Testing Conditional Asset Pricing in Pakistan: The Role of Value-at-risk and Illiquidity Factors

Abstract

This article investigates performance of conditional and unconditional Capital Asset Pricing Model and Fama–French model augmented with a downside risk, that is, the value-at-risk (VaR) factor and an illiquidity factor as additional risk factors using the discount factor methodology of Cochrane (1996). Using monthly portfolio data as test assets from the Pakistani stock market from January 1993 to January 2013 we provide empirical evidence on the efficacy of the VaR and illiquidity factors in asset pricing. We find that these factors improve the efficacy of the Fama–French model and including these factors reduces the explanatory power of co-kurtosis factor.

Keywords

Conditional asset pricing models stochastic discount factor (SDF)value-at-risk

1. Introduction

Emerging markets are observed to be more volatile than the developed markets due to institutional, political and economic conditions which cause higher market concentration, lower liquidity, higher volatility and greater extent of infrequent trading than the developed markets (Bailey & Chung, 1996; Bekaert & Harvey, 1997; Harvey, 1995). Iqbal, Brooks and Galagedera (2010) stipulate that due to these circumstances, modelling prices of risky assets become a more challenging task in emerging markets. Many previous studies investigate the usefulness of higher order co-moments including Dittmar (2002) and Hwang and Satchell (1999). Iqbal et al. (2010) show that the excess kurtosis factor is helpful in explaining Pakistani emerging market returns over and above the three Fama–French factors. However, the excess kurtosis factor is not associated with a meaningful economic risk.

Two widely stated interpretations of Kurtosis include: (a) the ‘peakedness’ of distribution and (b) a measure of the phenomenon that infrequent large deviations being more prevailing in returns distribution than the frequent but small deviations. We argue that relevance of co-kurtosis factor in emerging markets may be either an artefact of infrequent trading, for which the real cause is the illiquidity, or a downside risk associated with large extreme deviations. In emerging markets, many stocks do not trade for many days resulting in an excess of zero returns which in turn result in large kurtosis. When these stocks are formed into portfolios, the portfolio returns could be explained by co-kurtosis factor. Thus, illiquidity induced by infrequent trading might be a possible economic reason behind the earlier findings of importance of co-moment-based factor especially the kurtosis factor. In a more modern approach to kurtosis, co-kurtosis factor can be associated with high tail risk as extreme large movements are more probable in emerging markets. Some measure of downside risk, for example, the value at risk (VaR) can capture this feature of data. Consequently, we explicitly introduced an illiquidity factor and a VaR factor in asset pricing and examine whether these two factors subsume some of the explanatory power of the excess kurtosis factor.

Bali, Gokcan and Liang (2007) argue that VaR provides a better characterisation of market risk because it takes into account higher-order moments such as skewness and kurtosis along with the standard deviation. The VaR has now become a popular measure of market risk exposure among practitioners and researchers. Unlike beta and standard deviations, this measure is theoretically more consistent with notion of downside risk. However, as Ernst, Stange and Kaserer (2009) point out that an often criticised downside of the VaR is its inability to capture illiquidity risk, its computation generally relies solely on market prices. Due to the neglect of illiquidity risk, the real risk is generally underestimated. In an empirical study of Spanish stock market, Martınez, Nieto, Rubio and Tapia (2005) found, using both the unconditional and conditional frameworks, that systematic illiquidity risk is priced. They also suggest that the liquidity factor can be the missing factor in asset pricing and the HML factor of the Fama and French might be a proxy of the liquidity risk. It is well known that the economic origin of HML factor in the Fama–French model is not identified although it is usually associated with distress factor. We consider VaR and illiquidity as additional risk factors to compare the performance of unconditional and the CAPM and the Fama–French models.

The VaR and illiquidity are important factors since they are associated with sound economic meaning. So unlike co-moment factors of skewness and excess kurtosis, these risk factors are expected to convey more useful information for investors, policymakers and portfolio managers. However, despite its practical appeal, the VaR has not been employed as a risk factor in asset pricing studies especially in discount factor framework.

