# Research Methods In Finance Asset Pricing Models Finance Essay

Since the development of the Capital Asset Pricing Model (hereafter CAPM), there has been several attempts to develop an alternative and superior asset pricing model in order to determine expected returns with a higher degree of accuracy. The motivation for this derives from the statistical and therefore theoretical rejection of the CAPM in a strict sense. According to statistical tests by Miller and Scholes (1972) the low beta measured was significantly inconsistent with the higher explained variable, which suggested that the market portfolio was not mean-variance efficient. Methodically, benchmark errors have been pervasively problematic in statistical tests due to the simple fact that the market portfolio is usually represented as a proxy, since it is practically unobservable and as a result most proxies may be misleading/non-representative.

In order to overcome the benchmark error, more recent models have attempted to determine the explanatory variables for risk premium more explicitly rather than simply relying on an overarching market portfolio. Specifically, the new pricing models discussed in this report are chronologically the Chen, Roll and Ross (1986), Fama French (1992) and Chen and Zhang (2009) models. The rest of the paper is organised as follows. In Section I the authors briefly discuss each model. Section II gives a data description. Secion III outlines the data and methodology used. Section IV provides a discussion of the empirical results. Section V concludes.

## I. Literature Review

## A. Capital Asset Pricing Model

The CAPM was developed in a series of articles by William Sharpe (1964), John Lintner (1965) and Fischer Black (1972) following the publication of Harry Markowitz’s portfolio selection model in 1952. CAPM measures the relationship between risk of an asset and its expected return. One of its main insights is that expected returns increase linearly with β, a measure of systematic risk. However, CAPM is based on assumptions that ignore real world complexities. The two main critiques of CAPM are its use of a market proxy as mentioned in the introduction, and the measurement error of β (slope coefficient). If the right-hand-side of an equation is measured with error then β is biased downward and α (intercept coefficient) is biased upward. This led to the use of portfolios rather than individual securities in order to mitigate these statistical problems. Despite this, the single variable expected return-beta relationship predicted by either the risk-free rate or the zero-beta version of the CAPM is not fully consistent with empirical observations (BKM 2008). In summary, empirical tests do not validate CAPM quantitative observations but it offers us insight with reasonable accuracy. For this reason, we will use it as the benchmark asset pricing model in our analysis.

## B. Chen, Roll and Ross

Chen, Roll and Ross (hereafter CRR) (1986) hypothesised that daily changes in prices are predominantly driven by unanticipated events related to macroeconomic variables. Their focus was, similarly to the CAPM, focusing on systematic or market risk commonly known as beta and particularly on the individual factors that affect it. Further intuition from CRR (1986) suggested that should these systematic variables, whatever they may be, influence dividends a plausible assumption would be that they also influence equity market returns. Deriving these factors was therefore based on fundamental theory that forms a given share price in the first place: The aggregate time-value discounted summation of dividends (CRR 1986). This led to the development of a multifactor approach, incorporating the following set of factors and signs:

(i) The growth rate of industrial production; MP(t).

(ii) Unexpected inflation; UI(t).

(iii) Change of expected inflation; DEI(t).

(iv) Term premium; UTS(t).

(v) Default premium; UPR(t).

## C. Fama and French

As a response to the traditional CAPM, Fama and French (hereafter FF) (1992) evaluated the empirical evidence surrounding the inconsistencies between theory and results of the CAPM considering other factors statistically recorded as having affected price behaviour. They hypothesised that the mean-variance deficiency of the CAPM to explain expected returns relying on a univariate approach with the unobservable market portfolio was not sufficient but rather multidimensional (Fama and French 1992). The two additional factors they identify are based on previous academic studies that have found strong cross-sectional relationships between average returns. These factors were size (measured by market capitalisation) and book-to-market ratio. Also with the inclusion of the market factor, they proceeded with the following factors and signs:

(i) The market risk premium; R(Mkt).

(ii) Size Factor: Market capitalisation, calculated as small minus big; SMB(t).

(iii) Value factor: Book-to-market ratio, calculated as high minus low; HMB(t).

