Empirical investigation of state-of-the-art mean reversion strategies for equity markets
aa r X i v : . [ q -f i n . P M ] S e p E MPIR ICAL INVESTIGATION OF STATE - OF - THE - ART MEANR EVER SION STR ATEGIES FOR EQUITY MAR KETS
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Seung-Hyun Moon
School of Computer Science & EngineeringSeoul National University1 Gwanak-ro, Gwanak-gu, Seoul 08826Republic of Korea [email protected]
Yong-Hyuk Kim
School of SoftwareKwangwoon University20 Kwangwoon-ro, Nowon-gu, Seoul 01897Republic of Korea [email protected]
Byung-Ro Moon
School of Computer Science & EngineeringSeoul National University1 Gwanak-ro, Gwanak-gu, Seoul 08826Republic of Korea [email protected]
September 11, 2019 A BSTRACT
Recent studies have shown that online portfolio selection strategies that exploit the mean reversionproperty can achieve excess return from equity markets. This paper empirically investigates the per-formance of state-of-the-art mean reversion strategies on real market data. The aims of the study aretwofold. The first is to find out why the mean reversion strategies perform extremely well on well-known benchmark datasets, and the second is to test whether or not the mean reversion strategieswork well on recent market data. The mean reversion strategies used in this study are the passive ag-gressive mean reversion (PAMR) strategy, the on-line moving average reversion (OLMAR) strategy,and the transaction cost optimization (TCO) strategies. To test the strategies, we use the historicalprices of the stocks that constitute S&P 500 index over the period from 2000 to 2017 as well as well-known benchmark datasets. Our findings are that the well-known benchmark datasets favor meanreversion strategies, and mean reversion strategies may fail even in favorable market conditions,especially when there exist explicit or implicit transaction costs. K eywords Online portfolio selection, Rebalancing investments, Mean reversion.
Mean reversion is a tendency to move back to the average. It implies that a stock with a recent above-average returnis subject to produce a below-average return during the next period, and vice versa. Whether or not such a tendencyexists in stock price movements has been controversial in recent decades. Fama and French (1988) find negativeautocorrelations of stock returns, Balvers et al. (2000) discover strong evidence of mean reversion in national equityindexes, and Gropp (2004) finds evidences of mean reversion in portfolio prices. However, Kim et al. (1991) find noevidence of mean reversion after the Second World War, and Booth et al. (2016) show that there is no momentum-reversal anomaly in the U.S. stock returns from 1962 to 2013.Meanwhile, there have been a significant improvement in online portfolio selection. The online portfolio selectioninvolves selling securities that are expected to fall in price and buying securities that are expected to rise within aportfolio. Only past price information can be used to determine the proportion of assets in the portfolio. Althoughmpirical investigation of state-of-the-art mean reversion strategies for equity markets
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REPRINT the framework of well-performing strategies can be used to predict the price movements of multiple assets (Roch,2013), the natural objective of online portfolio selection is to maximize the cumulative wealth of an investor. Refer toLi and Hoi (2014) for a comprehensive survey on online portfolio selection.The most widely used dataset for online portfolio selection is the NYSE(O) dataset, which includes the historicaldaily prices of 36 stocks listed in the New York Stock Exchange (NYSE) for 22 years. Various strategies have beenproposed and tested using the dataset (Agarwal et al., 2006; Borodin et al., 2004; Gy¨orfi et al., 2006; Helmbold et al.,1998; Huang et al., 2018; Levina and Shafer, 2008; Li et al., 2012, 2013, 2015, 2018; Singer, 1997). However, it isworthwhile to test the strategies with as many datasets as possible to see how well they perform on unseen data.Recently, three kinds of mean reversion strategies that outperform other state-of-the-art strategies have been developedusing machine learning techniques. One is the passive aggressive mean reversion (PAMR) strategy (Li et al., 2012),another is the online moving average reversion (OLMAR) strategy (Li et al., 2015), and the other is the transactioncost optimization (TCO) strategies (Li et al., 2018). The PAMR exploits the single-period mean reversion propertyand the OLMAR exploits the multi-period mean reversion property. The TCO can exploit the single-period or themulti-period mean reversion property depending on the settings. While a simple buy-and-hold strategy increases itswealth by 14 times, the three strategies increase the wealth by over ten trillion times on the NYSE(O) dataset.In this paper, we first attempt to find out why mean reversion strategies work extremely well on well-known benchmarkdatasets. For this aim, we devise simple mean reversion strategies that behave similarly to the state-of-the-art meanreversion strategies. The strategies we devised are much easier to comprehend than their counterparts so that we canrealize the property of the datasets on which mean reversion strategies perform well. The performance of the simplestrategies on the benchmark datasets reveal the characteristic feature of the datasets.On well-known benchmark datasets, Li and Hoi (2015) report that OLMAR can outperform the other strategies with ahigh confidence, and Li et al. (2018) state that TCO can effectively handle reasonable transaction costs. However, it iswondered whether or not the strategies can be successful on other datasets. If they work well on most datasets, it is athreat to the efficient-market hypothesis, which implies that it is impossible to beat the market. Thus, we test whetheror not the mean reversion strategies work well on recent market data. Our empirical results show the potential pitfallsof the state-of-the-art mean reversion strategies.
