8. Regression details

statistics in order to determine how good the different asset pricing models performed in explaining portfolio returns. In addition to these statistics, average absolute alpha spread was added for a more complete picture of the alpha results. Furthermore, different models' explan-

If a capital asset pricing model (CAPM, three-factor model or five-factor model) completely captures expected returns, the intercept (alphas) is indistinguishable from zero in a regression

Table 5 shows the GRS statistics of [24] that tests this hypothesis for combinations of LHS portfolios and factors. The GRS test easily rejects all models considered for all LHS portfolios and RHS factors. The probability, or p-value, of getting a GRS statistic larger than the one observed if the true intercepts are all zero is shown in column 'pGRS'. One can see from Table 5 that except CAPM in Panel A and Panel C, sets of left-hand-side returns, the p-values round to zero to at least three decimals. Only five-factor model in Panel C has a p-value of 0.30, and it is

CAPM 0.95 2.74 0.57 0.25 0.06 Three-factor (S/BM) 2.13 0.00 1.32 0.31 0.34 Five-factor (S/BM) 2.47 0.00 0.73 0.72 0.49

CAPM 4.23 0.00 0.77 0.49 0.07 Three-factor (S/OP) 5.63 0.00 1.41 0.44 0.33 Five-factor (S/OP) 6.15 0.00 0.91 0.87 0.50

CAPM 0.94 2.66 0.66 0.18 0.06 Three-factor (S/Inv) 2.93 0.00 1.38 0.19 0.33 Five-factor (S/Inv) 1.33 0.30 0.73 0.74 0.50

CAPM (all) 3.20 0.00 0.67 0.30 0.06 Three-factor (all) 13.37 0.00 1.15 0.57 0.39 Five-factor (all) 2.66 0.00 0.79 0.78 0.49

average absolute deviation from the average intercept; and the average R<sup>2</sup>

The table tests the ability of CAPM, 3F-FF and 5F-FF models to explain monthly excess returns on 16 size-B/M portfolios (Panel A), 16 size-OP portfolios (Panel B), 16 size-Inv portfolios (Panel C) and a joint sample of all 48 portfolios (Panel D). For each panel, the table shows the tested model; the GRS statistic testing whether the expected values of all 16 or 48 intercept estimates are zero; the p-value for the GRS statistic; the average absolute value of the intercepts, Avg |α|; the

Table 5. Summary of statistics for model comparison tests: summary of statistics for tests of CAPM, 3F-FF and 5F-FF

. Bolded and shaded GRS statistics indicate the

fGRS pGRS Avg |α| Avg j j <sup>α</sup> � <sup>α</sup> Avg R2

atory power was measured using R<sup>2</sup> values.

80 Financial Management from an Emerging Market Perspective

still significant at the 5% level.

Panel A: size-B/M portfolios

Panel B: size-OP portfolios

Panel C: size-Inv portfolios

Panel D: all portfolios

significance at the 5% level.

models (June 2000–May 2017, 204 months).

of an asset's excess returns on the model's factor returns.

In this part, I give individual regression alphas, the coefficients that were defined in Eqs. (5)–(7), their corresponding t-values and R-squared values in order to provide a more detailed picture of model performance. I concentrate on the significance of alphas. Significant alpha patterns between models are compared and further analysed by looking at regression slopes in Tables 6–8.



Dep. variables

Five-factor

α

> S1BM1

Coeff.

t-Stat

S1BM2

Coeff.

t-Stat

S1BM3

Coeff.

t-Stat

S1BM4

Coeff.

t-Stat

> S2BM1

Coeff.

t-Stat

S2BM2

Coeff.

t-STAT

> S2BM3

Coeff.

t-Stat

S2BM4

Coeff.

t-Stat

S3BM1

Coeff.

t-Stat

S3BM2

Coeff.

t-Stat

S3BM3

Coeff.

t-Stat

S3BM4

Coeff.