## D. Chen and Zhang

The Chen and Zhang (hereafter CZ) three factor model (also known as the q-theory factor model) is a response to both the CAPM and the Fama-French three factor model. Q-theory provides the basis of the model and is a theory of investment behaviour which compares the market value of a company’s shares and the replacement cost of each of the shares, as developed by Tobin (1969) and Cochrane (1991). Chen and Zhang claim their model outperforms traditional asset pricing anomalies associated with short-term prior returns, financial distress, net stock issues, asset growth, earnings surprises, and valuation ratios. Similarly to Fama and French, a market factor is included from the consumption side and new factors (ROA and INV) are motivated from the production side of the economy and constructed in a similar way to Fama and French’s size and value factors. The three factors are as follows:

(i) The market risk premium; R(Mkt).

(ii) Return on assets factor, (high-minus-low); ROA(t)

(iii) Investment factor, (low-minus-high); INV(t)

## II. Data Description

In order to determine which model provides the strongest explanatory power of expected returns/best one-size-fits-all result, adjusted R2, derived from Ordinary Least Square (OLS) regression analyses, was used as an indicator of explanatory power. Each model was tested using left-hand side (LHS) and right-hand side (RHS) data. For the LHS data, the authors were provided with pre-determined, average monthly returns for ten decile portfolios constructed according to size. The portfolios, constructed by French (2010), were based on US equity markets NYSE, AMEX and NASDAQ. In addition, a pre-determined, monthly risk-free rate was provided, which was subtracted from each decile portfolio’s return. This led to the ten explained variables for each month and, or risk-premiums. The returns are calculated as simple arithmetic returns as this is more appropriate for cross-sectional data. Cross-sectional data includes 204 observations for each ten portfolios, equalling a total of 2040 observations and ranging from January 1990 to December 2006.

For the RHS data, the authors were provided with appropriate pre-determined values for each factor of the respective asset pricing models. This amounted to a total of ten different explanatory variables with the market risk premium appearing in the CAPM, Fama and French and Chen and Zhang models. This data allowed the authors to complete the first order OLS regression analyses. Naturally, RHS data correspond with the time range used for the LHS data.

The validity of all data given was taken at face value but a double-check was performed for the portfolio returns, which are publicly available on Kenneth French’s website.

## III. Data and Methodology

## A. Descriptive Statistics

Upon organising the data and before proceeding to the regression stage, basic descriptive statistics of monthly returns for each decile portfolio were obtained and are presented in Table I. Over the period decile 1 portfolio achieves the greatest monthly mean return of 1% in contrast to decile 10 with a 0.6% monthly return. This supports the conventional wisdom that small capitalisation (cap) stocks perform better than large cap stocks on average. A linear relationship between risk and return is evident from the higher average return of decile 1 corresponding with a high standard deviation. This is relative to larger decile portfolios whose standard deviation decreases as portfolio size increases. On this basis large decile portfolios are less volatile. The difference in standard deviations is, however, minimal exhibiting the effects of diversification.

Jarque-Bera (JB) is a formal test of normality whose value is derived from skewness and kurtosis. A distribution is normal if its skewness equals 0 and its kurtosis equals 3. The positive skewness of small decile portfolios suggests a greater probability of negative returns and “a greater likelihood of large increases in returns than falls” (Worthington and Higgs 2005). All portfolios with the exception of deciles 1 and 2 exhibit negative skewness. These portfolios have a higher probability of positive albeit lower returns. The risk/return trade-off is also exemplified by the kurtosis, illustrated in Figures 1 and 2. The larger kurtosis of 5.915 in decile 1 (Figure 1) is as a result of greater variance due to extreme values in the tail of the distribution. This in contrast to decile 10 (Figure 2) with a kurtosis of 3.955 which is closer to a mesokurtic distribution of 3. The implications of these results are that over the period small cap stocks performed better than large cap stocks on average but were subject to a greater level of risk. JB is a null hypothesis that is used to check if the distribution of data is normal. We reject the null hypothesis for each portfolio at a 99% significance level. In other words, the distribution of returns for each portfolio does not follow a standard normal distribution.

## Table I

## Descriptive Statistics of Portfolio Returns

This table shows the descriptive statistics of monthly returns for ten value-weighted size portfolios over the period January 1990 to December 2006. Max. stands for Maximimum. Min. stands for Minimum. Std. Dev. stands for Standard Deviation. JB stands for Jarque-Bera. Obs. stands for Observations.

