Empirical Results of an obverservation of Stocks

An overview of the return values of all ascertained stocks that have been divided into five decile portfolios are presented in the tabular arraies below. Different statistics variables are observed in all four Schemes employed to prove the Momentum Investing ( 3×3, 6×6, 9×9, and 12×12 ) . Variables and readings are discussed in item for every scheme used in this paper as follows:

A. Descriptive Statistics for 3×3 Schemes

Table I

Descriptive Statisticss for 3×3 Schemes

This tabular array shows the descriptive statistics of the mean monthly returns for FTSE100 over the period of July 21, 2000 to July 21, 2010. Sample stocks are ranked into 5 portfolio decile while P1 is the lowest decile and P5 is the highest decile. In order to detect the profitableness of impulse schemes, investing portfolios are generated by keeping long place of P5 and short place of P1. Std. Dev. bases for Standard Deviation. Probability refers to Jarque-Bera p-value.

Panel A: Consequences for 3×3 Schemes ( Arithmetic Returns )

P1

P2

P3

P4

P5

Mean

0.015704

0.008924

0.007359

0.010452

0.010364

Median

0.014693

0.018943

0.017387

0.016854

0.015222

Maximum

0.510300

0.210688

0.198576

0.300627

0.155031

Minimum

-0.348384

-0.231827

-0.235480

-0.176693

-0.206974

Std. Dev.

0.097360

0.064428

0.061588

0.058180

0.064353

Lopsidedness

0.682350

-0.681631

-0.645830

0.385922

-0.649583

Kurtosis

10.028640

6.233348

6.024749

8.417499

4.241197

Jarque-Bera

252.0492

60.5390

53.1860

147.2298

15.8730

Probability

0.000000

0.000000

0.000000

0.000000

0.000357

Sum

1.853085

1.053016

0.868371

1.233329

1.222942

Sum Sq. Dev.

1.109033

0.485661

0.443789

0.396036

0.484532

Observations

118

118

118

118

118

Panel B: Consequences for 3×3 Schemes ( Logarithmic Returns )

P1

P2

P3

P4

P5

Mean

-0.000064

0.002614

-0.000116

0.004007

0.003936

Median

0.009656

0.018393

0.013462

0.012520

0.012995

Maximum

0.200689

0.177795

0.165481

0.144549

0.139704

Minimum

-0.456970

-0.298006

-0.261096

-0.213709

-0.235241

Std. Dev.

0.090199

0.067036

0.060813

0.055201

0.063848

Lopsidedness

-1.708914

-1.525795

-1.439723

-0.998759

-1.002620

Kurtosis

9.448332

7.512529

7.073282

5.714944

5.189270

Jarque-Bera

261.8741

145.9026

122.3406

55.8582

43.3350

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

Sum

-0.007567

0.308485

-0.013677

0.472819

0.464451

Sum Sq. Dev.

0.951892

0.525774

0.432686

0.356515

0.476965

Observations

118

118

118

118

118

Table I shows the descriptive statistics for 3×3 Schemes which contain 118 observations. For the methodological analysis that employed Arithmetic Returns ( Panel A ) , it shows that the mean of the returns of all five decile portfolios are positive, with P1 as the highest ( 0.015704 ) and P3 as the lowest ( 0.007359 ) mean generated group of stocks. All portfolios have positive median returns which are ranked from the highest to the lowest value as P2 ( 0.018943 ) , P3 ( 0.017387 ) , P4 ( 0.016854 ) , P5 ( 0.015222 ) and P1 ( 0.014693 ) , severally. Both utmost value of maximal returns ( 0.510300 ) and minimal returns ( -0.348384 ) belong to the stock which is grouped in P1, while P5 has the lowest maximal return ( 0.155031 ) and P4 has the highest minimal return ( -0.176693 ) . This is consistent with consequences indicated by value of standard divergence, which shows the volatility and uncertainness of securities. The bigger figure of standard divergence indicates the greater volatility of the stock motion. Therefore this can be inferred that P1 is the most volatile group of stocks ( 0.097360 ) and P4 is the least volatile decile portfolios ( 0.058180 ) . The value of lopsidedness tells the grade of dissymmetry and peakedness of distributions of return series. Positive lopsidedness means that the return distributions of these securities have higher chance to bring forth more additions ( fewer losingss ) than securities whose returns follow the normal distribution. In contrast, negative lopsidedness implies a higher chance of gaining negative returns and generates more losingss ( fewer additions ) than those which are usually distributed. From all ascertained portfolios, it shows that merely P1 ( 0.682350 ) and P4 ( 0.385922 ) are positively skewed, while the remainder ( P2 ( -0.681631 ) , P3 ( -0.645830 ) and P5 ( -0.649583 ) ) are negatively skewed. The positive extra kurtosis is observed for all portfolios, hence it points out that return distributions are peaked ( leptokurtic ) relation to the normal return distribution which has kurtosis value of 3. Consistent with the determination from the measuring of lopsidedness and extra kurtosis, the Jarque-Bera trials reveal p-value of nothing, which means that, at any significance degrees, the void hypothesis of normalcy in Arithmetic returns has been rejected for all portfolios.