 1.20

0.01

 1.48

 0.88

0.76

0.16

 0.55

 1.95

 0.04

 0.59

 0.91

 0.37

 1.11

 0.01

 0.01

 0.32

 0.34

5.37

 3.73

2.71

2.02

1.26

 1.01

 1.61

3.35

0.38

 0.60

 0.22

 0.02

 1.32

 0.73

0.51

0.37

 0.45

 0.86

 0.07

 0.29

 0.70

 0.25

 0.26

 0.04

 0.04

 0.32

 0.29

5.37

 3.97

5.24

1.37

1.94

 1.06

2.55

 3.52

 0.24

 0.02

 1.44

 0.84

1.07

0.27

 0.50

 1.48

 0.08

 0.52

 0.82

 0.28

 0.73 0.95

 0.68

 0.05

 0.05

0.19

 0.00

5.67

 4.24

6.18

1.26

1.64

 0.86

2.75

 3.67

0.14

 0.00

 1.43

 0.84

1.19

0.23

 0.54

 1.21

 0.06

 0.54

 0.82

 0.30

 0.44 0.60

 0.47

 0.03

 0.03

 0.55

0.48

7.51

 4.81

5.44

1.71

2.02

0.50

3.10

 4.15

 0.41

0.03

 1.98

 1.00

1.09

0.33

 0.58

 1.59

 0.04

 0.65

 1.00

 0.33

 0.67 0.83

 0.07

 0.01

 0.00

 0.49

0.64

6.19

 3.65

3.38

1.32

1.43

 0.11

2.68

 3.41

 0.35

0.04

 1.57

 0.73

0.65

0.25

 0.48

 1.02

 0.01

 0.51

 0.75

 0.28

 0.33 0.46

0.23

0.02

 0.02

2.00

1.14

6.24

 2.45

0.99

0.04

2.43

0.88

7.13

 3.41

 1.40

0.07

 1.54

 0.48

0.19

0.01

 0.44

 1.51

0.05

 1.18

 0.65

 0.48

 0.71 1.04

0.88

0.06

 0.06

 0.52

1.20

6.00

0.97

1.74

 0.37

 0.37

0.07

 1.51

 0.19

0.33

 0.07

 0.41

 0.55 0.86

0.89

6.55

1.80

0.02

0.73

0.05

 1.12

 0.35

 0.43

0.02

0.05

 0.05

2.23

0.61

5.57

 3.91

1.46

 0.97

 1.76

0.04

 1.55

 0.86

0.31

 0.20

 0.45

 1.75 2.47

0.37

5.95

 4.77

0.03

 1.13

 1.04

 0.46

 0.68 0.88

0.66

0.05

 0.05

 1.94

0.83

5.47

 4.72

1.50

 0.77

 1.46

0.05

 1.45

 0.99

0.30

 0.15

 0.47

 1.51 2.23

0.58

5.81

 5.54

0.04

 1.05

 1.15

 0.48

 0.37 0.50

0.94

0.07

 0.07

 1.76

0.85

7.50

 3.48

1.50

 0.28

 1.42

0.06

 2.13

 0.78

0.33

 0.06

 0.51

 1.50 2.05

0.46

7.46

 4.43

0.03

 1.46

 0.99

 0.51

 0.36 0.45

0.61

0.05

 0.04

82 Financial Management from an Emerging Market Perspective

 1.69

1.15

6.58

1.88

1.18

 0.79

 1.26

0.07

 1.73

 0.39

0.24

 0.15

 0.45

 1.18 1.74

0.88

6.60

 2.79

0.06

 1.20

 0.58

 0.44

 0.42 0.58

0.84

0.06

 0.06

Rm-Rf

 SMB

 HML

 RMW

 CMA

 R2

α

Rm-Rf

 SMB

 HML

 R2

α

Rm-Rf

 R2

 size-B/M models

Three-factor

 size-B/M models

CAPM

Table 6. CAPM, 3F-FF and 5F-FF regressions for 16 value-weighted size-B/M portfolios (June 2000–May 2017, 204 months).

portfolios.

coefficients

 at the 5% level of

 The factors are constructed

 using

significance.