1

2

3

4

5

6

7

8

9

10

Mean

0.010

0.009

0.009

0.007

0.008

0.007

0.009

0.008

0.008

0.006

Median

0.012

0.011

0.013

0.015

0.016

0.012

0.011

0.011

0.013

0.008

Max.

0.292

0.255

0.180

0.166

0.145

0.110

0.130

0.127

0.113

0.113

Min.

-0.213

-0.232

-0.215

-0.200

-0.206

-0.204

-0.191

-0.175

-0.152

-0.149

Std. Dev.

0.061

0.063

0.058

0.055

0.054

0.049

0.046

0.048

0.042

0.041

Skewness

0.243

0.014

-0.503

-0.490

-0.530

-0.668

-0.530

-0.461

-0.506

-0.460

Kurtosis

5.915

4.832

3.945

3.842

3.807

4.096

4.191

4.045

3.699

3.955

JB

74.251

28.548

16.194

14.193

15.094

25.377

21.596

16.529

12.877

14.942

JB p-value

0.0000

0.0000

0.0003

0.0008

0.0005

0.0000

0.0000

0.0003

0.0016

0.0006

Obs

204

204

204

204

204

204

204

204

204

204

## B. Ordinary Least Square Regression Analysis

In order to obtain the necessary output, standard OLS regressions were conducted using Eviews 6. Expected return for each portfolio (minus the risk-free rate) was regressed against the asset factors of each model. The regression equations for each model are as follows with explanations of each variable included in the Appendix:

## B.1. CAPM

(1)

## B.2. Chen, Roll and Ross

## (2)

## B.3. Fama and French

## (3)

## B.4. Chen and Zhang

## (4)

## C. R-Squared

As outlined earlier, this report incorporates the adjusted R2 in measuring model superiority. R-squared measures the ratio of the explained variation relative to the total variation or in other words, how much of the sample variation in the LHS is explained by the RHS (Wooldridge 2002). Mathematically, it is calculated as:

(5)

A higher R2 between 0 and 1 hence indicates how closely the regression line fits in relation to the sample data. For our results, we would naturally expect to determine a higher R2 to correspond with a better choice of explanatory variables. Other advantages in using R2 as the sole measure of statistical model quality are that it conveniently displays the linear explanatory power of the RHS factor(s) and thus may help to corroborate theoretical arguments set. It is also useful for assessing the predictive capability of these RHS factor(s), which is precisely the goal of this report. As a standardised indicator varying from 0 to 1 (0% and 100%), it is therefore easily comparable between individual regressions with the same LHS sample.

Despite the usefulness of R2 in OLS regressions, there are a variety of drawbacks to be taken into account during interpretations. Specifically, R-squared will never decrease and generally will increase when adding more explanatory variables, regardless of their true explanatory power. Also, it only applies to linear models and hence does not identify non-linear relationships. Furthermore, a high R2 can result from spurious regressions, where a third unaccounted variable affects both the LHS and RHS factors. The indicator is by definition, sample-specific and for instance using non-US explanatory or explained variables unlike in this study could and probably would yield different results.

As the CRR model incorporates five separate RHS factors, the problem with the conventional R2 indicator would be magnified especially in this case. In order to overcome this statistical problem, the adjusted R2 was used as the measure of goodness of fit. The adjusted R2 has been specifically developed to tackle the problem of artificially higher results simply due to a higher number of explanatory factors.

Finally, conventional wisdom states that correlation should not be confused with causality. With that said, the results in this study were interpreted with a strong emphasis on particularly the economic rationale in addition to statistical logic.

## IV. Discussion of Empirical Results

## A. Chen, Roll and Ross

## A.1. Statistical Interpretation

For each of the ten size decile portfolios, the adjusted R2 obtained from the CAPM is much higher than that from the CRR model. As can be seen from the Figure 1, the R2 from the CAPM steadily increases from 0.42 to 0.93 as the market capitalisation of the portfolio constituents increases. In contrast, the adjusted R2 from the CRR model remains around 0.01 for all of the portfolios except for decile 2, for which the R2 surprisingly becomes negative. Such a huge difference implies that the market premium factor has a significantly higher explanatory power than macroeconomic factors, for the given sample of monthly returns. Moreover, as the size of stocks in the portfolio rises, the market factor explains an increasing proportion of portfolio returns. In other words, stock returns of larger firms can be better described by the CAPM model than that of small firms. On the other hand, the adjusted R2 from the CRR model does not change much among different size-sorted decile portfolios, which leads to a larger gap between the explanatory power of the CAPM and CRR models for large size stock portfolios.

