	0.00207	0.00006	0.00207	0.00006
5	2,090	0.01120	0.00030	0.00224	0.00006
10	2,085	0.02091	0.00050	0.00209	0.00005
20	2,045	0.03880	0.00082	0.00194	0.00004
30	1,999	0.05736	0.00122	0.00191	0.00004
50	1,841	0.09011	0.00167	0.00180	0.00003
100	1,429	0.16861	0.00336	0.00169	0.00003
1	2,102	0.00200	0.00005	0.00200	0.00005
5	2,090	0.01083	0.00027	0.00217	0.00005
10	2,085	0.02045	0.00047	0.00204	0.00005
20	2,045	0.03827	0.00081	0.00191	0.00004
30	1,999	0.05655	0.00121	0.00189	0.00004
50	1,841	0.08903	0.00167	0.00178	0.00003
100	1,429	0.16716	0.00336	0.00167	0.00003

Notes:

				Significant CCRV sample
	(a) Whole sample			(b) Original return			(c) CCRV- Orthogonalized return		(d) Difference: (c) – (b)
VR(interval)		Average	Std.err		Average	Std.err	Average	Std.err	Average	-value
VR(1)	2,452	1.0669	0.0146	2,090	0.9548	0.0063	0.9542	0.0049	−0.0006	−0.22
VR(10)	2,434	0.9614	0.0026	2,085	0.9626	0.0030	0.9642	0.0030	0.0017	4.12
VR(20)	2,388	0.9625	0.0041	2,045	0.9500	0.0045	0.9535	0.0045	0.0035	7.23
VR(30)	2,350	0.9354	0.0052	1,999	0.9376	0.0054	0.9425	0.0054	0.0049	11.14
VR(50)	2,223	0.9373	0.0061	1,841	0.9252	0.0061	0.9320	0.0062	0.0068	16.94
VR(100)	1,830	0.8788	0.0076	1,429	0.9116	0.0084	0.9216	0.0084	0.0100	21.91

Notes: ( ) = (5/ )* ( )/ ( ), where denotes returns over a -day measurement interval. Average indicates the average of variance ratios for each stock within the given group. Std.err stands for the standard error. is the number of observations. This table contains the whole sample result, the results for the original return of significant CCRV stocks and the results for CCRV-orthogonalized return of significant CCRV stocks. The CCRV-orthogonalized return signifies the regression residual of the daily return on change in daily turnover. This table also reports the difference between the variance ratios of CCRV-orthogonalized return and original return

This table reports the estimated approximate entropy for original return and CCRV-orthogonalized return in the Korean stock market from 2000 to 2015. Values in parentheses are standard errors

More precisely, the stock price is not a martingale in the dividend discount model because

                              E t [ P t + 1 ] = ( 1 + R ) P t − E t [ D t + 1 ] .

However, we can obtain a martingale process if we substitute P t with V t , which is the dividend-adjusted total value:

                          V t = N t P t ( 1 + R ) t   a n d   N t + 1 = N t ( 1 + D t + 1 P t + 1 )

For the testing of mean reversion of the stock price, the methodologies of Fama and French (1988) , Jegadeesh (1991) and Poterba and Summers (1988) are usually applied. We choose the last one because building the pseudo return path is only possible by this methodology.

According to Kang and Chae (2019a) , the half-life, which is the time necessary for half of the deviated amount of trading volume to decay, is about two to three days. Thus, we suppose that five days is enough time for trading volume shock to almost disappear.

Refer to the difference in CCRV distribution according to size and liquidity level in Table 10 of Kang and Chae (2019b) .

This tendency is more extreme in out-of-the-money (OTM). For example, when the volatility is 30% for a one-year 120% OTM call option, a 1% change in volatility has a 6% effect on the option price; in a 150% OTM call option, it has an 18% effect or higher.

If the market is efficient, the VR value will be 1.

For the pre-2000 data, even when we adjust the base interval to 10 or 20 days instead of 5 days, we do not find the short-term mean reversion patterns.

As for CCRV, more research is still needed on its cause and we expect this analysis to be useful for discussion.

Alternately, it is possible that the difference in mean reversion in the one-to-five-day interval will be one of the determinants of the sign of the CCRV.

This result is also observed in Table 6 . In Tables 3 , 4 and 5 , without the CCRV sign classification, the result is not observed.

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Acknowledgements

We are grateful to the anonymous referees for their valuable comments.

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PAMR: Passive aggressive mean reversion strategy for portfolio selection

• Published: 21 February 2012
• Volume 87 , pages 221–258, ( 2012 )

Cite this article

• Peilin Zhao 1 ,
• Steven C. H. Hoi 1 &
• Vivekanand Gopalkrishnan 2  

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This article proposes a novel online portfolio selection strategy named “Passive Aggressive Mean Reversion” (PAMR). Unlike traditional trend following approaches, the proposed approach relies upon the mean reversion relation of financial markets. Equipped with online passive aggressive learning technique from machine learning, the proposed portfolio selection strategy can effectively exploit the mean reversion property of markets. By analyzing PAMR’s update scheme, we find that it nicely trades off between portfolio return and volatility risk and reflects the mean reversion trading principle. We also present several variants of PAMR algorithm, including a mixture algorithm which mixes PAMR and other strategies. We conduct extensive numerical experiments to evaluate the empirical performance of the proposed algorithms on various real datasets. The encouraging results show that in most cases the proposed PAMR strategy outperforms all benchmarks and almost all state-of-the-art portfolio selection strategies under various performance metrics. In addition to its superior performance, the proposed PAMR runs extremely fast and thus is very suitable for real-life online trading applications. The experimental testbed including source codes and data sets is available at  http://www.cais.ntu.edu.sg/~chhoi/PAMR/ .

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1 Introduction

Portfolio Selection (PS) is a practical financial engineering problem that requires determining a strategy of investing wealth among a set of assets in order to achieve certain objectives, such as maximizing cumulative wealth or risk-adjusted return, in the long run. In this article, we investigate sequential portfolio selection (also termed online portfolio selection ) strategies, which sequentially determine portfolios based on publicly available information.

Traditionally in finance, portfolios are often selected according to mean-variance theory (Markowitz 1952 , 1959 ) or its variants, to trade off between return and risk. In recent years, this problem has also been actively studied from a learning to select portfolio perspective, with roots in the fields of machine learning, data mining, information theory and statistics. Rather than trading with a single stock using computational intelligence techniques, learning to select portfolio approach focuses on a portfolio, which consists of multiple assets/stocks. Several approaches for online portfolio selection, often characterized by machine learning formulations and effective optimization solutions, have been proposed in literature (Kelly 1956 ; Breiman 1961 ; Cover 1991 ; Ordentlich and Cover 1996 ; Helmbold et al. 1996 ; Borodin and El-Yaniv 1998 ; Borodin et al. 2000 , 2004 ; Stoltz and Lugosi 2005 ; Hazan 2006 ; Györfi et al. 2006 ; Blum and Mansour 2007 ; Levina and Shafer 2008 ; Györfi et al. 2008 ). Despite being studied extensively, most approaches are limited in some aspects or the other.

Our goal of this work is to investigate a new online portfolio selection strategy that employs online learning techniques to exploit the financial markets. Some existing strategies adopt the trend following approach, that is, they assume that price relative will follow its historical trading days. However, this philosophy fails when price relatives do not go in any particular direction, but rather actively move within a range. So in this study, we exploit another well-known principle in finance, viz., mean reversion (Jegadeesh 1990 ), through an online machine learning framework. To this end, we propose a novel portfolio selection strategy named “Passive Aggressive Mean Reversion” (PAMR), which exploits the mean reversion property of financial markets by online passive aggressive learning (Crammer et al. 2006 ). PAMR’s key idea is to formulate a new loss function that can effectively exploit the mean reversion property, and then adopt passive aggressive online learning to search for optimal portfolio among the asset pool to maximize the cumulative return.

Under different scenarios, the proposed PAMR strategy either passively keeps last portfolio or aggressively approaches a new portfolio by following the mean reversion principle. By solving three well formulated optimization problems, we arrive at three simple portfolio update rules. It is interesting to find that the final portfolio update scheme reaches certain trade-offs between portfolio return and volatility risk, and explicitly reflects the mean reversion trading rule. Moreover, we propose a mixture algorithm, which mixes PAMR and other strategies, and show that the mixture can be universal if one universal strategy is included. The key advantages of PAMR are its highly competitive performance and fairly attractive computation time efficiency. Our extensive numerical experiments on various real datasets show that in most cases the proposed PAMR strategy is quite performance efficient in comparison to a number of state-of-the-art portfolio selection strategies under a variety of performance metrics. At the same time, the proposed strategy costs linear time with respect to the product of the number of stocks and trading days, and its computational time in back tests is orders of magnitude less than its competitors, showing its applicability to real-world large scale online applications.

As a summary, our contributions in this article include:

We propose a new algorithm for online portfolio selection, named “Passive Aggressive Mean Reversion” (PAMR). To the best of our knowledge, it is the first portfolio selection strategy that exploits both the mean reversion property in finance and the powerful online passive aggressive learning technique in machine learning.

We propose a mixture algorithm to mix the proposed PAMR algorithms and other universal strategies, resulting in a theoretically guaranteed universal mixture strategy.

We analyze the final portfolio update scheme of PAMR and show that it is essentially related to certain trade-offs between portfolio return and volatility risk.

We conduct an extensive set of numerical experiments on a number of up-to-date datasets from various markets. The results show that in most cases the proposed PAMR strategy not only outperforms the benchmarks (including market index, best stock and challenging best constant rebalanced portfolio (Cover 1991 ) in hindsight), but also outperforms various state-of-the-art strategies under various performance metrics tested.

We also extend the proposed strategy to handle some practical issues for a real-life portfolio selection task, viz., transaction cost and margin buying, and show its practical viability through the extensive empirical study.

We show that the time complexity of the proposed algorithm is linear with respect to the number of stocks per trading day, and its empirical computational time in the back tests is quite competitive compared with the state of the arts, indicating the proposed strategy is suitable for online large-scale real applications.

The rest of the article is organized as follows. Section  2 formally states online portfolio selection problem. Section  3 reviews existing state-of-the-art approaches tackling this problem, and highlights their limitations. Section  4 presents our proposed PAMR strategy and analyzes the algorithm. Section  5 validates the effectiveness of PAMR by extensive empirical studies on historical financial markets. Finally, Sect.  6 summarizes this article and indicates future directions.

2 Problem setting

Let us consider a financial market with m assets, over which we wish to invest. The changes of asset prices for n trading periods are represented by a sequence of non-negative, non-zero price relative vectors \({\mathbf{x}}_{1}, \ldots, {\mathbf{x}}_{n}\in{\mathbb{R}}_{+}^{m}\) . Let us use x n to denote such a sequence of vectors. The i th component of the t th vector x ti denotes the ratio of closing price to last closing price of the i th asset on the t th trading day, thus an investment in asset i on the t th trading day increases by a factor of x ti .