 (For the definitions

independent

 2 3 sorts on size and each of the B/M, OP and Inv portfolios.

 of dependent variables, see Table 1).

 Shaded t-statistics indicate the significance

 of



Dep. variables

Five-factor

α

> S1OP1

Coeff.

t-Stat

> S1OP2

Coeff.

t-Stat

> S1OP3

Coeff.

t-Stat

> S1OP4

Coeff.

t-Stat

S2OP1

Coeff.

t-Stat

S2OP2

Coeff.

t-Stat

S2OP3

Coeff.

t-Stat

S2OP4

Coeff.

t-Stat

S3OP1

Coeff.

t-Stat

S3OP2

Coeff.

t-Stat

S3OP3

Coeff.

t-Stat

S3OP4

Coeff.

 0.07

 0.00

 1.39

 0.82

1.09

0.16

 0.56

 1.26

 0.06

 0.63

 0.83

 0.35

 0.48

 0.03

 0.04

 1.04

 0.60

5.76

 3.40

4.56

0.01

2.01

1.23

2.68

 3.23

 0.68

 0.03

 1.32

 0.62

0.80

 0.00

 0.48

 1.30

 0.08

 0.46

 0.64

 0.28

 0.70 1.08

 0.89

 0.06

 0.06

1.03

0.18

5.74

 4.70

4.89

2.16

0.59

 0.70

 1.85

3.98

0.45

 0.23

0.74

0.01

 1.44

 0.93

0.94

0.40

 0.53

 0.43

 0.05

 0.36

 0.89

 0.29

0.33

 0.02

 0.02

 0.71

0.53

4.00

 4.00

4.60

2.17

2.55

0.25

2.12

 3.51

 0.53

0.03

 1.06

 0.84

0.93

0.43

 0.46

 1.88

 0.02

 0.42

 0.79

 0.27

 1.18 1.59

0.13

0.01

 0.01

 0.10

0.36

6.76

 3.57

6.21

1.18

1.79

 0.58

3.10

 3.09

 0.08

0.02

 1.76

 0.73

1.24

0.23

 0.56

 1.38

 0.04

 0.64

 0.73

 0.28

 0.67 0.86

 0.30

 0.02

 0.02

 1.52

0.49

6.29

 3.99

5.66

0.03

2.65

0.35

2.28

 3.55

 1.09

0.03

 1.59

 0.80

1.09

 0.01

 0.54

 1.96

 0.02

 0.45

 0.80

 0.27

 1.24 1.67

0.03

 0.00

 0.00

 1.93

1.79

5.88

 4.60

1.07

0.01

2.35

1.54

6.54

 5.51

 1.33

0.11

 1.42

 0.88

0.20

 0.00

 0.48

 1.44

0.09

 1.07

 1.03

 0.51

 0.38 0.56

1.74

0.11

 0.12

 1.62

1.25

5.80

 2.82

0.49

 0.00

 1.13

0.08

 1.42

 0.55

0.09

 0.00

 0.42

 1.12 1.81

1.05

6.53

 3.74

0.06

 1.08

 0.71

 0.46

 0.30 0.45

1.12

0.07

 0.08

 0.51

0.53

6.46

 3.56

1.99

 0.45

 0.38

0.03

 1.69

 0.74

0.40

 0.09

 0.47

 0.55 0.81

0.14

6.19

 4.29

0.01

 1.13

 0.89

 0.45

0.42

0.57

0.38

0.03

 0.03

2.36

0.66

4.47

 3.28

1.87

 1.64

 1.81

0.04

 1.21

 0.70

0.39

 0.33

 0.40

 1.76 2.49

0.45

4.28

 3.89

0.03

 0.81

 0.84

 0.36

 0.91 1.25

0.73

0.05

 0.05

2.36

1.64

6.86

 4.85

1.98

 1.34

 1.81

0.11

 1.86

 1.04

0.41

 0.27

 0.53

 1.80 2.58

1.33

6.98

 5.81

0.09

 1.30

 1.24

 0.53

 0.52 0.66

1.55

0.12

 0.11

84 Financial Management from an Emerging Market Perspective

3.83

0.89

6.99

 3.82

0.13

 0.49

 2.85

0.06

 1.83

 0.79

 0.03

 0.09

 0.50

 2.51 3.72

0.69

6.98

 4.84

0.04

 1.26

 1.00

 0.50

 1.43 1.91

0.89

0.06

 0.06

Rm-Rf

 SMB

 HML

 RMW

 CMA

 R2

α

Rm-Rf

 SMB

 HML

 R2

α

Rm-Rf

 R2

 size-OP models

Three-factor

 size-OP models

CAPM


portfolios.

coefficients

 at the 5% level of significance

 The factors are constructed

 using

 (for the definitions

independent

 2 3 sorts on size and each of B/M, OP and Inv portfolios.

 of dependent variables, see Table 1).

 Shaded t-statistics

 indicate the significance

 of



Dep. variables

Five-factor

α

> S1INV1

Coeff.

t-Stat

S1INV2

Coeff.

t-Stat

S1INV3

Coeff.

t-Stat

> S1INV4

Coeff.

t-Stat

S2INV1

Coeff.

t-Stat

S2INV2

Coeff.

t-Stat

> S2INV3

Coeff.

t-Stat

S2INV4

Coeff.

t-Stat

S3INV1

Coeff.

t-Stat

S3INV2

Coeff.

t-Stat

S3INV3

Coeff.

t-Stat

S3INV4

Coeff.

