Detecting stock market seasonality A period mining approach Stephane Cheung Yukinobu Hamuro
Katsuhiko Okada
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Detecting stock market seasonality A period mining approach Stephane Cheung* /Yukinobu Hamuro* / Katsuhiko Okada * ** *Kwansei Gakuin University Business School **Magne-Max Capital Management, Japan FSA registered Investment Advisor
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Market seasonality Fact finding • Agrawal and Tandon (1994) JIMF 18 countries Seasonality, daily, weekend effect, last trading day of the month, large preand inter holiday return. Jan, return.
• Bouman and Jacobsen (2002) AER 36 global markets out of 37 examined have “sell in May effect”, or “Halloween effect”, including Japan
• Kamstra, Kramer and Levi (2003) AER Markets in northern hemisphere demonstrate “sell in May” but not in southern hemisphere. SAD effect
• Sakakibara, Yamasaki and Okada (2013) IRF Japan is unique as Jun. is a good month while most other financial markets demonstrate lower June return -> “Dekansho-bushi” effect
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Market Seasonality • Mark Twain famously observed that October is the most dangerous month to invest in stock market. …The tragedy of pudd’nhead Wilson. 1894. • The Stock Exchange world is in a sort of twilight state at the moment. The potential buyers seem to have “sold in May and gone away”….. Financial Times May,30 1964 • ‘Sell in May and Go away’, famous Wall Street adage is once again in focus. Would the market behave as the old saying goes? Many investors in the market have anxiety in a corner of their minds……Nihon Keizai Shimbun, April 30, 2013
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316 years 82 years Jacobsen Global Fin. Dow Jones index Data index
50 years Dow Jones index
1693-2009
1929-2011
1961-2011
Jan
0.69
1.0
1.2
Feb
0.09
0.0
0.0
Mar
-0.03
0.4
1.1
April
0.49
1.4
2.0
May
0.02
-0.2
-0.1
June
-0.12
0.5
-0.6
July
-0.31
1.5
0.9
Aug
0.44
0.8
0.2
Sep
-0.49
-1.3
-0.8
Oct
-0.5
0.0
0.5
Nov
0.35
0.8
1.2
Dec
0.81
1.5
1.5
Nov-April
2.42
5.2
7.2
May-Oct
-0.96
1.28
0.09
Diff
3.38
3.92
7.11 4
Mean monthly return of Nikkei 225 and TOPIX Jan 1971-Dec 2014
1.2
Mean Monthly Return (%)
1
0.8
0.6
0.4
0.2
0
TOPIX
-0.2
Nikkei 225
First-half Year Last-half Year
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If fund managers had followed the “words of wisdom”…
Japan’s Lost 2 decades 3500 10% 3000 0% 2500 -10% 2000
-20% -30%
1500
-40% 1000 -50% 500 -60%
1989/12 1990/09 1991/06 1992/03 1992/12 1993/09 1994/06 1995/03 1995/12 1996/09 1997/06 1998/03 1998/12 1999/09 2000/06 2001/03 2001/12 2002/09 2003/06 2004/03 2004/12 2005/09 2006/06 2007/03 2007/12 2008/09
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TOPIX
-70% -80%
Dekansho-bushi Investment
Full Year Investment
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Research question and expectations Seasonality found in an index level implies that seasonality does exist in individual stock level, or industry level. The best season for holding stock i may be different from stock j. Finding seasonality in individual stock level enable us to create a portfolio durable for trading throughout the year.
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Data centric approach • Traditional financial economist approach a. Researchers come up with some insights b. Create a model (Hypothesis building) c. Collect data d. Conduct an empirical test to prove or disprove the model (accept or reject the hypothesis) e. Conclusion and conjecture
• Data centric approach a. There is no model. We don’t even have the hypothesis b. Large scale data c. Methodologies to detect correlation, potential predictability, to handle sparse data structure. d. Pattern implies hypothesis
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The Period Mining Model • Look for stocks that has high propensity to perform well at a given date. (Period Mining) • Use previous -4 to -1 years for training data • Rolling window up to present • Mining universe is TOPIX 500. Minimum market value of its composite is 130bil yen ($1.2bil)
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Methodology (Image) Model Building Period
