Stephane Cheung Yukinobu Hamuro Katsuhiko Okada

Detecting stock market seasonalityA period mining approach

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Detecting stock market seasonalityA period mining approach

Stephane Cheung* /Yukinobu Hamuro* / Katsuhiko Okada * ***Kwansei Gakuin University Business School**Magne-Max Capital Management, Japan FSA registered Investment Advisor

1

Market seasonality

Fact finding• Agrawal and Tandon (1994) JIMF

18 countries Seasonality, daily, weekend effect, last trading day of the month, large pre-and inter holiday return. Jan, return.

• Bouman and Jacobsen (2002) AER36 global markets out of 37 examined have “sell in May effect”, or “Halloween effect”, including Japan

• Kamstra, Kramer and Levi (2003) AERMarkets in northern hemisphere demonstrate “sell in May” but not in southern hemisphere. SAD effect

• Sakakibara, Yamasaki and Okada (2013) IRFJapan is unique as Jun. is a good month while most other financial markets demonstrate lower June return -> “Dekansho-bushi” effect

2

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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4

316 yearsJacobsen Global Fin.

Data index

82 yearsDow Jones index

50 yearsDow Jones index

1693-2009 1929-2011 1961-2011Jan 0.69 1.0 1.2Feb 0.09 0.0 0.0Mar -0.03 0.4 1.1April 0.49 1.4 2.0May 0.02 -0.2 -0.1June -0.12 0.5 -0.6July -0.31 1.5 0.9Aug 0.44 0.8 0.2Sep -0.49 -1.3 -0.8Oct -0.5 0.0 0.5Nov 0.35 0.8 1.2Dec 0.81 1.5 1.5

Nov-April 2.42 5.2 7.2May-Oct -0.96 1.28 0.09

Diff 3.38 3.92 7.11

Nikkei 225

TOPIX

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

First-half YearLast-half Year

Mea

n M

onth

ly R

etur

n (%)

Mean monthly return of Nikkei 225 and TOPIXJan 1971-Dec 2014

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If fund managers had followed the “words of wisdom”…

0

500

1000

1500

2000

2500

3000

3500

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

TOPIX

-80%

-70%

-60%

-50%

-40%

-30%

-20%

-10%

0%

10%

Dekansho-bushi Investment Full Year Investment

Japan’s Lost 2 decades

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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 approacha. Researchers come up with some insightsb. Create a model (Hypothesis building)c. Collect datad. Conduct an empirical test to prove or disprove the model (accept or reject the hypothesis)e. Conclusion and conjecture

• Data centric approacha. There is no model. We don’t even have the hypothesisb. Large scale datac. 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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10

Methodology (Image)

2001-2004

2005

Model Building Period

2002-2005

2006

Portfolio formation based on the model created in 2001-4

Rolling window up to present

Portfolio simulation up to 2015.

Model Building Period

Period Mining: 4 steps in model building

Historical data of Stock Price for a stock (eg. Toyota)

1. Enumeration PhaseEnumerating 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 PhaseFiltering items by aggregated statistics which match the given conditions (eg. lower bound of duration days).

4. Organizing PhaseSelect 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

2. Aggregation PhaseAggregating the statistics on each item for last n years (n = 4 in this exp.)

Selected Items (Periods)to hold

term: 125days starting from Jan 1st or July 1st

28 terms from 2001 to 2014

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Enumeration Phase

1 2 3 4 5 6 7 8 .... 124 125 126 127 250

date

……

…

…

term 1

term 2…

starting date: 1,2,...,125 holding period: 1,2,...,125

125 × 125 × 500 equities ≒ 7.8M items per term

Enumerate stock price statistics on period ps,e

s=1, e=2s=1, e=3s=1, e=4

:s=1, e=125s=2, e=3s=2, e=4

:s=2, e=125s=3, e=4s=3, e=5

s=3, e=127

s=125, e=126s=125, e=127

s=125, e=250

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Aggregation Phase

a) Holding periodb) Average of abnormal returnc) SD of abnormal returnd) Average of zigzag rate

:

:

:

:

:

Aggregate all the items based on the following four

criteria

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Filtering Phase

notation min max Rationale

lh lower bound for holding period 0 20 Lower bound: Too short holding period is not favorable due to transactions cost. Upperbound :Index level seasonality is 6 monthuh upper bound for holding period 10 125

lr lower bound for average of abnormal return 0.01 0.1 Abnormal return is higher the better. but

extremely high abnormal return may be due to other reasons than seasonality.ur upper bound for average of abnormal

return 0.11 2.1

us upper bound for SD of abnormalreturn 0.01 1.0 Prefer items that generate stable abnormal

return in the four year training period

lz lower bound for zigzag rate 0 1It is preferable if average trend of stock price movement is monotonously increase in the holding period.