 0.67

0.02

 1.50

 0.66

0.77

0.45

 0.51

 1.76

 0.03

 0.58

 0.64

 0.28

 1.13

 0.02

 0.02

 0.30

 0.25

5.97

 4.90

5.60

2.85

2.36

1.15

2.83

 4.12

 0.19

 0.01

 1.31

 0.85

0.94

0.46

 0.56

 1.52

 0.07

 0.49

 0.82

 0.32

 0.78 1.19

 0.69

 0.04

 0.05

 0.19

0.16

4.85

 3.78

5.29

0.88

1.65

 0.63

 1.55

3.27

0.80

 0.24

 0.14

0.01

 1.26

 0.77

1.05

0.17

 0.48

 1.22

 0.04

 0.31

 0.74

 0.24

 0.59

 0.02

 0.02

 0.55

0.29

6.02

 4.16

6.34

0.21

2.02

0.57

2.36

 3.61

 0.37

0.02

 1.42

 0.78

1.14

0.04

 0.55

 1.40

 0.04

 0.44

 0.77

 0.28

 0.71 1.02

 0.18

 0.01

 0.01

0.26

0.78

6.04

 3.95

5.19

2.49

1.63

 0.19

3.18

 3.43

0.19

0.05

 1.56

 0.80

1.02

0.47

 0.53

 1.22

 0.01

 0.63

 0.78

 0.30

 0.46 0.61

0.13

0.01

 0.01

 0.01

0.25

6.12

 3.56

4.29

2.11

1.53

 0.57

3.29

 3.24

 0.01

0.02

 1.64

 0.75

0.88

0.42

 0.50

 1.17

 0.04

 0.67

 0.75

 0.30

 0.42 0.55

 0.28

 0.02

 0.02

2.36

1.01

6.46

 2.97

2.58

1.19

2.87

0.65

6.77

 3.85

 1.62

0.06

 1.56

 0.57

0.47

 0.21

 0.47

 1.79

0.04

 1.13

 0.73

 0.47

 0.93 1.38

0.75

0.05

 0.05

 1.80

1.31

5.91

 3.63

2.63

0.49

2.54

0.92

6.32

 4.39

 1.20

0.08

 1.39

 0.67

0.47

 0.09

 0.46

 1.53

0.05

 1.02

 0.81

 0.46

 0.65 1.00

1.09

0.07

 0.08

 1.88

1.49

5.49

 4.20

0.92

0.49

2.30

1.18

5.36

 4.82

 1.42

0.10

 1.47

 0.89

0.19

0.10

 0.45

 1.59

0.08

 0.99

 1.02

 0.44

 0.56 0.76

1.44

0.10

 0.10

2.38

1.33

5.78

 3.96

0.43

 0.95

 1.70

0.08

 1.45

 0.78

0.08

 0.18

 0.46

 1.46 2.29

1.20

6.27

 4.94

0.07

 1.07

 0.97

 0.48

 0.45 0.65

1.41

0.09

 0.10

 1.75

0.73

6.65

 3.67

2.63

1.14

2.10

0.31

6.06

 4.37

 1.31

0.05

 1.76

 0.76

0.53

 0.22

 0.49

 1.46

0.02

 1.13

 0.93

 0.45

 0.46 0.61

0.55

0.04

 0.04

86 Financial Management from an Emerging Market Perspective

 1.64

0.83

6.50

 2.84

2.07

1.77

1.69

0.55

6.67

 3.80

 1.28

0.06

 1.79

 0.62

0.43

 0.36

 0.47

 1.20

0.04

 1.27

 0.83

 0.46

 0.24 0.31

0.66

0.05

 0.05

Rm-Rf

 SMB

 HML

 RMW

 CMA

 R2

α

Rm-Rf

 SMB

 HML

 R2

α

Rm-Rf

 R2

 size-Inv models

Three-factor

 size-Inv models

CAPM

Table 8. CAPM, 3F-FF and 5F-FF regressions for 16 value-weighted size-Inv portfolios (June 2000–May 2017, 204 months).

coefficients

 at the 5% level of significance

 (for the definitions