Surprisingly, the consequences for Logarithmic Returns ( Panel B ) are different from Arithmetical Returns. Two portfolios ( P1 ( -0.000064 ) and P3 ( -0.000116 ) ) show negative average returns while the remainder show positive Numberss. The highest average return belongs to P4 which has value of 0.004007. All average values remain positive which gives apathetic consequence as the old methodological analysis with Arithmetic returns. P2 ( 0.018393 ) still has the highest average returns while P1 ( 0.009656 ) still has the lowest average returns every bit good. Highest upper limit and lowest maximal returns are still for the stock in P1 ( 0.200689 ) and P5 ( 0.139704 ) , severally. However the consequence of maximal returns of all portfolios calculated by Logarithmic returns is less than half of what used Arithmetical returns. All minimal returns are negative for all portfolios. Furthermore, the most volatile and least volatile portfolios are still the same as in Panel A. All observed decile portfolios have negative lopsidedness, which can be implied that they all have more opportunity to bring forth losingss.

To reason, Table I provides sufficient grounds that all portfolios employed 3×3 Schemes do non follow the normal distribution, no affair which computation method is used to calculate the returns. Consequences besides show that simple returns have higher value of the statistics variables than those of log returns ; nevertheless, the consequences of the two are insignificantly different.

B. Descriptive Statistics for 6×6 Schemes

Table II

Descriptive Statisticss for 6×6 Schemes

This tabular array shows the descriptive statistics of the mean monthly returns for FTSE100 over the period of July 21, 2000 to July 21, 2010. Sample stocks are ranked into 5 portfolio decile while P1 is the lowest decile and P5 is the highest decile. In order to detect the profitableness of impulse schemes, investing portfolios are generated by keeping long place of P5 and short place of P1. Std. Dev. bases for Standard Deviation. Probability refers to Jarque-Bera p-value.

Panel A: Consequences for 6×6 Schemes ( Arithmetic Returns )

P1

P2

P3

P4

P5

Mean

0.016593

0.006603

0.008087

0.008360

0.011895

Median

0.017160

0.016148

0.014287

0.017057

0.014160

Maximum

0.745327

0.178404

0.184876

0.140579

0.345263

Minimum

-0.286731

-0.225284

-0.190709

-0.198466

-0.295008

Std. Dev.

0.106421

0.062378

0.054101

0.055589

0.078365

Lopsidedness

2.515138

-0.777199

-0.715031

-1.114148

-0.065870

Kurtosis

21.364760

5.602921

5.694569

5.365219

6.825013

Jarque-Bera

1737.3050

44.0419

44.5902

50.5979

70.1887

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

Sum

1.908232

0.759317

0.930061

0.961362

1.367907

Sum Sq. Dev.

1.291096

0.443570

0.333664

0.352270

0.700076

Observations

115

115

115

115

115

Panel B: Consequences for 6×6 Schemes ( Logarithmic Returns )

P1

P2

P3

P4

P5

Mean

0.001243

0.000558

0.001152

0.003181

0.002634

Median

0.015723

0.013566

0.012982

0.016087

0.010964

Maximum

0.258661

0.164763

0.148964

0.139733

0.141086

Minimum

-0.377278

-0.269786

-0.226819

-0.263494

-0.329403

Std. Dev.