Suppose that we have m stocks to trade for n days. The closing prices of the stocks on the t th day is denoted by p t =( p t (1) , p t (2) , ..., p t ( m )) ∈ R m + , where p t ( j ) is the closing price of the j th stock, and the daily return of the stocks onthe t th day is denoted by a price relative vector, x t = ( x t (1) , x t (2) , ..., x t ( m )) ∈ R m + , where x t ( j ) = p t ( j ) /p t − ( j ) .At the beginning of the t th day, we specify a portfolio vector, b t = ( b t (1) , b t (2) , ..., b t ( m )) ∈ R m + , where b t ( j ) is theproportion of the j th stock to our wealth and k b t k = 1 . After x t is revealed, the daily return on the t th day is b t · x t ,and compound return for n days is Q nt =1 b t · x t . The aim of online portfolio selection is to maximize the compoundreturn. It is assumed that all stocks are arbitrarily divisible and we can trade them at their last closing prices.Let BAH b denote a buy-and-hold (BAH) strategy, which allocates investor’s initial wealth according to the portfoliovector b on the first trading day and does not make any transactions until the last trading day. When the portfolio vectorof BAH is uniform, i.e., b = (1 /m, ..., /m ) , the BAH strategy is denoted by BAH U . Let CRP b denote a constantrebalanced portfolio (CRP), which reallocates investor’s wealth according to the portfolio vector b at the beginningof every trading day. When the portfolio vector of CRP is uniform, i.e., b = (1 /m, ..., /m ) , the CRP is denoted by CRP U . In spite of its simplicity, CRP U often outperforms BAH U and other more sophisticated rebalancing strategiessuch as universal portfolios (Agarwal et al., 2006; Borodin et al., 2004; Cover, 1991). The CRP U exploits the meanreversion property of stock market since it sells stocks that performed above average and buys stocks that performedbelow average on the previous trading day. The performances of BAH U and CRP U provide benchmarks for thisstudy.The PAMR is a mean reversion strategy that exploits the single-period mean reversion. The PAMR utilizes the passiveaggressive online learning algorithm (Crammer et al., 2006) to find the mean revertible portfolio vector that is as closeas possible to the current vector, i.e., b t +1 = arg min b ∈ R m + k b − b t k subject to b · x t ≤ ǫ and k b k = 1 , (1)where ǫ ∈ R + is a user-defined parameter. We set ǫ to 0.5 which is the default value in Li et al. (2012).2mpirical investigation of state-of-the-art mean reversion strategies for equity markets A P
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The PAMR is a single-period mean reversion strategy since its portfolio vector depends on x t , the latest price relativevector. The strategy finds a portfolio vector whose latest return is confined to ǫ in order to benefit from the single-period reversion ( b · x t ≤ ǫ ). To better understand why the strategy works well, we devise a simple mean reversion(SMR) strategy that exploits the single-period mean reversion in the simplest way. The SMR puts all the money intothe worst performer of the previous day. If there are multiple stocks that satisfy the investment criterion, we evenlybuy all the stocks that satisfy the criterion, i.e., b t +1 = arg min b ∈ R m + b · x t subject to m X i =1 m X j =1 | b ( i ) − b ( j ) | I [ x t ( i ) = x t ( j )] = 0 and k b k = 1 , (2)where I [ · ] is the indicator function.Another state-of-the-art mean reversion strategy is OLMAR which exploits the multi-period mean reversion property.Instead of using the latest price relative vector, the OLMAR employs a simple moving average (SMA) to determinea portfolio vector. The simple moving average is the unweighted mean of the closing prices. The SMA t ( w ) ∈ R m + represents the SMA on the t th day, i.e., SMA t ( w ) = 1 w t X i = t − w +1 p i , (3)where w ∈ Z + is the window size. The OLMAR finds the mean revertible portfolio vector that is as close as possibleto the current vector, and thus has a similar formulation to PAMR, i.e., b t +1 = arg min b ∈ R m + k b − b t k subject to b · ˜ x t +1 ≥ ǫ and k b k = 1 , (4)where ǫ ∈ R + is a user-defined parameter and ˜ x t +1 ∈ R m + is the predicted price relative which can be calculated when p t is revealed. The OLMAR uses the SMA to calculate the predicted price relative, i.e., ˜ x t +1 ( w ) = SMA t ( w ) ⊘ p t , (5)where ⊘ denotes an element-wise division. We set ǫ to 10 and w to 5, which are the default values used in Li et al.