 of dependent variables, see Table 1). Looking explicitly at the number of significant alpha values in Tables 6–8 (at the 5% level of confidence), both the CAPM and the 3F-FF model perform poorly, while 5F-FF model's performance although is not the best but much better. The highest number of significant alpha values belongs to 3F-FF model. This shows that 3F-FF model leaves a high proportion of unexplained part in the behaviour of LHS variables. From the regression tables, it seems that CAPM model has no any significant values for alphas; interestingly enough, none of the coefficients of RM-Rf is significant at the 5% level of confidence, and R2 terms are too low in CAPMs. In addition to this observation, in none of the models, RM-Rf has no statistically significant coefficient except in a few models (see rows 13, 14 and 15 in Tables 6–8).

On the other hand, inspection of alphas in 5F-FF models for size-B/M, size-OP and size-Inv portfolios reveals that very few of alpha values are significant at the 5% level confidence and I think the best performing model (but imperfect) for the Turkish data is the 5F-FF Fama-French model.

There is an interesting pattern in the results of the regression tables (Tables 6–8). As the size of the companies in portfolios increases, more factors become statistically significant, and the explanatory power of the 5F-FF model rises. This makes me to think that monthly excess returns of portfolios, covering high-market-cap companies, are more sensitive to SMB, HML, RMA and CMA factors.

Before I present the regression details, readers should be aware that in an OLS regression, Rsquared values will always increase with the inclusion of more factors. Another point is that R<sup>2</sup> increases and become much stronger for almost any explanatory power metric when the regressions' explanatory factors are created from return differences in the data itself. In other words, correlation between explanatory variables, which are created from return differences, and dependent variables has to be highly correlated. Hence, R<sup>2</sup> s in the equations become stronger and true.

Sundqvist in [22] states that:

'The method to construct the factor will hence automatically bring the augmented model's explanatory power in small portfolio regressions closer to the R-squared values found in big portfolio regressions, boosting the average explanatory power more than it would for regressions of randomly picked portfolio sets. This same phenomenon is apparent in all other risk factors, which are included as sorting variables and simultaneously used as explanatory variables created from return differences in the sample data'.