0.089027

0.065374

0.055877

0.058864

0.071487

Lopsidedness

-1.054903

-1.385703

-1.341294

-1.554918

-1.334156

Kurtosis

6.878643

6.577331

6.474675

7.523455

6.717988

Jarque-Bera

93.4143

98.1237

92.3337

144.3860

100.3534

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

Sum

0.142981

0.064132

0.132458

0.365864

0.302934

Sum Sq. Dev.

0.903537

0.487216

0.355934

0.395003

0.582592

Observations

115

115

115

115

115

Table II presents the descriptive statistics for 6×6 Schemes which contain 3 observations less than the old schemes because of utilizing 3 months less when calculating J-month lagged returns. Panel A shows the statistics description of ascertained informations concentrating in Arithmetic Returns, while Panel B shows consequences of Logarithmic Returns. For Panel A, all the average returns for every decile portfolios are positive with the highest of 0.016593 for P1 and the lowest of 0.006603 for P2. Median can be ranked from highest to lowest as P1 ( 0.017160 ) , P4 ( 0.017057 ) , P2 ( 0.016148 ) , P3 ( 0.014287 ) and P5 ( 0.014160 ) , severally. The highest maximal return is for P1 ( 0.745327 ) which accounts for more than five times as of portfolio that has the lowest maximal return, P4 ( 0.140579 ) . All five portfolios have negative minimal returns with highest minimal return for P3 ( -0.190709 ) and lowest minimal return for P5 ( -0.295008 ) . By sing the criterion divergence value, it is clearly seen that P1 ( 0.106421 ) is the most volatile and P3 ( 0.054101 ) is the least volatile portfolios. Lopsidedness shows positive figure merely for P1 ( 2.515138 ) , which mean that P1 has higher opportunity to bring forth more additions, while the remainder have higher opportunity to gain losingss when comparison to returns that follow usually distribution form. Positive extra kurtosis have been observed for all five decile portfolios, where P1 ( 21.364760 ) has the highest value of all. The return of P1 leptokurtic distribution has more values around the mean and in dress suits than the normal distribution has ; hence, this portfolio has higher frequence of holding big positive or negative returns. Thus this consequence of kurtosis is consistent with standard divergence stating that P1 has the highest volatility among all ascertained portfolios. The Jarque-Bera trial of zero p-value Tells that void hypothesis of normalcy in Arithmetic returns has been rejected for every decile portfolios at any significance degrees.

For Panel B, the statistics consequences are non that different between two methods of ciphering returns, unlike those of 3×3 schemes ( Table I ) which gives a clear unsimilarity of consequences in Panel A and Panel B. Highest average return is founded in P4 ( 0.003181 ) , while lowest average return is in P2 ( 0.000558 ) . The spread between medians for all decile portfolios is non large, with the highest average value of 0.016087 for P4 and the lowest value of 0.010964 for P5. The consequences for the highest maximal return is still for P1 ( 0.258661 ) , whereas the lowest maximal return is for P4 ( 0.139733 ) , therefore this consequences is precisely the same as when employ Arithmetic Returns for computation. All minimal returns are negative values which give the same way of consequences as in Panel A but in more utmost value. Minimum returns off all five portfolios is in P1 ( -0.377278 ) , therefore this is unvarying with the consequence of standard divergence bespeaking that P1 ( 0.089027 ) is the most volatile portfolio. In contrast, the least volatile of all is P3 with standard divergence value of 0.055877. All observed portfolios show negative lopsidedness, hence this consequence is inconsistent with the consequence in Panel A. Negative lopsidedness indicates a higher chance of gaining negative returns and bring forth more losingss than the securities followed normal distribution. The kurtosis values show that all portfolios have positive extra kurtosis over the value of theoretical normal distribution ( value of 3 ) . This indicates that the returns are peaked, therefore all portfolios may hold more frequence of bring forthing utmost positive and negative returns compared to usually distributed return portfolio.

In short, all 6×6 schemes portfolios do non exhibit normal distribution form of returns in both Arithmetic Returns and Logarithmic Returns. In add-on, sing to the chance of p-value, all the returns have zero chance which suggests that the consequence is non significantly different for both in Panel A and Panel B.

