FADE THE GAP: ODDS FAVOR MEAN REVERSION First Draft: July 2014 This Draft: July 2014 Jia-Yuh Chen and Timothy L. Palmer Abstract When a stock opens a day s trading at a lower price than its previous day s low price, it is said to gap down. It gaps up when the stock opens higher than the previous high price. In this paper, we examine a popular technical trading rule called fade the gap. To fade the gap, a trader buys a stock when it gaps down or short sells when up. The strategy is a bet on a stock s short-term mean-reverting behavior. In our setup, all positions are closed at the end of the trading day and the operation is repeated throughout a period from January 1988 to June 2014. We show that fade-the-gap portfolios with various gap thresholds not only have large Sharpe ratios, but also deliver positive, statistically and economically significant alphas after controlling for Fama and French s three risk factors. Our result adds another anomaly to the effi cient market hypothesis literature. JEL classification: G02; G11; G14 Keywords: Technical Trading; Filter Rule; Mean Reversion; Overreaction; Market Effi ciency; Gap Up; Gap Down; Fade the Gap; Momentum; Contrarian Corresponding author, Assistant Professor of Finance, Coe College, 1220 First Ave NE, Cedar Rapids, IA 52402, USA; Phone: (319)399-8247; Email: jchen@coe.edu We are grateful for the financial support from Spellman Research Fund at Coe College.
1 Introduction When a stock opens a day s trading at a lower price than its previous day s low price 1, it is said to gap down. It gaps up when the stock opens higher than the previous high price. In this paper, we examine a popular technical trading rule called fade the gap. To fade the gap, a trader buys a stock when it gaps down or short sells when up. We construct various portfolios to exploit such a trading strategy. These portfolios have abnormal returns that are not explained by Fama and French (1996) s three-factor model. Technical traders use public stock information to forecast price movements. For example, a filter rule may say, "If the daily closing price of a particular security moves up at least x per cent, buy and hold the security until its price moves down at least x per cent from a subsequent high, at which time simultaneously sell and go short." Fama and Blume (1966) show such a rule does not earn the trader a statistically or economically significant profit. Theirs and other work, such as Jensen and Benington (1970), conclude that market effi ciency renders most trading rules useless. However, researchers keep coming up with evidence showing this conclusion might be premature. Sweeney (1988) revisits the filter rule and finds that a subset of sample used by Fama and Blume (1966) perform significantly better than a simple buy-and-hold strategy. Brock et al. (1992) scrutinize two technical rules moving average and trading range break and find supportive results for technical traders. De Bondt and Thaler (1985) claim with evidence that investor overreaction results in stock price s reverting to the mean "extreme movements in stock prices will be followed by subsequent price movements in the opposite direction." These and others are considered not only as anomalies, but a challenge to the effi cient market hypothesis. However, Fama and French (1996) show that the risk premium represented by their three risk factors can explain these abnormal returns well. Nevertheless, new anomalies keep popping up to challenge the effi cient market hypothesis. Our paper adds one more anomaly to the literature. Our research is in a narrower field of technical rules. We study whether stocks follow a trend or reverse the course when overshoot 2. Trend following, sometimes known as time-series momentum, is supported by Moskowitz et al. (2012). This line of literature is different from the so-called momentum trading, first discovered by Jegadeesh and Titman (1993). In momentum trading, portfolios are formed cross-sectionally. A stock in the momentum portfolio is selected based on its relative performance to the overall market, not on its own time trend. As Moskowitz et al. (2012) point out, "time series momentum focuses purely on a security s own past return." Our study is in line with this definition. But their work is on futures market, while we are more interested in stocks. With different securities and time-frame from Moskowitz et al. (2012), we reach a different conclusion. We observe mean reversion in our data. Such a mean reversion phenomenon is often attributed to investors overreaction. Brown et al. (1988) find stock investors overreact to bad news and underreact to good ones. That is, stocks 1 This is known as a full gap. For a down gap, if it opens lower than its previous close, but higher than the previous low, it is called a partial gap. Naturally, there are more partial-gap events than full-gap ones. We focus on full gaps in our research. 2 The distinction is also known to be momentum vs. contrarian by traders. 1