An investment in the market is specified by a portfolio vector b t =( b t 1 ,…, b tm ), where b ti represents the proportion of wealth invested in the i th asset. Typically, we assume portfolio is self-financed and no margin/short is allowed, therefore each entry of a portfolio is non-negative and adds up to one, that is, b t ∈Δ m , where \({\Delta}_{m}= \{{\mathbf{b}} : {\mathbf {b}}\in {\mathbb{R}}_{+}^{m}, \sum_{i=1}^{m} b_{i} = 1 \}\) . The investment procedure is represented by a portfolio strategy , that is, a sequence of mappings \(\mathbf{b}_{1}= (\frac{1}{m}, \dots, \frac{1}{m} ), {\mathbf {b}}_{t}: {\mathbb{R}}_{+}^{m(t-1)}\rightarrow\Delta_{m}, t=2, 3, \ldots\) , where b t = b t ( x 1 ,…, x t −1 ) is the portfolio used on the t th trading period given past market price relatives x t −1 ={ x 1 ,…, x t −1 }. Let us denote by b n the portfolio strategy for n trading periods.

For the t th trading day, an investment according to portfolio b t results in a portfolio daily return s t , that is, the wealth increases by a factor of \({\mathbf{s}}_{t}={\mathbf{b}}_{t}^{\top}{\mathbf{x}}_{t}=\sum_{i=1}^{m}b_{ti}x_{ti}\) . Since we use price relative, the investment results in multiplicative cumulative return. Thus, after n trading days, the investment according to a portfolio strategy b n results in portfolio cumulative wealth S n , which is increased by a factor of \(\prod_{t=1}^{n}{\mathbf{b}}_{t}^{\top} {\mathbf{x}}_{t}\) , that is,

where S 0 denotes the initial wealth, and is set to $1 in this article for convenience.

Finally, we formulate the online portfolio selection problem as a sequential decision problem. The portfolio manager is a decision maker whose goal is to make a portfolio strategy on financial markets to satisfy certain requirements. In this study, his target is to maximize the portfolio cumulative wealth. He computes his portfolios in a sequential fashion. On each trading day t , the portfolio manager has access to all previous sequences of price relative vectors x t −1 ={ x 1 ,…, x t −1 }, and previous sequences of portfolio vectors b t −1 ={ b 1 ,…, b t −1 }. On the basis of these historical information, the portfolio manager computes a new portfolio vector b t for coming price relative vector x t . Note that without historical information, the initial portfolio is set to uniform. The resulting portfolio is evaluated by its portfolio daily return. This procedure is repeated until the end of the trading periods, and the portfolio is finally evaluated according to the portfolio cumulative wealth achieved. Figure  1 models the portfolio selection problem as a sequential decision problem.

Portfolio selection as a sequential decision problem

In the above portfolio selection model, we make several general assumptions:

Transaction cost: we assume no transaction cost or taxes exists in this portfolio selection model;

Market liquidity: we assume that one can buy and sell required quantities at last closing price of any given trading period;

Impact cost: we assume that market behavior is not affected by a portfolio selection strategy in our study.

3 Related work

In this section, we review some popular portfolio selection approaches, and some machine learning and trading philosophies that inspire the proposed approach.

3.1 Benchmark approaches

The most common baseline is Buy-And-Hold (BAH) strategy, that is, one invests his/her wealth among a pool of assets with an initial portfolio and holds the portfolio all the time. The BAH strategy with a uniform initial portfolio is referred to as uniform BAH strategy, which is adopted as market strategy producing the market index in our study. Contrary to the static BAH strategy, active trading strategies usually change portfolios regularly during the entire trading periods. A classical active strategy is Constant Rebalanced Portfolios (CRP) (Cover and Gluss 1986 ), which keeps a fixed fraction of a investor’s wealth in each underlying asset every trading day. The best possible CRP strategy is often called Best CRP (BCRP), which apparently is only a hindsight strategy. The CRP strategy can take advantage of market fluctuations for active trading, and its underlying idea is based on the mean reversion principle, or known as “ Buy Low, Sell High ”. To handle transaction cost issue for CRP strategy, Blum and Kalai ( 1999 ) proposed semi-CRP that partially balances between potential return and potential transaction cost and rebalances to initial portfolio at the end of any subset of the trading periods rather than every trading day.

3.2 Online learning

In this section, we briefly introduce the related work on online machine learning (Rosenblatt 1958 ; Crammer and Singer 2003 ; Cesa-Bianchi et al. 2004 ; Crammer et al. 2006 ; Fink et al. 2006 ) to have the learning inspiration for our work. Perceptron algorithm (Rosenblatt 1958 ; Freund and Schapire 1999 ) is one important online approach which updates the learning function by adding a new example with a constant weight when it is misclassified. Recently a number of online learning algorithms have been proposed based on the criterion of maximum margin (Li and Long 1999 ; Gentile 2001 ; Kivinen et al. 2001 ; Crammer and Singer 2003 ; Crammer et al. 2006 ; Zhao et al. 2011 ). For example, Relaxed Online Maximum Margin (ROMMA) (Li and Long 1999 ) algorithm repeatedly chooses the hyper-planes that correctly classify the existing training examples with the maximum margin. Passive Aggressive (PA) (Crammer et al. 2006 ) algorithm updates the classification function when a new example is misclassified or its classification score does not exceed some predefined thresholds. As empirical studies show, the maximum margin based online learning algorithms are generally more effectively than the Perceptron algorithm. In this article, we mainly adopt the idea of Passive Aggressive learning since it is suitable for our motivations as further illustrated in Sect.  4.1 .

3.3 Learning to select portfolio

Learning to select portfolio has been extensively studied in information theory and machine learning. Generally, a strategy selects one optimal strategy (it can be market strategy, challenging BCRP strategy, or even Oracle strategy which chooses the best stock every trading day) and tries to obtain the same cumulative return. The regret of a strategy is defined as the gap between its logarithmic cumulative wealth achieved and that of the optimal strategy.

One important type of learning to select portfolio is regret minimization approach, which chooses BCRP strategy as the optimal strategy. Cover ( 1991 ) proposed Universal Portfolios (UP) strategy, where the portfolio is historical performance weighted average of all constant rebalanced portfolio experts. The regret achieved by Cover’s UP is O( m log n ), and its run time complexity is O( n m ), where m denotes the number of stocks and n denotes the number of trading days. The implementation is exponential in the number of stocks and thus restricts the number of assets used in experiments and real applications. Kalai and Vempala ( 2002 ) presented a time-efficient implementation of Cover’s UP based on non-uniform random walks that are rapidly mixing, which requires poly running time O( m 7 n 8 ). Following their work, Cover and Ordentlich ( 1996 ) developed universal procedures when side information Footnote 1 is taken into account as a finite number of values. Cross and Barron ( 2003 ) proposed a new universal portfolio strategy tracking the best in-hindsight wealth achievable within target classes of linearly parameterized portfolio sequences, which are more general than the standard CRP class and permit the portfolio to display a continuous form of dependence on past prices or other side information. Belentepe ( 2005 ) presented a statistical view of Cover’s UP, showing that it is approximately equivalent to a constrained sequential portfolio optimization, which connects Cover’s UP with traditional mean-variance portfolio theory.

Another famous strategy is Exponential Gradient (EG) strategy (Helmbold et al. 1997 , 1996 ) for online portfolio selection using multiplicative updates. In general, EG strategy tries to maximize the expected logarithmic portfolio daily return (approximated using the last price relative), and minimize the deviation between next portfolio and last portfolio. The regret achieved by EG is O( \(\sqrt{n \log m}\) ) with O( mn ) running time. While its regret is not as tight as Cover’s UP, its linear time complexity is substantially less than the latter.

Recently, convex optimization has been applied to resolve the portfolio selection problem (Agarwal et al. 2006 ; Agarwal and Hazan 2005 ; Hazan 2006 ; Hazan et al. 2007 ). Examples include Online Newton Step (ONS) strategy (Agarwal et al. 2006 ), which aims to maximize the expected logarithmic cumulative wealth (approximated using historical price relatives) and to minimize the variation of the expected portfolio. ONS exploits the second order information of the log wealth function and applies it to the online scenario. It theoretically achieves a regret of O( m log n ) which is the same as Cover’s UP, and has running time complexity of O( m 3 n ). Following ONS, Hazan and Seshadhri ( 2009 ) recently proposed a new adaptive-regret approach with more decent theoretical results, which essentially is an ONS based strategy.

Another promising direction for portfolio selection is wealth maximization approach, which is based on the notion of approaching the Oracle as the optimal strategy. This idea was followed by Borodin et al. ( 2004 ) in their proposal of a non-universal portfolio strategy named Anti-Correlation (Anticor). Unlike the regret minimization approaches, Anticor strategy takes advantage of the statistical properties of financial market. The underlying motivation is to bet on the consistency of positive lagged cross-correlation and negative autocorrelation. It exploits the statistical information from the historical stock price relatives and adopts the classical mean reversion trading idea to transfer the wealth in the portfolio. Although it does not provide any theoretical guarantee, its empirical results (Borodin et al. 2004 ) showed that Anticor can outperform all existing strategies in most cases. Unlike the greedy algorithm by the Anticor strategy, Li et al. ( 2011b ) very recently proposed Confidence Weighted Mean Reversion (CWMR) strategy to actively exploit the mean reversion property and the second order information of a portfolio, which produces better performance than Anticor.

In addition, Györfi et al. ( 2006 ) recently introduced a framework of Nonparametric Kernel-based Moving Window (B K ) learning strategies for portfolio selection based on nonparametric prediction techniques (Györfi and Schäfer 2003 ). Their algorithm first identifies a list of similar historical price relative sequences whose Euclidean distances with recent market windows are smaller than a threshold, then optimizes the portfolio with respect to the list of similar sequences. Under the same framework, Györfi et al. ( 2007 ) proposed another variant called Nonparametric Kernel-based Semi-log-optimal strategy, which is actually an approximation of the B K strategy, mainly to improve the computational efficiency. Replacing log utility function by Markowitz-type utility function, Ottucsák and Vajda ( 2007 ) proposed Nonparametric Kernel-based Markowitz-type strategy, which connects return and risk (or mean and variance) with the online portfolio selection strategy. Following the same framework as B K strategy, Nonparametric Nearest Neighbor learning (B NN ) strategy proposed in Györfi et al. ( 2008 ) aims to search for the ℓ nearest neighbors in historical price relative sequences rather than search price relatives within a specified Euclidean ball. This method has been empirically shown to be a robust trading strategy. Along this direction, Li et al. ( 2011a ) recently proposed Correlation-driven Nonparametric learning (CORN) strategy to search for similar price relatives via correlation coefficient and considerably boosted the empirical performance of nonparametric learning approach.

Besides the main stream of learning to select portfolio, another type of trading strategy is based on switching between various strategies, that is, maintaining a probability distribution among the strategies. Singer ( 1997 ) proposed Switching Portfolios (SP), which aims to deal with changing markets by taking into account the possibility that the market changes its behavior after each trading day. It switches among a set of basic investment strategies and assumes the a priori duration of using one basic strategy is geometrically distributed. Levina and Shafer ( 2008 ) proposed Gaussian Random Walk (GRW) strategy, which is a Markov switching strategy. GRW switches among the basic investment strategies as a Gaussian random walk in the simplex of portfolios.

Last, we note that our work is very different from another great body of existing work in literature (Kimoto et al. 1993 ; Tay and Cao 2001 ; Cao and Tay 2003 ; Tsang et al. 2004 ; Lu et al. 2009 ), which attempted to make financial time series forecasting and stock price predictions by applying machine learning techniques, such as neural networks (Kimoto et al. 1993 ), decision trees (Tsang et al. 2004 ), and support vector machines (SVM) (Tay and Cao 2001 ; Cao and Tay 2003 ; Lu et al. 2009 ), etc. The key difference between these work and ours is that their learning goal is to make explicit predictions of future prices/trends while our learning goal is to directly optimize portfolio without predicting prices explicitly.

