In the portfolio management field, Eugene Fama and Kenneth French developed the Fama-French three factor model to describe market behavior. The FF three-factor model for the expected return on share is similar to the CAPM but with two extra factors. Fama and French (1992) find that asset-pricing model of Sharpe (1964), Lintner (1965), and Black (1972) has some drawbacks, empirical contradictions, concerning average returns. Their research fails to support the central prediction of the CAPM model, that average stock returns are positively related to market ï¢ beginning from 1963 till mid-1990s. Furthermore, variables like size (market equity), leverage, book-to-market equity, earnings-price ratios (E/P) can be regarded as various ways to scale stock prices. They can be considered as different ways of extracting information from stock prices about the cross-section of expected stock returns (Ball (1978); Keim (1988)). Fama and French (1992) also indicate that the combination of size and book-to-market equity seems to absorb the roles of leverage and E/P in average stock returns, as they seem to be redundant for explaining average returns. All in all Fama and French noticed that two classes of stocks have tended to do better than the market as a whole: (i) small caps and (ii) stocks with a high book-to-market ratio. They then added two factors to CAPM to reflect a portfolio's exposure to these two classes:
Rit - Rft = ï¡im + ï¢M,t ( Rmt - Rft ) + ï¢SMB,t (SMBt) + ï¢HML,t (HMLt) + ï¥it
Therefore, Ri is the portfolio's rate of return, Rf is the risk-free return rate, and Rm is the return of the whole stock market. The "three factor" βM is analogous to the classical β but not equal to it, since there are now two additional factors to do some of the work. SMB stands for "small (market capitalization) minus big" and HML for "high (book-to-price ratio) minus low"; they measure the historic excess returns of small caps over big caps and of value stocks over growth stocks. These factors are calculated with combinations of portfolios composed by ranked stocks and are available historical market data. Historical values may be accessed on Kenneth French's web page. [1]
Fama and French attempted to better measure market returns and, through research, found that firms with poor prospects, defined by low stock prices and high ratios of book-to-market equity, have higher expected stock returns than firms with strong prospects. By including these two additional factors, the model adjusts for the outperformance tendency, which is thought to make it a better tool for evaluating manager performance.
There is a lot of debate about whether the outperformance tendency is due to market efficiency or market inefficiency. On the efficiency side of the debate, the outperformance is generally explained by the excess risk that high book-to- market equity and small cap stocks face as a result of their higher cost of capital and greater business risk. On the inefficiency side, market participants mispricing the value of these companies, which provides the excess return in the long run as the value adjusts, explain the outperformance. [2]
The Chen-Zhang Three-factor Model
Chen and Zhang (2009) have proposed a new model that explains a number of confusing aspects about stock returns. Over the past twenty years it has become clear that Fama-French model is not able to explain many cross-sectional patterns. Cheng and Zhang (2009) motivate a new three-factor model from q-theory. In the new model the expected returns on a portfolio in excess of risk-free rate is described by the sensitivity of its return to three factors: the market factor, a low-minus-high investment factor, and a high-minus-low returns on assets factor:
Ri,t - Rf,t = ï¡i,t + ï¢M,t ( RM,t - Rft ) + ï¢INV,t (INVt) + ï¢RAO,t (RAOt) + ï¥it
The performance of the model is outstanding. The new three-factor model outperforms traditional asset pricing models in explaining anomalies associated with short-term prior returns, financial distress, net stock issues, asset growth, earnings surprises, and valuation ratios. Chen and Zhang do not interpret the investment and ROA factors as risk factors. Moreover factors are constructed on economic fundamentals that are less likely to be affected by mispricing, at least directly. The evidence of Chen and Zhang suggests that low-investment stocks and high-ROA stocks have high average returns whether or not they have similar return patterns of other low-investment and high-ROA stocks.
