Long-Term Price Overreactions: Are Markets Inefficient?

This paper examines long-term price overreactions in various financial markets (commodities, US stock market and FOREX). First, t-tests are carried out for overreactions as a statistical phenomenon. Second, a trading robot approach is applied to test the profitability of two alternative strategies, one based on the classical overreaction anomaly, the other on a so-called “inertia anomaly”. Both weekly and monthly data are used. Evidence of anomalies is found predominantly in the case of weekly data. In the majority of cases strategies based on overreaction anomalies are not profitable, and therefore the latter cannot be seen as inconsistent with the EMH.


Introduction
The Efficient Market Hypothesis (EMH) is one of the central tenets of financial economics (Fama, 1965). However, the empirical literature has provided extensive evidence of various "anomalies", such as fat tails, volatility clustering, long memory etc. that are inconsistent with the EMH paradigm and suggests that it is possible to make abnormal profits using appropriate trading strategies. A well-known anomaly is the so-called overreaction hypothesis, namely the idea that agents make investment decisions giving disproportionate weight to more recent information (see De Bondt and Thaler, 1985). Clements et al. (2009) report that the overreaction anomaly has not only persisted but in fact increased over the last twenty years. Its existence has been documented in several studies for different markets and frequencies such as monthly, weekly or daily data (see, e.g., Bremer and Sweeny, 1991;Clare and Thomas, 1995;Larson and Madura, 2006;Mynhardt and Plastun, 2013;Caporale et al. 2014).
This paper analyses long-term overreactions by (i) carrying out t-tests to establish whether overreaction anomalies exist using both weekly and monthly data, and (ii) using a trading robot method to examine whether they give rise to exploitable profit opportunities, i.e. whether price overreactions are simply a statistical phenomena or can also be seen as evidence against the EMH. The analysis is carried out for various financial markets: the US stock market (the Dow Jones Index and 10 companies included in this index), FOREX (10 currency pairs) and commodity markets (gold and oil). A similar investigation was carried out by Caporale et al. (2014); however, their analysis focused on short-term (i.e., daily) overreactions, whilst the present study considers a longer horizon, namely a week or a month.
The paper is structured as follows. Section 2 briefly reviews the existing literature on the overreaction hypothesis. Section 3 outlines the methodology. Section 4 discusses the empirical results and Section 5 summarises the main findings.

Literature review
The seminal paper on the overreaction hypothesis is due to De Bondt and Thaler (DT, 1985), who followed the work of Kahneman and Tversky (1982), and showed that the best (worst) performing portfolios in the NYSE over a three-year period tended to under (over)perform over the following three-year period. Their explanation was that significant deviations of asset prices from their fundamental value occur because of agents' irrational behaviour, with recent news being given an excessive weight. DT also reported an asymmetry in the overreaction (it is bigger for undervalued than for overvalued stocks), and a "January effect", with a clustering of overreactions in that particular month.
Other studies include Brown, Harlow and Tinic (1988), who analysed NYSE data for the period 1946-1983 and reached similar conclusions to DT; Ferri and Min (1996), who confirmed the presence of overreactions using S&P 500 data for the period 1962-1991; Larson and Madura (2003), who used NYSE data for the period 1988-1998 and also showed the presence of overreactions. Clement et al. (2009) confirmed the original findings of DT using CRSP data for the period 1926-1982, and also showed that the overreaction anomaly had increased during the following twenty years.
In addition to papers analysing stock markets (Alonso and Rubio, 1990, Brailsford, 1992, Bowman and Iverson, 1998, Antoniou et. al., 2005, Mynhardt and Plastun, 2013 among others), some consider other markets such as the gold (Cutler, Poterba, and Summers (1991)), or the options market (Poteshman, 2001). Finally, Conrad and Kaul (1993) showed that the returns used in many studies (supporting the overreaction hypothesis) are upwardly biased, and "true" returns have no relation to overreaction; therefore this issue is still unresolved.
The other aspect of the overreaction hypothesis is its practical implementation, i.e.
the possibility of obtaining extra profits by exploiting this anomaly. Jegadeesh and Titman (1993) and Lehmann (1990) found that a strategy based on overreactions can indeed generate abnormal profits. Baytas and Cakiki (1999) also tested a trading strategy based on the overreaction hypothesis, and showed that contrarian portfolios on the long-term horizons can generate significant profits.
The most recent and thorough investigation is due to Caporale et al. (2014), who analyse different financial markets (FOREX, stock and commodity) using the same approach as in the present study. That study shows that a strategy based on countermovements after overreactions does not generate profits in the FOREX and the commodity markets, but it is profitable in the case of the US stock market. Also, it detects a brand new anomaly based on the overreaction hypothesis, i.e. an "inertia" anomaly (after an overreaction day prices tend to move in the same direction for some time). Here we extend the analysis by considering long-term overreactions and the possibility of making extra profits over weekly and monthly intervals.