reverse after initial big drops but follow a trend when they trend up. Bremer and Sweeney (1991) also show that after extremely large negative movements, stocks have unexpected large positive returns. Neither study is the same as our intraday fade-the-gap strategy. Our paper is the first study on this strategy. In our findings, when a stock opens a day s trading with a large gap from the previous day, the reversal phenomenon is substantial. For the baseline -4% down gap, the stocks have an average daily return of 1.21% (buy at open and sell at close). For short-selling on gap-ups more than 4%, the daily return average is 0.012%. Both are statistically significant. We then use the fade-the-gap strategy to construct a trading portfolio. When there are more than one gap event in a day, we allocate our investment equally among these trading opportunities. The pure gap-down portfolio (-4% gap) has a Sharpe ratio of 0.6617. The pure gap-up portfolio (4% gap) was a Sharpe ratio of 1.095. Both compare well with the benchmark S&P 500 Total Return Index s 0.525. An Up+Down combined strategy (±4% gap)has an even higher Sharpe ratio of 1.3355. To further understand these portfolios risk-adjusted performance, we run regressions of Fama and French (1996) s threefactor model. The pure gap-down portfolio (-4% gap) has an unexplained monthly return of 2.75%, gap-down (4%) 3.90%, and Up+Down (±4% gap) 4.58%. All are statistically significant at 1% level. Our result shows that stocks opening with an abrupt departure from the previous day s trading tend to reverse the course later on during the day. Since these large gaps are typically results of news of stocks in question, we believe our finding is consistent with the investor overreaction literature, such as De Bondt and Thaler (1985) and Chopra et al. (1992). 2 Data Analysis 2.1 Data We scan 623 stocks that currently have averaged over 2 million daily trading volume. Data started from January 1, 1988 3 to June 30, 2014. Data source is Yahoo! Finance. We calculate the gaps difference between the previous day s low (high, for gap-up) and the trading day s open. A gap event is defined as a gap greater than the threshold, say, ±4% in our baseline scenarios. We further filter out gap events that do not see the average volume of the previous three days exceeding 2 million shares. Finally, we calculate daily returns of these events. Daily returns summary statistics is shown in Table 1. In addition to gap events happening to individual stocks, we also take a simple average of all the event stock s daily returns if the event day has more than one stock gapping up or down. This simple averaging is the same as constructing portfolios that employ the fade-the-gap strategy repeatedly. These average daily returns are recorded in the Portfolio columns in Table 1. We have more discussion on portfolio construction in the next section. Almost all the daily returns are statistically different from zero. The reverse phenomenon is 3 This date is picked for three reasons. We observe more and tradable opportunites over time. Gaps were rare before 1970s. We are also more interested in post-1987 s Black Monday. More importantly, our benchmark S&P 500 Total Return Index can only trace back to January 1988. 2
