The improvement of the stock selection process for
portfolios with Blume’s beta adjustment technique:
A comparison between the market model and the Fama and French three factor modelStudent: Chiel Jansen of Lorkeers S2056844 Supervisor: Dr. A. Plantinga
Study programme: Msc. Finance
Abstract
Blume’s adjusting technique is a well-known method to adjust betas to increase their forecasting power. The adjustment technique is mostly performed on U.S stocks and only on single index models. This paper uses Blume’s adjustment technique to estimate future betas of stocks traded on the Japanese stock market for the Fama-French model and the market model over the period 1980-2010. I show that the adjustment technique has a beneficial effect for the market model and an opposite effect for the Fama-French model.
JEL Classification: C20, G11, G17
1 1. Introduction
To derive the efficient frontier given a set of stocks, one needs to know the minimum variance portfolio and the tangency portfolio. The minimum variance and the tangency portfolio require several inputs. The minimum variance portfolio is determined by the covariance matrix of the stocks, while the tangency portfolio also requires the expected excess returns of the stocks. Although the number of necessary input seems small, calculations to obtain the covariance matrix increases substantially with the number of stocks in the portfolio. For example to calculate the minimum variance portfolio of only ten stocks, one needs 45 covariance coefficients, a portfolio of 100 stocks requires already 4950 covariance coefficients. The total number of estimated parameters for the covariance matrix is equal to 𝑁(𝑁−1)
2 where 𝑁 is the total number of stocks. In order to decrease the necessary amount of covariance coefficients a few methods are proposed which greatly decreases the number of parameters.
Elton et al. (1978) investigates a few methods to simplify the estimations for the correlation and covariance matrices. They test the historical correlation matrix, the correlation matrix based on the betas of a single index period, the correlation matrix based on Blume’s adjustment technique (Blume, 1971) and finally a correlation matrix based on Vasicek’s adjustment technique (Vasicek, 1973). The historical method is the most demanding method in terms of input data and computational power, yet according to Elton et al. the covariance matrix based on historical covariances between stocks yields the lowest forecasting power. All other methods yields a better forecasted covariance matrix and even requires less data. I build further on the findings in the paper of Elton, Gruber and Urich by comparing the forecasting power of two different models together with Blume’s adjusting technique.
2 model also have a mean-reversing tendency.
The mean of the betas corresponding with the size and value factor of the Fama-French model should be around zero, since both the HML and SMB factors are zero investment portfolios with a long and short position in the stocks. Especially the size factor should have a mean of approximately zero, since the median of the market capitalization of the firms is used to classify ‘big’ and ‘small’ firms. Given the negative relationship between size and return, it can be expected that the SMB factor also has a mean-reversing property. Small firms tend to generate higher returns than big firms. This should imply that small firms grow faster than large firms and this results in a lower premium for the small firm over time. This in turn should lower the beta of the size index. The opposite should be true for firms with a high market capitalization, due to the lower returns the beta of the big firms will increase over time towards the mean of zero. For the HML factor the expected properties are less open for interpretation, since the construction of the factor is not based on the difference between the stocks above and beneath the median. The book-to-market ratio should be more stable over time and should display a less mean-reversing character in comparison with the SMB factor. The HML factor seeks the difference in return between value firms and growth firms. A value firm has a low book-to-market ratio and requires a lower return than high book-to-market firms. The latter group are growth firms which require a high return. Theory depicts that eventually the growth opportunities reduces and that therefore the growth firms will gradually change in value firms. Based on this reasoning a mean-reversing characteristic can be expected, yet the reasoning is not necessary applicable on the value firms.
This paper contributes the existing literature in two different manners. First additional research for the market index is provided and second the regression technique is tested for the Fama-French model which might provide new insights about the forecasting abilities of the Fama-French model.
The results suggests that adjusting the betas increases the forecasting power for the market model, but decreases the forecasting power of the Fama-French model. This suggests that the betas of the size and value components in the Fama-French three factor model are not stable over time and therefore cannot be reliable estimated by linear regressions based on past data. Grouping them together yields a beneficial effect on the beta adjustment technique. Finally the market model is preferred above the Fama-French model for forecasting betas and minimum variance portfolios.