Data and methodology
We analyse the following weekly and monthly series: for the US stock market, the Dow

Student's t-tests
Student's t-tests are carried out for the overreaction hypothesis, according to which an overreaction should be followed by a correction, i.e. price counter-movements, and they should be bigger than after normal periods for as long as it takes the market to process new information.
The two hypotheses to be tested are therefore: H1: Counter-reactions after overreactions differ from those after normal periods.
H2: Price movements after overreactions in the direction of the overreaction differ from such movements after normal periods.
The null hypothesis is in both cases that the data after normal and overreaction periods belong to the same population. Given the size of our data set, the Central Limit Theorem (Mendenhall, Beaver and Beaver, 2003) can be invoked to justify the assumption of normality required for the t-tests.
As already mentioned, we focus on long-term overreactions, so the period of analysis is one week or one month. The parameters characterising price behaviour over such a time interval are maximum, minimum, open and close prices. In most studies price movements are measured as the difference between the open and close price. In our opinion the weekly (monthly) return, i.e. the difference between the maximum and minimum prices during the week (month), is more appropriate. This is calculated as: High is the maximum price, and i Low is the minimum price for week (month) і.
We consider three definitions of "overreaction": when the current weekly (monthly) return exceeds the average plus two standard deviations, i.e., 3) when the current weekly (monthly) return exceeds the average plus three standard deviations, i.e., The next step is to determine the size of the price movement during the following week (month). For Hypothesis 1 (the counter-reaction or counter-movement assumption), we measure it as the difference between the next period's open price and the maximum deviation from it in the opposite direction to the price movement in the overreaction period.
If the price increased, then the size of the counter-reaction is calculated as: where 1 i cR + is the counter-reaction size, and l i Open + is the next period's open price.
If the price decreased, then the corresponding definition is: The null hypothesis to be tested is that they are both drawn from the same population.

Trading robot analysis
The trading robot approach considers the long-term overreactions from a trader's viewpoint, i.e. whether it is possible to make abnormal profits by exploiting the overreaction anomaly, and simulates the actions of a trader using an algorithm representing a trading strategy. This is a programme in the MetaTrader terminal that has been developed in MetaQuotes Language 4 (MQL4) and used for the automation of analytical and trading processes. Trading robots (called experts in MetaTrader) allow to analyse price data and manage trading activities on the basis of the signals received.
MetaQuotes Language 4 is the language for programming trade strategies built in the client terminal. The syntax of MQL4 is quite similar to that of the C language. It allows to programme trading robots that automate trade processes and is ideally suited to the implementation of trading strategies. The terminal also allows to check the efficiency of trading robots using historical data. These are saved in the MetaTrader terminal as bars and represent records appearing as TOHLCV (HST format). The trading terminal allows to test experts by various methods. By selecting smaller periods it is possible to see price fluctuations within bars, i.e., price changes will be reproduced more precisely. For example, when an expert is tested on one-hour data, price changes for a bar can be modelled using one-minute data. The price history stored in the client terminal includes only Bid prices. In order to model Ask prices, the strategy tester uses the current spread at the beginning of testing. However, a user can set a custom spread for testing in the "Spread", thereby approximating better actual price movements.
We examine two trading strategies: - The results of the trading strategy testing and some key data are presented in the "Report" in Appendix A. The most important indicators given in the "Report" are: -Total net profit: this is the difference between "Gross profit" and "Gross loss" measured in US dollars. We used marginal trading with the leverage 1:100, therefore it is necessary to invest $1000 to make the profit mentioned in the Trading Report. The annual return is defined as Total net profit/100, so, for instance, an annual total net profit of $100 represents a 10% annual return on Stop Loss =stop*sigma_dz.