3.4 Analysis of existing work

One popular trading idea in reality is trend following or momentum strategy, which assumes that historically better-performing stocks would still perform better than others in future. Some existing algorithms, such as EG and ONS, approximate the expected logarithmic daily return and logarithmic cumulative return respectively using historical price relatives. Though this idea is easy to understand and makes fortunes to many of the best traders and investors in the world, trend following is very hard to implement effectively. In addition, in the short-term, the stock price relatives may not follow previous trends as empirically evidenced by Jegadeesh ( 1990 ) and Lo and MacKinlay ( 1990 ).

Besides the trend following approach, another widely adopted approach in the learning community is mean reversion  (Cover and Gluss 1986 ; Cover 1991 ; Borodin et al. 2004 ), which is also termed as contrarian approach. This approach stems from the CRP strategy (Cover and Gluss 1986 ), which rebalances to the initial portfolio every trading day. The idea behind this approach is that if one stock performs worse than others, it tends to perform better than others in the next trading day. As a result, the defining characteristic of a contrarian strategy is the purchase of securities that have performed poorly in the past and the sale of securities that have performed well, or quite simply, “ Sell the Winner, Buy the Loser ”. According to Lo and MacKinlay ( 1990 ), the effectiveness of mean reversion is a consequence of positive cross-autocovariances across securities. Among existing algorithms, CRP, UP, and Anticor adopt this trading idea. However, CRP and UP passively revert to the mean, while empirical evidence from Anticor algorithm (Borodin et al. 2004 ) shows that active reversion to the mean may better exploit the fluctuation of financial markets and is likely to obtain a much higher profit. On the other hand, although Anticor actively reverts to the mean, it is a heuristic method based on statistical correlations to transfer the wealth within the portfolio. In other words, it may not effectively exploit the mean reversion property.

In between, pattern matching based nonparametric learning algorithms (B K and B NN , etc.) can identify many market conditions including both mean reversion and trend following. However, when locating similar price relatives, the nonparametric learning approaches may locate both mean reversion and trend following price relatives, whose patterns are essentially opposite, thus weakening the maximization of the expected cumulative wealth.

In a word, both trend following and mean reversion can generate profit in the financial markets, if appropriately used. In the following, we will propose an active mean reversion based portfolio selection method. Though simple in update rules, it empirically outperforms the above existing portfolio selection strategies in most cases. The success of the proposed portfolio selection strategy indicates that it appropriately takes advantage of the mean reversion trading idea and generates significantly high profits in the back tests with real market data.

4 Passive aggressive mean reversion approach for portfolio selection

4.1 intuition and overview.

The proposed approach is motivated by Constant Rebalanced Portfolios (Cover and Gluss 1986 ), which adopts the mean reversion trading idea. A simple but convincing example showing the mean reversion idea is illustrated in Table  1 . Consider a fluctuating market with two stocks (A, B), and the stock price relative sequence is \((\frac{1}{2}, 2 ), (2, \frac{1}{2} ), \ldots\)  , where each stock is not going anywhere but actively moving within a range. Obviously, in a long-term period, market strategy cannot achieve any abnormal return from this sequence since the cumulative wealth of each stock remains the same after 2 n trading days. However, Best CRP in hindsight can achieve a growth rate of \((\frac{5}{4})^{n}\) for a n -trading period. Now let us analyze the BCRP strategy on the stock price relative sequence to show the underlying mean reversion trading idea. Suppose the initial portfolio is \((\frac{1}{2},\frac{1}{2} )\) and at the end of the 1st trading day, the closing price adjusted wealth distribution becomes \((\frac{1}{5}, \frac{4}{5} )\) with corresponding cumulative wealth increasing by a factor of \(\frac{5}{4}\) . At the beginning of the 2nd trading day, portfolio manager rebalances the portfolio to initial portfolio \((\frac{1}{2},\frac{1}{2} )\) by transferring the wealth from better-performing stock (B) to worse-performing stock (A) in the previous trading day. At the beginning of the 3rd trading day, the wealth transfer with the mean reversion trading idea continues. Although the market strategy gains nothing, BCRP can achieve a growth rate of \(\frac{5}{4}\) per trading day using the mean reversion trading idea, which assumes that if one stock price performs worse, it tends to perform better in the subsequent trading day.

Another motivation of the proposed PAMR algorithm is inspired by the fact that in financial crisis, all stocks drop synchronously or certain stocks drop significantly. Under these situations, actively rebalancing may not be appropriate since it puts too much wealth on “mine” stocks, such as Bear Stearns during the recent financial crisis. To avoid the potential risk concerning such “mine” stocks, it is a good choice to stick to the previous portfolio, which constitutes the CRP strategy. Here, the reason to choose the passive CRP strategy is that identifying these “mine” stocks a priori is almost impossible, which are usually known in hindsight. Thus, to avoid suffering too much from such situations, PAMR alternates the strategy between “aggressive” and “passive” reversion depending on the market conditions. The passive mean reversion strategy avoids the high risk of the aggressive approach that would put almost all wealth on these “mine” stocks when they drop significantly.

In this article, we propose a novel trading strategy named “Passive Aggressive Mean Reversion”, or PAMR for short. On the one hand, the underlying assumption of our approach is that better-performing stocks would perform worse than others in the next trading day. On the other hand, if the market drops too much, we would stop actively rebalancing the portfolio to avoid certain “mine” stocks and their associated risk. In order to exploit these intuitions, we suggest to adopt Passive Aggressive (PA) online learning (Crammer et al. 2006 ), which was originally proposed for classification tasks. Loosely speaking, the basic idea of PA for classification is that it passively keeps previous solution if loss is zero, while it aggressively updates the solution whenever the suffering loss is nonzero.

Let us now describe the basic idea of the proposed strategy in detail. Firstly, if the portfolio daily return is below a certain threshold, we will try to keep the previous portfolio such that it passively reverts to the mean to avoid the potential “mine” stocks. Secondly, if the portfolio daily return is above the threshold, we will actively rebalance the portfolio to ensure that the expected portfolio daily return is below the threshold in the belief that the stock price relatives will revert in the next trading day. This sounds a bit counter-intuitive, but it is indeed reasonable, because if the stock price relative reverts, keeping the expected portfolio daily return below the threshold is able to maintain a high portfolio daily return in the next trading day. Here, the expected portfolio return is calculated with respect to the historical price relatives, for example, in our study, the last price relative, which is consistent with EG algorithm (Helmbold et al. 1997 , 1996 ).

To further illustrate why aggressive reversion to the mean can be more effective than a passive one, let us continue the example in Table  1 that has a market going to nowhere but actively fluctuating. We show that in such markets, the proposed strategy is much more powerful than BCRP in hindsight, a passive mean reversion trading strategy. Table  2 compares the two trading strategies. As the motivating example shows, the growth rate of BCRP is \((\frac{5}{4})^{n}\) for a n -trading period, while at the same time, the growth rate of the proposed PAMR strategy is \(\frac{5}{4}\times(\frac{3}{2})^{n-1}\) (the details of the calculation/algorithm will be presented later). We intuitively explain the success of PAMR below.

Assume the threshold for PAMR update is set to 1, that is, if portfolio daily return is below 1, we do nothing but keep the existing portfolio. Our strategy begins with a portfolio \((\frac{1}{2}, \frac {1}{2} )\) . For the 1st trading day, the return is \(\frac{5}{4}>1\) . Then at the beginning of the 2nd trading day, we rebalance the portfolio to satisfy the condition that approximate portfolio daily return based on last price relatives is below the threshold 1, and the resulting portfolio is \((\frac {2}{3}, \frac{1}{3} )\) . Although it seems that we build a portfolio such that the approximate portfolio return is below the threshold, in practice, as the reversion to the mean suggests, we are maximizing the portfolio return in the next trading day. As we can observe, the return for the 2nd trading day is \(\frac{3}{2}>1\) . Then following the same rule, we will rebalance the portfolio to \((\frac{1}{3}, \frac{2}{3} )\) . As a result, in such a market, the growth rate of the proposed strategy is \(\frac{5}{4}\times (\frac{3}{2} )^{n-1}\) for a n -trading period, which is much more superior to that of BCRP, that is, \((\frac{5}{4} )^{n}\) .

4.2 Formulations

Now we shall formally devise the proposed Passive Aggressive Mean Reversion (PAMR) strategy for portfolio selection problem. The PAMR strategy is based on the mean reversion idea as described in Sect.  4.1 , and is equipped with Passive Aggressive (PA) online learning technique (Crammer et al. 2006 ).

First of all, given a portfolio vector b and a price relative vector x t , we define a ϵ -insensitive loss function for the t th trading day as

where ϵ ≥0 is the sensitivity parameter which controls the mean reversion threshold. Since typically portfolio daily return fluctuates around 1, we often empirically choose ϵ ≤1 in order to buy worse performing stocks. The ϵ -insensitive loss is zero when return is less than the reversion threshold ϵ , and otherwise grows linearly with respect to the daily return. For conciseness, let us use \(\ell_{\epsilon}^{t}\) to denote ℓ ϵ ( b ; x t ), that is, the ϵ -insensitive loss of the t th trading day. By defining this loss function, we can distinguish the two motivating cases described in Sect.  4.1 .

In the following parts, we will formulate three variants of the proposed strategy, and will propose specific algorithms to solve them in the subsequent section. Recalling that b t denotes the portfolio vector on the t th trading day, the first proposed method for Passive Aggressive Mean Reversion (PAMR) is formulated as the constrained optimization below:

Optimization Problem 1

The above formulation attempts to find an optimal portfolio by minimizing the deviation from last portfolio b t under the condition of satisfying the constraint of zero loss. On the one hand, the above approach passively keeps the last portfolio, that is, b t +1 = b t whenever \({\ell}_{\epsilon}^{t} = 0\) that means the portfolio daily return is below the threshold ϵ . On the other hand, whenever the loss is nonzero, it aggressively updates the solution by forcing it to strictly satisfy the constraint ℓ ϵ ( b t +1 ; x t )=0. It is clear that this formulation is able to address the two motivations.

Although the above formulation is reasonable to address our concerns, it may have some undesirable properties in situations with noisy price relatives, which are common in real-word financial markets. For example, a noisy price relative appearing in some trending sequences may suddenly change the portfolio in a wrong direction due to the aggressive update. To avoid such problems, we propose two variants of PAMR that are able to trade off between aggressiveness and passiveness. The idea of formulating the two PAMR variants is similar to soft margin support vector machines by introducing some non-negative slack variables into optimization. Specifically, for the first variant, we modify the objective function by introducing a term that scales linearly with respect to ξ , which results in the following optimization:

Optimization Problem 2

where C is a positive parameter to control the influence of the slack variable term on the objective function. We refer to this parameter as the aggressiveness parameter similar to PA learning (Crammer et al. 2006 ) and call this variant “PAMR-1”.

Instead of using a linear term of slack variable, in the second variant, we modify the objective function by introducing a slack variable term that scales quadratically with respect to ξ , which results in the following optimization problem:

Optimization Problem 3

Note that in the above formulation we do not need to enforce the constraint ξ ≥0 as ξ 2 is always non-negative. We refer to this variant as “PAMR-2”.

