*3.3. Model Specification*

In order to test the significance and existence of the diverse factors on asset pricing in the Pakistani stock market (KSE), we employ numerous pricing models and follow a stepwise approach. We start with a standard CAPM:

$$E(R\_i) - R\_f = \alpha\_i + \beta\_i \left[ E(R\_m) - R\_f \right] + \epsilon\_i \tag{1}$$

Next, we add SMBL and HML factors into the CAPM:

$$E(R\_i) - R\_f = a\_i + \beta\_i \left[ E(R\_m) - R\_f \right] + s\_i(SMB) + h\_i(HML) + \varepsilon\_i \tag{2}$$

where, *E*(*Ri*) − *Rf* is the portfolio *i's* return in excess of risk-free rate *Rf* , *αi* is the intercept of the regression equation representing the non-market return component, *E*(*Rm*) − *Rf* is the market risk premium (market portfolio return in excess of risk-free rate), SMB (small minus big) is the return on small size stocks minus return on big size stocks captures size premium, HML (high minus low) incorporates value premium that is the difference between returns of value stocks (high B/M ratio) and growth stocks (low B/M ratio). *β<sup>i</sup>*, *si* and *hi*, are the slopes of expected risk premium of portfolio *i* to the market, size and value factors in the regression, respectively, while *i* represents the random return component due to unexpected events related to a particular portfolio. It is supposed that *i* has a multivariate normal distribution and is identically and independently distributed over time.

#### *3.4. Variable Construction and Portfolio Formation*

In order to examine the three factors for Pakistani stocks, we experiment with three ways of constructing size and value factors to explore the impact of the special features in the Pakistani stock market. The three portfolio construction methods (baskets of stocks) are: 'fixed basket', 'non-financial basket' and 'variable basket'. By following the previous studies on KSE, we construct the fixed basket; which includes only those stocks which have survived the entire sample period. Next, we follow Fama and French (1993) and construct the non-financial basket. The non-financial basket excludes stocks of financial companies, however, every year new companies are included into the basket upon meeting the sample selection and criteria limitations. The variable basket is based on the special features of KSE, such as: liquidity of the financial companies, active participation, and fraction of the market value of the financial firms to the total market value of the index. It includes both non-financial and financial companies into the basket every year upon meeting the sample selection and criteria limitations. The variable and the non-financial baskets include delisted firms in the sample up to the delisting year to control the survivorship bias, whilst the fixed basket does not include.

The dependent variable of the three factor model is the excess return on equal weighted six portfolios. 2 pcs (small and big) portfolios are determined for size effect and 3 pcs (high, medium, low) portfolios are determined for value effect. A total of six intersection portfolios (SL, SM, SH, BL, BM, BH) are created with the following criteria: shares classified according to market value have been subdivided using the median market cap as breakpoint, while shares classified according to book-to-market have been divided by the 30th and 70th percentiles as breakpoints. In the case of risk-based portfolios, i.e., constructing portfolio by categorizing the stock's sensitivity to market movements, six portfolios on the basis of risk profile of the stocks are carefully considered as the dependent variables.

The independent variables consist of market, size and value premiums. For the market risk premium, we find the difference between the return on market portfolio and risk free rate, and show that it exists in both Fama-French three factor model and CAPM. Size premium (SMB) is the average return on three small portfolios SL, SM and SH minus the average return on the three big portfolios BL, BM and BH, while HML is the average return on two value portfolios SH and BH minus the average return on the two growth portfolios SL and BL. SMB and HML are computed as follows:

$$\text{SMB} = \frac{(\text{SL} + \text{SM} + \text{SH})}{3} - \frac{(\text{BL} + \text{BM} + \text{BH})}{3} \tag{3}$$

$$\text{HML} = \frac{(\text{SH} + \text{BH})}{2} + \frac{(\text{SL} + \text{BL})}{2} \tag{4}$$

#### **4. Empirical Results and Discussion**

The descriptive statistics in Table 1 report that the average monthly returns on the SMB are statistically insignificant in the fixed basket, and significant at 10% and 5% levels in non-financial and variable baskets, respectively. In contrast, value premium is significant at 5% level across all the methodologies. To understand the changes in the results caused by construction methodologies, we start with a detailed analysis of the variable basket, because it is significant for both factors at 5% level. Afterwards, we compare the performance of these factors obtained by three different portfolio construction methodologies. Table 3 reports the descriptive statistics of the monthly excess return and volatility of the six size-B/M sorted portfolios from January 2002 to December 2015 obtained by using variable basket.

**Table 3.** Descriptive statistics on the excess return (and volatility).


Note: Author's calculation. The table reports the descriptive statistics on monthly average excess returns between six size-B/M portfolios for the Pakistani stock market. The values of standard deviation are reported in parentheses. The sample period is 2002:01–2015:12 (168 monthly observations). Source: the official website of the Pakistan stock exchange (https://www.psx.com.pk/) and the official website of the State Bank of Pakistan (http://sbp.org.pk/).

Holding group size constant, the average return and volatility of the portfolios increase with the portfolio's B/M ratio. The average monthly return on portfolios containing low B/M is 1.41% and the standard deviation is 8.15%, whereas stocks with high B/M ratio have an average return of 2.43% and a standard deviation of 10.76%. Conversely, when the B/M ratio is constant, the average return on small capitalization firms is 2.42% and the standard deviation is 9.69%, whereas stocks with big capitalization have an average return of 1.41% and a standard deviation 9.22%. Average returns on all portfolios are positive in our study, but are contradictory to the results reported by Mirza and Shahid (2008). Their results for KSE during the bull rally between 2003 and 2007 report negative average monthly returns on portfolios SH, BH and BM. The monthly average returns are the highest in the small value category (SH), approximately 3.46%, while the lowest in the small growth category (SL) is approximately 1.38%. The monthly standard deviation is the highest in the big value category (BH), approximately 11.15% and the lowest in the small medium-B/M category (SM), approximately 7.19%.

Table 4 represents the summary statistics of all three portfolios for time period from January 2002 to December 2015. The monthly average returns of the three explanatory variables are all positive and significant. The annual average return on the market, size and value factors is approximately 18.22%, 9.15% and 12.27%, respectively, whereas the standard deviation is approximately 7.60%, 5.32% and 6.53%, respectively. It is evident from the results that small stocks and values stocks outperform the big stocks and growth stocks, respectively.


**Table 4.** Summary statistics of independent variables (factors).

Note: Author's calculation. The table reports the summary statistics of the market risk premium (*Rm* − *Rf* ), size premium (SMB) and value premium (HML). The sample period is 2002:01–2015:12 (168 monthly observations). Source: the official website of the Pakistan stock exchange (https://www.psx.com.pk/) and the official website of the State Bank of Pakistan (http://sbp.org.pk/).

Table 5 reports the correlation coefficients among the independent variables. We did not notice any excessively high values of the correlation coefficients that may arise a concern about any multicollinearity problem. The observed correlation shows that SMB and HML can be regarded as separate measures of risk premium, which are not dependent on market risk premium. The correlation between SMB and HML also shows a valid justification for considering size and value risk factors separately.

**Table 5.** Correlation coefficients of monthly factor returns.


Note: Author's calculation. The table reports the correlation coefficients between market, size and value factors. The sample period is 2002:01–2015:12 (168 monthly observations). Source: the official website of the Pakistan stock exchange (https://www.psx.com.pk/) and the official website of the State Bank of Pakistan (http://sbp.org.pk/).