(2015).Under the assumption of the multi-period mean reversion, current price tends to revert to the mean. Thus, the valueof the predicted price relative represents the likelihood of gaining a profit. The higher the value of the predicted pricerelative, the higher the chance of profit. To test whether the assumption of the OLMAR holds, we devise a simplemoving average reversion (SMAR) strategy, which invests into the stock that has the highest predicted price relative.If there are multiple stocks that satisfy the criterion, we evenly buy all the stocks that satisfy the criterion, i.e., b t +1 = arg max b ∈ R m + b · ˜x t +1 subject to m X i =1 m X j =1 | b ( i ) − b ( j ) | I [˜ x t +1 ( i ) = ˜ x t +1 ( j )] = 0 and k b k = 1 . (6)The SMAR uses the SMA to calculate ˜x t +1 in the same way as the OLMAR.The other state-of-the-art mean reversion strategy is TCO, which is devised to perform well in the case of non-zerotransaction costs. Since insignificant trades can incur unnecessary transaction costs, the TCO only accepts changeslarger than a certain threshold in the portfolio vector. Let ˆ b t ∈ R m + be the last price adjusted portfolio vector and ˜ b t + ∈ R m be the weights to be adjusted after the t th trading day, then the TCO uses the predicted price relative todetermine ˜ b t + , i.e., ˜ b t + = η (cid:18) ˜ x t +1 ˆ b t · ˜ x t +1 − m · ˜ x t +1 ˆ b t · ˜x t +1 (cid:19) , (7)3mpirical investigation of state-of-the-art mean reversion strategies for equity markets A P
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Table 1: Summary of the first set of historical daily data.
Name Period Days Assets a Max( x t ( j ) ) b Min( x t ( j ) ) c NYSE(O) 1962 - 1984 5651 36 1.3529 0.7500NYSE(N) 1985 - 2010 6431 23 1.8146 0.4545DJIA 2001 - 2003 507 30 1.2012 0.4027SP500 1998 - 2003 1276 25 1.2439 0.6976TSE 1994 - 1998 1259 88 1.9392 0.3685MSCI 2006 - 2010 1043 24 1.1663 0.8274 a The number of assets in the portfolio. a The highest daily return of a single asset on the dataset. b The lowest daily return of a single asset on the dataset. where ˆ b t = ( b t ⊙ x t ) / ( b t · x t ) , ⊙ is an element-wise product, and η ∈ R + is a user-defined parameter. The negativecomponents of ˜ b t + have lower predicted price relatives than average, so their weights will be decreased on the nexttrading day, and vice versa, i.e., b t +1 ( j ) = ˆ b t ( j ) + (cid:16) I h ˜ b t + ( j ) ≥ i − I h ˜ b t + ( j ) < i(cid:17) · max (cid:16)(cid:12)(cid:12)(cid:12) ˜ b t + ( j ) (cid:12)(cid:12)(cid:12) − λ, (cid:17) , (8)for all ≤ j ≤ m , where λ ∈ R + is a user-defined parameter. Only changes greater than the threshold of λ areapplied to the next portfolio vector. We set η to 10 and λ to × η × γ , where γ denotes the rate of transaction costs. Since b t +1 may not satisfy the constraints of the portfolio vector, b t +1 is normalized, i.e., b t +1 = arg min b ∈ R m + k b − b t +1 k subject to k b k = 1 . (9)There are two versions of the TCO depending on how the predictive price relatives are calculated. The TCO-1 usesthe single-period mean reversion property, and the TCO-2 uses the multi-period mean reversion property, i.e., ˜ x t +1 = (cid:26) ⊘ x t in TCO-1 SMA t (5) ⊘ x t in TCO-2. (10) We use two sets of historical daily data to test mean reversion strategies. The first set of the data is summarized inTable 1. The name of each data represents the universe of the data, which is a stock market index or a stock exchange.The first set consists of the data that have been widely used by other researchers. Refer to Li et al. (2012) for thedetails of the data on the first set.For the second set of the data, we collect the historical prices of 505 stocks that were the components of the S&P 500on October 31, 2018. We obtain the data from Yahoo Finance. Out of the 505 stocks, we exclude the 111 stocks thatwere not listed on January 1, 2000 and 5 stocks whose data could not be retrieved correctly. We then sort the remaining389 stocks in alphabetical order by the ticker symbols of the companies and make 10 portfolios each of which contains38 or 39 stocks in the order of the sort. The second set of the data is summarized in Table 2. Experiments are conducted to investigate the performance of the strategies described in this paper. We implement SMRand SMAR in MATLAB and use the implementations of Li et al. (2015, 2016, 2018) for the rest of the strategies. Asstated earlier,
BAH U and CRP U provide benchmarks for rebalancing strategies since they can be easily applied inthe real stock market.In actual trading, traders have to pay transaction costs when they trade. Commissions and taxes are explicit transactioncosts, while bid-ask spreads and price-impact costs are implicit transaction costs. Keim and Madhavan (1998) find Li et al. (2018) states that they set λ to × γ . However, their implementation sets λ to × η × γ , which performs betterthan × γ and coincides with the result of their study. The second set of the data in csv and mat format will be available at the site: https://github.com/uramoon/mr_dataset . A P
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Table 2: Summary of the second set of historical daily data.