2001-2004
Portfolio formation based on the model created in 2001-4
2005 Rolling window up to present
Model Building Period
2002-2005
2006 Portfolio simulation up to 2015.
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Period Mining: 4 steps in model building Historical data of Stock Price for a stock (eg. Toyota)
2. Aggregation Phase Aggregating the statistics on each item for last n years (n = 4 in this exp.)
term: 125days starting from Jan 1st or July 1st 28 terms from 2001 to 2014
1. Enumeration Phase Enumerating statistics (eg. abnormal return) of all periods defined by combination of starting date and holding period for each term. (We call this period as "item")
3. Filtering Phase Filtering items by aggregated statistics which match the given conditions (eg. lower bound of duration days).
4. Organizing Phase Select items so that there is no overlap and maximize the total abnormal return.
starting date: 1,2,...,125 holding period: 1,2,...,125 125 × 125 = 15,625 items per stock per term
Selected Items (Periods) to hold 11
Enumeration Phase Enumerate stock price statistics on period ps,e starting date: 1,2,...,125 holding period: 1,2,...,125 125 × 125 × 500 equities ≒ 1 2 3 4
date term 2
… …
s=125, e=126 s=125, e=127
…
s=125, e=250
250
…
s=3, e=127
.... 124 125 126 127
…
s=1, e=2 s=1, e=3 s=1, e=4 : s=1, e=125 s=2, e=3 s=2, e=4 : s=2, e=125 s=3, e=4 s=3, e=5
5 6 7 8
7.8M items per term
term 1 12
Aggregation Phase :
Aggregate all the items based on the following four criteria
:
:
a) b) c) d)
Holding period Average of abnormal return SD of abnormal return Average of zigzag rate
:
:
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Filtering Phase Select periods matching the given 6 conditions. The optimal parameters will be estimated using machine learning technique (Bayesian Global Optimization). declare
notation
min
max
lh
lower bound for holding period
0
20
uh
upper bound for holding period
10
125
lr
lower bound for average of abnormal return
0.01
0.1
ur
upper bound for average of abnormal return
0.11
2.1
us
upper bound for SD of abnormal return
0.01
1.0
lz
lower bound for zigzag rate
0
1
Rationale Lower bound: Too short holding period is not favorable due to transactions cost. Upper bound :Index level seasonality is 6 month
Abnormal return is higher the better. but extremely high abnormal return may be due to other reasons than seasonality. Prefer items that generate stable abnormal return in the four year training period It is preferable if average trend of stock price movement is monotonously increase in the holding period. 14
Organizing Phase Objective : maximize the sum of all abnormal return Subject to : selected periods are not overlapped each other 1 2 3 4
5 6 7 8
.... 125 126 127 128 .....
256
1 2 3 4
5 6 7 8
.... 125 126 127 128 .....
256
date
date
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Creating calendar time portfolio Item set for Toyota Item set for Hitachi Item set for Sony Jan 4
Feb 4
Feb 9
Mar 19
Mar 25 Time line
*Toyota *Hitachi
*Toyota *Hitachi *Sony
*Hitachi *Sony
*Sony
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Results
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Performance summary 2005年1月ー2014年12月
Period Mining Model EW
Period Mining Model VW
TOPIX
TOPIX Mid 400
Annualized daily return
14.32%
9.25%
4.67%
5.65%
Annualized daily risk
22.80%
23.47%
22.95%
21.99%
Maximum daily gain
12.60%
12.39%
13.73%
12.87%
Maximum daily drawdown
-10.51%
-9.20%
-9.52%
-10.62%
0.628
0.394
0.204
0.257
Sharpe ratio
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20050105 20050303 20050428 20050628 20050823 20051020 20051216 20060215 20060412 20060609 20060804 20060929 20061128 20070126 20070326 20070523 20070718 20070911 20071108 20080110 20080307 20080507 20080701 20080826 20081023 20081219 20090220 20090417 20090617 20090812 20091009 20091208 20100205 20100405 20100603 20100729 20100924 20101122 20110121 20110318 20110519 20110713 20110907 20111107 20120105 20120301 20120426 20120625 20120820 20121016 20121211 20130213 20130410 20130607 20130802 20130930 20131126 20140128 20140326 20140523 20140717 20140911 20141111
Calendar time portfolio performance since inception. Benchmark index: TOPIX , Initial NAV:100, 2005-2014
350 490
300
Period Mining Model 390
250
200 290
Performance difference (right scale)
TOPIX
150 190
100 90
50
0 -10
19
100
20050105 20050304 20050506 20050701 20050829 20051027 20051227 20060224 20060424 20060622 20060818 20061017 20061214 20070215 20070413 20070613 20070809 20071009 20071205 20080207 20080407 20080605 20080801 20080930 20081128 20090130 20090331 20090601 20090728 20090925 20091125 20100126 20100325 20100526 20100722 20100916 20101117 20110119 20110317 20110519 20110714 20110909 20111110 20120112 20120308 20120509 20120704 20120830 20121029 20121226 20130228 20130426 20130626 20130822 20131022 20131218 20140221 20140421 20140619 20140815 20141015 20141212
Calendar time portfolio performance since inception.
