Exploring Stock Return Determinants: From Markowitz to Fama-French Model

1.0 Section A

1.1 Introduction

Markowitz developed a portfolio design hypothesis that should compensate investors with better risk returns (Markowitz, 1959). The Markowitz Model assessed risk as the deviation of its portfolio, its dispersion measurement or the overall risk. In the Capital Asset Pricing Model (CAPM), (Sharpe, 1964) and (Mossin, 1966) their view was that investors would be compensated not just for the overall risk, but also for their market risk or systematic risk, as measured by beta of the stock. Investors are not compensated for carrying the inventory risk, which in a portfolio setting can be diversified. The beta of a stock is the slope of return from the stock which is back from the return from the market. Modern capital theory has grown from a single beta expressing market risk to MFMs with 4 or more betas. A stock value is frequently analysed in a number of ways, such as earnings price (P/E), book value price (P/BV), sales price (P/S) but it should fall in some order. Security assessment and the generation of projected returns take numerous ways. (Graham et al., 1934) suggested buying stocks on the basis of the P/E ratio. They proposed that no stock be bought if its P/E ratio was 1.5 times higher than that of P/E on the market. The P/E criterion have therefore been defined. Interestingly, at the height of the Great Depression Graham and Dodd put forward the low P/E model. The data supporting the Low P/E model was provided (Basu, 1977). Scholars frequently prefer to test their reciprocal “high EP” method to the low P / E technique.

To understand how the Fama-French model is used to determine stock returns, stock returns for UK market listed companies were selected. The period of analysis was between January of 2008 and December of 2017. The stocks selected include XP Power Limited, Restore Plc, Alliance Pharma Plc, Craneware Plc, RWS Holdings Plc, Future Plc, BAE Systems, PPHE Hotel Group and Watkin Jones Ltd.  One discovers among accounting irregularities the price/earnings, price/book and sales (Levy, 1999). Levy also considers dividend rates to be (positive) anomalies in stock. (Malkiel, 1996) presents data to promote the purchase for outperformance of low P/B, low  P/E, and high D/P (dividend yield) shares, given that low stocks have modest chances for growth Malkiel talks of the ‘double bonus,’ that is, if growth happens, income will rise and multiple earnings will rise, and the price will continue to rise. Of course, both earnings and P/E multiple may decrease if growth does not materialise.


10 stocks listed on the UK market

 Mean  8.166667  10.00000  17.58333  34.86667  48.45833  42.63333  42.10833  4.416667  9.816667  17.34167 -0.503000 -0.101833  1.067833  0.045583
 Median  8.000000  10.00000  17.00000  32.00000  48.00000  43.00000  41.00000  4.500000  10.00000  17.00000 -0.470000  0.200000  1.240000  0.010000
 Maximum  12.00000  12.00000  25.00000  47.00000  61.00000  47.00000  47.00000  6.000000  12.00000  22.00000  8.220000  5.470000  13.65000  0.210000
 Minimum  5.000000  9.000000  13.00000  29.00000  38.00000  36.00000  38.00000  3.000000  8.000000  14.00000 -13.96000 -5.030000 -13.38000  0.000000
 Std. Dev.  1.605313  0.698137  2.328426  5.670109  6.472821  2.397244  2.592153  1.119599  0.925669  1.889804  2.660704  2.287689  4.107974  0.066063
 Skewness  0.277641  0.595196  0.551582  0.793085  0.328900 -0.973833  0.426829  0.029229  0.689636  0.835958 -0.583089  0.091692 -0.393720  1.244840
 Kurtosis  2.312169  3.781213  3.230511  2.124737  2.025166  4.618202  1.928516  1.633563  3.056630  2.929905  8.210026  2.531121  4.460158  3.020288
 Jarque-Bera  3.907252  10.13663  6.350522  16.41012  6.915005  32.05990  9.384059  9.352834  9.527984  14.00110  142.5217  1.267386  13.76062  30.99457
 Probability  0.141759  0.006293  0.041783  0.000273  0.031508  0.000000  0.009168  0.009312  0.008531  0.000911  0.000000  0.530629  0.001028  0.000000
 Sum  980.0000  1200.000  2110.000  4184.000  5815.000  5116.000  5053.000  530.0000  1178.000  2081.000 -60.36000 -12.22000  128.1400  5.470000
 Sum Sq. Dev.  306.6667  58.00000  645.1667  3825.867  4985.792  683.8667  799.5917  149.1667  101.9667  424.9917  842.4423  622.7890  2008.178  0.519359
 Observations  120  120  120  120  120  120  120  120  120  120  120  120  120  120