In conditional models, the parameters are allowed to vary with the variables which either measure business cycle or predict returns. We conducted empirical analysis for the Pakistani stock market for the period January 1993 to January 2013. The Karachi Stock Exchange (KSE) is the largest of the three stock markets of Pakistan. Iqbal (2012) reviews several features of this market and compares this with other emerging and developed markets. This market possesses the typical features of an emerging market, that is, high volatility, high frequency of extreme price movements due to political instability and institutional features. This article provides a comprehensive study of the CAPM, the Fama–French models and various extensions of these models in discount factor framework with constant and time-varying parameters. We use the generalised method of moment (GMM) to estimate the parameters of conditional and unconditional models. The selection of the conditioning variables is guided by the previous literature which suggests that the conditioning variables should either indicate the investors’ expectations about market returns or predict business cycle.

Following this introduction, the article is organised as follows: The second section provides brief literature review on the conditional asset pricing models in emerging markets. The methodology is discussed in the third section. The fourth section describes the data. Results and discussion of model estimation and comparison are presented in the fifth section. Finally, the sixth section provides conclusion.

2. Literature Review

The traditional asset pricing studies of Sharpe (1964) and Linter (1965), Black (1972) and Ross (1976) implicitly assume that the expected returns, factor risk premium and covariance of assets are time invariant. However, many researchers have argued, and provided empirical evidence, that the expected return and parameters of the asset pricing model may be time varying. Ferson and Harvey (1993) and Ferson and Korajczyk (1995) support the view that predictability of expected stock return is the result of time variation in expected returns. Jagannathan and Wang (1996) found that market beta varies over time. Ang and Chen (2005) found that in longer period, CAPM works efficiently and that the betas of test portfolios vary over time significantly. Keeping this evidence into consideration, several authors investigated whether the failure of unconditional CAPM could be overcome by incorporating models where the parameters vary with conditioning information available to investors.

The efficacy of condition information was demonstrated by Cochrane (1996) who found that conditional models perform better than the consumption-based models using the 10 size portfolios from NYSE. Lettau and Ludvigson (2001) study the US market by constructing 25 size and book-to-market portfolios in conditional framework and observe that the model with consumption–aggregate wealth ratio as the information variable performs as good as the Fama–French three-factor model. By using the same approach, Schrimpf and Schroder (2007) found that conditional CAPM performs well in the German stock market when term spread is used as an information variable.

Due to uncertain macroeconomic and political conditions, the risk–return relationship in emerging market is expected to vary over time. Drobetz, Stürmer and Zimmermann (2002) argue that the possible reason of expected return predictability in emerging markets is due to the time variation in economic risk premium. Considering Pakistani stock market, Iqbal et al. (2010) employed the discount factor methodology in testing asset pricing models with time-varying parameters. They found that the unconditional Fama–French model augmented with an excess kurtosis factor performs the best among the competing models.

Bali and Cakici (2004), Chen, Chen and Wu (2014) and Iqbal, Azher and Ijaz (2013) employed VaR as a risk factor in asset pricing using multiple regression approaches which assume constant model parameters and constant expected returns. In their empirical test on hedge fund returns, Bali et al. (2007) found a significant cross-sectional relation between hedge fund returns and VaR. During the period of January 1995 to December 2003, they found that the live funds with high VaR outperform those with low VaR by an annual return difference of 9 per cent.

There empirical studies on conditional asset pricing using the discount factor methodology are very rare especially using VaR and illiquidity as risk factors. This study intends to fill this gap in the literature by employing these factors which are expected to perform well in emerging markets.

3. Empirical Methodology

Conditional Asset Pricing Models

The pioneering asset pricing studies (Black, 1972; Linter, 1965; Ross, 1976; Sharpe, 1964) assume that the expected returns, factor risk premium and covariance of assets are time invariant. Several arguments are put forward against these assumptions. One view is that expected returns and risk premium vary over the business cycle. In a recession, investors are short of liquidity and require higher risk premiums for a given level of risk. In a boom, they have extra liquidity for investment and therefore the expected risk premium is less than otherwise. The investment opportunities also vary over time. From the firm’s point of view, Jagannathan and Wang (1996) argue that systematic risk of the firm varies over time. During a recession, financial leverage of the troubled firms increases causing their beta to increase. Brooks, Faff and Lee (1992) point out that the maturity and growth of firms also tend to change the riskiness of the firm over time. This is especially true for technological and communication firms which have shown tremendous growth over recent times in both developing and advanced countries. The relative share of different sectors may also change due to technological shocks. Hence, betas and expected returns would depend on the nature of the information available and may vary over time. In response to these arguments, many authors have concluded that the empirical failure of the unconditional CAPM might reflect the misspecification of the wrong assumption regarding the constancy of expected return, beta and the risk premium. Consequently, a strand of asset pricing literature has emerged which incorporates conditional information that is available to investors in the asset pricing model. Various methodologies have been suggested and used on conditional asset pricing in the literature. In this article, we use the scaled factor methodology proposed by Cochrane (1996, 2001) in which the parameters are allowed to vary with information variables. In this approach, the discount factors are the linear combination of the risk factors. This section briefly discusses the scaled factor methodology and discount factor methodology of Cochrane (2001). According to Cochrane (2001), there exists a positive stochastic discount factor (SDF) for the absence of arbitrage condition given by:

1 = E_{t} (M_{t + 1} R_{t + 1}^{j}) = E (M_{t + 1} R_{t + 1}^{j} | ν_{t}) j = 1, 2 \dots N

(1)

where $R_{t + 1}^{j}$ denote jth test asset return at time t+1 and v_t is the investor’s information set at time t and (1) holds for all assets j in the economy. As we focus on linear factor models, we can express the SDF as a linear function of factors f as:

M_{t + 1} = α_{t} + β^{'} ​_{t}^{​} ​ f_{t + 1}

(2)

where f_t₊₁ is the vector of k × 1 factors and β is the k × 1 coefficient vector which gives the information regarding the importance of a factor. The factors associated with traditional one-factor CAPM and three-factor Fama–French models are specified as follows:

f_{t + 1} = [R M_{t + 1}]

f_{t + 1} = [R M_{t + 1}, S M B_{t + 1}, H M L_{t + 1}]

In the literature, the time variation in parameters of the linear discount factor, the expected returns and risk premia are allowed to vary over time, for example, in Cochrane (2001), so that the SDF depends linearly on the time t information set and hence the conditional SDF models will be of the form:

M_{t + 1} = α_{0} + α_{1} z_{t} + β_{0} F_{t + 1} + β_{1} z_{t} F_{t + 1}

(3)

where z_t and F_t₊₁ are the vectors of conditioning variables and the risk factors, respectively. This approach is termed as the scaled factor methodology by Cochrane (1996). For example, the conditional SDF models corresponding to the CAPM and Fama–French three-factor models, respectively, scaled by information variable z_t are specified as follows:

M_{t + 1} = α_{0} + α_{1} z_{t} + β_{0} R M_{t}^{}_{+ 1}^{} + β_{1} (z_{t} R M_{t + 1})

(4)

\begin{matrix} M_{t + 1} = α_{0} + α_{1} z_{t} + β_{0} R M_{t}^{}_{+ 1}^{} + β_{1} S M B_{t + 1} + β_{2} H M L_{t + 1} \\ + β_{3} (z_{t} R M_{t + 1}) + β_{4} (z_{t} S M B_{t + 1}) + β_{5} (z_{t} H M L_{t + 1}) \end{matrix}

(5)

Models with other factors are similarly constructed. Plugging such SDF factors in the asset pricing Equation 1, we can estimate the models through the GMM and obtain the estimates of β_i and the standard errors. These estimates will then also enable us to test whether associated unscaled or scaled factors help in explaining the variation in pricing kernel. Moreover, we also evaluate whether any or scaled or unscaled factor earn a risk premium.

Model Specification Test

To compare the performance of different asset pricing models and to find out the efficacy of a model, we employ the following three model specification tests: (a) the Hansen (1982) J_T statistic, (b) the Hansen and Jagannathan (1997) HJ-distance measure and (c) the Andrews (1993) Sup-LM parameter stability test.

For the GMM overidentification test (J_T test) of Hansen (1982), we use the usual SDF equation to express the pricing error for the model given by (3) as follows:

θ (β) = E (β^{'} f_{t + 1} R_{t + 1} - 1)

(6)

The sample analogue of the pricing error is as follows:

θ_{T} (β) = \frac{1}{T} ​ \sum_{t + 1 = 1}^{T} (β^{'} f_{t + 1} R_{t + 1} - 1)

(7)

For true model, θ(β) = 0. The GMM estimates β such that the following weighted combination of the sample pricing errors is minimised.