Name Period Days Assets a Max( x t ( j ) ) b Min( x t ( j ) ) c SP500(0) 2000 - 2017 4527 39 1.6600 0.3921SP500(1) 2000 - 2017 4527 39 1.5782 0.5494SP500(2) 2000 - 2017 4527 39 1.6296 0.5402SP500(3) 2000 - 2017 4527 39 1.6037 0.4133SP500(4) 2000 - 2017 4527 39 2.0236 0.4844SP500(5) 2000 - 2017 4527 39 1.5425 0.3948SP500(6) 2000 - 2017 4527 39 1.8698 0.5873SP500(7) 2000 - 2017 4527 39 1.7545 0.3195SP500(8) 2000 - 2017 4527 39 1.5605 0.4096SP500(9) 2000 - 2017 4527 38 d a The number of assets in the portfolio. a The highest daily return of a single asset on the dataset. b The lowest daily return of a single asset on the dataset. d One less stocks than the others since the total number of stocks is 389.
Table 3: Cumulative wealth achieved by various strategies using the first set of historical daily data. Transaction costsare not considered. The top two results are indicated in bold type for each dataset.
Data
BAH U CRP U SMR SMAR PAMR OLMAR TCO-1 TCO-2NYSE(O) 14.50 27.08 4.39e+15
SP500 1.34 1.65 8.88
TSE 1.61 1.60 that explicit costs are at least .
13 % and implicit costs are at least .
11 % on institutional trades data. The lowesttotal costs are .
26 % for seller-initiated trades, and .
31 % for buyer-initiated trades in NYSE and NYSE American.To incorporate all expenses that can be incurred in actual trading, transaction costs are taken into account in theexperiments. We adopt the proportional commission model used by Borodin et al. (2004) and Li et al. (2012, 2013,2015). Given a rate of transaction costs γ ∈ [0 , , the model incurs transaction costs at a rate of γ for each buy andfor each sell so that the daily return on the t th day becomes ( b t · x t ) × (1 − γ · k b t − b t − k ). Since transaction costsdepend on trading volume, strategies with higher volatility suffer more from transaction costs.Table 3 shows the cumulative wealth of each strategy using the first set of data. The initial wealth of each strategy is1, and transaction costs are not considered. The results of SMR and SMAR are newly obtained ones, and the resultsof the other strategies are consistent with their original papers. In Table 3, the results of BAH U indicates that the firstset of data includes both upward and downward trends. All of the mean reversion strategies work better than BAH U and CRP U on all dataset except DJIA; i.e., the SMR and PAMR fail to outperform BAH U on DJIA. Still, the overallperformances of mean reversion strategies are very impressive, and it is notable that in all datasets except TSE, thetop two strategies are based on the multi-period mean reversion property. It is also noteworthy that very simple meanreversion strategies we devise (SMR and SMAR) produce the best results on five datasets. The success of the verysimple strategies shows that there are exceptionally strong mean reversion tendencies in the well-known benchmarkdataset, which might have caused mean reversion strategies to perform well.In Li et al. (2018), the state-of-the-art mean reversion strategies (PAMR, OLMAR, and TCOs) are tested under therates of transaction costs at .
25 % and . , and it is claimed that TCOs are able to withstand reasonable transactioncosts. Table 4 shows the cumulative wealth when transaction costs are set to .