3 2. Theory
2.1 Market model
The market model assumes that returns are solely explained by a single market index. The returns of a stock and the market are related through a linear relationship and the sensitivity between the market index and the individual stock or portfolio of stocks determines the return. Sharpe (1964) introduces the following model:
𝑅𝑖 = 𝛼𝑖+ 𝛽𝑖𝑅𝑚+ 𝜀𝑖 (1)
Where 𝑅𝑖is the return of the stock, 𝑅𝑚 is the return of the benchmark, 𝛼𝑖 and 𝛽𝑖 are regression coefficient and 𝜀𝑖 is the error term.
The beta in Eq. (1) is equal to the covariance of the returns between the stock and the market index divided by the variance of the index returns. For stocks analyzed with the market model, the covariance matrix is estimated with the betas of the stocks and the variance of the market. The covariance between two stocks is:
𝜎𝑖,𝑗= 𝛽𝑖𝛽𝑗𝜎𝑀2 + 𝛽
𝑖 𝐸(𝜀𝑀𝜀𝑖) + 𝛽𝑗 𝐸(𝜀𝑀𝜀𝑗) + 𝐸(𝜀𝑖𝜀𝑗) (2) Where 𝜎𝑀2 is the variance of the market index and E() is the expected value operator. The formula is obtained from Elton and Gruber (1973). By employing a single least square regression the second and third term on the right-hand side are zero by construction, since the expected error terms between the dependent and independent terms are zero. The last term on the right-hand side is not expected to be equal to zero due to some reasons which are discussed below.
The market model assumes that the error term is zero due to construction and the assumption that all covariances are only subjected to co-movements between the stocks and the index. The error term allows that not all co-movements between stocks is explainable by only the co-movements with the market index but that there are also other indices that can explain a part of the covariance between the stocks. The error term captures the part of the covariance which is explained by other indices than the market index, for example industry effects, and is therefore most likely unequal to zero.
4
2.2 Multi Index Model
The market model assumes that the entire variation in stock returns is driven by the variation of a single market index. This is a strong assumption, and is quite often not true. Multiple studies show more factors, or indices, that explains stock returns for firms listed on the Japanese stock market (Lau et al., 1974 and Chan et al. 1991)
The most popular multi-index model is proposed by Fama and French (1992, 1993, 1996), who argue in a series of papers that there are anomalies in the returns that cannot be explained by the market model. They find two other risk measures that explains the excess returns of stocks over the risk-free return. Fama and French (1996) presents the asset pricing model based on multiple indices.
𝐸(𝑅𝑖) − 𝑅𝑓 = 𝛼𝑖+ 𝛽𝑖(𝑅𝑚) + 𝛽𝑆𝑀𝐵,𝑖𝑆𝑀𝐵 + 𝛽𝐻𝑀𝐿,𝑖𝐻𝑀𝐿 + 𝜀𝑖 (3)
where is 𝐸(𝑅𝑖) the expected return of the stock, 𝑅𝑓 the risk-free return, HML is the value factor calculated as the difference in return of high book-to-market firms and low book-to-markets firms, SMB is the size factor and is calculated as the difference in returns between firms with a low market capitalization and firms with a high market capitalization and 𝛼, the betas and 𝜀𝑖 are regression coefficients. The betas are again equal to the covariance of the excess returns of the stocks and their respective index returns divided by the variance of the index returns. Estimating the covariance matrix for multi-index models is similar to Eq. (2) with the addition of the two additional indices.
𝜎𝑖,𝑗= 𝛽𝑖𝛽𝑗𝜎𝑀2 + 𝛽
𝑖,𝑠𝑚𝑏𝛽𝑗,𝑠𝑚𝑏𝜎𝑠𝑚𝑏2 + 𝛽𝑖,ℎ𝑚𝑙𝛽𝑗,ℎ𝑚𝑙𝜎ℎ𝑚𝑙2 + 𝐸(𝜀𝑖𝜀𝑗) (4) The covariance calculations are equal to the method used by the single index model with the difference that the covariance of the other two beta factors, 𝛽𝑆𝑀𝐵 and 𝛽𝐻𝑀𝐿 is added. The cross products in error terms with the corresponding betas are not included since they are zero by construction due to the simple linear regression technique used to explain the excess returns in the Fama-French model. The expected products of the residuals are included in line with the same reasoning as the market model.