J_{T} = θ_{T} (β)^{'} ω_{T} θ_{T} (β)

(8)

where ω_T is the weighting matrix. Following the literature, ω_T is specified as the optimal weighting matrix, that is, $ω_{T}^{*} = S_{T}^{- 1}$ (Hansen, 1982) or the weighing matrix ω_T = E(RR')^–1 advocated by Hansen and Jagannathan (1997). Here, S = TVar(θ_T). Using $ω_{T}^{*}$ matrix, the tests for the null hypothesis that all pricing errors $θ_{T} (\hat{β})$ are zero are carried out by the J-statistic of the overidentifying restrictions:

T J_{T} (\hat{b}) ~ χ^{2} (n u m b e r o f ​ ​ m o m e n t c o n d i t i o n s - n u m b e r o f p a r a m t e r s)

(9)

Hansen and Jagannathan (1997) argue that J_T test might not differentiate between models that result really in small pricing error or a model which gives small J-statistic value since the estimation errors captured by matrix S are large. The J-statistic gives higher weight to the moment conditions which are measured with least estimation error. The Hansen–Jagannathan distance measure is the minimum distance from the pricing kernel of the model under consideration to the true pricing kernels. This distance measure is named as HJ-distance. For correctly specified model, the HJ-distance is zero.

The advantage of using HJ-distance measure is that the weighting matrix ω_T = E[RR']^–1 remains unchanged across the models that employ the same set of test assets, while the optimal weighting matrix $ω_{T}^{*}$ changes with different models. The chi-square asymptotic p-values are not applicable for ω_T; therefore, the p-values are obtained by simulation using E[RR']^–1 as weighting matrix to estimate β and construct the test statistic $T J_{T} (\hat{β})$ . Jagannathan and Wang (1996) show that $T J_{T} (\hat{b})$ is distributed as weighted sum of χ²(1) random variables.

Garcia and Ghysels (1998) argue that the J_T test for the overall validity of conditional asset pricing model may have low power especially from the emerging markets for which the structural stability of the economic relationships may be questionable. They advocate using a Sup-LM of Andrews (1993) for testing the structural stability of the SDF parameters which may be a more powerful test of the model specification. In this test, the null hypothesis is that the SDF parameter β is constant and the alternative hypothesis assumes that there is a single break at some unknown point π in the sample. Then, Sup-LM (π) is computed usually over the sample 0.2T to 0.8T. The p-values of the test are tabulated in the Andrews (1993). For details, refer Appendix A in Garcia and Ghysels (1998).

4. The Data, Portfolios and Factors Construction

The Price Data

The strategy of selecting the data was to cover the longest possible time series on closing prices available in Datastream for Pakistani stock market. We used monthly closing prices for 214 listed firms and the KSE-100 Index over the period January 1993 to January 2013. The sample period covers 20 years and 1 month and includes 241 monthly observations. The monthly returns are obtained by continuous compounding as:

R_{i t} = \ln (\frac{P_{t}}{P_{t - 1}}) ×100

(10)

Construction of Portfolios

The construction of portfolios involves market value of equity (ME: product of stock’s price and shares outstanding) and book-to-market equity (BE/ME) of each included stock for which the data are gathered from the Datastream database. We construct 16 size and BE/ME portfolios and the Fama–French factors (SMB and HML) by adopting the Fama and French (1993) mimicking portfolio strategy.

We use monthly returns of the KSE value weighted index as the proxy for market portfolio (RM) returns. To construct the VaR factor, we follow the Bali and Cakici (2004) and devise a factor HVaRL. The HVaRL factor is constructed by taking the difference between the simple average of the returns of the high VaR and low VaR portfolios. The construction of 95 per cent VaR portfolios is similar to the portfolio strategy of Fama–French (1993), that is, in December of each year t from 1993 to January 2013, we rank 241 stocks on 95 per cent VaR and the median value is used to split the stocks into two groups, that is, high VaR and low VaR groups. Then, the HVaRL factor is obtained as the difference between the simple average of high VaR and low VaR portfolios. The VaR of each stock was obtained by historical simulation method. As compared to the variance–covariance method, the historical simulation method does not require the assumption of normally distributed returns and constant correlations.