As the Panel B of Table II shows, the R2 derived from the CRR model are significantly higher than the adjusted R2 for all of the ten portfolios. Since there are five independent variables in the RHS of the CRR model, the relatively high R2 compared with adjusted R2 implies that the model may contain one or more redundant variable(s) that has poor explanatory power. The adjusted R2, which penalises the use of such variables, provides a more accurate tool for the assessment of this model. Alternatively, assessing regression coefficients produced little valuable insight as none of the decile portfolios reported significant coefficients for any explanatory variable.

Figure 1: CAPM versus CRR

## A.2. Economic Interpretation

In order to gain more insight into the abysmally low adjusted R2 results, economic reasoning was required. If we follow the logic in Chen, Roll and Ross (1986) to adopt crude oil price as a proxy of macroeconomic conditions, we will detect great volatility as well as several market crashes over the sample period from January 1990 to December 2006. Specifically, these events include market crashes caused by the Gulf War in 1994, the Asian Crisis of 1998 and the erosion of OPEC excess oil production capacity in the following years up to 2002. Therefore, instead of analysing the entire sample alone, sub-periods were identified with market crashes as cut-off points as an attempt to retain relatively stable return distributions over sub-periods. The familiar CRR regression was performed for each sub-period. The adjusted R2 for the entire period was subtracted from the adjusted R2 for each sub-period in order to observe the differences. The results are presented in figure 1.

Focusing on adjusted R2 reveals considerable sub-period variation when comparing the arithmetic adjusted R2 difference between the respective sub-period and the full sample period. Evidently, the period from 1990 to 1994 exhibits the largest disparity of R2 results in that the difference is approximately 25% more for most deciles. The other sub-periods also improve but to a lesser extent with 5% to 10% positive changes in R2 compared to the full sample. The interval from 1999 to 2002 exhibits the lowest R2 in contrast to the other three intervals with deciles 4 to 10 posting negative R2. The implication is that there are redundant variables in the regression equation.

Retrospectively however and according to IMF World Economic Outlook (1999), the Asian crisis precipitated most of the countries to adhere to more disciplined monetary and fiscal policies, which led to record low inflation levels since the 1980s. Stronger government interventions restricted the fluctuation of macro variables such as inflation rate and interest rate, and therefore limited the role they could play in the CRR model during the sample period of 1999 to 2002. It is these very fundamental changes in macroeconomic conditions that have been the likely reason for such drastic changes in the explanatory power of the CRR model. Even allowing for sub-period analysis, adjusted R2 remains at levels of less than 30%, which is exceptionally low for an asset pricing model.

## B. Fama and French

The adjusted R2 for the CAPM and Fama-French models begin at approximately 40% and 60% respectively and increase as the size of the portfolios increase. Both models explain a high proportion of the variation of returns in the tenth decile with the Fama-French being slightly more superior in terms of R2, to the CAPM at 95.7% as opposed 92.6%. These results suggest that the market factor arguably contributes a significant proportion of the total R2 for each model and as said, results are almost on par for larger decile portfolios. Yet the additional explanatory power in the Fama-

## Table II

## Regression Results of Asset Models

The table presents the results from regressing each decile portfolio’s monthly excess returns against the explanatory variables of each particular model over the period January 1990 to December 2006. α – intercept. β – slope coefficient (factor loading). Mkt - market risk premium. MP - growth rate of industrial production. UI - unexpected inflation. DEI - change in expected inflation. UTS - term premium. UPR - default premium. HML – value factor (book-to-market). SMB – size factor (market capitalisation). ROA – return on assets. IA – investment factor. Significance of α and β is represented by an asterisk (*) at 0.05 level of significance and a double asterisk (**) at 0.01 level of significance.