2.3 Literature review
5 time frame of the paper from Lau only includes five years, it shows that the CAPM model can be used to explain returns of (portfolios of) stocks. The CAPM model and the market model are nearly similar to each other and therefore I expect that the results for the CAPM model are also applicable for het market model.
Chan et al. (1991) find that size, based on the market capitalization, and book-to-market ratio of a firm both have an impact on the stock returns. They find that especially the book-to-market ratio has a strong relationship with the expected stock returns. Furthermore they find that size also has an impact on the stock returns, similar to Fama and French in their papers. Size has a negative impact on the cross-sectional returns of stocks. This implies that indeed small firms tend to outperform larger firms, which is in line with the SMB factor in the Fama-French three factor model. Daniel et al. (2001) find for the Japanese stock market there is a value premium and they cannot reject the Fama-French model, although they suggest using a characteristic model instead of the Fama-French three factor model. Based on these two papers it is assumed that the Fama-French three factor model can be used to explain stock returns for the Japanese market.
Both models strongly rely on the betas and their underlying properties. First to calculate the minimum variance portfolio for the next period both models assume that the beta obtained from the equation(s) are unbiased estimators for the systematic risk in the forecasted portfolio. So the process of constructing a portfolio assumes that the historical beta is a reasonable accurate prediction for the beta in the estimation period. This follows that the betas should be stationary over time, or at least reasonable predictable based on historical information in order to forecast the betas. Blume (1971) uses a regression technique to estimate betas, which will be described later.
Unfortunately there is some debate about the stationary of betas over time. Fabozzi and Francis (1978) rejects the method used by Blume and suggest that historical betas cannot be used as an estimator for future betas, since betas move randomly through time. They underpin their findings with a quantitative analysis. They conclude that a ‘significant minority’ of stocks traded on the NYSE have a beta with a random coefficient. Gordon and Benson (1982) believe that Fabozzi and Francis overstate the effect of betas as random coefficients.
6 therefore the findings might not be representable for this model.
Betas obtained from any estimation technique are subjected to estimation errors and this results in a sub-optimal stock selection since the estimated covariance coefficients are calculated with the betas obtained from the estimation techniques. To construct an optimal portfolio the true or fundamental beta of a stock have to be known or calculated. Beaver et al. (1970) try to make a link between the beta of a company and seven firm specific variables, while Thompson (1978) links the fundamental beta of a company to 43 variables and Rosenberg and Marathe (1975) even use 101 variables to explain the beta of a company. Rosenberg en Marathe use different groups of variables based on market variability and earnings variability. Although this provides a better estimate of the true beta of a firm it has some serious drawbacks. First the amount of data necessary to compute the beta of a company is tremendous. Over 100 variables are necessary to calculate the beta of a single firm and it is not uncommon that not all data is available which leads to a reduced estimate of the beta. Therefore the estimated beta obtained from return series are a good alternative since this method greatly reduces the amount of necessary input.
Blume (1971) investigates a few assumption underlying the market model. He primarily focuses on the beta and the properties of this risk measure. He finds that the beta is indeed a risk measure by two different approaches, a portfolio approach and the equilibrium approach. Blume shows by using the following equation that the beta is indeed a risk measure.
𝑉𝑎𝑟(𝑅𝑝) = 𝛽𝑝2 𝑉𝑎𝑟(𝑅𝑚) +𝑉𝑎𝑟(𝜀)𝑁 (5)
Where Rp and βp are respectively the return and the systematic risk of the portfolio, 𝑅𝑚 is the market risk premium, ε the error term of the market model, and N is the number of stocks in the portfolio. It is clear that an increase in N will decrease the second part of the equation to zero. Evans and Archer (1968) show that with a portfolio of ten stocks the second part of the equation is approximately zero, therefore the variance of the portfolio return is, in a well-diversified portfolio, only subjected to the first part of the equation. Since the variance of the market risk premium is constant, the beta should be a measure of risk, because it is the only variable in this equation to increase or decrease the variance of the returns. In the equilibrium approach Blume (1971) shows that the risk premium of an individual stock is proportional to the market risk premium. The proportion is equal to the beta of the individual stock.