Empirical results
The first step is to set the basic overreaction parameters/criterions by choosing the number of standard deviations (sigma_dz) to be added to the average to form the "standard" period interval for price fluctuations and the averaging period to calculate the mean and the standard deviation (symbol: period_dz).
For this purpose we used the Dow Jones Index data for the time period 1991-2014.
The number of abnormal returns detected in the period 1991-2014 is reported in Table 2 (for weekly data) and Table 3 (for monthly data).  As can be seen from the above tables, both parameters (averaging period and number of standard deviations added to the mean) affect the number of detected anomalies.
Changes in the averaging period have relatively small effect on the number of detected anomalies (the difference between the results when the period considered is 5 and 30 respectively is less than 20%). By contrast, each additional standard deviation significantly decreases the number of observed abnormal returns. Therefore 2-4% of the full sample (the number of abnormal returns in the case of 3 sigmas) is not sufficiently representative to draw conclusions. To investigate whether sigma_dz equal to 1 is most appropriate we carry out t-tests of long-term counter-reactions for the Dow Jones index over the period 1991-2014 (see Tables 4 and 5 for weekly and monthly data respectively). As can be seen, the anomaly is most easily detected in the case of sigma_dz= 1 (the t-stat is the biggest), and therefore we set sigma_dz equal to 1.   (Tables 6 and 7 for weekly and monthly data respectively) suggest that the optimal averaging period is 30, their corresponding t-statistics being significantly higher than for other averaging periods.  Therefore the key parameters for the t-tests of long-term overreaction in different financial markets analysis are set as follows: the period_dz (averaging period) is set equal to 30 and sigma_dz (the number of standard deviations added to mean used as a criterion of overreaction) equal to 1.
The results for H1 are presented in Tables 8 -12. In the case of the commodity markets (Table 8), this hypothesis is rejected for both assets with weekly data (this is evidence supporting the existence of an anomaly) but cannot be rejected for oil with monthly data. The results from testing Hypothesis 1 for the US stock market (see Tables 9 and 10) are unstable across frequencies: the anomaly is found in the case of weekly but not of monthly data.  By contrast, the results from testing Hypothesis 1 for the FOREX (Tables 11 and 12) are relatively stable, and no anomaly is detected with either dataset.  Overall, it appears that in the case of H1 the best frequency to detect the counterreactions after long-term overreactions is weekly. H1 cannot be rejected for the US stock market (in all cases with weekly data) and commodity markets. FOREX is not subject to the anomaly described in H1. Therefore the classical long-term counter-movement after overreactions is confirmed in US stock market and commodities markets, but only with weekly data.
The results for H2 are presented in Tables 13 -17. This hypothesis cannot be rejected for the commodity markets (see Table 13) for both data sets (weekly and monthly). The results from testing Hypothesis 2 for the US stock markets (Tables 14 and 15) are less stable and are mixed. The anomaly is detected for the Dow Jones and Microsoft data in the weekly but not in the monthly case. For Boeing the opposite conclusion is reached. Overall, there is evidence of an "inertia" anomaly in the US stock market but this is true only for weekly data  The results from testing Hypothesis 2 for the FOREX (Tables 16 and 17) are mixed. No anomaly is detected for the EURUSD (for both data sets), there is evidence of an anomaly with monthly but not weekly data for USD CHF, and this is found in both cases for the AUDUSD.  The general conclusions from the t-test are as follows: an anomaly is generally detected using weekly but not monthly data; FOREX is mostly immune to the "inertia" anomaly; the US stock and commodity markets are most affected by the overreaction anomalies.
Next, we analyse whether these anomalies give rise to exploitable profit opportunities. If they do not, we conclude that they do not represent evidence inconsistent with the EMH. We expand the list of assets in order to provide more extensive results. The The parameters of the trading strategies 1 and 2 are set as follows: -Period_dz = 30 (see above for an explanation); -Time_val = week (see above); -Sigma_dz=1 (see above).
-Profit_koef = 1 sigma_dz (1 standard deviation as a measure of the current volatility of the asset).
-Stop = 10 sigma_dz (to prevent a total loss of the investment in case of a market crash).
The results of the trading robot analysis are presented in Table 18 (Strategy 1) and  Strategy 1, based on the classical overreaction hypothesis, trades on counterreactions after periods of abnormal price dynamics. In general, it is unprofitable for FOREX (7 pairs out of 10 produce negative results) and commodities market (in the case of Gold). For the US stock market the results are mixed (50% of profitable assets), but in general this anomaly does not seem to be exploitable. The assets to be traded on the basis of the classical overreaction hypothesis with weekly data are therefore: GBPCHF (ROI=27% per year), GBPJPY (25%), USDJPY (12%), Boeing (36.6%) and ExxonMobil (8.6%). Strategy 2, based on the so-called "inertia anomaly"), trades on price movements in the direction of the overreaction in the following period. In general it is unprofitable for the US stock market (7 assets out of the 10 analysed produce negative results), whilst the results are mixed for the FOREX (6 pairs out of 10 yield negative results). There is evidence of profit opportunities in the commodity market. The assets to be traded on the basis of the inertia anomaly with weekly data are therefore: USDCAD (ROI=13% per year), USDCHF (5%), EURUSD (6%), AIG (27%), Alcoa (10%) and Gold (11%).