1

2

3

4

5

6

7

8

9

10

Panel A: Capital Asset Pricing Model

α

0.004

0.002

0.002

0.000

0.001

0.000

0.002

0.001

0.002

## *

0.000

β (Mkt)

0.961

## **

1.171

## **

1.139

## **

1.123

## **

1.146

## **

1.072

## **

1.041

## **

1.089

## **

0.979

## **

0.957

## **

R-Squared

0.428

0.584

0.669

0.705

0.768

0.827

0.860

0.890

0.926

0.926

Adjusted R-Squared

0.425

0.582

0.668

0.703

0.766

0.826

0.859

0.889

0.926

0.926

S.E. of regression

0.046

0.041

0.033

0.030

0.026

0.020

0.017

0.016

0.011

0.011

Panel B: Chen, Roll and Ross Macroeconomic Factors Model

α

0.004

0.015

0.014

0.013

0.008

0.010

0.014

0.013

0.016

0.023

β (MP)

-1.065

-1.401

-1.446

-1.413

-1.236

-1.299

-1.112

-1.106

-0.941

-0.848

β (UI)

-1.869

-2.567

-4.431

-4.453

-4.862

-3.841

-4.776

-3.836

-3.169

-2.014

β (DEI)

6.337

4.445

7.720

4.830

6.161

2.232

7.979

3.827

2.473

-3.344

β (UTS)

0.685

0.370

0.437

0.495

0.455

0.453

0.306

0.276

0.178

0.023

β (UPR)

-0.492

-1.102

-1.160

-1.310

-0.610

-0.917

-0.952

-0.931

-1.047

-1.849

R-Squared

0.026

0.018

0.030

0.039

0.038

0.047

0.036

0.033

0.031

0.042

Adjusted R-Squared

0.002

-0.007

0.005

0.015

0.014

0.023

0.011

0.008

0.006

0.018

S.E. of regression

0.061

0.063

0.057

0.055

0.054

0.048

0.046

0.047

0.042

0.041

Table II - Continued

1

2

3

4

5

6

7

8

9

10

Panel C: Fama French Three Factor Model

α

0.002

0.000

0.000

-0.002

-0.001

-0.001

0.000

-0.001

0.000

0.001

β (Mkt)

0.991

## **

1.185

## **

1.158

## **

1.163

## **

1.168

## **

1.113

## **

1.104

## **

1.153

## **

1.052

## **

0.939

## **

β (HML)

0.417

## **

0.383

## **

0.354

## **

0.377

## **

0.278

## **

0.260

## **

0.263

## **

0.244

## **

0.211

## **

-0.138

## **

β (SMB)

0.798

## **

0.806

## **

0.714

## **

0.636

## **

0.517

## **

0.362

## **

0.239

## **

0.189

## **

0.059

## **

-0.218

## **

R-Squared

0.617

0.762

0.838

0.850

0.868

0.889

0.896

0.915

0.944

0.957

Adjusted R-Squared

0.611

0.758

0.835

0.848

0.866

0.887

0.895

0.914

0.943

0.957

S.E. of regression

0.038

0.031

0.023

0.022

0.020

0.016

0.015

0.014

0.010

0.009

Panel D: Chen and Zhang Factors Model

α

0.011

## **

0.007

## *

0.006

## *

0.004

0.004

0.002

0.003

0.002

0.001

-0.002

β (Mkt)

0.664

## **

0.952

## **

0.961

## **

0.980

## **

1.024

## **

1.003

## **

1.022

## **

1.045

## **

1.001

## **

1.015

## **

β (ROA)

-0.553

## **

-0.393

## **

-0.249

## **

-0.168

## **

-0.135

## **

-0.029

-0.026

-0.051

## *

0.047

## **

0.081

## **

β (INV)

0.299

## *

0.179

-0.094

-0.179

-0.178

-0.260

## **

-0.014

-0.059

-0.045

0.029

R-Squared

0.598

0.662

0.709

0.728

0.785

0.837

0.860

0.893

0.929

0.934

Adjusted R-Squared

0.592

0.657

0.704

0.724

0.782

0.835

0.858

0.891

0.928

0.933

S.E. of regression

0.039

0.037

0.031

0.029

0.025

0.020

0.017

0.016

0.011

0.011

French model is particularly evident in low-capitalisation stocks, which benefit from the supplementary size SMB and value HML factors.