From the analysing descriptive statistics of the portfolios and their related factors, mean returns for the portfolios and their related factors are as shown in the above table.

1.2 Estimated values

In evaluating the stock returns basing on the Fama-French model, ten portfolios were used considering their sizes, momentum and the percentage of book-value to market-value. By analysing the book-to-market ratios and the momentum of the portfolios, there is no perfect negative correlation existing between returns and risks. All the main portfolios featured in the analysis contain many stocks, a factor that has reduced the non-systematic risk. However, in consideration of the B/M, the portfolios contain fewer stocks thus increasing the inherent risk in the portfolios.


 Mean  8.166667  10.00000  17.58333  34.86667  48.45833  42.63333  42.10833  4.416667  9.816667  17.34167 -0.503000 -0.101833  1.067833  0.045583
 Median  8.000000  10.00000  17.00000  32.00000  48.00000  43.00000  41.00000  4.500000  10.00000  17.00000 -0.470000  0.200000  1.240000  0.010000
 Maximum  12.00000  12.00000  25.00000  47.00000  61.00000  47.00000  47.00000  6.000000  12.00000  22.00000  8.220000  5.470000  13.65000  0.210000
 Minimum  5.000000  9.000000  13.00000  29.00000  38.00000  36.00000  38.00000  3.000000  8.000000  14.00000 -13.96000 -5.030000 -13.38000  0.000000
 Std. Dev.  1.605313  0.698137  2.328426  5.670109  6.472821  2.397244  2.592153  1.119599  0.925669  1.889804  2.660704  2.287689  4.107974  0.066063
 Skewness  0.277641  0.595196  0.551582  0.793085  0.328900 -0.973833  0.426829  0.029229  0.689636  0.835958 -0.583089  0.091692 -0.393720  1.244840
 Kurtosis  2.312169  3.781213  3.230511  2.124737  2.025166  4.618202  1.928516  1.633563  3.056630  2.929905  8.210026  2.531121  4.460158  3.020288
 Jarque-Bera  3.907252  10.13663  6.350522  16.41012  6.915005  32.05990  9.384059  9.352834  9.527984  14.00110  142.5217  1.267386  13.76062  30.99457
 Probability  0.141759  0.006293  0.041783  0.000273  0.031508  0.000000  0.009168  0.009312  0.008531  0.000911  0.000000  0.530629  0.001028  0.000000
 Sum  980.0000  1200.000  2110.000  4184.000  5815.000  5116.000  5053.000  530.0000  1178.000  2081.000 -60.36000 -12.22000  128.1400  5.470000
 Sum Sq. Dev.  306.6667  58.00000  645.1667  3825.867  4985.792  683.8667  799.5917  149.1667  101.9667  424.9917  842.4423  622.7890  2008.178  0.519359
 Observations  120  120  120  120  120  120  120  120  120  120  120  120  120  120


1.3 Discussions

1.3.1 Estimating results

A regression analysis was conducted basing on the 3-factor model in which case independent variables were stock portfolios whereas the dependent variable was exposure to the broad market (MKT_RF). Results, in the below table, show a positive and significant intercept for all included portfolios at 0.0357 with a Standard Deviation of the dependent variable of 4.107. Based on exposure to the broad market (MKT_RF), beer, books hshld smoke, soda and toys reported negative results.