7 papers by Blume (1971, 1975, 1979) he shows that betas of a portfolio with only a few stocks are less stationary than the beta of a portfolio with more stocks. Blume provides an explanation for the increased stationary of the betas. He states that betas are subjected to estimation errors and that on average these estimation errors, either positive or negative, tend to cancel each other out in a portfolio with more betas. Furthermore it appears that the betas have a tendency to converge to one. Since the market beta is assumed to be one, the convergence is explained by a mean-reversing property of betas. This characteristic can be used for predicting future betas or adjust the betas which are observed in the market. Blume proposes a scheme to adjust and predict future betas by linear regressing the betas of a stock or portfolios of stocks over time. Blume uses the following linear regression
𝛽𝑡 = 𝑎 + 𝑏 ∗ 𝛽𝑡−1+ 𝜀𝑖 (6)
Where βt is the forecasted beta, βt-1 the beta a period prior of the forecasted period, a and b are regression coefficients and ε is the residual term with E(ε) is equal to 0 and ε ̴N(0,σ2), which will be
diversified away in a well-diversified portfolio. In general the values for a and b in eq. (6) is set equal to 0.343 and 0.677. In his paper Blume acknowledges that the values of a and b are not constant and can deviate between time and markets. The values given for a and b results in some interesting properties of the beta.
Extreme betas, either high or low, will converge stronger to the mean of one than betas which are less extreme. Blume (1975) provides two possible explanations for this phenomenon. First the risk of existing projects tend to become less extreme over time and the second explanation is that new projects of firms have a less extreme risk and therefore the risk of the company will be less extreme. Vasicek (1973) states that the betas of stocks listed on the NYSE concentrates between the values 0.5 and 1.5. Although this range may not be representable for the Japanese stock market it does give a certain range about the value of betas. Vasicek further argues, that based on probabilities, betas beneath this range are most likely underestimated and betas above this range are most likely overestimated and therefore they require a higher adjustment.
8 mean-reversion property as suggested by Blume. Gangemi et al. (1991) find that for the Japanese stock market the betas on an aggregated scale show a mean-reversing tendency over the period 1975-1994. Klemkosky and Martin (1975) also find that constructing portfolios increases the estimating power of betas and that the adjustment proposed by Blume indeed decreases the estimation errors.
Following the described theory I form the following hypotheses to test in this paper. The first hypothesis concerns the forecasting technique by Blume. The second hypothesis challenges the accuracy of the adjusted betas versus the historical betas and the final hypothesis relates to the construction of minimum variance portfolios for the adjusted betas and the historical betas. All hypothesis are applicable for both the market model and the Fama-French model and uses a single stock approach and an approach with portfolios of ten stocks. The hypothesis are as follows:
Hypothesis 1: The linear regression technique of Blume provides stable regressions coefficients to estimate future betas.
Hypothesis 2: Adjusting the betas with Blume’s regression technique enhances the accuracy of betas to predict future betas.
Hypothesis 3: Forecasted portfolios based on adjusted betas will provide a better estimate for the actual minimum variance portfolio of a given period in comparison with the portfolio formed with the use of historical betas.
9 3. Data
The data sample consist of 1996 Japanese stocks of the period January 1981 until December 2010. The stocks are listed during the entire period, got listed during the period or were eventually delisted due to mergers, defaults or other reasons. The aim of including the stocks which delisted during the period is to avoid the survivorship bias (Brown et al., 1992). I use monthly data of the return index of the firms and annual data for the market capitalization and the book to market ratio of the firms. Furthermore the monthly return on the Japanese government bond yield return index is used as a proxy of the risk free rate for that period. The market index is a self-constructed value weighted index. The number of stocks in the sample fluctuates over time, new firms enter the stock market while other firms leave the stock market through delisting. The total number of stocks per period and model differs due to available information and the performance of the stocks. All data is obtained from Thompson Datastream.