Bali and Cakici (2004) method is also followed for construction of the illiquidity factor (HILLIQL). Following Lesmond, Ogden and Trzcinka (1999), we measure stock illiquidity as the proportion of days with zero returns in a calendar month, that is:

Z_{t} = \frac{N_{i, t}}{T_{t}}

(11)

where T_t is a number of trading days in month t and N_i_,t is the number of zero return days of stock i in month t. The construction of HILLIQL portfolios was based on monthly illiquidity measure for each stock. The procedure is same as the one used for HVaRL portfolios. In each month t from 1993 through 2013, we ranked all stocks according to their illiquidity and used the median illiquidity (ILLIQ) to split stocks into two groups, that is, high ILLIQ and low ILLIQ. The HILLIQL factor is obtained as the difference between the simple average of the returns for the high illiquidity and low illiquidity portfolios. Following Iqbal et al. (2010), we also include the cubic market return factor (RM³) to capture kurtosis as recent studies show that excess kurtosis is especially useful in modelling emerging market returns (see Hwang & Satchel, 1999).

A preliminary analysis of the portfolios shows that the market return distribution is negatively skewed. Portfolio returns show high volatility as the standard deviations are high. It was found that in 15 out of 16 portfolios, normality was rejected at 1 per cent level of significance and excess kurtosis seems to be the reason of non-normality.

The Conditioning Variables

The selection of the appropriate conditioning variables is based on the criteria suggested in the literature that these variables should measure investors’ expectations about future market returns or should be associated with business cycle conditions. Following previous studies, we select six conditioning variables: (a) term spread (TERM) difference between long maturity bond and a short maturity bill; (b) short-term interest rate (RF) proxy by 30 days repo rate; (c) aggregate dividend yields (DY); (d) cyclical component of industrial production (CY); (e) aggregate trading volume (VOL) on the market; and (f) January dummy (JAN) to see whether a January effect plays a role on the Pakistani stock market.

To test the predictive ability of conditioning variables, we apply the Wald test to test H₀ (b = 0) in the following regression where z_t is the vector of conditioning variables.

R_{i, t +1} = a + b^{'} z_{t} + e_{t +1}

(12)

The Wald test is distributed as the chi-square with degrees of freedom 6 (the number of conditioning variables). Table 1 reports the Wald test, with p-values in parenthesis, for 16 test assets for the period of January 1993 to January 2013. All the portfolios except smallest size and lowest book-to-market portfolio indicate the significant predictability for the conditioning variables at the 5 per cent level.

Table 1.

Predictive Ability of Conditioning Variables

Size Quartiles	Book-to-market Quartiles
Size Quartiles	B1	B2	B3	B4
S1	4.001	14.636	13.894	16.401
	(0.677)	(0.023)	(0.031)	(0.012)
S2	17.136	17.583	16.607	28.555
	(0.009)	(0.007)	(0.011)	(0.000)
S3	26.450	14.263	16.111	18.708
	(0.000)	(0.027)	(0.013)	(0.005)
S4	12.791	16.092	16.260	22.609
	(0.047)	(0.013)	(0.012)	(0.001)

Source: Authors’ calculations.

Notes: This table reports the Wald test of predictive ability of the conditioning variables employed in this article with p-values in parenthesis. The Wald test is the test of joint restriction (H_o: b = 0) in the regression R_i,t = a + b^’z_t+ e_t+1. The test statistic is asymptotically distributed as chi-square with the number of degrees of freedom as number of restriction in each equation which is six in this case. The sample period is from January 1993 to January 2013. The conditioning variables are mean-centred.

5. Results and Discussion

Overview of the SDF Models through Model Specification Tests and Graphical Analysis

We analyse several specifications of conditional and unconditional CAPM and FF model with and without the additional risk factors (HVaRL, HILLIQL and RM³) for the period of January 1993–January 2013. We study 24 models and obtain parameter of the SDF model by the GMM. We obtain risk premia by OLS and the sequential GMM method and evaluate these models by using specifications tests. Our study aims to identify the risk factors that help explaining the portfolio returns significantly. We illustrate the goodness of fit of the models by plotting the realised average return versus fitted expected return. After evaluating performances of different models and identifying the potential risk factors, we discuss the parameter estimates of the models.

Tables 2–4 summarise the overview of SDF models goodness of fit for both the unconditional and conditional CAPM and the FF models. These tables present two model specification tests J_T test and HJ distance, with p-values in parenthesis, and a model stability test Sup-LM whose critical values are obtained from Table 1 of Andrews (1993).