Select periods matching the given 6 conditions. The optimal parameters will be estimated using machine learning technique (Bayesian

Global Optimization).declare

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Organizing Phase

1 2 3 4 5 6 7 8 .... 125 126 127 128 ..... 256 date

1 2 3 4 5 6 7 8 .... 125 126 127 128 ..... 256 date

Objective : maximize the sum of all abnormal returnSubject to : selected periods are not overlapped each other

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Item set for Toyota

Item set for Hitachi

Item set for Sony

Time line

Creating calendar time portfolio

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Jan 4

*Toyota*Hitachi

*Toyota*Hitachi*Sony

*Hitachi*Sony *Sony

Feb 4 Feb 9 Mar 19 Mar 25

Results

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Performance summary

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PeriodMining Model

EW

PeriodMining Model

VWTOPIX TOPIX Mid

400Annualized 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%

Sharpe ratio 0.628 0.394 0.204 0.257

2005年1月ー2014年12月

Calendar time portfolio performance since inception. Benchmark index: TOPIX , Initial NAV:100, 2005-2014

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Performance difference (right scale)

Period Mining Model

TOPIX

-10

90

190

290

390

490

0

50

100

150

200

250

300

350

2005

0105

2005

0303

2005

0428

2005

0628

2005

0823

2005

1020

2005

1216

2006

0215

2006

0412

2006

0609

2006

0804

2006

0929

2006

1128

2007

0126

2007

0326

2007

0523

2007

0718

2007

0911

2007

1108

2008

0110

2008

0307

2008

0507

2008

0701

2008

0826

2008

1023

2008

1219

2009

0220

2009

0417

2009

0617

2009

0812

2009

1009

2009

1208

2010

0205

2010

0405

2010

0603

2010

0729

2010

0924

2010

1122

2011

0121

2011

0318

2011

0519

2011

0713

2011

0907

2011

1107

2012

0105

2012

0301

2012

0426

2012

0625

2012

0820

2012

1016

2012

1211

2013

0213

2013

0410

2013

0607

2013

0802

2013

0930

2013

1126

2014

0128

2014

0326

2014

0523

2014

0717

2014

0911

2014

1111

Calendar time portfolio performance since inception. Benchmark index: TOPIX Mid 400, Initial NAV:100, 2005-2014

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Performance difference (right scale)

Period Mining Model

TOPIX Mid 400

-10

90

190

290

390

490

0

50

100

150

200

250

300

350

2005

0105

2005

0304

2005

0506

2005

0701

2005

0829

2005

1027

2005

1227

2006

0224

2006

0424

2006

0622

2006

0818

2006

1017

2006

1214

2007

0215

2007

0413

2007

0613

2007

0809

2007

1009

2007

1205

2008

0207

2008

0407

2008

0605

2008

0801

2008

0930

2008

1128

2009

0130

2009

0331

2009

0601

2009

0728

2009

0925

2009

1125

2010

0126

2010

0325

2010

0526

2010

0722

2010

0916

2010

1117

2011

0119

2011

0317

2011

0519

2011

0714

2011

0909

2011

1110

2012

0112

2012

0308

2012

0509

2012

0704

2012

0830

2012

1029

2012

1226

2013

0228

2013

0426

2013

0626

2013

0822

2013

1022

2013

1218

2014

0221

2014

0421

2014

0619

2014

0815

2014

1015

2014

1212

Controlling for “size” and “book-to-market ratio”Three factor model

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Rp,t Rf ,t i i (Rm,t Rf ,t ) siSMBt hiHML 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%

The daily performance against the benchmark

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-2.00%

-1.50%

-1.00%

-0.50%

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

2005

0105

2005

0307

2005

0510

2005

0706

2005

0902

2005

1104

2006

0106

2006

0307

2006

0509

2006

0705

2006

0901

2006

1101

2007

0104

2007

0306

2007

0508

2007

0704

2007

0831

2007

1101

2008

0107

2008

0306

2008

0508

2008

0704

2008

0902

2008

1104

2009

0107

2009

0309

2009

0512

2009

0708

2009

0904

2009

1109

2010

0112

2010

0311

2010

0514

2010

0712

2010

0908

2010

1110

2011

0113

2011

0314

2011

0517

2011

0713

2011

0909

2011

1111

2012

0116

2012

0313

2012

0515

2012

0711

2012

0907

2012

1107

2013

0111

2013

0313

2013

0515

2013

0711

2013

0909

2013

1111

2014

0115

2014

0314

2014

0516

2014

0714

2014

0910

2014

1112

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.

• The difficulty lies in the potential arbitrage profit is only guaranteed in the years, not in the months.

• We endeavored to look for patterns of seasonal investor behavior in large 500 firms listed in TSE 1st.

• To detect the pattern, we used period mining technique and other related techniques commonly used in the computational science.

• Portfolio of stocks in their best season of the year outperform the benchmark index by a substantial margin.

• This “seasonal arbitrage” will remain to be profitable as few participants are playing in this market.

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