also obvious. All gap-down events show positive daily returns, while gap-up ones negative. For example, buying at open and selling at close on an event day with -4% gap-down gives a trader on average 1.21% daily return, 0.31% if following a portfolio strategy. This compares well to 0.04% daily return of the S&P 500 Total Return Index during the same period. Since the median values are not far off the mean ones, the results are not too skewed by extreme values. A couple of observations are in order. First, there are more gap-up events than gap-down, in terms of the number of stocks, gap count, and days. But trading on gap-down is more profitable than gap-up 4. For our baseline ±4% gap, trading on gap-down is 1.21%, but gap-up only 0.12%. For portfolios, fading a down gap is 0.3%, while up 0.15%. These are true across different thresholds. Secondly, the size of the gap has asymmetric impact on the two strategies profitability. As the size of the gap increases, fading a down gap has decreasing profits, while gap-ups see profits increasing. This is true for both individual gap events and portfolios. Next we will build hypothetical trading portfolios to see whether there is an anomaly to the effi cient market hypothesis. 2.2 Portfolios In our hypothetical trading portfolios, a trader at the opening of each trading day scans the stock universe to find stocks with gap events. In a gap-down portfolio, he buys stocks that gap down more, in absolute term, than the threshold, say -4% for the baseline scenario. If there are more than one gap event on the trading day, he divides his cash equally among these trading opportunities. At the end of the trading day, he closes out positions and converts his holdings back to cash. If there isn t any gap events, the trader sits out and keeps his wealth in cash. For the gap-up portfolios, the operation is the same, except that the trader short sells at the opening and buys back at closing. We ignore transaction costs and interests accrued. Portfolio performance is shown in Table 2. As a benchmark, returns on a portfolio fully invested in the S&P 500 Total Return Index are also listed. We also construct a "super" fade-the-gap portfolio by combining both gap-up and gap-down events. In this Up+Down portfolio, the trader splits his portfolio wealth equally among all gap trading opportunities. As seen in Table 2, baseline -4% gap-down has a monthly return of 3.3%, exceeding the S&P 500 Total Return s 0.9%. In terms of annualized Sharpe ratio, the baseline scenario also outperforms the benchmark. For gap-up portfolios, their monthly returns and Sharpe ratios are all well above the S&P 500 Total Return. What is really remarkable is the Up+Down portfolio. It has 3.9% monthly return with a Sharpe ratio of 1.3355, which is about 2.5 times of that of the S&P 500. In Figure 1, we plot various portfolios value over time to illustrate fade-the-gap s stellar performance. 4 Since the trader short sells when the stock gaps up, returns must change signs from negavite to positive if we look at gap-up trading. 3
2.3 Risk-Adjusted Performance Although the differences in Sharpe ratios indicate that the fade-the-gap strategy s performance stands even if we account for risks involved, the true test is whether we can find alpha after controlling for Fama and French (1996) s three risk factors. We use the monthly returns of various portfolios in question to estimate the following model. R t R ft = α + β M (R Mt R ft ) + β SMB SMB t + β HML HML t + ε t, where R t is a portfolio s monthly return at time t, R ft the risk-free rate at time t, R Mt market s monthly return at time t, SMB t Fama and Fench s small stock effect premium at time t, and HML t value stock effect premium at time t 5. Results are reported in Table 3. For the baseline gap-down -4% portfolio, the alpha is 2.75% (monthly), significant at 1% level. But for all other fading-a-down-gap portfolios, only -5% gap-down delivers significant alpha. On the other hand, all the gap-up portfolios have very large and significant alphas. The super Up+Down portfolio has a 4.58% (monthly) alpha with a t-statistic equalling 10.9. The results are mixed, but in general show that the fade-the-gap strategy provide great returns to the trader not by taking on a large risk 6. It appears that we find another anomaly to the effi cient market hypothesis, which calls for no alpha for any trading strategy. 2.4 Discussion Constructing a hypothetical portfolio ex post inevitably faces the question of whether such a trading strategy is realistic in the real world. We offer here some of our thoughts on feasibility on the fadethe-gap strategy. 