Table II shows that the intercept α hovers around zero for each decile portfolio and is also statistically insignificant compared to those of CAPM, where decile 9 has a significant intercept. The encouraging results for the Fama-French model are corroborated by all of the factor coefficients for both SMB and HML, which are significant at least at a 99% confidence level. Both of the coefficients are larger for small decile portfolios and negatively related to decile size, eventually turning negative in the tenth decile. The mean of the initial SMB factor data used for the regressions is negative (-0.02157) and is hence likely to take a negative sign when the market factor exerts such dominance as an explanatory variable.

Figure 2: CAPM versus. Fama-French

In comparison to the CAPM, Fama and French (1992) argued a rather pessimistic future for the unvariate CAPM in that no matter what approach is incorporated, the market coefficient will empirically remain flat unless a true proxy is obtained. With this said, the near-impossible task of compiling this true market proxy would still not yield any better explanatory results as studied by Stambaugh (1982). Specifically, Stambaugh (1982) concluded that different compositions of market proxies yielded virtually identical results, which suggests that benchmark errors are not the root of the problem. As a result, the additional explanatory variables serve an important purpose.

## C. Chen and Zhang

The explanatory power of both models increases linearly and converges toward equal explanatory power as the size of portfolios increase. This is potentially because both models include the market risk premium and the explanatory power of the additional factors in CZ reduce for larger decile portfolios. For portfolio deciles 6, 7, 8, 9 and 10 the difference in adjusted R2 is less than 1%. Similar to the Fama-French model where the relative importance of the market model increases in larger portfolios, we argue that the negligible difference in adjusted R2 is in part due to the declining coefficient for the ROA factor as decile size increases. Additionally, for our sample data all except two ROA values are significant. The IA factor coefficients also reduce but as Table II shows, only two of these values are significant. This would indicate that IA is somewhat of a redundant variable. It must be noted that heteroskedasticity is not accounted for, which may affect the significance of all coefficients in the regression. This may be because the negative relation between IA and return is conditional on ROA. It would result in autocorrelation and therefore, a reduction in the significance of variables. As Chen and Zhang (2009) put it:

“ROA is not disconnected to investment: more profitable firms tend to invest more than less profitable firms”

What is clear is that based on adjusted R2, the CZ model performs substantially better than the CAPM for the lower decile portfolios as shown in Figure 3. The CZ model takes into account anomalies common to small firms and not captured by the CAPM. Firstly, the CZ model includes a ROA factor, which is negative for small decile portfolios and increases along with portfolio size. A plausible assumption could be, that lower decile portfolios are more financially distressed making them less profitable which leads to low ROA and finally lower expected returns as compared to firms in high decile portfolios. The ROA factor can also be used to explain anomalies related to short-term prior returns and earnings surprises.

Secondly, the CZ model includes an investment factor (IA). The negative relation between IA and return can be used to explain several anomalies not captured by the CAPM, including the value premium, net stock issues and asset growth anomalies for example. Overall, the familiar market factor appears to be the dominant variable among the two models but including additional factors of ROA and IA means the CZ accounts for anomalies not captured in the single-factor CAPM, and thus exhibits greater explanatory power than the CAPM.

There are a number of fallacies with the CZ model worth considering. As mentioned the IA factor is significant for all but two deciles indicating that it may be a redundant variable. The intercept for portfolio deciles 1, 2 and 3 are all significant which violates the assumption of APT which stipulates that the intercept must be zero. In the next sub-section we will discuss the reasons why the CZ may be inferior to the Fama-French three factor model.

Figure 3: CAPM versus Chen-Zhang

## D. Comparative Overview

Figure 4 illustrates the explanatory power among each asset pricing model. The CRR model consistently performs the poorest while the FF model is superior over the others throughout and especially for small decile portfolios on the basis of adjusted R2. The difference between the FF and CZ models is minimal for decile 1 but this spread grows through deciles 2 to 6. The authors attribute this to the insignificance of CZ’s IA factor, which applies for all portfolios except decile 1 and 6. It suggests that this explanatory variable is unnecessary in the model, which could have not been inferred simply by assessing R2 values. For both models the size of the factor coefficients decreases as the size of the decile increases. This illustrates the declining marginal explanatory power from added variables for both models in large decile portfolios. Converging to the same degree of explanatory power as the CAPM indicates the importance of the market factor used in all three models. This is the key differentiating feature between all three models and the CRR.