Table 1: Descriptive statistics of the betas per period for the market model.
Period 1981-1985 1986-1990 1991-1995 1996-2000 2001-2005 2006-2010 Mean 0.513 0.930 1.078 0.893 0.922 1.032 Median 0.483 0.931 1.098 0.874 0.847 1.011 Maximum 2.294 1.913 2.346 3.333 3.472 2.925 Minimum -0.865 -0.189 -0.092 -0.172 -0.246 -0.626 Std. Dev. 0.512 0.331 0.396 0.478 0.517 0.542 Skewness 0.260 -0.149 -0.217 0.533 0.723 0.318 Kurtosis 2.689 2.855 2.935 3.882 3.708 2.812 Observations 832 893 1,582 1,731 1,758 1,596
10 negative betas are rare. The percentage of non-extreme betas, that is betas with a value between 0.5 and 1.5 as proposed by Vasicek (1973) deviates between 60 and 85 percent of the total samples in each period. An exception is the first period where only 45% of the betas are considered non-extreme.
The literature describes that size has a negative impact on the return of a stock. The data here provides no inconclusive evidence for this relationship. The beta and the firm size, measured as the market capitalization, are positively related for the first two periods and for the fore last period. The other periods show a negative relation between size and beta. The positive relationship between beta size is especially strong in the first period, while in the other periods with a positive relationship the slope of the regression is slightly positive.
Table 2: Descriptive statistics of the betas per period for the Fama-French three factor model. MRP is the beta corresponding with the market risk premium, SMB the beta of the size index and HML the beta of the book-to-market index.
Period 1981-1985 1986-1990 1991-1995 MRP SMB HML MRP SMB HML MRP SMB HML Mean 0.515 -0.023 -0.334 0.933 0.468 -0.400 1.077 1.113 -2.158 Median 0.483 0.034 -0.332 0.934 0.666 -0.397 1.098 1.079 -2.155 Maximum 2.30 3.416 2.396 1.912 2.291 1.700 2.224 4.333 1.762 Minimum -0.848 -3.376 -2.895 -0.183 -2.258 -2.596 -0.083 -0.840 -5.851 Std. Dev. 0.510 1.009 0.756 0.329 0.874 0.699 0.394 0.797 1.150 Skewness 0.261 -0.359 -0.061 -0.149 -0.829 0.014 -0.232 0.338 -0.030 Kurtosis 2.690 3.384 3.338 2.859 3.371 2.875 2.898 2.819 2.891 Observations 832 832 832 894 894 894 1,581 1,581 1,581 1996-2000 2001-2005 2006-2010 MRP SMB HML MRP SMB HML MRP SMB HML Mean 0.893 1.413 0.128 0.921 0.643 -0.602 1.068 1.289 0.027 Median 0.874 1.310 0.251 0.847 0.488 -0.483 1.047 1.141 0.057 Maximum 3.332 6.257 1.743 3.472 6.037 1.645 3.024 8.560 2.541 Minimum -0.172 -1.258 -5.375 -0.247 -1.921 -4.651 -0.644 -1.349 -3.000 Std. Dev. 0.478 1.101 0.667 0.517 1.014 0.712 0.560 1.074 0.585 Skewness 0.533 0.547 -1.41 0.723 0.996 -1.141 0.320 0.986 -0.283 Kurtosis 3.882 3.263 7.950 3.708 4.682 5.554 2.813 5.158 4.767 Observations 1,731 1,731 1,731 1,758 1,758 1,758 1,596 1,596 1,596
11 factor in the period 2006-2010 all other SMB and HML factors show a mean different than zero. Noteworthy is the highly fluctuating mean of especially the high-minus-low factor and in a lesser degree the fluctuations of the mean of the small-minus-big factor.
Grouping the stocks in portfolios of ten-stocks based on the betas corresponding with the market risk premium does not alter the characteristics of the data sets, except for the normality of the data per period. Grouping the stocks has a beneficial effect on the normality. By grouping the stocks the extreme betas are compensated by less extreme betas and therefore the distribution have less extreme outliers which enhances the normal distribution of the dataset.