2.4.1 Tradability If a trader were to construct a portfolio like us, he had to consider whether these opportunities are truly tradable. One issue is on how long the opening price lasts. If the price adjusts rapidly, it leaves no room for the strategy to implement. We do not have access to tick-by-tick price data, but a random sampling of recent gap events convinces us that the stocks trade rather close to the opening prices five minutes into trading. Given the fast speed of trade platforms available to the trader, he should be able to execute trades relatively close to the opening price. Another issue is on market depth. Without a liquid market, the trader might be able to see, but not able to touch the opening price. We foresee the problem and restrict out gap events to be included only after we see the average trading volume in the last three trading days exceeding 2 million shares. We believe this prefiltering takes care of the market depth issue. The last issue on tradability is transaction costs. Unfortunately, we are not able to collect bid-ask spread for the opening prices, but given 5 All these premiums and risk-free rates are pulled from Ken French s website. We appreciate Prof. French s generosity to make them available to the public. 6 Since fading a up gap requires short-selling, it is not surprise to see that the portfolio s market beta is negative. In a sense, fading a up gap actually reduces the trader s market risk. 4
the wide performance gap seen in Figure 1 and the large alphas shown in Table 3, we believe the impact of transaction cost on the fade-the-gap strategy is minimal. 2.4.2 Structural Change Returns from fading the gap might not be consistent over time. This is alarming for a trader who wants to implement this strategy going forward. In Figure 1 when we show the portfolio values over time, we see the steady growth of the gap-down (-4%) strategy has reversed in 2009 and continued till the end of our sample period. It seems that fading a down gap is not profitable anymore after 2009. However, all the gap-up portfolios, together with the super Up+Down, still consistently deliver fairly outstanding performance. To further understand the possible change in the gap-down strategy, we plot the number of gap events in a month over time in Figure 2. We already know from Table 1 that up-gaps happen more frequently than down-gaps. We see the same thing in Figure 2. Intuitively, there should more gap events during the time when the market is volatile. We also observe that. Both strategies have similar up-and-down patterns. We next look at fading-the-gap portfolios daily performance. We select the monthly median value of these daily returns to see how they evolve over time. Just as the structure break observed in Figure 1, we also see the gap-down strategy turns negative after 2009 in Figure 3. Fading an up gap shows a consistent patter throughout our sample period. We then break down the down-gap series into two regimes and find that the daily average return before 2009 is 0.55%, and -0.3% after. The reason for this unprecedented structure break is beyond the scope of this paper. 2.4.3 Survivorship Bias Our research design builds in a survivorship bias, which causes some problems but should not change our results significantly. We screen stocks by looking at current trading volume. We make sure gap events only happen to stocks that are heavily traded now and around the time gap events happened. In a sense, we select only survivors. If a trader had carried out the fade-the-gap strategy on a stock that was traded over 2 million shares in the past but failed to pass that threshold today, the result would not have been included. Due to the limitation of our data, we could not include these socalled "fallen angels." They become fallen angels for various reasons. For example, they might be caused by the decline in business, bankruptcy, or simply being acquired by other companies. How serious is this problem? It would be a serious problem if there was a structural difference between the fallen angels and survivors. Since all the trades involved are intraday, it is hard for us to believe the long-term business prospect has a serious impact on our short-term stock trading strategy. Furthermore, our sample also include a good number of future fallen angels because some of them will disappear from our radar in a decade or two. Therefore, we do not see the survivorship bias is detrimental to our result. 5