Descriptive Statistics
TABLES
Over the 16 years, the highest mean return is for portfolio 1 (0.010316), while the lowest mean return is for portfolio 10 (0.005699). As far as median are concerned, the highest comes to portfolio 5 (0.01625) and the lowest comes for portfolio 10 (0.00795). The standard deviation of all 10 deciles portfolio reveals that portfolio10 is the least volatile (0.041051) with portfolio2 being the least volatile (0.06326). The lowest minimum returns are for portfolio2 (-0.2324) and the highest minimum returns are for portfolio10 (-0.1486), while the highest maximum returns are for portfolio 1 (0.2916) and the lowest maximum returns are for portfolio 6 (0.1097).
Talking about the skewness, it is observed that portfolio 1 has positive skewness(0.242876), which means that return distribution of that stock has a higher probability of earning positive returns and that gains are likely to be greater than anticipated by the normal distribution. On the contrary, portfolios 6 as well as portfolio 10 have negative skewness, which indicates the higher probability of earning negative returns. The positive excess kurtosis are spotted in all ten portfolios namely the distribution is peaked or leptokurtic relative to the normal which is 3. Thus, it means that the portfolios can have more frequent large positive or negative returns.
Method
We imported the data in EVIEWS after organizing them. We then ran a regression for each portfolio using the ordinary least squares (OLS) method with the dependent variable being the excess returns of the share over the risk free rate and the independent variables being the factors that each model specifies as contributing to those returns. We then collected the adjusted coefficients of determination () for every regression and constructed graphs showing the adjusted R2 on the y-axis and the market capitalization of the portfolio on the x-axis. Graphs comparing the CAPM with each other model and one encompassing them all together were constructed.
R2
The purpose of this exercise was to use the R2 of the regressions to determine the power of each model to interpret the variability in the excess returns following the example of Roll . EVIEWS calculates the adjusted R2 as:
The adjusted R2 is a better measure of goodness of fit than the simple R2 because the latter does not decrease when the number of independent variables or regressors increases. This would create some doubt as some of the models that we use incorporate more factors than others. An extreme case would be to have a R2=1 if the regressors are as many as the sample observations and it has been observed that it can be as high as 0.9 for time series. As we can see from the formula above the R2-adjusted can decrease when we add regressors to the equation and could even turn out to be negative for some models with limited capabilities. However as Roll mentions the measure cannot be considered as a conclusive test as we cannot establish whether the factors used are pervasive and related to the risk rewards. (Roll, 1988). For this reason it is considered to be a "soft" measure. Furthermore there is no distribution for which will enable us to study its behavior. Finally the can be artificially high when there is a trend in the data. Obviously this cannot be a problem as our variables are not in levels but in differences.
The Adjusted R-square Distribution in Four Models
Distribution of Adjusted R-square
It is obvious just by eyeballing the data that all of our models have an upward trend of adjusted R-square, that is increases when the portfolio size increases. This is consistent with Roll`s results(Roll, 1988). High capitalisation portfolios can be considered as a collection of smaller ones. These portfolios are diversified and hence they will have higher R2. In the spirit of Roll we ran a regression between R2 and size for all the models to test for statistical significance
CAPM versus Chan and Zhang
. The Compared with CAPM, Chen and Zhang model results in a larger adjusted R-square in the first six portfolios, with a decreasing difference from p1 to p6. It is then followed by converging, finally reaching around 93% for both models. However, Chen and Zhang is superior to CAPM in this regression, for only have one smaller adjusted R-square (p7) than CAPM.
CAPM versus Fama French
Compared with CAPM, Fama French shows strong capacity to explain excess return, whose R-square keeps larger for each of the portfolios. As we see, the lowest adjusted R-square is above 60% and the highest to be 96%, keeps in a high level all the time. However, in common with Chen and Zhang, this dominant position, for FF is threatened by the increasing size of portfolio, which means, with the rise in size, the gap between two models is declining.