Dependent Variable: MKT_RF
Method: Least Squares
Date: 08/24/21   Time: 09:45
Sample (adjusted): 1 120
Included observations: 120 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 9.572105 16.81624 0.569218 0.5704
AGRIC 0.327559 0.652651 0.501890 0.6168
BEER -1.185971 1.121048 -1.057913 0.2924
BOOKS -0.546367 0.478997 -1.140648 0.2565
CLTHS 0.092115 0.377937 0.243730 0.8079
FOOD 0.207346 0.409123 0.506805 0.6133
FUN 0.239614 0.326451 0.733997 0.4645
HSHLD -0.118414 0.443185 -0.267189 0.7898
SMOKE -1.236483 2.165190 -0.571074 0.5691
SODA -0.081519 0.634317 -0.128514 0.8980
TOYS -0.111868 0.512729 -0.218182 0.8277
R-squared 0.035798     Mean dependent var 1.067833
Adjusted R-squared -0.052661     S.D. dependent var 4.107974
S.E. of regression 4.214752     Akaike info criterion 5.802248
Sum squared resid 1936.291     Schwarz criterion 6.057768
Log likelihood -337.1349     Hannan-Quinn criter. 5.906016
F-statistic 0.404679     Durbin-Watson stat 2.303012
Prob(F-statistic) 0.941888

1.4 Small Stock, Value Premium and Momentum Effects

There are many cross-sectional anomalies regarding stock returns (Harvey et al, 2014). Some preliminary evidence of value and momentum effects in stocks is shown in Tables 1 and 2. Table 1 presents average monthly excess rates-of-return (meaning the difference between the rate of return and the one-month Treasury bill rate) of UK exchange-listed stocks sorted into portfolios by size (market capitalization = share price multiplied by the number of shares outstanding) and by book-to-market ratio (book value of equity per share divided by share price). Within each of the 10 categories, the stocks are weighted by market capitalization to form a portfolio each month. The portfolio returns and the risk-free rate are from Kenneth French’s Data Library. Book-to-market is one measure of value: high book-to-market stocks have low prices compared to their book values and are considered value stocks, whereas low book-to-market stocks have high prices compared to their book values and are considered growth stocks (anticipated growth being one reason they might have high prices). The first row of the table presents average returns for the 20% of stocks having the lowest market capitalization (microcap stocks), and the bottom row presents average returns for the 20% of stocks having the highest market capitalization (mid-cap and large-cap stocks). Value beats growth by almost 1%per month for the smallest quintile and by 0.22% (22 basis points) per month for the largest quintile. Also, returns are generally decreasing as market capitalization increases, except for the quintile of growth stocks. The data shown in Table 1 are not evidence of an anomaly or puzzle, because no attempt is made in Table 1 to adjust for the different risks of different groups of stocks. Table 1 presents data parallel to that of Table 1 but sorting on market capitalization and momentum, measured as the cumulative return over the first 11 of the prior 12 months (that is, the prior year skipping the most recent month). These returns also come from Kenneth French’s data library. Momentum has an even stronger influence on returns than does value/growth. Within the smallest quintile of stocks, the average difference between the high-momentum and low-momentum portfolios is almost 1.4% per month. Small stocks that have been falling have historically been very poor investments, failing even to beat Treasury bills. The average difference between the high-momentum and low-momentum portfolios exceeds 0.6% per month even within the largest quintile of stocks. (Asness et al, 2013) document the pervasiveness of the following effects across asset classes: Value: assets that are cheaper in the sense of having a lower price compared to share earnings or Book per share value or in the sense of having suffered long-term price declines have higher returns that are not explained by standard measures of risk. Momentum: In future, assets that rose in price compared to other similar assets will continue to succeed in ways that have not been explained by typical risk assessments.

Table 1. Average Monthly Excess Returns (in %) of Size- and Book-to-Market-Sorted Portfolios, January of 2008 through December of 2017.

  Growth 2 3 4 Value
SMALL 0.11 0.75 0.81 0.98 1.10
2 0.43 0.71 0.88 0.93 0.98
3 0.47 0.77 0.78 0.84 1.08
4 0.60 0.60 0.72 0.82 0.86
BIG 0.45 0.60 0.52 0.58 0.67


Table 2.  Average Monthly Excess Returns (in %) of Size- and Momentum-Sorted Portfolios, January of 2008 through December of 2017.