The strong difference between the first period and the other periods, especially for the market model, supports the assumption that the first period lacks a good data set. As a consequence the emphasis for drawing conclusions is on the second part of the results. The first period will have an impact until the third period for the adjusted beta forecast due to the regression technique of Blume. Therefore the emphasis of the results of the adjusted beta will be primarily based on the last three periods while for the historical betas the results will be primarily based on all but the first forecasted period.
4. Methodology
The log returns are calculated with the following equation: 𝑅𝑖 = ln ( 𝑆𝑡
𝑆𝑡−1) (7)
where 𝑅𝑖is the return of stock i, 𝑆𝑡 is the return index value of stock i on time t and 𝑆𝑡−1 is the return index value of stock i on time t-1. The value-weighted market index is constructed as the weighted returns of the stocks based on their market capitalization. Only the period prior to the estimation period is used to estimate the expected return, standard deviation and other characteristics of the stocks and the market. Therefore, the expected return of stock i is equal to the average return of stock i in the previous period. A stock is included when over the period there are at least 57 returns observed of the in total 60 observations. This ensures that the average return, the standard deviation and the beta of a stock is reasonable estimated. Finally there is a check if stocks are at least listed for two consecutive periods of five years in order to estimate the linear regression technique of Blume.
4.1 market model
12
𝑅𝑖 = 𝛼𝑖+ 𝛽𝑖𝑅𝑚+ 𝜀𝑖 (8)
It is common in the literature that the beta is estimated with monthly data and Bartholdy and Peare (2005) find in their paper that an estimation period of five years with monthly data is recommended to estimate the beta of a stock. The alpha and beta are estimated by solving Eq. (8).
After both the beta and the alpha are known the standard deviation of the residuals is estimated. This is necessary since the standard deviation of the error terms explain a apart of the (co)variances of stocks and are therefore required to estimate the covariance matrices. The error term is calculated using the following equation:
𝜀𝑖,𝑡= 𝑅𝑖,𝑡− (𝛼̂𝑖+ 𝛽̂𝑖𝑅𝑚,𝑡) (9)
Which is simply the true return minus the estimated return by the market model. The variance and the standard deviation of the error terms are calculated by taking the variance or standard deviation of the error terms. Both Eq. (8) and Eq. (9) are obtained from Elton et al. (2011).
4.2 Fama French three factor model
The construction of the Fama-French factors follows the procedure of Fama and French (1996). The size factor uses the median of the market capitalization at June of each year. The month of which the market capitalization is chosen is arbitrary and depends on the end of the fiscal year of listed firms in a specific country. Since April is the end of the fiscal year for listed firms in Japan, the month June can be used to obtain the market capitalization of the firms. The additional two months gives an margin of error for the data to get updated. The firms with a market capitalization beneath the median value are labeled as small firms and firms with a market capitalization above the median value are defined as big firms.
13 stocks in the high book-to-market portfolios. Once the factors are known, the betas and the alpha of each stock are calculated in a similar fashion as the beta and the alpha for the market model. A simple addition to Eq. (9) is used to calculate the error terms of the Fama-French three factor model.
For the estimations of Blume’s regression technique, the evaluation of this regression technique and for the construction of the minimum variance portfolio I use single stocks and portfolios of ten stocks. The portfolio of stocks is constructed as an equally weighted portfolio based on the betas corresponding with the market index. The ten highest betas are grouped in the first portfolio, the second ten highest betas are grouped in the second portfolio and so further.
Once the betas are known, the weights for the constant and the coefficient of Blume’s adjustment method is estimated by Eq. (6). By using the regression the betas of the first period will be adjusted according to the values of the constant and the coefficients for each period. This implies that the regression found between the first two periods are used to adjust the betas of the second period to use as forecast for the third period. After calculating both the historical and the adjusted betas they are compared based on the accuracy of these betas with the actual beta in the forecasted period. The comparison is made by comparing the mean square error. Blume (1971) and Klemkosky and Martin (1975) use the mean square error as a tool to compare estimated betas. The mean square error is calculated as follows
𝑀𝑆𝐸 =1
𝑁∑(𝛽𝐹− 𝛽𝐴)2 (10)
where 𝛽𝐹 is the beta used in the forecast and 𝛽𝐴 is the actual beta and N is the number of observations included in the sample.