C. Descriptive Statistics for 9×9 Schemes

Table III

Descriptive Statisticss for 9×9 Schemes

This tabular array shows the descriptive statistics of the mean monthly returns for FTSE100 over the period of July 21, 2000 to July 21, 2010. Sample stocks are ranked into 5 portfolio decile while P1 is the lowest decile and P5 is the highest decile. In order to detect the profitableness of impulse schemes, investing portfolios are generated by keeping long place of P5 and short place of P1. Std. Dev. bases for Standard Deviation. Probability refers to Jarque-Bera p-value.

Panel A: Consequences for 9×9 Schemes ( Arithmetic Returns )

P1

P2

P3

P4

P5

Mean

0.015586

0.006871

0.007279

0.011115

0.012768

Median

0.015958

0.020000

0.020361

0.017943

0.016808

Maximum

0.762235

0.192867

0.177421

0.158054

0.295204

Minimum

-0.289228

-0.241632

-0.200523

-0.179433

-0.285305

Std. Dev.

0.110518

0.064374

0.055161

0.053406

0.076961

Lopsidedness

2.584174

-0.948440

-1.013508

-0.850616

-0.278617

Kurtosis

21.068740

5.596226

5.980951

5.121797

6.121630

Jarque-Bera

1648.2260

48.2466

60.6427

34.5157

46.9237

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

Sum

1.745608

0.769578

0.815228

1.244866

1.430064

Sum Sq. Dev.

1.355777

0.459989

0.337749

0.316589

0.657457

Observations

112

112

112

112

112

Panel B: Consequences for 9×9 Schemes ( Logarithmic Returns )

P1

P2

P3

P4

P5

Mean

0.000592

0.000369

0.001606

0.004772

0.003405

Median

0.012148

0.014405

0.015480

0.015534

0.013471

Maximum

0.261757

0.173377

0.152345

0.113868

0.155009

Minimum

-0.362977

-0.319699

-0.223188

-0.220102

-0.325558

Std. Dev.

0.090268

0.070396

0.056957

0.054222

0.071875

Lopsidedness

-0.977591

-1.602322

-1.363893

-1.461904

-1.339729

Kurtosis

6.461173

7.662063

6.405404

6.527345

7.037447

Jarque-Bera

73.7448

149.3547

88.8421

97.9571

109.5755

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

Sum

0.066299

0.041283

0.179830

0.534489

0.381347

Sum Sq. Dev.

0.904453

0.550065

0.360098

0.326348

0.573423

Observations

112

112

112

112

112

The descriptive statistics for 9×9 Schemes are described in Table III. The Numberss of observations are 112 in sum, which are 3 observations less than 6×6 Schemes and 6 observations less than 3×3 Schemes due to the difference in informations used to calculate J-month lagged returns. Panel A and B show statistics description of informations that employed two different methods in return computation: Arithmetical Returns and Logarithmic Returns, severally. For Panel A, average returns of all portfolios show positive values. The highest average returns belong to P1 ( 0.015586 ) while the lowest average returns belong to P2 ( 0.006871 ) . The medians of all observed informations set are shown in positive Numberss, which can be ranked in falling order as P3 ( 0.020361 ) , P2 ( 0.020000 ) , P4 ( 0.01794 ) , P5 ( 0.016808 ) , and P1 ( 0.015958 ) . Sing the upper limit and minimal value of returns, P1 has the highest maximal return ( 0.762235 ) every bit good as the lowest minimal return ( -0.289228 ) . On the contrary, the P4 exhibits the lowest maximal return ( 0.158054 ) and highest minimal return ( -0.179433 ) among all observations. Standard divergence tells how volatile the motion of the securities is when comparing to intend value. The greater figure shows that the stocks move off from average and generate wider scope between the lowest and the highest value, in other words, it shows the utmost fluctuation of stocks motion. By sing standard divergence of all five decile portfolios, the portfolio that is most volatile is P1 ( 0.110518 ) , whereas the portfolio that is least volatile is P4 ( 0.053406 ) . Therefore, this consequence is unvarying with the reading of upper limit and minimal returns mentioned earlier. Four out of five portfolios shows negative figure of lopsidedness, except P1 ( 2.584174 ) which is the lone portfolio that is positively skewed. Consequently, when comparison these consequences to the value of normal distributed returns, P2, P3, P4 and P5 have fewer opportunities to originate additions, but tend to meet more of negative net incomes which will make losingss for the keeping portfolios. Every decile portfolios have extra positive kurtosis from what of normal return distribution ( value of 3 ) . The consequence for P1 ( 21.068740 ) display the highest value among the group while the remainder shows values around 6.