4.3 Algorithms

We now derive the approximate solutions for the above three PAMR formulations using standard techniques from convex analysis (Boyd and Vandenberghe 2004 ), and present the proposed PAMR algorithms for portfolio selection task. Specifically, the following three propositions summarize the solutions to the PAMR methods.

Proposition 1

The solution to the Optimization Problem  1 ( PAMR ) without considering the non - negativity constraint ( b ⪰0) is expressed as :

where \(\bar{x}_{t}=\frac{\mathbf{x}_{t}\cdot\mathbf{1}}{m}\) denotes the market return , and τ t is computed as :

The proof can be found in Appendix  A . □

Proposition 2

The solution to the Optimization Problem  2 ( PAMR -1) without considering the non - negativity constraint ( b ⪰0) is expressed as :

The proof can be found in Appendix  B . □

Proposition 3

The solution to the Optimization Problem  3 ( PAMR -2) without considering the non - negativity constraint ( b ⪰0) is expressed as :

where \(\bar{{x}}_{t}=\frac{\mathbf{x}_{t}\cdot{\mathbf{1}}}{m}\) denotes the market return , and τ t is computed as :

The proof can be found in Appendix  C . □

Figure  2 summarizes the details of the proposed PAMR algorithms. Firstly, with no historical information, the initial portfolio is set to uniform portfolio \({\mathbf{b}}_{1}= (\frac{1}{m}, \ldots, \frac{1}{m} )\) . At the beginning of t th trading day, we rebalance according to the portfolio determined at the end of last trading day. At the end of t th trading day, the market reveals a stock price relative vector, which represents the stock price movements. Since both the portfolio and the stock price relatives are already known, portfolio manager is able to measure the portfolio daily return b t ⋅ x t and the suffering loss ℓ ϵ ( b t ; x t ) as defined in ( 1 ). Then, we calculate an optimal step size τ t based on last portfolio and stock price relatives. Given the optimal step size τ t , we can update the portfolio for next trading day. Finally, we perform a normalization step to obtain the final portfolio by projecting the updated portfolio into the simplex domain.

The proposed Passive Aggressive Mean Reversion (PAMR) strategies

4.4 Analysis and interpretation

To reflect the mean reversion trading idea, we are interested in analyzing the resulting update rules of the proposed PAMR algorithms, which mainly involve the portfolio b t +1 and the step size τ t . In particular, we want to examine how the update rules are related to return and risk—the two most important concerns in a portfolio selection task.

First of all, we analyze the resulting portfolio update rule in ( 5 ) for the three PAMR algorithms, that is, \(\mathbf{b}_{t+1} = \mathbf{b}_{t}-\tau_{t} ({\mathbf {x}_{t}-\bar{x}_{t}\mathbf{1}} )\) . In the update rule, the step size τ t is non-negative, and \(\bar{{x}}_{t}\) is the mean return or market return. For term \({\mathbf{x}_{t}-\bar{x}_{t}\mathbf{1}}\) , we can see it represents stock abnormal returns with respect to the market on the t th trading day. More precisely, we can interpret it as the directional vector for the weight transfer. The negative sign before the term indicates that the resulting update scheme is consistent with the motivation, that is, the weights shall be transferred from better performing stocks (with positive abnormal returns) to worse performing stocks (with negative abnormal returns) at the beginning of next day.

Besides, another important update is the step size τ t calculated as ( 6 ), ( 7 ), and ( 8 ), for three PAMR methods, respectively. The step size τ t adaptively controls the weights to be transferred by taking effect on the directional vector. One interesting term in common for the three updates of τ t is \(\frac{\ell_{\epsilon}^{t}}{\Vert \mathbf{x}_{t}-\bar{x}_{t}\mathbf {1}\Vert ^{2}}\) . The numerator of the term equals to the t th portfolio daily return minus the mean reversion threshold. Assuming other variables are constant, if the return is high (low), it leads to a large (small) value of τ t , which would more (less) aggressively transfer the wealth from better performing stocks to worse performing stocks. The denominator is essentially the market quadratic variability, that is, the number of stocks times the market variance of the t th trading day. In modern portfolio theory, variance of stock return is typically regarded as a volatility risk term for a portfolio (Markowitz 1952 ). As indicated by the denominator, if the risk is high (low), the step size τ t would become small (large). As a result of small (large) step size, the weight transfer made by the update scheme will be weakened (strengthened), which is consistent with our intuition that prediction would be not accurate in drastically dropping markets, and we opt to make relatively less transfer in order to reduce risk. Moreover, PAMR-1 caps the step size by a constant C , while PAMR-2 decreases the step size by adding a constant \(\frac{1}{2C}\) to its denominator. Both measures can prevent drastic weight transfer in case of noisy price relatives, which is consistent with their motivations.

From the above analysis on the updates of direction and step size, we can conclude that PAMR nicely balances between return and risk and clearly reflects the mean reversion trading idea. To the best of our knowledge, this important trade-off between return and risk has been considered by only one existing approach, that is, nonparametric kernel-based Markowitz-type strategy (Ottucsák and Vajda 2007 ). While the kernel-based Markowitz-type strategy trades off the return and risk with respect to similar historical price relatives, the proposed PAMR explicitly trades off the return and risk with respect to last price relatives. This nice property distinguishes the proposed approach from most existing approaches that often cater to return, but ignore the risk concern, and are therefore undesirable according to modern portfolio theory (Markowitz 1952 ).

Now let us briefly analyze the time complexity of the proposed PAMR algorithms. From Fig.  2 , we can see that besides the normalization step, PAMR strategy takes O( m ) per trading day, where m denotes the number of assets. Moreover, the normalization or projection step (Step 7 in Fig.  2 ) can be efficiently implemented (Michelot 1986 ; Duchi et al. 2008 ). In our implementation, we adopt the projection Footnote 2 according to Duchi et al. ( 2008 ), which takes linear time with respect to m . Thus, the total time complexity is O( mn ), where n is the total number of trading days. Such time complexity is the same as that of EG algorithm and is much superior to other existing methods. Linear time complexity enables the proposed algorithm to handle transactions in certain scenarios where low latency is of crucial importance, such as high frequency trading (Aldridge 2009 ).

4.5 Discussions

4.5.1 discussion on intuitions.

Although the motivating example in Sect.  4.1 demonstrates the effectiveness of PAMR over BCRP strategy, PAMR may not always outperform BCRP. In general, PAMR is an online algorithm while BCRP is offline optimal for an i.i.d. market (see Cover and Thomas 1991 , Theorem 15.3.1). Next, we discuss some possible situations where PAMR may fail to outperform BCRP.

Consider a special case where one stock crashes and the other explodes, e.g., a market sequence of two stocks as \((\frac{1}{2}, 2 ),(\frac{1}{2}, 2 ), \ldots\)  . Assuming the same parameter settings as the motivating example, BCRP will increase at an exponential rate 2 n as it wholly invests in the 2nd asset, while PAMR will keep a fixed wealth on \(\frac{5}{4}\) over the trading period. Obviously, in such a situation, PAMR performs much worse than BCRP does, i.e., PAMR produces a cumulative wealth of \(\frac{5}{4}\) against 2 n achieved by BCRP over a n trading period. Though not shiny in such situations, PAMR still bounds its losses. Moreover, such a market, which violates the mean reversion assumption, is occasional, at least from the view point of our empirical studies.

4.5.2 Discussion on loss function

In our definition of loss function, that is, ( 1 ), we use the original portfolio expected return b ⋅ x t , while it is possible to use log utility (Latané 1959 ) on the return, that is, log( b ⋅ x t ). With this log utility, the optimization problems ( 2 ), ( 3 ), and ( 4 ) are all non-convex and nonlinear, and thus difficult to solve. One way to solve these non-convex optimization problems is to use log’s first-order Taylor expansion at last portfolio and ignore the higher order terms, that is, \(\log (\mathbf{b}\cdot\mathbf{x}_{t}) \approx\log (\mathbf{b}_{t}\cdot\mathbf{x}_{t})+\frac{\mathbf{x}_{t}}{\mathbf{b}_{t}\cdot\mathbf{x}_{t}} (\mathbf {b}-\mathbf{b}_{t} )\) . After linear approximation, the optimization problems can be solved using the same techniques used in our derivation. However, such linear approximation of loss function may have some drawbacks. First of all, linear approximation yields a upper bound on regret in terms of a log utility loss function. There is no way to justify the goodness of the linear approximation. Moreover, if we use log utility, then the loss function is flat, then sharply rises and finally flattens out. While linear approximation is good in the two flat regimes, it is typically terrible at the point of non-differentiability and sub-par in the sharply rising region.

On the other hand, for the loss function in form of ( 1 ) without log utility or with linear approximation of log utility, the best possible regret in a minimax sense is at most O( \(\sqrt{n}\) ) (Abernethy et al. 2009 ), while true log loss minimization algorithm can routinely achieve O(log n ). However, although our loss function is non-differentiable and it would achieve a potential regret of O( \(\sqrt{n}\) ), it is not a traditional loss function maximizing return (like traditional loss function, −log( b ⋅ x t )), but only a tool to realize mean reversion. Thus the regret achieved using our loss function does not represent a regret about return, which may not be meaningful as traditional regret bound is.

Anyway, the potential worse bound may have unknown weaknesses, which may not be elicited by the following empirical evaluations. Though on our experiments PAMR works well, anyone who cares about its theoretical aspects should be notified about the possible worse bound.

4.5.3 Discussion on formulation

Although our formulations mainly focus on the portfolio daily return without explicitly dealing with risk (e.g., volatility of daily returns), the final derived algorithms can be nicely interpreted as certain trade-offs between risk and return, as discussed in Sect.  4.4 . Such interesting observation is further verified by our empirical evaluation in Sect.  5.4.2 , which shows that the proposed PAMR algorithms achieve good risk-adjusted return in terms of two risk-related metrics (i.e., volatility risk and drawdown risk, respectively).

Similar to previous studies, we avoid incorporating transaction cost in the original formulations, which simplifies the formulations and clearly highlights PAMR’s key ingredients. To further show the impact of transaction costs, it is not difficult to evaluate the effect of transaction costs, as shown in Sect.  5.2.2 . In the following empirical study, we present results on both cases: with and without transaction costs. From the empirical results in Sect.  5.4.5 , we find that in most markets, the proposed PAMR algorithms work well without or even with moderate transaction costs.

Besides, it is important to note that there are two key parameters in the proposed PAMR algorithm and its variants, viz., the sensitivity parameter ϵ and the aggressiveness parameter  C . In practice, the choice of these parameters could affect the performance of the proposed algorithms. To achieve a good performance in a specific market, these parameters have to be finely tuned. We will thoroughly examine the effects of the two parameters on real-life datasets in Sect.  5.4.4 , and make suggestions for the empirical selection of their values.