Benchmark index: TOPIX Mid 400, Initial NAV:100, 2005-2014
350 490
300
Period Mining Model
200
0 390
250
290
Performance difference (right scale)
150 190
TOPIX Mid 400 90
50
-10
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Controlling for “size” and “book-to-market ratio” Three factor model
Rp,t R f ,t i i (Rm,t R f ,t ) si SMBt hi HML i,t Coefficient
Standard error
t-value
p-value
Intercept
0.00026
5.675E-05
4.63
0.00000
Rm-Rf
0.01061
4.674E-05
226.92
0.00000
SMB
0.00131
1.009E-04
12.998
0.00000
HML
0.00169
1.492E-04
7.834
0.00000
Annualized alpha 6.50% 21
20050105 20050307 20050510 20050706 20050902 20051104 20060106 20060307 20060509 20060705 20060901 20061101 20070104 20070306 20070508 20070704 20070831 20071101 20080107 20080306 20080508 20080704 20080902 20081104 20090107 20090309 20090512 20090708 20090904 20091109 20100112 20100311 20100514 20100712 20100908 20101110 20110113 20110314 20110517 20110713 20110909 20111111 20120116 20120313 20120515 20120711 20120907 20121107 20130111 20130313 20130515 20130711 20130909 20131111 20140115 20140314 20140516 20140714 20140910 20141112
The daily performance against the benchmark
2.50%
2.00%
1.50%
1.00%
0.50%
0.00%
-0.50%
-1.00%
-1.50%
-2.00%
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Period Mining Portfolio, Composite of stocks and holding period
Number of shares in the portfolio and average holding period
Approximately 400 stocks in the portfolio. There is a little difference between in the first half and the second half
Average holding period is about a week
Returns are not so concentrated in the first half.
Abnormal return
Raw return
Are we holding more stocks in the earlier month? Maybe...
Where does the profit (abnormal return) come from?
When and where does the profit come from?
Longer the holding period, better the return? Not really.
Abnormal return
Raw return
How long will the strategy work?
Pairs trading (Relative value arbitrage strategy) demonstrate “profitability” beyond various risk measures, short-sale constraints and transaction costs. Gatev et.al. RFS, 2004 However, the profitability has been waned as more and more hedge funds employ similar “pairs trading” strategies. Seasonality trading is an unexplored approach; chances of enjoying hefty profit could potentially be high.
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Conclusion •
Seasonal anomaly in the stock market has been documented in the literature for quite a while, and yet, the anomaly hasn’t been arbitraged away by professionals even today.
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The difficulty lies in the potential arbitrage profit is only guaranteed in the years, not in the months.
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We endeavored to look for patterns of seasonal investor behavior in large 500 firms listed in TSE 1st.
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To detect the pattern, we used period mining technique and other related techniques commonly used in the computational science.
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Portfolio of stocks in their best season of the year outperform the benchmark index by a substantial margin.
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This “seasonal arbitrage” will remain to be profitable as few participants are playing in this market.
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