Panel B nowadayss similar way of consequences where all portfolios have positive mean returns. The two extreme values are changed when utilizing different methods of returns computation. Logarithmic Returns shows that the highest average returns are for P4 ( 0.004772 ) , while the lowest average return is still for P2 ( 0.000369 ) . All five portfolios deciles have positive medians which can be placed in falling order as P4 ( 0.015534 ) , P3 ( 0.015480 ) , P2 ( 0.014405 ) , P5 ( 0.013471 ) , and P1 ( 0.012148 ) . Maximal returns are described in the tabular array demoing the highest value of 0.261757 which belongs to P1 and the lowest value of 0.113868 which belongs to P4. Minimum returns exhibit the opposite way where P4 earns the highest value of -0.220102 and P1 earns the lowest value of -0.362977. This statistics result is compatible with the consequences for standard divergence which shows that P1 is the most volatile portfolio with highest standard divergence ( 0.090268 ) and P4 is the least volatile portfolio with the lowest standard divergence ( 0.054222 ) of all. Unlike Panel A which uses Arithmetical mean to calculate returns, lopsidedness are negative for all portfolio deciles. This can be inferred that every keeping portfolios seem to execute ill and bring forth losingss more than additions if comparison to the consequence of theoretical normal distributed returns. Sing the kurtosis values, all five portfolios display similar consequences which are about in the scope between 6 and 8 ; unlike consequence in Panel A that P1 is the lone portfolio that has utmost figure of kurtosis.

In drumhead, chance of p-value still shows the value of zero which means that statistics consequences are undistinguished at any assurance degree of Jarque-Bera trial. For this ground, it can be explained that in 9×9 schemes, both Arithmetic Returns and Logarithmic Returns, all portfolios returns are non usually distributed.

D. Descriptive Statistics for 12×12 Schemes

Table IV

Descriptive Statisticss for 12×12 Schemes

This tabular array shows the descriptive statistics of the mean monthly returns for FTSE100 over the period of July 21, 2000 to July 21, 2010. Sample stocks are ranked into 5 portfolio decile while P1 is the lowest decile and P5 is the highest decile. In order to detect the profitableness of impulse schemes, investing portfolios are generated by keeping long place of P5 and short place of P1. Std. Dev. bases for Standard Deviation. Probability refers to Jarque-Bera p-value.

Panel A: Consequences for 12×12 Schemes ( Arithmetic Returns )

P1

P2

P3

P4

P5

Mean

0.014627

0.008069

0.007276

0.009527

0.012824

Median

0.016773

0.017832

0.015046

0.017977

0.017205

Maximum

0.513662

0.272998

0.153691

0.138341

0.202901

Minimum

-0.290740

-0.234688

-0.199093

-0.180477

-0.299651

Std. Dev.

0.102956

0.070319

0.055812

0.052699

0.071444

Lopsidedness

0.969122

-0.387531

-0.910109

-0.818627

-0.924584

Kurtosis

8.966178

6.198143

5.323964

4.822208

6.234244

Jarque-Bera

178.7239

49.1810

39.5761

27.2547

63.0372

Probability

0.000000

0.000000

0.000000

0.000001

0.000000

Sum

1.594376

0.879555

0.793109

1.038458

1.397858

Sum Sq. Dev.

1.144797

0.534038

0.336417

0.299936

0.551253

Observations

109

109

109

109

109

Panel B: Consequences for 12×12 Schemes ( Logarithmic Returns )

P1

P2

P3

P4

P5

Mean

0.000626

0.000129

0.000797

0.003997

0.003666

Median

0.014170

0.013154

0.013753

0.015509

0.015450

Maximum

0.210297

0.161204

0.132433

0.125832

0.159859

Minimum

-0.373744

-0.285237

-0.225001

-0.229752

-0.340607

Std. Dev.