  DOWN 2 3 4 UP
SMALL 0.02 0.62 0.84 0.99 1.35
2 0.12 0.63 0.80 0.98 1.19
3 0.28 0.60 0.70 0.80 1.14
4 0.19 0.60 0.69 0.80 0.98
BIG 0.13 0.52 0.41 0.58 0.74



2.1 Selected Stocks

2.1.1 Discussions and Results

The stocks selected cover a 5-year time beginning in 2015 and ending in 2019.

The stocks analysed include: AVL, BGSC, BP, DGE, IMB, JMI, PROV, PRU, SBDG, SSE, SVT, TNA, ULVR, VBR, 0P000131W0.L. Empirical evidence from UK and other countries in emerging and developed markets shows that the CAPM is gradually becoming weak in explaining how stocks perform in capital markets. The above views are echoed by Chan and Chui (1996) whose study conducted from 1971 to 1990 indicated a decline in the significant of beta in offering and explanation of stock performance within the UK.


Dependent Variable: C
Method: Least Squares
Date: 08/25/21   Time: 15:51
Sample (adjusted): 1 51
Included observations: 51 after adjustments
Stocks Coefficient Std. Error t-Statistic Prob.
AVL 0.000814 0.000534 1.523794 0.1361
BGSC 2.79E-05 0.001947 0.014350 0.9886
BP 0.000119 0.000342 0.348823 0.7292
DGE 0.000197 7.52E-05 2.619124 0.0127
IMB 2.71E-05 6.28E-05 0.431055 0.6689
JMI 0.000371 0.000357 1.040908 0.3047
PROV -0.010295 0.009663 -1.065360 0.2936
PRU 8.69E-05 0.000159 0.547898 0.5871
SBDG -0.145252 0.155029 -0.936931 0.3549
SSE 0.000532 0.000161 3.313328 0.0021
SVT 5.87E-05 0.000109 0.536407 0.5949
TNA -0.003057 0.001210 -2.526166 0.0159
ULVR -9.43E-05 4.07E-05 -2.313509 0.0264
VBR -0.002819 0.001692 -1.665755 0.1042
0P000131W0.L -0.001819 0.001092 -1.365755 0.0461
Mean dependent var 1.000000     S.D. dependent var 0.000000
S.E. of regression 0.039661     Akaike info criterion -3.388763
Sum squared resid 0.058202     Schwarz criterion -2.858458
Log likelihood 100.4135     Hannan-Quinn criter. -3.186118
Durbin-Watson stat 1.370560

2.1.2. Econometric issues related to estimations

The analysis can make the following econometric conclusions. First, throughout the period the Durbin Watson (DW) statistic was 1.37 from 2015 to 2019 which is autocorrelation in the residuals from the statistical model or regression analysis. This conclusion shows that the variability of portfolio returns can be explained by its betas (coefficient) on average. Secondly, across time, the variability was far higher. That conclusion is due to the assumption that the return variability for high-beta portfolios (for example, portfolios with one beta) is bigger than the return variance portfolios of low-beta (zero-beta portfolios).


2.1.3 Summary and Discussion

The results show a significant difference from zero in certain periods. This can show that the risk/return ratio was not linear during these times. The return on a portfolio, however, can fluctuate from year to year. The t-statistic is way above 0.05 for some stocks Where for α = 0.05 is not statistically significant. Our empirical results therefore sustain that the risk-return relation is linear. Therefore, we conclude that expected return is a linear function of beta. (Blume and Friend, 1973). The findings of this study indicate that the Sharpe-Lintner CAPM is a valid description of the risk-return relation. The conflicts in the results between our study and previous studies may be due to the differences in the statistical techniques or the market portfolio proxies employed. The results (Hawawini, 1983) indicate that tiny market cap securities could seem less hazardous when assessing betas over return ranges of arbitrary length, whereas otherwise small market cap securities with a relatively large market cap appear to be less risky.