When the t-values in Table 6 are analysed, CAPM's results (column labelled with CAPM) are far away from being reliable and informative. None of the t-statistic of RM-Rf (except for S4BM3) is significant at the 5% level of confidence. This might have been due to the fact that BIST-100 Index, which approximates the monthly excess return of the market, is heavily financial stock weighted. However, using different market indices in all models did not improve the results (more information on this issue is given in the discussion of the results in Table 7).

In this study, I include only nonfinancial sector stocks registered with the Istanbul stock exchange since the profitability and investment variables' definition is not comparable to the financial stocks. Therefore, I think the coefficient of RM-Rf variable, namely, β<sup>i</sup> does not reflect the market risk properly, and CAPM performs poorly.

Looking explicitly at the number of significant alpha values in Tables 6–8 (at the 5% level of confidence), both the CAPM and the 3F-FF model perform poorly, while 5F-FF model's performance although is not the best but much better. The highest number of significant alpha values belongs to 3F-FF model. This shows that 3F-FF model leaves a high proportion of unexplained part in the behaviour of LHS variables. From the regression tables, it seems that CAPM model has no any significant values for alphas; interestingly enough, none of the coefficients of RM-Rf is significant at the 5% level of confidence, and R2 terms are too low in CAPMs. In addition to this observation, in none of the models, RM-Rf has no statistically significant coefficient except

On the other hand, inspection of alphas in 5F-FF models for size-B/M, size-OP and size-Inv portfolios reveals that very few of alpha values are significant at the 5% level confidence and I think the best performing model (but imperfect) for the Turkish data is the 5F-FF Fama-French

There is an interesting pattern in the results of the regression tables (Tables 6–8). As the size of the companies in portfolios increases, more factors become statistically significant, and the explanatory power of the 5F-FF model rises. This makes me to think that monthly excess returns of portfolios, covering high-market-cap companies, are more sensitive to SMB, HML,

Before I present the regression details, readers should be aware that in an OLS regression, Rsquared values will always increase with the inclusion of more factors. Another point is that R<sup>2</sup> increases and become much stronger for almost any explanatory power metric when the regressions' explanatory factors are created from return differences in the data itself. In other words, correlation between explanatory variables, which are created from return differences,

'The method to construct the factor will hence automatically bring the augmented model's explanatory power in small portfolio regressions closer to the R-squared values found in big portfolio regressions, boosting the average explanatory power more than it would for regressions of randomly picked portfolio sets. This same phenomenon is apparent in all other risk factors, which are included as sorting variables and simultaneously used as explanatory vari-

When the t-values in Table 6 are analysed, CAPM's results (column labelled with CAPM) are far away from being reliable and informative. None of the t-statistic of RM-Rf (except for S4BM3) is significant at the 5% level of confidence. This might have been due to the fact that BIST-100 Index, which approximates the monthly excess return of the market, is heavily financial stock weighted. However, using different market indices in all models did not improve the results

In this study, I include only nonfinancial sector stocks registered with the Istanbul stock exchange since the profitability and investment variables' definition is not comparable to the

(more information on this issue is given in the discussion of the results in Table 7).

s in the equations become

in a few models (see rows 13, 14 and 15 in Tables 6–8).

88 Financial Management from an Emerging Market Perspective

and dependent variables has to be highly correlated. Hence, R<sup>2</sup>

ables created from return differences in the sample data'.

model.

RMA and CMA factors.

stronger and true.

Sundqvist in [22] states that:

The 3F-FF regressions find larger intercept t-values on most portfolios compared to the 5F-FF regressions, of which three are more than five standard errors from zero. I can find no exceptional characteristics that might shed light into the reasons behind the significant alpha value.