4.5.4 Discussion on PAMR variants

In this section, we will show an example to illustrate different behaviors of the three update rules, viz., PAMR, PAMR-1, and PAMR-2. As discussed in Sect.  4.2 , one objective for PAMR-1 and PAMR-2 is to prevent the portfolio being affected too much from noisy price relatives, which might drastically change the portfolio. Let us assume the environments and parameter settings as follows. Let the t th price relative x t =(1.00,0.01), which represents the situations that the 2nd price relative is a noise, and the t th portfolio b t =(1,0). Setting the parameters ϵ =0.30 and C =1.00, let us calculate next portfolio b t +1 . This market environment describes the situations where certain price relatives drop significantly, which is similar to some stocks during recent financial crisis. Without tuning, the original PAMR algorithm would transfer a large proportion of wealth to the 2nd asset in the next trading day. This can be verified by examining the portfolio calculated by PAMR, viz., PAMR calculates the update step size τ t =1.43 and obtains the subsequent portfolio b t +1 =(0.29,0.71). However, a natural choice of avoiding such noisy price relatives is to put less proportion of wealth to the second asset. Now, when calculating the next portfolios by PAMR-1 and PAMR-2, we obtain the update step size τ t =1.00 and τ t =0.71, respectively, which are smaller than the update step size of the original PAMR, that is, τ t =1.43. Accordingly, we obtain the next portfolios b t +1 =(0.50,0.50) and b t +1 =(0.65,0.35) for PAMR-1 and PAMR-2, respectively. Clearly, PAMR-1 and PAMR-2 transfer less wealth to the 2nd asset than the original PAMR does. Thus, PAMR-1 and PAMR-2 in general suffer relatively less from noisy price relatives, though we cannot completely avoid such suffering situation.

4.6 Mixture algorithm

One theoretical result desired by existing online portfolio selection algorithms is universal property (Cover 1991 ). Since mean reversion trading idea is counter-intuitive (Borodin et al. 2004 ), we find it is hard to prove the universality of PAMR. Alternatively, we present a general mixture algorithm, which guarantees worst-case performance, not for PAMR itself but for the mixture algorithm.

Briefly speaking, the proposed mixture algorithm frames PAMR as one “expert” in a mixture-of-experts setting, while at least one universal algorithm serves as other “experts”. Then, the proposed mixture adopts no-regret expert learning (Cesa-Bianchi and Lugosi 2006 ) to bound the regret of the overall system with respect to the best of these experts. If the mixture algorithm contains at least one universal algorithm, Footnote 3 then the universality of the mixture algorithm can be straightforwardly proved according to Cesa-Bianchi and Lugosi ( 2006 ) (see Example 10.3 and Theorem 10.3 for rigorous proofs). In our implementation, we adopt uniform buy and hold (BAH) mixture strategy, that is, we give equal proportion of portfolio wealth to each expert, let them run, and finally pool them again. We denote the BAH mixture algorithm as “MIX”. Other expert learning methods, such as exponential weighted, can also replace the buy and hold strategy, and they can also provide provable guarantees and get potentially stronger empirical performance. Though MIX seems trivial since it has a more involved mixing rule, one can make it nontrivial by extending the setting in a more general setting, such as the framework proposed by Akcoglu et al. ( 2002 ) and Das and Banerjee ( 2011 ). Obviously, such a mixture algorithm can be applied to any portfolio selection algorithm, either universal or not.

Though it is convenient to propose a mixture model consisting of PAMR such that the mixture model can achieve universality, PAMR’s universal consistency is still an open question and deserves further exploration.

5 Numerical experiments

To examine the empirical efficacy of the proposed PAMR strategy, we conduct an extensive set of numerical experiments on a variety of real datasets. In our experiments, we adopt six real datasets, which were collected from several diverse financial markets. The performance metrics include cumulative wealth and risk-adjusted returns (volatility risk and drawdown risk). We also compare the proposed PAMR algorithms with all existing algorithms stated in the related work section.

5.1 Experimental testbed on real data

In this study, we focus on historical daily prices in stock markets which are easy to obtain from public domains (such as Yahoo Finance and Google Finance), and thus publicly available to other researchers. Data from other types of markets, such as high frequency intra-day quotes and Forex markets, are either too expensive or hard to obtain and process, and thus may reduce the experimental reproducibility. In general, we employ six real and diverse datasets from several types of financial markets, Footnote 4 which are summarized in Table  3 .

The first one is NYSE dataset, one “standard” dataset pioneered by Cover ( 1991 ) and followed by several other researchers (Singer 1997 ; Helmbold et al. 1996 ; Borodin et al. 2004 ; Agarwal et al. 2006 ; Györfi et al. 2006 , 2008 ). This dataset contains 5651 daily price relatives of 36 stocks Footnote 5 in New York Stock Exchange (NYSE) for a 22-year period from Jul. 3rd 1962 to Dec. 31st 1984. We denote this dataset by “NYSE (O)” for short.

The second dataset is the extended version of the above NYSE dataset. For consistency, we collected the latest data in New York Stock Exchange (NYSE) from Jan. 1st 1985 to Jun. 30th 2010, which consists of 6431 trading days. We denote this new dataset as “NYSE (N)”. Footnote 6 It is worth noting that this new dataset consists of 23 stocks rather than the previous 36 stocks owing to amalgamations and bankruptcies. All self-collected price relatives are adjusted for splits and dividends, which is consistent with the previous “NYSE (O)” dataset.

The third dataset “TSE” is collected by Borodin et al. ( 2004 ), which consists of 88 stocks from Toronto Stock Exchange (TSE) containing price relatives of 1259 trading days, ranging from Jan. 4th 1994 to Dec. 31st 1998. The fourth dataset “SP500” is collected by Borodin et al. ( 2004 ), which consists of 25 stocks with the largest market capitalizations in the 500 SP500 components. It ranges from Jan. 2nd, 1998 to Jan. 31st 2003, containing 1276 trading days.

The fifth dataset is “MSCI”, a collection of global equity indices which are the constituents of MSCI World Index. Footnote 7 It contains 24 indices which represent the equity markets of 24 countries around the world, and consists of a total of 1043 trading days, ranging from Apr. 1st 2006 to Mar. 31st 2010. The final dataset is the “DJIA” dataset collected by Borodin et al. ( 2004 ), which consists of Dow Jones 30 composite stocks. DJIA contains 507 trading days, ranging from Jan. 14th 2001 to Jan. 14th 2003.

Besides the above six real market data, in the experiments, we also ran each dataset in their reverses (Borodin et al. 2004 ). For each dataset, we created a reversed dataset, which reverses the original order and inverts the price relatives. We denote these reverse datasets using a ‘−1’ superscript on the original dataset names. In nature, these reverse datasets are quite different from the original datasets, and we are interested in the behaviors of the proposed algorithm on these artificial datasets.

Unlike the previous studies, the above testbed covers much longer trading periods from 1962 to 2010 and much more diversified markets, which enables us to examine how the proposed PAMR strategy performs under different events and crises. For example, it covers several well-known events in the stock markets, such as dot-com bubble from 1995 to 2000 and subprime mortgage crisis from 2007 to 2009. The five stocks datasets are mainly chosen to test the capability of the proposed PAMR on regional stock markets, while the “MSCI” dataset aims to test PAMR’s capability on global indices, which may be potentially applicable to “Fund on Fund” (FOF). Footnote 8 As a remark, although we numerically test the PAMR algorithm on stock markets, we note that the proposed strategy could be generally applied to any type of financial markets.

5.2 Experimental setup and metrics

Regarding the parameter settings, there are two key parameters in the proposed PAMR algorithms. One is the sensitivity parameter ϵ and the other is the aggressiveness parameter C . Roughly speaking, the best values for these parameters are often dataset dependent. In the experiments, we simply set these parameters empirically without tuning for each dataset separately. Specifically, for all datasets and experiments, we set the sensitivity parameter ϵ to 0.5 in the three algorithms, and set the aggressiveness parameter C to 500 in both PAMR-1 and PAMR-2, with which the cumulative wealth achieved tends to be stable for the proposed PAMR on most datasets. It is worth noting that these choices for parameters are not always the best. Our experiments on the parameter sensitivity in Sect.  5.4.4 show that the proposed PAMR algorithms are quite robust with respect to different parameter settings.

For the proposed mixture algorithm (MIX), we set the expert pool Footnote 9 as initial uniform combination of PAMR, ONS, Anticor, and B NN , and individual experts are set according to their respective studies.

We adopt the most common metric, cumulative wealth , to primarily compare different trading strategies. In addition to the cumulative wealth, we also adopt annualized Sharpe Ratio (SR) to compare the performance of different trading algorithms. In general, the higher the values of the cumulative wealth, and the annualized Sharpe Ratio, the better the performance of the compared algorithm. Besides, we also adopt Maximum Drawdown (MDD) and Calmar Ratio (CR) for analyzing the downside risk of the PAMR strategy. The lower the MDD value, the more preferable the trading algorithm concerning the downside risk. The higher the CR value, the more performance efficient the trading algorithm concerning the downside risk. The performance criteria are detailed in the following section.

5.2.1 Performance criteria

One of the standard criteria to evaluate the performance of a strategy is portfolio cumulative wealth achieved by the strategy until the end of the whole trading period. In our study, we simply set the initial wealth S 0 =1 and thus the notation S n also denotes portfolio cumulative return at the end of the n th trading day, which is the ratio of the portfolio cumulative wealth divided by the initial wealth. Another equivalent criterion is annualized percentage yield (APY) which takes the compounding effect into account, that is, \({\mathrm{APY}}=\sqrt[y]{\mathbf{S}_{n}}-1\) , where y is the number of years corresponding to n trading days. APY measures the average wealth increment that one strategy could achieve compounded in a year. Typically, the higher the value of portfolio cumulative wealth or annualized percentage yield, the more performance preferable the trading strategy is.

For some process-dependent investors (Moody et al. 1998 ), it is important to evaluate risk and risk-adjusted return of portfolios (Sharpe 1963 , 1994 ). One common way to achieve this is to use annualized standard deviation of daily returns to measure the volatility risk and annualized Sharpe Ratio (SR) to evaluate the risk-adjusted return. For portfolio risk, we calculate the standard deviation of daily returns, and multiply by \(\sqrt{252}\) (here 252 is the average number of annual trading days) to obtain annualized standard deviation. For risk-adjusted return, we calculate annualized Sharpe Ratio according to, \({\mathrm{SR}}=\frac{{\mathrm{APY}}-R_{f}}{\sigma_{p}}\) , where R f is the risk-free return (typically the return of Treasury bills, fixed at 4% in this work), and σ p is the annualized standard deviation of daily returns. Basically, higher annualized Sharpe Ratios indicate better performance of a trading strategy concerning the volatility risk.

The investment community often analyzes DrawDown (DD) (Magdon-Ismail and Atiya 2004 ) to measure the decline from a historical peak in the cumulative wealth achieved by a financial trading strategy. Formally, let S (⋅) denote the process of cumulative wealth achieved by a trading strategy, that is, { S 1 ,…, S t ,…, S n }. The DrawDown at any time t , is defined as DD( t )=max[0,max i ∈(0, t ) S ( i )− S ( t )]. The Maximum DrawDown for a horizon  n , MDD( n ) is defined as, MDD( n )=max t ∈(0, n ) [DD( t )], which is an excellent way to measure the downside risk of different strategies. Moreover, we also adopt Calmar Ratio (CR) to measure the return relative of the drawdown risk of a portfolio, calculated as \({\mathrm{CR}} = \frac{\mathrm{APY}}{\mathrm{MDD}}\) . Generally speaking, the smaller the Maximum DrawDown, the more downside risk tolerable the financial trading strategy. Higher Calmar Ratios indicate better performance of a trading strategy concerning the drawdown risk.