2.1.4 Beta Estimation

Each beta is calculated in accordance with the conventional market model by the OLS method. A varied beta estimate is obtained, depending on the time interval employed (daily, monthly). Moreover, the beta of each portfolio is computed as the average of the beta of the component securities.


2.1.5 Empirical Results

In this section the empirical results for intervalling-effect bias are reported in beta estimates using the ordinary Least squares (OLS) procedure. For the calculation of returns, two distinct intervals (daily and monthly) are used. The sample is from 2015 to 2019 and is computed by 1,200, 240 and 50, respectively for each stock during this period of study, in daily and monthly results. The following table 1 provides summary information regarding a beta estimate for each of the three reference time intervals for the three portfolios under consideration.


Summary Statistics of Beta of the three Portfolios with regard to Composite UK Index

  Large Stocks Medium Stocks Small Stocks
  Daily Monthly Daily Monthly Daily Monthly
Mean Beta 0.1445 0.9540 0.2834 0.6831 0.0656 0.3260
Stdev 0.0672 0.3793 0.1805 0.2895 0.2114 0.4628
Max Beta 0.2680 0.4257 0.4240 0.1020 0.2002 0.6687
Min Beta 0.0208 0.3042 1.2250 0.1258 0.3654 0.1024
Range 0.2640 0.2016 0.3040 0.3487 0.3859 0.0029
Skewness -0.5280 0.0847 0.2012 0.2604 0.2198 0.8004
Kurtosis 0.0625 0.1259 1.0587 0.0862 0.0268 0.2458
Std error B 0.0451 1.9874 0.1254 0.4620 0.2468 0.0864
Mean 2 R 0.0128 0.3254 0.2280 0.2964 1.0014 1.0298


2.1.6 Conclusion

This study empirically examines the intervalling effect bias of the OLS beta estimate within the context of the stock market in the UK. In the five years sample period from January 2015 to December 2019, the UK stock market is analysed. During this sample period it has been shown that the intensity of the intervalling effect bias was highly pronounced for all features of the phenomenon reported by similar research and compared with the results of this paper. One additional speculative observation would be that, if the model of the UK stock market is true, the intensity of the intervalling effect impact would appear to represent the depth of market inefficiency for some time. If so, then certain functions that reflect this intensity can be designed to measure market inefficiency. In all situations the evolution of the mean beta when we go from daily to monthly data has not been improved at all, save from some slightly improved explanatory power of the proxies in daily and monthly cases. Even in this situation, the intervallling effect was therefore still intense, which enables us to infer that the impact of the changing economy rather obscured any improvements that could be envisaged in the price adjustment process. (Ho & Tsay, 2001) gave delivered proof that listing option  decreases the intervalling effect, thus supporting the view that option dealing has some fast-tracking effects in the process of adjusting  price.



3.1 Strengths associated with models of panel

3.1.1 Introduction


Panel data is used in describing observations of individuals concerning different cross-sections over a given period of time (Ahn et al., 2013). Panel data is subset of longitudinal data. Observations made within panel data are composed of two main dimensions. One dimension is the cross-sectional, denoted by I while the second is a statistic dimension denoted by t.

3.1.2 Strengths


There is  higher levels of accuracy within the model associated with panel data. Additionally, panel data exhibit higher variability in comparison to cross-sectional data considered as panel with T=1 or statistic data considered as a panel with N=1, thereby improving econometric estimates ‘efficiency. Panel data is also beneficial because it represents human behavior complexity in a more accurate manner relative to statistic data or cross-sectional data. For instance, panel data can accurately develop and test hypotheses regarding complex human behavior (Islam, 1995).  Panel data is also valuable in as it can be used to assess the effectiveness of social programs as well as providing a comprehension of dynamic relationships. Rather making use of data gathered from an individual, to predict individual behavior and outcomes, panel data pools together information thereby yielding predictions with an enhanced accuracy level. Other benefits of panel data include setting a basis for analyzing aggregate data on a micro-scale level. Nevertheless, when carrying an analysis of micro units that are heterogeneous in nature, data that is aggregated has significantly different statistical properties compared to the non-aggregated data. Additionally, panel data reduces complexity of computation and coming up with inferences in cases such as analyzing measurement errors and non-stationary statistics. According to Zhou (1994) errors in measurement make it difficult to identify econometric models. For instance, the truncating of Tobit models over a period of time, results in the loss of the realized value.