25 % in the proportional commissionmodel. We can see that the results of strategies that do not consider transaction costs in determining portfolio vectorsare very poor on all datasets except NYSE(O) and TSE. However, the performances of TCOs are very promising evenin the presence of transaction costs; i.e., the TCOs produce the top two results on five datasets.To test if mean reversion strategies perform well on various and recent datasets, experiments are conducted on thesecond set of data. Table 5 shows the cumulative wealth achieved by each strategy. The result of
BAH U shows thatevery dataset on the second set has an upward trend in the long term. The BAH U increases the wealth by over sixtimes on every dataset. A sensible strategy is expected to profit from these datasets. However, all the mean reversionstrategies fail to surpass BAH U and CRP U on SP500(0) and SP500(6). Three strategies (SMR, SMAR, and OLMAR)5mpirical investigation of state-of-the-art mean reversion strategies for equity markets A P
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Table 4: Cumulative wealth achieved by various strategies using the first set of historical daily data. Transaction costsare set to .
25 % in the proportional commission model. The top two results are indicated in bold type for each dataset.
Data
BAH U CRP U SMR SMAR PAMR OLMAR TCO-1 TCO-2NYSE(O) 14.46 22.93 1.47e+04 1.62e+08 2.09e+05 2.98e+07
NYSE(N) 18.01 25.92 0.00 0.47 0.00 0.05
DJIA 0.76 0.80 0.05 0.47 0.09 0.33
SP500 1.34
TSE 1.61 1.52 3.74 1.12 2.11 4.29
MSCI 0.90 0.90 0.08 0.40 0.15 0.34
Table 5: Cumulative wealth achieved by various strategies using the second set of historical daily data. Transactioncosts are not considered. The top two results are indicated in bold type for each dataset.
Data
BAH U CRP U SMR SMAR PAMR OLMAR TCO-1 TCO-2SP500(0)
SP500(3) 6.68 9.09 47.01
SP500(6)
SP500(8) 8.90 12.38 16.29 even suffer a serious loss on SP500(0) while
BAH U increases its wealth by over nine times. Nevertheless, the meanreversion strategies perform well on many datasets. The multi-period mean reversion strategies generally work betterthan the single-period counterparts. It is noteworthy that simple strategies work as well as the state-of-the-art meanreversion strategies on the second set of data; the SMAR produces the best results on three datasets.Mean reversion strategies are tested on the second set of data in the presence of transaction costs. Table 6 shows thecumulative wealth when transaction costs are set to .
25 % . All the mean reversion strategies, excluding TCOs, spendall assets due to transaction costs. Even TCOs, strategies that takes transaction costs into account, fail to get excessreturns from most datasets. We can see that applying mean reversion strategies to actual investments based on theresult of well-known benchmark datasets can be very dangerous especially when there exist transaction costs.
We investigate the performance of mean reversion strategies on various datasets. Unlike the other mean reversionstrategies used in this work,
CRP U generally shows consistent improvement over BAH U regardless of transactioncosts. Although, on the other hand, the others overall work well without transaction costs, they show poor results withTable 6: Cumulative wealth achieved by various strategies using the second set of historical daily data. Transactioncosts are set to .
25 % in the proportional commission model. The top two results are indicated in bold type for eachdataset.
Data
BAH U CRP U SMR SMAR PAMR OLMAR TCO-1 TCO-2SP500(0)
SP500(6)
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REPRINT transaction costs especially on the datasets we collected. Given a portfolio, PAMR and OLMAR should be used withcaution unless there is a good reason to believe that their counterparts (SMR and SMAR) will also perform well onthe portfolio. Our empirical results also imply that extra caution should be taken in using the strategies in the case thatthere exist transaction costs.On the first set of data, it is very surprising that SMR and SMAR, very simple mean reversion strategies which wedevise for this study, perform as well as more sophisticated mean reversion strategies. We also discover that if weinvested our wealth to the company “kin ark” in NYSE datasets only when its price had fallen in the previous day,our wealth would have been multiplied by 2.15e+9 in the NYSE(O) and 1.90e+4 in the NYSE(N) at the end of theperiod. The BAH of the single asset “kin ark”, however, increases initial wealth . times in NYSE(O) and . times in NYSE(N). Thus, we conclude that the datasets NYSE(O) and NYSE(N) do not have representative natureto be standard benchmarks. They are biased to the mean reversion characterstics. The second set of data, whichwe collect, can be alternative benchmark datasets for online portfolio selection to develop more robust rebalancingstrategies. Acknowledgments
We are grateful to Dr. Seung-Kyu Lee for his helpful comments.
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