To test whether simple luck can generate the return of the proposed strategy, we can also conduct a statistical test to measure the probability of this situation, as is popularly done in the fund management industry (Grinold and Kahn 1999 ). First, we separate the portfolio daily returns into two components: one benchmark-related and the other non-benchmark-related by regressing the portfolio excess returns Footnote 10 against the benchmark excess returns. Formally, s t − s t (F)= α + β ( s t (B)− s t (F))+ ϵ ( t ), where s t stands for the portfolio daily returns, s t (B) denotes the daily returns of the benchmark (market index) and s t (F) is the daily returns of the risk-free assets (here we simply choose Treasury bill and set it to 1.000156, or equivalently, annual interest of 4%). This regression estimates the portfolio’s alpha ( α ), which indicates the performance of the investment after accounting for the involved risk. Then we conduct a statistical t -test to evaluate whether alpha is significantly different from zero, by using the t statistic \(\frac{\alpha}{\mathrm{SE} (\alpha )}\) , where SE( α ) is the standard error for the estimated alpha. Thus, by assuming the alpha is normally distributed, we can obtain the probability that the returns of the proposed strategy are generated by simple luck. Generally speaking, the smaller the probability, the higher confidence the trading strategy.

5.2.2 Practical issues

While our model described in Sect.  2 is concise and not complicate to understand, it omits some practical issues in the portfolio management industry. We shall now relax some constraints in our model to address these issues.

In reality, an important and unavoidable issue is transaction cost . Generally, there are two ways to handle the transaction costs. The first, commonly adopted by learning to select portfolio strategies, is that the portfolio selection process doesn’t take into account the transaction cost while the following rebalancing incurs transaction costs. The second is that the transaction cost is directly involved in the portfolio selection process (Györfi and Vajda 2008 ). In this work, we take the first way and adopt proportional transaction cost model proposed in Blum and Kalai ( 1999 ) and Borodin et al. ( 2004 ). To be specific, rebalancing the portfolio incurs a transaction cost on every buy and sell operation, based upon a transaction cost rate γ ∈(0,1). At the beginning of the t th trading day, the portfolio manager rebalances the portfolio from the previous closing price adjusted portfolio \({\hat{\mathbf{b}}}_{t-1}\) to a new portfolio b t , incurring a transaction cost of \(\frac{\gamma}{2} \times \sum_{i}{\vert b_{(t,i)}-\hat{b}_{(t-1, i)}\vert }\) , where the initial portfolio is set to (0,…,0). Thus, the cumulative wealth achieved by the end of the n th trading day can be expressed as:

Another practical issue in portfolio selection is margin buying , which allows the portfolio managers to buy securities with cash borrowed from security brokers. Following previous studies (Cover 1991 ; Helmbold et al. 1996 ; Agarwal et al. 2006 ), we relax this constraint in the model and evaluate it empirically in Sect.  5.4.5 . In this study, the margin setting is assumed to be 50% down and 50% loan, at an annual interest rate of 6%, so the interest rate of the borrowed money, c is set to 0.000238. Thus, for each security in the asset pool, a new asset named “Margin Component” is generated. Following the down and loan percentage, the price relative for the “Margin Component” of asset i would be 2∗ x ti −1− c , where x ti is the price relative of the i th asset for the t th trading day. In cases of \(x_{ti}\leq\frac {1+c}{2}\) , that is, certain stocks drop more than half, we simply set “Margin Component” to 0. By adding this “Margin Component”, we magnify both the potential profit and loss of the trading strategy on the i th asset.

5.3 Comparison approaches

In our experiments, we implement the proposed PAMR strategy and its two variants, viz., PAMR-1 and PAMR-2. We compare them with a number of benchmarks and existing strategies as described in Sect.  3 . Below we summarize the list of compared algorithms, whose parameters are set according to the recommendations from their respective studies.

Market: Market strategy, that is, uniform Buy-And-Hold (BAH) strategy;

Best-Stock: Best stock in the market, which is a strategy in hindsight;

BCRP: Best Constant Rebalanced Portfolios strategy in hindsight;

UP: Cover’s Universal Portfolios implemented according to Kalai and Vempala ( 2002 ), where the parameters are set as δ 0 =0.004, δ =0.005, m =100, and S =500;

EG: Exponential Gradient (EG) algorithm with the best parameter η =0.05 as suggested by Helmbold et al. ( 1996 );

ONS: Online Newton Step (ONS) with the parameters suggested by Agarwal et al. ( 2006 ), that is, η =0, β =1, \(\gamma=\frac{1}{8}\) ;

SP: Switching Portfolios with parameter \(\gamma=\frac{1}{4}\) as suggested by Singer ( 1997 );

GRW: Gaussian Random Walk strategy with parameter σ =0.00005 recommended by Levina and Shafer ( 2008 );

M0: Prediction based algorithm M0 with parameter β =0.5 as suggested by Borodin et al. ( 2000 );

Anticor: BAH 30 (Anticor(Anticor)) as a variant of Anticor to smooth the performance, which achieves the best performance among the three solutions proposed by Borodin et al. ( 2004 );

B K : Nonparametric kernel-based moving window (B K ) strategy with W =5, L =10 and threshold c =1.0 which has the best empirical performance according to Györfi et al. ( 2006 );

B NN : Nonparametric nearest neighbor based strategy (B NN ) with parameters W =5, L =10 and \(p_{\ell}=0.02+0.5\frac{\ell-1}{L-1}\) as the authors suggested (Györfi et al. 2008 ).

5.4 Experimental results

5.4.1 experiment 1: evaluation of cumulative wealth.

We first compare the performance of the competing approaches based on their cumulative wealth. From the experimental results shown in Table  4 , we can draw several observations below.

First of all, we observe that learning to select portfolio strategies generally perform better than three common benchmarks, which shows that it is promising to investigate learning algorithms for portfolio selection. Second, we find that although the cumulative wealth achieved by the regret minimization approaches (UP, EG and ONS) is higher than market strategy, their performance is significantly lower than that achieved by the wealth maximization approaches (Anticor, B K and B NN ). This shows that to achieve better investment return, it is more powerful and promising to exploit the wealth maximization approaches for portfolio selection. Third, from the top two results indicated on each original dataset, it is clear that the proposed PAMR strategy (PAMR, PAMR-1, and PAMR-2) significantly outperforms most (except DJIA datasets) competitors including Anticor, B K and B NN , which are the state of the arts. The encouraging results in cumulative wealth validate the importance of exploiting the mean reversion property in the financial markets by an effective online learning strategy. On the other hand, though MIX beats the benchmarks on the DJIA dataset, PAMR algorithms perform bad on the DJIA dataset. This may be attributed to the reason that the motivating mean reversion does not exist in this dataset. This raises an important question, “How to select the portfolio pool such that the motivating mean reversion exists on target portfolio?” Sect.  5.5.2 provides some discussions on this question.

Further examining the details, we find that the most impressive performance is achieved by PAMR on the standard NYSE (O) dataset, where its initial wealth grows by a factor of more than 5 quadrillion at the end of the 22-year period. We note that the main reason PAMR achieved such exceptional results is that it is powerful to exploit highly volatile price relatives. To verify this, we examine the detailed performance of PAMR in Table  4 by looking into individual stocks, and we find that it relies considerably on one single stock (“Kin Ark”) which has the highest volatility in terms of standard deviation. After removing this stock from the portfolio, we find that the cumulative wealth significantly reduces to 1.27E+08. We will investigate the volatility issue in more details by another experiment on dataset sensitivity in Sect.  5.4.3 .

On the reverse datasets, though not performing as shiny as the original datasets, PAMR also performs well. Though some algorithms fail badly, in all cases, PAMR beats the benchmarks, including the market and BCRP strategies. In certain cases, it beats all competitors. It is worth noting these reverse datasets are artificial datasets, which never exist in real markets. PAMR’s performance on these datasets provides strong evidences that mean reversion does exist in even reverse market datasets and PAMR can successfully exploit it.

In addition to the final cumulative wealth, we are also interested in examining how the cumulative wealth changes over different trading periods. Figure  3 shows the trends of the cumulative wealth by the proposed PAMR algorithm and four algorithms (two benchmarks and two state-of-the-art algorithms). From the results, we can see that the proposed PAMR strategy consistently surpasses the benchmarks and the competing strategies over the entire trading period on most datasets (except DJIA dataset), which again validates the efficacy of the proposed technique.

Trends of cumulative wealth achieved by various strategies during the entire trading periods on the stock datasets

Finally, to measure whether the excess return can be simply obtained by luck, we conduct a statistical t -test as described in Sect.  5.2.1 . Table  5 shows the statistical results, which clearly show that the observed excess return is impossible to obtain by simple luck in most datasets. To be specific, the probabilities for achieving the excess returns by luck are almost 0 on datasets except DJIA. However, the statistics on DJIA dataset show that in this dataset, the assumption of mean reversion may not exist. Nevertheless, the results show that the PAMR strategy is a promising and reliable portfolio selection technique to achieve high return with high confidence.

5.4.2 Experiment 2: evaluation of risk and risk-adjusted return

We now evaluate the risk in terms of volatility risk and drawdown risk, and the risk-adjusted return in terms of annualized Sharpe ratio and Calmar ratio. Figure  4 shows the evaluation results on the six datasets. In addition to the proposed PAMR, we also plot two benchmarks (Market and BCRP) and two state-of-the-art algorithms (Anticor and B NN ) for comparison. As shown in Fig.  4 , Figs.  4 (a) and 4 (b) depict the volatility risk (standard deviation of daily returns) and the drawdown risk (maximum drawdown) on the six stock datasets. Figures  4 (c) and 4 (d) compare the corresponding Sharpe ratio and Calmar ratio.

Risk and risk-adjusted performance of various strategies on the six different datasets. In each diagram , the rightmost bars represent the results achieved by PAMR

In previous cumulative wealth results, we find that PAMR achieved the highest cumulative return on most original datasets. Of course, high return is associated with high risk, which is commonly acceptable in finance, as no real financial instrument can guarantee a high return without risk. The volatility risk in Fig.  4 (a) shows that PAMR almost achieves the highest risk in terms of volatility risk. On the other hand, the drawdown risk in Fig.  4 (b) shows that PAMR achieves modest drawdown risk in most datasets. These results validate the above notion that high return is often associated with high risk.

To further evaluate the return and risk, we examine the risk-adjusted return in terms of annualized Sharpe ratio and Calmar ratio. The results shown in Figs.  4 (c) and  4 (d) clearly show that PAMR achieves excellent performance in most cases, except DJIA dataset. These encouraging results show that PAMR is able to reach a good trade-off between return and risk, even though we do not explicitly consider risk in our problem formulation.

5.4.3 Experiment 3: dataset sensitivity

As observed in Sect.  5.4.1 , it is interesting that PAMR gained the excess return from the stock markets. In this section, we aim to examine how the dataset sensitivity affects the proposed PAMR strategy by evaluating performance on datasets of different volatilities.

To examine the effect of the dataset volatility, we create two datasets each consisting of 5 stocks, chosen from NYSE (N) dataset according to their volatility values. To be specific, we ranked the 23 stocks based on their daily volatility values measured by standard deviation of the logarithm of the price relatives (Hull 2008 ). Then we created two datasets of different volatility: NYSE (H) and NYSE (L), each consisting of 5 stocks of the highest and lowest volatility values, respectively. Table  6 shows the results achieved by various strategies on these two datasets.