3 Conclusion We document the performance of hypothetical portfolios constructed according to the fade-the-gap strategy. These hypothetical, yet realistic, portfolios have abnormal returns not explained by known risk factors. Academically, this result present another anomaly to the effi cient market hypothesis. Practically, it also gives traders certain confidence if they want to exploit this strategy. However, our result also indicates that there might be a structural change happening in the last few years. Fading a down gap does not look like a winning proposition anymore. Future researchers might want to look into other risk factors to explain this anomaly and the structural break. 6
Tables and Figures Table 1 : Summary Statistics of the Gap Event s Daily Returns Panel A:Gap Down Gap Size -4% -5% -6% -7% Number of stocks 582 564 550 519 Individual Portfolio Individual Portfolio Individual Portfolio Individual Portfolio Gap Count 9305 3343 6409 2839 4670 2444 3658 2157 Return Mean 0.0121*** 0.0031*** 0.0132*** 0.0028*** 0.0133*** 0.0027** 0.0122*** 0.0017 (16.8697) (3.8117) (14.1594) (2.9135) (11.4531) (2.4228) (8.8949) (1.3872) Std 0.0692 0.0470 0.0746 0.0517 0.0794 0.0555 0.0831 0.0585 Min -0.4580-0.2810-0.4368-0.4257-0.4368-0.4257-0.4368-0.4257 25% -0.0225-0.0207-0.0241-0.0239-0.0248-0.0250-0.0260-0.0260 Median 0.0070 0.0017 0.0063 0.0009 0.0053 0.0000 0.0039 0.0000 75% 0.0409 0.0263 0.0440 0.0275 0.0444 0.0285 0.0425 0.0273 Max 1.0270 0.2353 1.0270 0.2813 1.0270 0.2813 1.0270 0.2813 Panel B: Gap Up Gap Size 4% 5% 6% 7% Number of stocks 618 617 613 610 Individual Portfolio Individual Portfolio Individual Portfolio Individual Portfolio Gap Count 234071 6598 153859 6494 105357 6260 74953 5992 Return Mean -0.0012*** -0.0015*** -0.0015*** -0.0021*** -0.0019*** -0.0026*** -0.0022*** -0.0031*** (-12.5686) (-6.6228) (-12.1856) (-7.5781) (-11.2757) (-7.9440) (-10.2184) (-8.0473) Std 0.0449 0.0188 0.0498 0.0221 0.0548 0.0259 0.0593 0.0301 Min -0.5111-0.1292-0.5111-0.1444-0.5111-0.2100-0.5111-0.2472 25% -0.0230-0.0117-0.0262-0.0144-0.0293-0.0170-0.0324-0.0197 Median -0.0019-0.0009-0.0027-0.0017-0.0035-0.0028-0.0041-0.0033 75% 0.0192 0.0089 0.0212 0.0101 0.0231 0.0114 0.0252 0.0133 Max 1.3182 0.2895 1.3182 0.2895 1.3182 0.2895 1.2222 0.2895 Note : (1) t-statistics are reported in parentheses. *,** and ** denote significance level at 10%,5% and 1%, respectively. (2) Daily returns are calculated based on daily close and open prices. Once there is a big enough gap (an event day), we record the daily return. A gap is defined as the difference between a day s open and the previous day s low (gap down) and that between a day s open and the day before s high (gap up.) (3) Data is from Yahoo! Finance. The period runs from 1/1/1988 to 6/30/2014 (4) "Individual" columns represent all the gap incidents. There are potentially more than 1 incident in a trading day. "Portfolio" is the simple weighted average of daily returns in each event day 7
Table 2 : Performance of Various Fade-the-Gap Portfolios Panel A: Gap Down Gap Size -4% -5% -6% -7% Portfolio Monthly Return 0.0332 0.0252 0.0221 0.0126 Standard Deviation 0.1595 0.1581 0.1657 0.1623 Sharpe Ratio 0.6617 0.4921 0.4032 0.2091 Panel B: Gap Up Gap Size 4% 5% 6% 7% Portfolio Monthly Return 0.0326 0.0440 0.0538 0.0616 Standard Deviation 0.0937 0.1125 0.1329 0.1472 Sharpe Ratio 1.0950 1.2622 1.3230 1.3793 Panel C: Others Portfolio Strategy S&P 500 Total Return Up+Down (±4% Gap) Portfolio Monthly Return 0.0092 0.0392 Standard Deviation 0.0418 0.0940 Sharpe Ratio 0.5247 1.3350 Note : (1) Portfolios in Panels A and B are formed according to the fade-the-gap strategy. Monthly returns and their standard deviation are calculated based on hypothetical portfolios end-of-the-month values. Transactions costs are ignored.. (2) Sharpe ratio is calculated by monthly excess return/standard deviation of excess return, where excess return is monthly return minus risk-free rate taken from Ken French s website. (3) Data used in Panel A and B is from Yahoo! Finance. The period runs from 1/1/1988 to 6/30/2014. The number of months included is 318. (4) In Panel C, Portfolio S&P 500 Total Return uses S&P 500 Total Return Index (obtained from Global Financial Data) to conduct a buy-and-hold strategy and Portfolio Up+Down is a mixed strategy combining both gap-up and gap-down. For the latter, the trader splits his portfolio wealth equally among all gap trading opportunities, regardless of gap-up or gap-down. 8