The CRR performs the worst of all models with very little or even negative explanatory power. The negative values illustrate the benefit of adjusted R2 regarding accumulating variables mentioned earlier. The notion of irrelevant variables is particularly evident when comparing the differences between adjusted R2 and R2, which is the highest for any of the models despite the low initial explanatory power. The notoriously low adjusted R2 can potentially be attributed to the absence of the market factor. In addition, as reasoned in Section IV A.2. by analysing sub-periods, the CRR model seems to be sensitive to fundamental changes in macroeconomic conditions and consequently does not cope with the test of time well.

Figure 4: Comparative overview of adjusted R-Square for each model.

## V. Summary and Conclusion

Despite being used for a variety of tasks, such as calculating cost of capital and estimating returns, the CAPM is not fully consistent with empirical observations (BKM 2008). It does, however, offer us an insight with reasonable accuracy and is therefore a good benchmark to compare the explanatory power of newer asset pricing models. The results of our test show that the Chen, Roll and Ross model performs the worst of all. Based on systematic macroeconomic factors which influence dividends, it was assumed there would also be an effect on equity market returns. The authors conclude that the CRR model is inferior stemming from not only the abysmal adjusted R2 regardless of decile size of sub-period, but also and perhaps consequently from the fact that its explanatory variables are redundant and do not stand the test of time. The absence of the market factor is also substantial and is the dominant force in explaining returns in the Chen-Zhang and Fama-French models.

Chen and Zhang (2009) argue that their asset pricing model is superior to all previous models. For example, they suggest that all previous studies fail to recognise the link between distress and ROA and the positive ROA-expected return relation and therefore find the negative distress-expected return anomalous. However, based on the analysis of the explanatory power of the CZ against the Fama-French model, the authors reach a different conclusion. Fama-French has stronger explanatory power for lower decile portfolios and is in fact better than the CZ model for every decile portfolio in terms of adjusted R2. We cite the reason for this as being the insignificance of the CZ investment factor (IA) for the majority of decile portfolios and especially in deciles 2 -6 where the largest spread in explanatory power exists between CZ and Fama-French. An additional contrast to the Fama-French model is the fact that Fama-French explanatory variables are significant at the 0.01 level for each decile portfolio. Investment factor insignificance may be due to the correlation with the other Chen-Zhang explanatory variable, ROA.

Although Chen and Zhang’s assertions of superiority are based on the analysis of a longer time interval, the authors conclude that based on the results presented in this paper, the Fama-French three factor model provides the strongest explanatory power for cross-sectional variation in expected return.

## Appendix

## CAPM

R(Mkt) – denotes the market premium factor, which acts as a proxy for the market return.

## Chen, Roll and Ross Model

MP – denotes the index of industrial production from time t-1 to t.

UI – denotes the unexpected inflation rate from time t-1 to t, where the seasonally adjusted Consumer Price Index (I) from time t-1 to t is measured as:

DEI – denotes the change in expected inflation rate, which is another inflation variable that could affect the nominal interest rate as well as nominal expected cash flows.

UTS – denotes the term premium from time t-1 to t, which is the yield spread between the long-term and the one-year Treasury bonds. The term-structure spreads across different maturities could have an impact on the discount rate.

UPR – denotes the default premium, which is the yield spread between Moody’s Baa and Aaa corporate bonds. The risk premium could influence the discount rate, thereby affecting the stock returns.

## Fama and French Model

HML – denotes the difference in average return between two value investment portfolios and two growth investment portfolios on the basis of Market Equity (ME).

SMB – denotes the difference in average return between three small investment portfolios and three large investment portfolios on the basis of Book-to-Market (B/M).

## Chen-Zhang Model

ROA(t) - defined as income before extraordinary items divided by last quarter’s total assets

INV(t) - defined as annual change in gross property, plant and equipment plus the annual change in inventories divided by the lagged book value of assets

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