From the results, we find that different strategies perform diversely on these two datasets. The regret minimization approaches (UP, EG and ONS), perform well regardless of the market volatilities as the theoretical universal property shows, while the wealth maximization approaches (Anticor, B K and B NN ) and the proposed PAMR strategy achieved significantly higher cumulative wealth on NYSE (H), the high-volatility dataset. These results show that the volatility of datasets does considerably affect some algorithms, including the wealth maximization approaches and the proposed PAMR strategy. Specifically, we find that the proposed PAMR strategy could benefit much from a high-volatility dataset. For example, on the NYSE (L) dataset, the cumulative wealth achieved by PAMR algorithm is about 132, which is significantly boosted to 1.35E+05 on the NYSE (H) dataset. To further examine which algorithm can benefit most from high-volatility dataset, we calculate the “H/L ratio” value, which is the ratio of cumulative wealth achieved on the high-volatility dataset over that achieved on the low-volatility dataset. From the ratios, we can observe that the PAMR strategy obtained the highest H/L ratio, indicating that PAMR can benefit most from the high-volatility dataset among all the competing methods.

5.4.4 Experiment 4: parameter sensitivity

We now evaluate how different choices of parameters affect the performance of the proposed PAMR strategy. All three PAMR algorithms require to set sensitivity parameter ϵ , while aggressiveness parameter C is needed for PAMR-1 and PAMR-2.

First, we examine the effect of the sensitivity parameter ϵ on the cumulative wealth achieved by PAMR. As ϵ becomes greater than 1, PAMR degrades to uniform CRP strategy and the wealth stabilizes at the wealth achieved by uniform CRP. Thus, we evaluate the effect of ϵ in the range of [0,1.5]. Figure  5 shows the cumulative wealth achieved by PAMR with varying ϵ and those of the two benchmarks, that is, Market and BCRP strategies. Most results, besides DJIA dataset, show that the cumulative wealth achieved by PAMR grows as ϵ approaches 0, that is, the more sensitive the higher the wealth, which validates that the motivating mean reversion does exist on the stock markets. Moreover, in most cases, the cumulative wealth achieved by PAMR tends to stabilize as ϵ crosses certain dataset dependent thresholds. As stated before, we choose ϵ =0.5 in the experiments, with which the cumulative wealth becomes stabilized in most cases. We also note that on some datasets PAMR with ϵ =0 achieves the best. Though ϵ =0 means moving more weights to the worse performing stocks, it may not mean moving everything to the worst stock. On the one hand, the objectives in the formulations would prevent next portfolio far from last portfolio. On the other hand, PAMR-1 and PAMR-2 are designed to alleviate the huge changes. In a word, this experimental results clearly show that the proposed algorithm is robust with respect to the mean reversion sensitivity parameter. On the other side, for the failing case, DJIA, the mean reversion effect is different. As ϵ approaches 0, the cumulative wealth achieved by PAMR drops. This phenomena can be interpreted as that the motivating mean reversion does not exist in the DJIA dataset, at least in the sense of our motivation.

Parameter sensitivity of the cumulative wealth achieved by PAMR with respect to sensitivity parameter ϵ

Second, we evaluate the other important parameter for both PAMR-1 and PAMR-2 algorithms, that is, aggressiveness parameter C . Figures  6 and 7 show the effects on the cumulative wealth with varying sensitivity parameter ϵ from 0 to 1.5 and aggressiveness parameter C from 50 to 5000, on PAMR-1 and PAMR-2, respectively. Each heat map indicates the cumulative wealth achieved by PAMR with different C and ϵ combination. The indication bar on the right side of each heat map illustrates that each color represents a level of cumulative wealth achieved. It is clear that in most cases, except DJIA, we observe that as ϵ decreases and C increases, the cumulative wealth increases and then stabilizes as ϵ and C cross certain data-dependent thresholds. Moreover, we find C does not have a significant effect on the cumulative wealth achieved. We also find that the proposed PAMR algorithms are not so parameter sensitive, since a wide range of values correspond to the highest cumulative wealth. This again exhibits that the proposed PAMR strategy is robust with respect to its parameters. Similarly, the heat map on DJIA again shows that the mean reversion effect does not exist on the dataset, in the sense of our motivation.

Parameter sensitivity of the cumulative wealth achieved by PAMR-1 with respect to sensitivity parameter ϵ and aggressiveness parameter C

Parameter sensitivity of the cumulative wealth achieved by PAMR-2 with respect to sensitivity parameter ϵ and aggressiveness parameter C

5.4.5 Experiment 5: evaluation of practical issues

For a real-world application, there are some important practical issues for portfolio selection, including the issues of transaction cost and margin buying. This experiment aims to examine how these practical issues affect the proposed PAMR strategy.

First, transaction cost is an important and unavoidable issue that should be addressed in practice. In our experiment, we adopt proportional transaction cost model stated in Sect.  5.2.2 to test the effect of the transaction cost on the proposed PAMR strategy. Figure  8 depicts the effect of proportional transaction cost when PAMR is applied on the six datasets, where the transaction cost rate γ varies from 0 to 1%. We only present the results achieved by PAMR since the effect of its variants, that is, PAMR-1 and PAMR-2, is quite similar to that of PAMR. For comparison, we also plot the results achieved by two state-of-the-art strategies (Anticor and B NN ) and the cumulative wealth achieved by the two benchmarks (BCRP and Market). Since BCRP is the target strategy for regret minimization approaches (UP, EG and ONS) and for consistency, we do not plot the results achieved by these approaches.

Scalability of the cumulative wealth achieved by PAMR with respect to transaction cost rate ( γ ). The break-even transaction cost rates to the market index are about 0.7%, 0.4%, 0.1%, 0.3% and 0% on the six datasets, respectively

From the results shown in the figure, we can observe that PAMR can withstand reasonable transaction cost rates. For example, with a transaction cost rate of 0.2%, PAMR can beat the BCRP strategy on the four datasets. The break-even transaction cost rates with respect to the market index ranges from 0.1% to 0.7% on the datasets, except DJIA. Since PAMR more actively reverts to the mean and thus results in more drastic portfolio changes, it surpasses Anticor with low or medium transaction costs while it underperforms Anticor with high transaction costs, On the other hand, it outperforms B NN in most cases. Note that the transaction cost rate in real market is low. Footnote 11 This experiment clearly shows the practical applicability of the proposed PAMR strategy when we take transaction cost into consideration.

Second, margin buying is another practical concern for a real-world portfolio selection task. In the following, we evaluate the performance of the approaches when margin buying is allowed with the model described in Sect.  5.2.2 . Table  7 presents the cumulative wealth achieved by the competing approaches without/with margin loans on the six stock datasets. As we can observe, when margin buying is allowed, the profitability of PAMR increases, and in most cases, it achieves higher cumulative wealth than other competing approaches. These results clearly demonstrate that the proposed PAMR strategy can be extended to handle margin buying issue and benefit from margin buying, and thus has a better practical applicability.

5.4.6 Experiment 6: evaluation of computational time cost

Our last experiment is to evaluate the computational time costs of different approaches, which is also an important issue in developing a practical online trading strategy. As stated in Sect.  4.3 , the proposed PAMR algorithm enjoys linear time complexity per iteration, which is comparable to EG algorithm. Table  8 presents the computational time cost (in seconds) of the performance comparable approaches (Anticor, B K and B NN ) on the six stock datasets. All the experiments were conducted on an Intel Core 2 Quad 2.66 GHz processor with 4 GB RAM, using Matlab 2009b on Windows XP.

From the results, we can clearly see that in all cases the proposed PAMR takes significant less computational time than the three performance comparable strategies. Even though the computational time in the back tests, especially per trading day, is small, it is important in certain scenarios such as high frequency trading (Aldridge 2009 ), where transactions may occur in a fraction of a second. Nevertheless, the results clearly demonstrate the computational efficiency of the proposed PAMR strategy, which is also an important concern for real-world large-scale applications.

5.5 Discussions and threads to validity

5.5.1 discussion on model assumption.

Any statement about such encouraging empirical results would be incomplete without acknowledging the simplified assumptions made in Sect.  2 . To recall, we had made several assumptions regarding transaction cost, market liquidity and market impact, which would affect the practical deployment of the proposed algorithm.

The first assumption is that no transaction cost exists. In Sect.  5.4.5 we have already examined the effect of varying transaction costs, and the results show that the proposed algorithm can withstand moderate transaction costs. Currently, with the wide-spread adoption of electronic communication networks (ECNs) and multilateral trading facilities (MTFs) on financial markets, various online trading brokers charge very small transaction cost rates, especially for large institutional investors. They also use a flat-rate, Footnote 12 based on the volume threshold one reaches. Such measures can facilitate the portfolio managers to lower their transaction cost rates.

The second assumption is that the market is liquid and one can buy and sell any quantity at the quoted price. In practice, low market liquidity results in a large bid-ask spread —the gap between prices quoted for an immediate bid and an immediate ask. As a result, the execution of orders may incur a discrepancy between the prices sent by the algorithm and the prices actually executed. Moreover, stocks are often traded in multiples of lot , which is the standard trading unit containing certain number of stock shares. In this situation, the quantity of the stocks may not be arbitrary divisible. In the experiments, we have tried to minimize the effect of market liquidity by choosing the stocks that have large market capitalization, which usually have small bid-ask spreads and discrepancy, and thus have a high market liquidity.

The other assumption is that the portfolio strategy would have no impact on the market, that is, the stock market will not be affected by the trading algorithm. In practice, the impact can be neglected if the market capitalization of the portfolio is not too large. However, as the experimental results show, the portfolio wealth generated by PAMR increases astronomically, which would inevitably impact the market. One simple way to handle this issue is to scale down the portfolio, as done by many quantitative funds. Moreover, the development of algorithmic trading, which slices a big order into multiple smaller orders and schedules these orders to minimize the market impact, can significantly decrease the potential market impact of the proposed algorithm.

Here, we emphasize again that this study assumes a “perfect market”, which is consistent with previous studies in literature. It is important to note that even in such a perfect financial market, no algorithm has ever claimed such high performance, especially on the standard NYSE (O) dataset. Though it is common investment knowledge that past performance may not be reliable indicator of future performance, such high performance does provide us confidence that the proposed PAMR algorithm may work well in future unseen markets.

5.5.2 Discussion on PAMR assumption

Though the proposed algorithm performs well on most datasets, we can not claim that PAMR can perform well on arbitrary portfolio pools. It is worth noting that PAMR relies on the assumption that mean reversion exists in a portfolio pool, that is, buying worse performing stocks is profitable. Preceding experiments seem to show that in most cases mean reversion does exist in the market. However, it is still possible that this assumption fails to exist in certain cases, especially when portfolio components are wrongly selected. PAMR’s performance on DJIA dataset indicates that mean reversion may not exist in its portfolio components. Though both based on mean reversion, PAMR and Anticor are formulated with different time periods of mean reversion, which may interpret why Anticor achieves a good performance on DJIA. Thus before investing in real market, it is of crucial importance to ensure that the motivating mean reversion does exist among the portfolio pools. In academic, mean reversion property in single stock has been extensively studied (Poterba and Summers 1988 ; Hillebrand 2003 ; Exley et al. 2004 ), one natural way is to calculate the sign of auto-correlation (Poterba and Summers 1988 ). On the contrary, the mean reversion property among a portfolio lacks academic attention. Compared with mean reversion in single stock, for a portfolio, not only the mean reversion of single stock matters, but rather the interaction among stocks matters.

On the other hand, the mixture algorithm, that is, MIX, performs well on the DJIA dataset, beating three benchmarks. As we discussed in Sect.  4.6 , the mixture algorithm can provide a worst-case guarantee, which is lacked for the original PAMR algorithms. This can somehow solve the problem that PAMR itself does not have a worst-case guarantee. Moreover, it is worth noting that even with worst-case guarantee, some existing universal algorithms also perform poorly on the dataset.