3.2 Hausman’s test of Estimations of Panel data model


The Hausman test is conducted to find out the effective method between a specified fixed effect and a result that is random. The Hausman test is conducted following the reaching of post-chow phase and consequently, results are needed to make a decision regarding a particular effect. Tests are conducted sequentially beginning with the analysis of a set effect. P-value is evaluated whereby if the value is 0.000, evidently lower than 0.05, then the alternative estimates stands out. The Hausman test entails comparing null estimates with alternative estimates whereby one of them is consistent and aligned with zero estimates. A main difference between the two supports the chosen hypothesis (Bai and Ng, 2002). An endogeneity test is also useful in comparing coefficient values of the estimates. This section entails conducting he Hausman test with the incorporation of an auxiliary regression.

3.2.1 Conclusions-Hausman test


Following the conducting of Hausman’s tests and associated evaluations, various conclusions are made. Firstly, the tactic selected will result in a random effect as long as the Hausam test supports H0 or p value that is above 0.05. Thereafter, a lagrangrian multiplier test is carried out to find if the common effect or random effect persists. The tactic adopted in this case is fixed efficiency especially when the test results in H1 or 0 to 0.05 p-values.

3.2.2 Summary  of Hausman’s test:


If Result: H0: Select RE if (p> 0.05) – Random effects H1: Select FE if (p <0.05) – Fixed effects

3.2.3 Fixed Effect Model (FE)


The fixed effect model is based on the assumption that individual differences can be accommodated by different intercepts. The fixed effect model makes use of different intercept data with a variable dummy technique to determine variations in management, the culture of incentives and work (Bai and Ng, 2002). This model is analysed to find out differences between companies intercepting. Nevertheless, different companies’ intercept is the same.

3.2.4 Random Effect Model (RE)


The random effect model is useful in estimating panel data in a situation where there is connection of interference variables between times and individuals.  Each company’s error terms within random effect model accommodate the interception difference. Likewise, the heteroscedasticity removal advantage is contained within the model

3.3 Macroeconomic factors

The analysis entailed a study of 10 portfolios of industries between June 2010 and May 2020. The industry examined includes food, agriculture, soda, fun, books, toys, smoke, household clothing and beer. The farma-french model analyses stock returns using three variables including differences in the sizes of portfolios of stocks; different returns based on differences in the value of stocks and the extra returns from the market portfolio, also denoted as market premium (Bai, 2003). In the model, HML pitches and book-to-market stocks signify turbulence in the market. Based on the analysis; weak companies which have small but consistently growing earnings, report a positive HML path and a higher BE/ME. On the contrary, strong companies whose earnings are consistently exhibit a low BE/ME and an HML path which is negative. The SMB variable supports the dimensions effect. Likewise, based on the 3-factor model, small companies are associated with lower earnings on their assets relative to large companies as long as the book-to-market ratio remains constant. (Carhart, 1997) further expanded the 3-factor model to include a momentum as one of the determinants of stock returns.

Empirical analysis shows that the model is credible in explaining stock returns in mature markets. Following the modification of the Fama-French, model the authors proposed the inclusion of a floating ratio a measure for the governance of the company. Market premium in this case stands for an additional returns generated by a portfolio of broad market. The variation between returns of small-sized portfolio and returns of large portfolios (SMB) signifies a variation between portfolios of large stocks from the small stock portfolio. According to Pesaran et al. (1999) another factor within the model is a variation of income high book to market portfolio from low book to market portfolio (HML). The SMB variable supports the scale effect. According to the Fama-French model, small businesses are associated with lower earnings when compared with large companies as long as the book to market equity is kept constant.