CAPM versus CRR APT
It is astonishing that CRR has a rather weak adjusted R-square, which slightly fluctuates around zero. There's even a negative figure in p2 indicating bad regression results. Such a low R-square means those CRR factors has little capacity to explain excess return, There is also no pattern in graph compared with CAPM's upward trend which cannot reveal any relationship between size and R-square.
Comparison and Conclusion
From the graph above we can see that there is a pattern or rather, a positive correlation between portfolio size and R-square, with larger portfolios having higher R-squares with the only exception of the APT model where the R-square is not correlated with the size of the portfolio. For example the highest R-square in the APT model is for the decile 6. This behavior may suggest as proposed by Roll (1988) that big market capitalization stock portfolios are less sensitive to the factors that attempt to explain the excess returns in the APT model.
CRR APT works worst, with extremely low adjusted R-square
CRR is poorly worse than other three, for its R-square never goes beyond 3% through 10 portfolios. This indicates that there is only a minor portion of average excess return which can be explained by CRR factors. Thus, whether the regression is reliable should call for further testing.
For good models, adjusted R-square increases with the growth in size
As we see, all of the three good models have an uptrend in pace with the increase of portfolio size. There seems to be a statistically significant connection between adjusted R-square and size, which indicates that these models works better for larger size portfolios. Diversification may be a reansonable explanation for this, because larger firms generally have many divisions and often operate in more than one industry or market. Diversified portfolios should have high R-squares, for some risks have been diversified. [3]
Fama French, Chen & Zhang does well for large size portfolio as well as small size
Fama French has a very high starting point from the smallest size, going up sharply followed by smooth increase. Chen & Zhang's line appears slightly different, for it moves up in a more stable way. Overall, both of them have a great degree of explanatory power for large size as well as small size portfolio. Contrasted with them, CAPM is much weaker in explaining small size portfolio but works excellent for large size.
As can be seen the Fama & French model has the most explanatory power of all the models with the highest mean adjusted R-Square. The mean adjusted R-Squares were, respectively, 0.790 for Chan & Zhang, 0.757 for CAPM, 0.851 for Fama & French and 0.010 for APT. All the 10 size-sorted decile portfolios show a higher R-Square for the FF model. The APT model has a very low explanatory power for all the portfolios with even a negative adjusted R-Square for the second decile portfolio. A negative adjusted R-square is given because (explanation).
Economic reason that could affect the performance of the particular models in explaining the stock excess returns have to be explained taking into consideration the period of time that we are analyzing. Our data goes from 1990 to 2006, a period where the US stock markets and many other markets in the world experienced an extraordinary performance due to low interest rates, low and steady inflation, the boom in international trade due to globalization, the technology boom, and a general economic growth.
From 1980 both, the short term and long term interest rates have been decreasing whereas industrial production and corporate profits have been increasing as well as consumption but keeping inflation stable due to an increasing outsource production in emerging markets. This has reduced the cost of goods sold, cut payroll expenses and stabilized prices in the US market. Also the price of commodities remained rather low during this period with for example, oil trading at an annual average price per barrel of USD 23.19 in 1990, USD 27.39 in 2000 and USD 58.30 in 2006 (http://oilpatchresearch.com/Historical_Oil_Prices.html), really far away from the historic high of USD147.30 in July, 2008.
With all this said we think that the models such as Fama & French and Chan & Zhang did better in pricing theses assets because they used explanatory variables that focus more on company specific characteristics such as the ROA factor, the investment factor (INV), small (market capitalization) minus big and high (book-to-price ratio) minus low and not in macroeconomic factors such as inflation, return on long term government bonds, Treasury Bill rates etc as was the case with the APT model. We gave more importance to the factor used by the first two models mentioned because given the economic stability of the period of study, macroeconomic factors could be more easily predicted and reflected in the market making differences in US stock returns being better explained by more specific company related factors
References
Richard Roll (1998), R2, The Journal of Finance, Vol. XLIII, No.2