Conclusions
This paper examines long-term price overreactions in various financial markets (commodities, US stock market and FOREX). It addresses the issue of whether they should be seen simply as a statistical phenomenon or instead as anomalies giving rise to exploitable profit opportunities, only the latter being inconsistent with the EMH paradigm.
The analysis is conducted in two steps. First, t-tests are carried out for overreactions as a statistical phenomenon. Second, a trading robot approach is applied to test the profitability of two alternative strategies, one based on the classical overreaction anomaly (H1: counterreactions after overreactions differ from those after normal periods), the other on an "inertia" anomaly (H2: price movements after overreactions in the same direction of the overreaction differ from those after normal periods). Both weekly and monthly data are used. Evidence of anomalies is found predominantly in the case of weekly data.
More specifically, H1 cannot be rejected for the US stock market and commodity markets when the averaging period is 30, whilst it is rejected for the FOREX. The results for H2 are more mixed and provide evidence of an "inertia" anomaly in the commodity market and for some assets in the US stock market and FOREX. The trading robot analysis shows that in general strategies based on the overreaction anomalies are not profitable, and therefore the latter cannot be seen as inconsistent with the EMH. However, in some cases abnormal profits can be made; in particular this is true of (i) GBPCHF (ROI=27% per year), GBPJPY (25%), Boeing (36%), ExxonMobil (8.6%) in the case of the classical overreaction hypothesis and weekly data, and (ii) USDCAD (13%), USDCHF (5%), EURUSD (6%), AIG (27%), Alcoa (10%) and Gold (11%) in the case of the inertia anomaly and also with weekly data.
A comparison between these results and the daily ones reported in Caporale et al. (2014) suggests that the classic overreaction anomaly (H1) occurs at both short-and longterm intervals in the case of the US stock market and commodity markets. The results for the FOREX are mixed at both intervals, but mostly suggest no contrarian movements after overreactions. The findings concerning the "inertia" anomaly (H2) are more stable and consistent: it is detected for the commodity markets and US stock market at both short-and long-term horizons. As for the FOREX, there is a short-but not a long-term anomaly in most cases. The trading results imply that there is no single profitable strategy: the findings are quite sensitive to the specific asset being considered, and therefore it is necessary to investigate case by case whether it is possible to earn abnormal profits by exploiting the classical overreaction and/or inertia anomaly.

Appendix A
Example of strategy tester report: case of GBPJPY, period 2001-2014, H1 testing