Finally minimum variance portfolios are constructed based on the historical betas, the adjusted betas and the actual betas for each period. Only stocks with an expected positive return are considered for the portfolio. Recall that the expected return of a stock is the average return of the stock in the prior period. Since both the market model and the Fama-French three factor models are equilibrium models they assume that the investors have homogeneous expectations about the performance of stocks. So when past performance is indeed an indicator of the future performance of a stock, the expectation is that the stock will again have a similar return over the forecasted period. Therefore it is not advantageous to include stocks with a negative expected return in the portfolio when you also aim for a positive return of the portfolio.
14 residual risk and I the identity matrix. Furthermore let ‘ be the notation of a transposed matrix and −𝟏 the notation of an inverse matrix. The covariance matrix of the market model is then calculated as follows:
𝝎 = (𝜷′∗ 𝜷) ∗ 𝑉𝑎𝑟(𝑅
𝒎) + (𝝈𝜺′ ∗ 𝝈𝒆) ∗ 𝑰 (11)
The covariance matrix of the Fama-French three factor model is calculated with the following equation: 𝝎𝑭𝑭 = (𝜷′∗ 𝜷) ∗ 𝑉𝑎𝑟(𝑅𝒎) + (𝜷𝒔𝒎𝒃′∗ 𝜷𝒔𝒎𝒃) ∗ 𝑉𝑎𝑟(𝑅𝒔𝒎𝒃) + (𝜷𝒉𝒎𝒍′∗ 𝜷𝒉𝒎𝒍) ∗ 𝑉𝑎𝑟(𝑅𝒉𝒎𝒍) +
(𝝈𝜺′ ∗ 𝝈𝒆) ∗ 𝑰 (12)
The following two equations determine the composition of the minimum variance portfolio and the corresponding weights for each stock in the portfolio.
𝒁 = 𝝎−𝟏∗ 𝝉 (13)
𝑾 =∑ 𝒁𝒁 (14)
where Z is a score for each stock to minimize the variance of the portfolio and 𝝉 is a column vector ones. The corresponding weights for each stock is calculated by Eq. (14). By dividing the score of an individual stock by the sum of all scores the weight of a stock within the minimum variance portfolio, W, is obtained. Based on the obtained weights, the expected return and the variance of the portfolio are calculated with the following two equations:
𝐸[𝑅] = 𝑾 ∗ 𝑹 (15)
𝜎𝑝2= 𝑾′∗ 𝝎 ∗ 𝑾 (16)
Where R is the row vector with expected returns. The standard deviation is calculated by taking the square root of the outcome of Eq. (16).
The weights and their corresponding expected return and standard deviation for the tangency portfolio can be calculated with the same series of equations. To acquire the weights for the tangency portfolio simply replace the column vector of ones with a column vector with the expected excess returns of each stock.
15 5. Results
The results are discussed in the following order: First the Blume regressions are discussed, then the accuracy of the historical and Blume adjusted betas are presented and finally the performance of the constructed portfolios, based on historical and adjusted betas are discussed. The findings of the market model are first presented and then the results of the Fama-French three factor model in the similar order. Furthermore as argued in the data section, the emphasis of the results will be based on the latter periods due to the incomplete dataset for the first period.
5.1 Market model
Table 3: Regression of the beta coefficients between two consecutive periods to obtain the weights for the Blume adjustment method. The regressions are based on individual stocks and the 10-stock portfolios. All regression coefficients are significant at the 1%-level.