Table 3 : Estimates of Fama and French s Three-Factor Model Parameters α β M β SMB β HML R 2 Gap Down -4% 2.7540*** 0.5685*** 0.0371-0.3809 0.04 (3.0573) (2.6425) (0.1267) (-1.2254) -5% 2.0489** 0.3529 0.1084-0.2378 0.02 (2.2725) (1.6391) (0.3697) (-0.7644) -6% 1.5505 0.4494** 0.0865 0.2341 0.01 (1.6392) (1.9895) (0.2813) (0.7172) -7% 0.6157 0.4017* 0.0422 0.3426 0.01 (0.6638) (1.8132) (0.1399) (1.0703) Gap Up 4% 3.9035*** -1.2485*** -0.5075*** -0.0472 0.39 (9.2305) (-12.362) (-3.6915) (-0.3233) 5% 5.1164*** -1.3147-0.6230*** -0.0851 0.31 (9.4826) (-10.202) (-3.5517) (-0.4570) 6% 6.1313*** -1.4159*** -0.6679*** 0.0586 0.27 (9.3461) (-9.0371) (-3.1313) (0.2588) 7% 6.9617*** -1.4324*** -0.8803*** 0.0546 0.25 (9.4547) (-8.1452) (-3.6771) (0.2148) S&P 500 Total Return -0.0043 0.9949*** -0.1847*** 0.0243*** 0.99 (-0.1659) (162.33) (-22.142) (2.7471) Up+Down (±4%) 4.5840*** -1.2587*** -0.5497*** -0.0869 0.40 (10.910) (-12.543) (-4.0241) (-0.5991) Note : (1) The model being estimated is R t R ft = α + β M (R Mt R ft ) + β SMB SMB t +β HML HML t +ε t, where R t is a portfolio s monthly return at time t, R ft risk-free rate at time t (in percentage points), R Mt market s monthly return at time t, SMB t Fama and Fench s small stock effect premium at time t and HML t value stock effect premium. Except R t,all premiums are pulled from Ken French s website. (2) t-statistics are reported in parentheses. *,** and ** denote significance level at 10%,5% and 1%, respectively. (3) The period runs from 1/1/1988 to 6/30/2014. The number of months included is 318. 9
Figure 1 : Various Portfolios Value Over Time Note: Log values of portfolios are plotted, assuming all started in January 1988 with $1. 10
Figure 2 : Monthly Gap Count Over Time Note: These are series for total gap counts in a month. We record one count whenever there is a stock opens a day s trading lower than the previous day s low price (for the gap-down strategy) by the amount of 4%. For the gap-up strategy, one count is recorded when the stock opens higher than the previous day s high, also by the amount of 4%. 11
Figure 3 : Median Portfolio Daily Return Over Time Note: Here we look at two portfolios daily returns over time. We plot the median value of these daily returns in a month against time. The down -4% portfolio is constructed by dividing the trader s portfolio wealth equally among down-gap trading opportunities on each trading day. The trader closes out positions at the end of the day. The up 4% portfolio is constructed similarly, but using up-gap opportunities. 12
References Bremer, M. and Sweeney, R. J. (1991), The reversal of large stock-price decreases, The Journal of Finance, Vol. 46, Wiley for the American Finance Association, pp. pp. 747 754. Brock, W., Lakonishok, J. and LeBaron, B. (1992), Simple technical trading rules and the stochastic properties of stock returns, The Journal of Finance, Vol. 47, Wiley for the American Finance Association, pp. pp. 1731 1764. Brown, K. C., Harlow, W. and Tinic, S. M. (1988), Risk aversion, uncertain information, and market effi ciency, Journal of Financial Economics, Vol. 22, Elsevier, pp. 355 385. Chopra, N., Lakonishok, J. and Ritter, J. R. (1992), Measuring abnormal performance: do stocks overreact?, Journal of financial Economics, Vol. 31, Elsevier, pp. 235 268. De Bondt, W. F. M. and Thaler, R. (1985), Does the stock market overreact?, The Journal of Finance, Vol. 40, Wiley for the American Finance Association, pp. pp. 793 805. Fama, E. F. and Blume, M. E. (1966), Filter rules and stock-market trading, The Journal of Business, Vol. 39, The University of Chicago Press, pp. pp. 226 241. Fama, E. F. and French, K. R. (1996), Multifactor explanations of asset pricing anomalies, The Journal of Finance, Vol. 51, Wiley for the American Finance Association, pp. pp. 55 84. Jegadeesh, N. and Titman, S. (1993), Returns to buying winners and selling losers: Implications for stock market effi ciency, The Journal of Finance, Vol. 48, Wiley Online Library, pp. 65 91. Jensen, M. C. and Benington, G. A. (1970), Random walks and technical theories: Some additional evidence, The Journal of Finance, Vol. 25, Wiley for the American Finance Association, pp. pp. 469 482. Moskowitz, T. J., Ooi, Y. H. and Pedersen, L. H. (2012), Time series momentum, Journal of Financial Economics, Vol. 104, Elsevier, pp. 228 250. Sweeney, R. J. (1988), Some new filter rule tests: Methods and results, The Journal of Financial and Quantitative Analysis, Vol. 23, Cambridge University Press on behalf of the University of Washington School of Business Administration, pp. pp. 285 300. 13