Table 2 compares the performance of conditional Fama–French three-factor model with conditional CAPM augmented with HVaRL and HILLIQL and conditional CAPM augmented with HVaRL and RM³. We observe that in nearly all the cases, the conditional CAPM augmented with HVaRL and HILLIQL is found to perform better than the Fama–French factors (SMB and HML) and RM³ based on the HJ-distance. This is also the case with most of the cases using J_T statistic.

We also observe similar results (Table 3) for unconditional CAPM augmented with HVaRL and HILLIQL factors. This model performs better than the unconditional Fama–French three-factor model and the unconditional CAPM augmented with HVaRL and RM³ in terms of both the HJ-distance and the J_T test. The unconditional Fama–French model is found to have instable parameters.

Table 2.
A Comparison among Three-factor Conditional Models

Model Specification Test J_T test(p-value) HJ-distance(p-value) Sup-LM

Panel A: Scaled with TERM

(i) RM, SMB and HML 8.165 0.056 23.552*

0.518 0.828

(ii) RM, HVaRL and HILLIQL 8.603 0.052 16.796

0.475 0.836

(iii) RM, HVaRL and RM³ 11.074 0.084 18.929

0.271 0.078

Panel B: Scaled with RF

(i) RM, SMB and HML 15.116 0.072 27.725*

0.088 0.006

(ii) RM, HVaRL and HILLIQL 16.791 0.068 22.558*

0.052 0.286

(iii) RM, HVaRL and RM³ 8.358 0.067 18.313**

0.499 0.887

Panel C: Scaled with DY

(i) RM, SMB and HML 11.018 0.093 31.291*

0.275 0.000

(ii) RM, HVaRL and HILLIQL 11.462 0.075 18.997**

0.245 0.509

(iii) RM, HVaRL and RM³ 15.221 0.091 20.640*

0.085 0.043

Panel D: Scaled with CY

(i) RM, SMB and HML 9.983 0.053 12.772

0.352 0.865

(ii) RM, HVaRL and HILLIQL 9.876 0.041 14.504

0.361 0.971

(iii) RM, HVaRL and RM³ 14.665 0.048 13.523

0.101 0.774

Panel D: Scaled with VOL

(i) RM, SMB and HML 15.503 0.076 27.925*

0.078 0.084

(ii) RM, HVaRL and HILLIQL 11.25 0.069 23.545*

0.259 0.693

(iii) RM, HVaRL and RM³ 15.063 0.094 26.958*

0.089 0.021

Model Specification Test	J_T test(p-value)	HJ-distance(p-value)	Sup-LM
Panel A: Scaled with TERM
(i) RM, SMB and HML	8.165	0.056	23.552*
	0.518	0.828
(ii) RM, HVaRL and HILLIQL	8.603	0.052	16.796**
	0.475	0.836
(iii) RM, HVaRL and RM³	11.074	0.084	18.929**
	0.271	0.078
Panel B: Scaled with RF
(i) RM, SMB and HML	15.116	0.072	27.725*
	0.088	0.006
(ii) RM, HVaRL and HILLIQL	16.791	0.068	22.558*
	0.052	0.286
(iii) RM, HVaRL and RM³	8.358	0.067	18.313**
	0.499	0.887
Panel C: Scaled with DY
(i) RM, SMB and HML	11.018	0.093	31.291*
	0.275	0.000
(ii) RM, HVaRL and HILLIQL	11.462	0.075	18.997**
	0.245	0.509
(iii) RM, HVaRL and RM³	15.221	0.091	20.640*
	0.085	0.043
Panel D: Scaled with CY
(i) RM, SMB and HML	9.983	0.053	12.772
	0.352	0.865
(ii) RM, HVaRL and HILLIQL	9.876	0.041	14.504
	0.361	0.971
(iii) RM, HVaRL and RM³	14.665	0.048	13.523
	0.101	0.774
Panel D: Scaled with VOL
(i) RM, SMB and HML	15.503	0.076	27.925*
	0.078	0.084
(ii) RM, HVaRL and HILLIQL	11.25	0.069	23.545*
	0.259	0.693
(iii) RM, HVaRL and RM³	15.063	0.094	26.958*
	0.089	0.021

Source: Authors’ calculations.