Now let us briefly analyze the reason that PAMR failed on DJIA. To test whether mean reversion exists in the DJIA dataset, we propose a naïve trading strategy to test our motivating mean reversion in the dataset. The test strategy sets the weights proportional to differences between assets’ returns and that of last best stock, that is, last best stock will be given zero weight, while the worst performing stock will be given a maximum weight. We are interested in whether this simple algorithm produces positive return among existing datasets. If it produces positive daily return, then the assumption that buying worse stocks may work well. Otherwise, our motivating assumption fails. The test is conducted on all six datasets. We calculated their arithmetic average daily returns and their standard deviations of daily returns. Since we are interested in absolution return, we compare their average values with 1. From the statistics in Table  9 , we can find that the five successful datasets release average profit (>1.0), while DJIA releases average loss (<1.0). Thus, on DJIA dataset, it is expected to produce losses by purchasing worse performing stocks in the portfolio. Though expected daily loss is small, it would produce huge cumulative loss with a long trading period.

It is interesting to observe above results, however, we cannot claim that this method can definitely identify successful portfolio pools. Analyzing the mean reversion property in portfolio scenario and selecting portfolio components such that the portfolio satisfies mean reversion deserve further attention.

5.5.3 Discussion on back tests

Back tests in historical markets may suffer from “data-snooping bias” issue. One common “data-snooping bias” is dataset selection issue. On the one hand, we selected four datasets, that is, NYSE (O), TSE, SP500, and DJIA datasets, based on previous studies without consideration to the proposed approach. On the other hand, we developed the PAMR algorithm based solely on NYSE (O) dataset, while other five datasets (NYSE (N), TSE, SP500, MSCI and DJIA datasets) were obtained after the algorithm was fully developed. However, even we are cautious about the dataset selection issue, it may still appear in the experiments, especially for the datasets with relatively long history, that is, NYSE (O) and NYSE (N). The NYSE (O) dataset, pioneered by Cover ( 1991 ) and followed by other researchers, becomes one “standard” dataset in the learning community. Since it contains 36 large cap NYSE stocks that survived in hindsight for 22 years, thus it suffers from extreme survival bias. Nevertheless, it still has the merit to compare the performance among algorithms as done in all previous work. The NYSE (N) dataset, as a continuation of NYSE (O), contains 23 assets survived from previous 36 stocks for another 25 years. Therefore, it becomes even worse than the previous NYSE (O) dataset in terms of survival bias. In a word, even the experiment results on these datasets clearly show the effectiveness of the proposed PAMR algorithm, one can not make claims without noticing the deficiencies of these datasets.

Another common bias is asset selection issue. Four of the six datasets (NYSE (O), TSE, SP500, and DJIA) are collected by others, and to the best of our knowledge, their assets are mainly the largest blue chip stocks in their respective markets. As a continuation of NYSE (O) dataset, we self-collected NYSE (N) , which again contains several largest survival stocks in NYSE (O). The remaining dataset (MSCI) is chosen according to the world indices. In a word, we try to avoid the asset selection bias via arbitrarily choosing the representative stocks in their respective markets, which usually have large capitalization and thus high liquidity. Moreover, investing in these largest assets may reduce the market impact caused by the proposed portfolio strategy. Finally, following existing model assumption and experimental setting, we do not consider the assets of low quality, such as the bankrupt stocks and penny stocks. On the one hand, the bankrupt stock data is difficult to acquire, thus we cannot observe their behaviors and predict the behaviors of PAMR on datasets with bankrupt stocks. In reality, the bankruptcy situation happens rarely for the blue chip stocks as typically a bankrupt stock would be removed from the list of blue chip stocks before it actually goes bankruptcy. On the other hand, the penny stocks lack the required liquidity to support the trading frequency in current research. Besides, one could also explore many practical strategies to exclude the low quality stocks from the asset pool at some early stage, such as some financial methods via either technical or fundamental analysis.

6 Conclusion

In this article, we proposed a novel portfolio selection strategy, “Passive Aggressive Mean Reversion” (PAMR). Motivated by the idea of mean reversion and passive aggressive learning, PAMR outperforms all benchmarks and various existing strategies on a number of real datasets from different markets. PAMR can also be easily extended to handle certain practical issues, e.g., transaction cost and margin buying. At the same time, PAMR executes in much less time than existing approaches, making it suitable for online applications. We also find that the update scheme of PAMR is based on the trade-off between the return and volatility risk, which is ignored by most existing learning strategies. This interesting property connects the PAMR strategy with modern portfolio theory, which may provide further explanation from the aspect of finance.

Although in most cases the proposed PAMR strategy achieves encouraging empirical results, it is still far from perfect for a real investment task, and may be improved in the following aspects. First of all, though universality may not be required in real investment, PAMR’s universality is still an open question. Second, none of existing algorithms considers the bankrupt assets, which may happen in real investment. It is thus interesting to study the behaviors of the bankrupt assets and design strategies to exploit them. Besides, we note that PAMR sometimes fails when the mean reversion property does not exist in the portfolio components. Then it is crucial to propose efficient methods to test mean reversion. Finally, though PAMR handles the issue of transaction costs well, it is not formally addressed in our problem formulation. It would be interesting to incorporate the transaction cost issue when formulating the problem in order to improve the performance in case of high transaction costs and gain higher break-even ratios with respect to the market index.

Side information includes interest rates, consumer confidence figures, etc.

The precise matlab routine ProjectOntoSimplex can be found on http://www.cs.berkeley.edu/~jduchi/projects/DuchiShSiCh08/ .

Such statement also appeared in footnote 1 of Borodin et al. ( 2004 ).

All the datasets and their compositions can be downloaded from http://www.cais.ntu.edu.sg/~libin/portfolios . Borodin et al. ( 2004 )’s datasets can also be downloaded from http://www.cs.technion.ac.il/~rani/portfolios/ .

According to Helmbold et al. ( 1996 ), the dataset was originally collected by Hal Stern. The stocks are mainly large cap stocks in NYSE, however, we do no know the criteria of choosing these 36 stocks.

The dataset before 2007 was collected by Gábor Gelencsér ( http://www.cs.bme.hu/~oti/portfolio ), we collected the remaining data from 2007 to 2010 via Yahoo Finance.

The constituents of MSCI World Index can be found from MSCI Barra ( http://www.mscibarra.com ), accessed on 28 May 2010.

It is worth noting that not every index is tradable through exchange traded funds (ETFs).

One can arbitrarily select experts, however, at least one universal algorithm should be included in order to guarantee the worst-case performance of the mixture algorithm.

Excess return is daily return less risk-free return.

For example, without consideration taxes and bid-ask, Interactive Broker charges 0.005$ per share traded. Considering the average price of Dow Jones Composite is around 50$ (accessed on June 2011), the percentage is about 0.01%.

For example, for US equities and options, E ∗ Trade ( https://global.etrade.com/gl/home , accessed on 16 March 2011) charges only $9.99 for $50000+ or 30+ stocks per quarter.

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Acknowledgements

This paper was fully supported by Singapore MOE Tier-1 Research Grant (RG67/07).

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Appendix A: Proof of Proposition 1

First, if \(\ell_{\epsilon}^{t}=0\) then b t satisfies the constraint in ( 2 ) and is clearly the optimal solution.

Now let us focus on the case where \(\ell_{\epsilon}^{t} \neq0\) . To solve the problem, we define the Lagrangian of the optimization problem in ( 2 ) to be,

where τ ≥0 is a Lagrange multiplier related to the loss function, λ is the Lagrange multiplier associated with the simplex constraint, and 1 denotes the column vector of m 1s. Note that the non-negativity of portfolio b is not considered here since introducing this term causes too much complexity, and alternatively we project the resulting portfolio into a simplex to enforce the non-negativity constraint.

Setting the partial derivatives of \({\mathcal{L}}\) with respect to the elements of b to zero gives,

Multiplying both sides with 1 ⊤ , and b ⊤ 1 =1, we can get \(\lambda=-\frac{\tau}{m}{{\mathbf{x}}_{t} \cdot {\mathbf{1}}}\) . Moreover, since \(\bar{{{x}}}_{t}=\frac{{\mathbf{x}}_{t} \cdot{\mathbf{1}}}{m}\) , where \(\bar{{{x}}}_{t}\) is the mean of the t th asset price relative, or the market return, we can rewrite λ in the following form,

And the solution for \({\mathcal{L}}\) is,

Plugging ( 10 ) and ( 11 ) to ( 9 ), we get,

Note that in the derivation of the above formula, we used the following formula, that is,

Setting the derivative of \({\mathcal{L}}(\tau)\) with respect to τ to 0, we get,

Then τ can be set to the following formula,

Since τ ≥0, we project τ to [0,∞), thus,

Note that in case of zero market volatility, that is, \(\Vert {\mathbf{x}}_{t}-\bar{{{x}}}_{t}{\mathbf{1}} \Vert ^{2}=0\) , we just set τ =0. And we can state the update scheme for the case where \(\ell_{\epsilon}^{t}=0\) and the case where \(\ell_{\epsilon }^{t}>0\) by setting  τ . Thus, we simplify the notation according to ( 1 ) and show the unified update scheme. □

Appendix B: Proof of Proposition 2

We derive the solution of PAMR-1 following the same procedure as the derivation of PAMR. Let us consider the situation when the loss is not 0 and we get the Lagrangian,

Multiply both side with 1 ⊤ , and b ⊤ 1 =1, we can get, \(\lambda= -{\tau}\frac{{\mathbf{x}}_{t} \cdot {\mathbf{1}}}{m}=-\tau{\bar{{\mathbf{x}}}_{t}}\) . And the approximation solution is

Next, note that the minimum of the term ξ ( C − τ − μ ) with respect to ξ is zero whenever C − τ − μ =0. If C − τ − μ ≠0 then the minimum can be made to approach −∞. Since we need to maximize the dual we can rule out the latter case and pose the following constraint on the dual variables, C − τ − μ =0. The KKT conditions confine μ to be non-negative so we conclude that τ ≤ C . We can project τ to the interval [0, C ] and get,

Also note that we simplify the notation according to ( 1 ) and show the unified update scheme. □

Appendix C: Proof of Proposition 3

We derive the solution following the derivations of the PAMR and PAMR-1. Let us focus on the situation when the loss is not 0 and we can get the Lagrangian,

Multiply both side with 1 ⊤ , and b ⊤ 1 =1, we can get, \(\lambda= -{\tau}\frac{{\mathbf{x}}_{t} \cdot {\mathbf{1}}}{m}=-\tau{\bar{{x}}}\) . And the approximation solution is,

Setting the partial derivatives of \({\mathcal{L}}\) with respect to the elements of ξ to zero gives,

Expressing ξ as above and replacing b , we rewrite the Lagrangian as

Take the derivative with respect to τ and set it to zero, we can get,

Then we get the update scheme of τ , and project it to [0,∞)

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Li, B., Zhao, P., Hoi, S.C.H. et al. PAMR: Passive aggressive mean reversion strategy for portfolio selection. Mach Learn 87 , 221–258 (2012). https://doi.org/10.1007/s10994-012-5281-z

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DOI : https://doi.org/10.1007/s10994-012-5281-z

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Seasonality, Trend-following, and Mean reversion in Bitcoin

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