The 5F-FF regressions show high SMB slopes for the portfolio when compared to 3F-FF regressions' slopes. All of the coefficients of SMB variable are statistically significant at the 5% level of confidence in 5F-FF model, while the same metrics have 11 significant coefficients in 3F-FF models. It is somewhat unusual that none of the coefficients of RM-Rf in 5F-FF model regressions are significant at the 5% level of significance. However, as I already discussed above, this may be due to the fact that BIST-100 Index is heavily financially sector stock weighted.

From Table 6 it is easily seen that (last four rows in the table) excess monthly returns of the biggest-sized company portfolios are best explained by the 5F-FF model. However, as can be seen from the t-statistics, when the company size gets smaller (portfolios starting with the letter S1 and S2), RMW and CMA variables have no effect on excess monthly returns of smallsized portfolios, while SMB and HML variables have significant t-statistics. Assuming that the 5F-FF model is the true definition of the monthly excess returns of portfolios sorted by size and B/M, this result may imply that, when constructing portfolios, the market does not take into account the profitability or investing factors for relatively small-sized companies.

Fama and French [19] note however that one should not expect univariate characteristics and multivariate regression slopes connected to the characteristic to line up. The slopes estimate marginal effects holding constantly all other explanatory variables, and the characteristics are measured with lags relative to returns when pricing should be forward-looking.

The dependent variables are the monthly excess returns of the portfolios sorted by size and profitability (for the definitions of the variables, see Table 1). The results in Table 7 are not much different from the ones that are in Table 6. The best performing model is the 5F-FF model. The CAPM has no power to explain the cross-sectional variance of expected returns for the size-B/M, size-OP and size-Inv portfolios I examine. Peculiarly enough, as in the case of Table 6, none of the alphas in CAPM are significant at the 5% level of significance. This result could have been due to market return data (RM, represented by BIST-100 Index), which is composed of mainly financial sector companies' price data. To save space not presented in this study, I inform the reader that all of the models (CAPM, three-factor Fama-French model and five-factor Fama-French model) using BIST-TUM Index, which covers all stocks registered with the Turkish stock market, instead of BIST-100 Index, the power of the models in explaining the sorted portfolio monthly excess returns, did not improve. This shows that, at least in the Turkish case, market risk is not a preliminary and strong factor in determining monthly excess returns of portfolios.

When one concentrates on alpha coefficients of the 5F-FF model, it is seen that only 3 of the 16 alphas are statistically significant. But in the case of the 3F-FF model, 9 of the 16 alpha coefficients are significant at the 5% level, showing that three-factor model leaves a high percentage of unexplained part of cross-sectional variance of expected returns.

One of the main messages of the results of the 5F-FF model is that as the size of the companies under investigation increases, the explanatory power of the model rises. It seems from the results of both Tables 6 and 7 that the best explanatory factor for the monthly excess returns of portfolios is the size factor.

As the size of the companies in portfolios that have been sorted by size and profitability gets smaller, RMW and CMA become ineffective in determining the monthly excess returns of these portfolios (see the coefficients of RMW and CMA in the first six rows of Table 7). However, bigger size portfolios have significant RMW and CMA coefficients.

Table 8 gives the results of CAPM, 3F-FF model and 5F-FF model for portfolios that are sorted by size and investment. The CAPM has no power in explaining the returns of portfolios, while the coefficients of the factors in 3F-FF model seem powerful, 8 of the 16 alpha values are significant at the 5% level. This result indicates that the CAPM and 3F-FF model are not the true definition of a model to explain the variations in monthly excess returns of portfolios. However, as in the results of the previous tables (Tables 6 and 7), the 5F-FF model in general has statistically significant coefficients for the Fama-French factors, namely, SMB, HML, RMW and CMA. Also, only 2 of the 16 coefficients of intercepts (alphas) are significant at the 5% level of significance.

Variations of the monthly returns of portfolios constructed by big-sized companies are best explained by the 5F-FF model. However, CMA factor turns out to be insignificant when estimating the model for portfolios with relatively small-sized companies.