Notes: This table displays model specification test results for evaluating the goodness of fit for (a) conditional Fama–French three-factor model, (b) conditional CAPM augmented with HVaRL and HILLIQL, and (c) conditional CAPM augmented with HVaRL and RM³ and scaled by six conditioning variables. The J_T test is the overidentifying restriction test by Hansen (1982) and follows chi-square with degrees of freedom (number of moment conditions−number of parameter estimates). The HJ-distance is the popularly used measure for model evaluation by Hansen and Jagannathan (1997) and the p-values for HJ-distance are obtained by simulation following Jagannathan and Wang (1996). The Sup-LM is the Andrews (1993) parameter stability test and the critical values are obtained from Table 1 (Andrews, 1993). The sample period is from January 1993 to January 2013 which includes 241 monthly observations. * and ** show 1% and 5% level of significance.

Next we investigate which of the individual factors, that is, HVaRL or HILLIQL, provide greater improvement, that is, greater decrease in J_T test and HJ-distance. Iqbal et al. (2010) found that inclusion of RM³ improves performance of the CAPM and the Fama–French model. However, it is not clear which underlying risk characteristics are captured by this RM³. Table 4 shows that as compared to the Fama–French model augmented with the RM³, the HVaRL factor provides improvement in the performance of the unconditional Fama–French model in terms of HJ-distance (with stable parameters). Furthermore, according to J_T test statistic, illiquidity factor (HILLIQL) provides greater improvement in the performance of both conditional and unconditional models as compared to the RM³. Also in most of the cases with conditional models, inclusion of either illiquidity factor (HILLIQL) or value-at-risk factor (HVaRL) provides greater improvement in the performance of the model as compared to the RM³. The results for these variables are not reported in this article but are available on request.

Table 3.

A Comparison among Three-factor Unconditional Models

Model Specification Test	J_T test(p-value)	HJ-distance(p-value)	Sup-LM
(i) RM, SMB and HML	35.257	0.099	31.883*
	0.000	0.000
(ii) RM, HVaRL and HILLIQL	16.541	0.088	13.964
	0.167	0.383
(iii) RM, HVaRL and RM³	20.594	0.113	11.939
	0.057	0.000

Source: Authors’ calculations.

Overall, we observe that the inclusion of the HVaRL and HILLIQL risk factors improves the performance of almost all the conditional and unconditional CAPM and Fama–French models significantly as compared to RM³.

Figures 1–3 present the graphs of selected models with smaller pricing errors. Full results can be obtained from the authors on request. Pricing error graph is plotted between realised average portfolio returns and the expected returns (% per month) predicted by conditional or unconditional models. We select the graphs on the basis of HJ-distance measure, with stable parameters and small pricing errors. The closer the portfolios returns to the 45⁰ diagonal line, the better the asset pricing model, as the model predicted returns will be closer to the realised returns.

Figure 1a presents the pricing error plot for the conditional Fama–French three-factor model, and Figure 1b presents the conditional CAPM augmented with HVaRL and HILLIQL both scaled by CY. We observe that the graphical analysis supports the findings of Table 2 which shows the conditional CAPM augmented with the HVaRL and HILLIQL factors produces much smaller pricing error than the conditional Fama–French three-factor model.

Table 4.

A Comparison of Individual Risk Factor Performance on the Basis of Unconditional Fama–French and Unconditional CAPM

Model Specification Test	J_T test(p-value)	HJ-distance(p-value)	Sup-LM
Panel A: Unconditional CAPM Models
RM	15.457	0.137	16.533*
	0.348	0.000
RM and HVaRL	17.126	0.118	58.656*
	0.194	0.006
RM and HILLIQL	16.128	0.124	12.379
	0.242	0.000
RM and RM³	17.175	0.114	12.431
	0.191	0.000
Panel B: Unconditional Fama–French Models
RM, SMB and HML	35.257	0.099	31.883*
	0.000	0.000
RM, SMB, HML and HVaRL	29.769	0.088	21.821**
	0.002	0.114
RM,SMB,HML and HILLIQL	20.854	0.093	16.641
	0.076	0.012
RM, SMB, HML and RM³	26.446	0.089	19.927**
	0.015	0.000

Source: Authors’ calculations.