0.093297

0.070909

0.058292

0.054127

0.069955

Lopsidedness

-1.101329

-1.416211

-1.383271

-1.440117

-1.566581

Kurtosis

6.054122

6.682085

6.224538

6.574486

8.204093

Jarque-Bera

64.3980

98.0108

81.9835

95.7051

167.5843

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

Sum

0.068281

0.014018

0.086823

0.435674

0.399594

Sum Sq. Dev.

0.940061

0.543040

0.366978

0.316417

0.528513

Observations

109

109

109

109

109

Table IV presents the statistics description for the 12×12 Schemes with the entire figure of 109 observations. Consequences from Panel A are from Arithmetic Returns computation method, while consequences from Panel B are from Logarithmic Returns computation method. In Panel A, the mean values are positive for all the stocks observed. The greatest sum of average returns is for P1 ( 0.014627 ) , while the minimal average returns is for P3 ( 0.007276 ) . For the median, every portfolio has the same scope of median between 0.015 – 0.018. If rank from the highest median to the lowest, the order of portfolios are P4 ( 0.017977 ) , P2 ( 0.017832 ) , P5 ( 0.017205 ) , P1 ( 0.016773 ) and P3 ( 0.015046 ) , severally. The highest maximal return is for P1 ( 0.513662 ) which about four times more than the lowest minimal return of P4 ( 0.138341 ) , which gives the highest minimal return ( -0.180477 ) by contrast with the lowest minimal return as of P5 ( -0.299651 ) . These consequences are consistent with the criterion divergence which discloses that P1 has the greatest volatility in stock monetary values ( 0.102956 ) , while P4 has the least volatility in stock monetary values ( 0.052699 ) . Every portfolio show negative value of lopsidedness, except for P1. Therefore, it means that the distribution of returns series in P1 is more ailing and can be interpreted that P1 has a higher chance to bring forth more additions than losingss when comparison to the theoretical normal distribution. Furthermore, statistics present positive Numberss of kurtosis for all ascertained portfolios. The highest value of extra kurtosis belongs to P1 ( 8.966178 ) , whereas the lowest value of extra kurtosis belongs to P4 ( 4.822208 ) . Hence, consequences tell that P1 seems to hold higher frequence of holding utmost returns than P4.

Panel B shows similar consequences as in Panel A but the returns are far less. Positive mean returns are observed in all decile portfolios, with the highest of 0.003997 for P4 and the lowest of 0.000129 for P2. Median returns can be ranked in order from high to low as P4 ( 0.015509 ) , P5 ( 0.015450 ) , P1 ( 0.014170 ) , P3 ( 0.013753 ) and P2 ( 0.013154 ) , severally. The lowest maximal return is for P4 ( 0.125832 ) and the highest maximal return is for P1 ( 0.210297 ) , which gives the lowest minimal return of -0.373744. In contrast, the portfolio that has the highest minimal return is P3 ( -0.225001 ) . Furthermore, statistics show that stocks in P1 has the most utmost motion since it has highest value of standard divergence ( 0.093297 ) ; on the contrary, stocks in P4 exhibit comparatively more steady motion way since it has the lowest value of standard divergence ( 0.054127 ) . Negative lopsidedness are observed for all five portfolio deciles therefore this indicates that investing portfolios tend to hold higher opportunity to bring forth more losingss than those of returns with normal distribution form. Additionally, kurtosis values present that all portfolios have positive extra kurtosis compared to value of 3 of normal distribution returns. Hence, it shows that the portfolios returns distributions are peaked ( leptokurtic distribution ) and has higher frequence of bring forthing utmost returns in both positive and negative sides.

To sum up, consequences of 12×12 Schemes utilizing Arithmetical Returns ( Panel A ) and Logarithmic Returns ( Panel B ) are insignificantly different. Both methods show zero p-value at any assurance degree when we employ the Jarque-Bera trial.

In decision, no affair which methods of return computations are used in four impulse schemes applied in this research paper ( 3×3, 6×6, 9×9, and 12×12 schemes ) , all descriptive statistics consequences are insignificantly different as a effect of zero p-value observed in Jarque-Bera trials.

4.3 Evidence of Momentum Profitability

4.4 Discussion of Consequences