Notes: This table displays model specification test results for evaluating the goodness of fit for (a) conditional Fama–French three-factor model, (b) conditional CAPM augmented with HVaRL and HILLIQL, and (c) conditional CAPM augmented with HVaRL and RM³ and scaled by six conditioning variables. The J_T test is the overidentifying restriction test by Hansen (1982) and follows chi-square with degrees of freedom (number of moment conditions−number of parameter estimates). The HJ-distance is the popularly used measure for model evaluation by Hansen and Jagannathan (1997) and the p-values for HJ-distance are obtained by simulation following Jagannathan and Wang (1996). The Sup-LM is the 1 (1993) parameter stability test and the critical values are obtained from Table 1 (Andrews, 1993). The sample period is from January 1993 to January 2013 which includes 241 monthly observations. * and ** show 1% and 5% level of significance.

Figure 2 presents the unconditional CAPM and unconditional CAPM augmented with HILLIQL. We observe that the pricing errors for unconditional CAPM are too high except for some large size and book-to-market portfolios. When the unconditional CAPM is augmented with HILLIQL, the model performs better than the unconditional CAPM but a lot of discrepancy exists between realised and model predicted returns. We then plot the pricing errors for the conditional CAPM and conditional Fama–French model and examine the performance of the models augmenting HVaRL, HILLIQL and RM³ factors and scaled by CY.

Figure 1.

Source: Authors’ calculations.

Figure 2.

Source: Authors’ calculations.

Figure 3.

Source: Authors’ calculations.

Figure 3 presents the pricing errors for conditional CAPM and the conditional Fama–French model scaled by CY individually and augmenting these two models with HVaRL, HILLIQL and RM³ factors separately. When the conditional CAPM and the Fama–French three-factor models are augmented with HVaRL or with HILLIQL, the pricing errors are smaller (panels b, c, e and f). This indicates that these two risk factors have the ability to explain the portfolio returns than the RM³. Our graphical study supports the econometric results and reveals that the performance conditional CAPM and the Fama–French models improve when they are augmented with either HVaRL or HILLIQL factors.

Furthermore, we find that when the factors are scaled by CY, they result in smaller pricing errors and appear to have some success in explaining the portfolio returns. The use of ‘term spread’ and ‘trading volume’ as conditioning variables also tends to produce small pricing errors and explain some portfolio returns. Other conditioning variables are not found to be helpful in improving the model performance.

In the study of Iqbal et al. (2010), the unconditional and conditional CAPM and Fama–French models overestimate the realised portfolio returns and generally result in upward biased except for the unconditional Fama–French augmented with RM³. Our graphical analysis reveals that the risk factors HVaRL and HILLIQL help explain portfolio returns better than the RM³. This is an important finding since VaR as a downside risk factor and illiquidity factor have better economic meaning than the excess kurtosis factor which is statistical in construction. A possible reason of why the HILLIQL subsumes the explanatory power of excess kurtosis factor is that excess kurtosis is the result of clustering of large number of zero returns. The zero returns are in turn the result of sluggish price movement of illiquid stocks which constitute these portfolios. As Iqbal and Brooks (2007) point out, infrequent trading prevails in many stocks in emerging markets. Also VaR being the downside risk measure has already captured some higher moments in stock returns. Thus, it can be stated that the illiquidity and downside risk may be the possible causes of apparently significant excess kurtosis factor in earlier asset pricing studies. The robustness of this argument to other emerging markets may be an interesting area of research.

6. Conclusion

Using 16 size and book-to-market portfolios as test assets from Pakistani stock market, it is found that a downside risk factor based on VaR and an illiquidity risk factor help in explaining the realised returns better than the statistically motivated excess kurtosis risk factor. The CAPM augmented with HVaRL and HILLIQL as additional risk factors performs better than the CAPM augmented with HVaRL and RM³ and the Fama–French three-factor model for both conditional and unconditional version of the models. These two additional risk factors (HVaRL and HILLQIL) improve the performance of model as indicated by Hansen–Jagannathan distance measure and result in smaller pricing errors in most of the test portfolios. Some information variables such as term spread, trading volume and CY result in smaller pricing errors. However, the CY results in the smallest pricing error for most of the models. This shows that model parameters depend on the state of business cycle of economy. Moreover, the risk factors HVaRL and HILLIQL are significant in most of the cases. We argue that these two factors subsume some of the explanatory power of higher co-moments. As these two factors have sound economic meaning, they provide a useful guide for investors in formulating portfolios based on these characteristics of stocks.

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