Regression tendency implied between periods
𝜷𝒕= 𝒂 + 𝒃 ∗ 𝜷𝒕−𝟏 Individual stock 𝜷𝒕= 𝒂 + 𝒃 ∗ 𝜷𝒕−𝟏 10-stock portfolio 1981-1985 – 1986/1990 𝛽𝑡= 0.9895 − 0.0945 ∗ 𝛽𝑡−1 𝛽𝑡 = 0.9945 − 0.1025 ∗ 𝛽𝑡−1 1986/1990 – 1991/1995 𝛽𝑡= 0.9703 + 0.2276 ∗ 𝛽𝑡−1 𝛽𝑡 = 0.9741 + 0.2236 ∗ 𝛽𝑡−1 1991/1995 – 1996/2000 𝛽𝑡= 0.2234 + 0.6293 ∗ 𝛽𝑡−1 𝛽𝑡 = 0.2188 + 0.6345 ∗ 𝛽𝑡−1 1996/2000 – 2001/2005 𝛽𝑡 = 0.3458 + 0.6523 ∗ 𝛽𝑡−1 𝛽𝑡 = 0.3471 + 0.6498 ∗ 𝛽𝑡−1 2001/2005 – 2006/2010 𝛽𝑡= 0.4743 + 0.6120 ∗ 𝛽𝑡−1 𝛽𝑡 = 0.4538 + 0.6357 ∗ 𝛽𝑡−1
16
Table 4: The mean square error of the historical beta and adjusted beta with respect to the actual beta of the forecasted period for both the individual stocks and the ten stock portfolios. The mean square error is calculated using the following formula: 𝑴𝑺𝑬 = 𝑵𝟏∑(𝜷𝑭− 𝜷𝑨)𝟐 where 𝜷𝑭 and 𝜷𝑨 are the forecasted beta and the actual beta for a given period and N is the number of betas included in the sample.
Forecasted period
Mean Square Error individual stocks
Mean Square Error 10-stock portfolio
Historical beta Adjusted beta Historical beta Adjusted beta
1991/1995 0.2333 0.1943 0.1390 0.1041
1996/2000 0.2086 0.2821 0.0663 0.1408
2001/2005 0.1870 0.1794 0.0405 0.0335
2006/2010 0.2491 0.2037 0.0722 0.0327
Table 4 shows the mean square errors (MSEs) for the historical beta and the adjusted beta with respect to the actual beta of the forecasted period. There is not much difference between the forecasting power of the historical beta and the adjusted beta. For all but the second forecast period, the adjusted beta slightly outperforms the historical beta. A more striking result is the great reduction in estimation error when the stocks are grouped in to portfolios of ten-stocks. It appears that in line with the findings of Blume (1971) the stability of betas increases when portfolios are formed. Again the adjusted beta slightly increases the accuracy of the betas which suggests that Blume’s regression technique does indeed provide a better proxy for the future betas than the historical betas. Based on these findings the ten-stocks portfolios should outperform the single stocks in terms of forecasting power and the adjustment of the betas should enhance the forecasting power of both the single stock and ten-stock portfolios.
Table 5: Comparison of the standard deviation and the Sharpe ratio of the estimated minimum variance portfolios based on the historical beta and the Blume adjusted beta. Actual are the statistics of the actual Minimum Variance Portfolio of the estimated period where the betas of the stocks for that period are known. All portfolios are based on market model and uses individual stocks. The difference between the performances of the portfolios is only subjected to the allocation of weights to the stocks in the portfolios.
Forecast period Historical Adjusted Actual
St.Dev Sharpe St.Dev Sharpe St.Dev Sharpe
1986/1990 6.651% 0.226 1.024% 0.942
1991/1995 5.589% 0.221 11.001% 0.010 0.736% 1.961
1996/2000 1.193% 0.893 14.683% 0.145 0.621% 1.344
2001/2005 1.415% 0.230 0.774% 0.488 0.442% 0.463
17 The results for the individual stocks using the market model are displayed in table 5. The table shows the difference between the standard deviation and Sharpe ratio of the portfolios. The historical portfolio uses the historical beta, the adjusted portfolio uses the adjusted beta and the actual portfolio uses the beta of the stocks in the forecasted period. The actual portfolio is the optimal minimum variance portfolio of the forecasted portfolio given a set of stocks. All portfolios in a period use the same set of stocks and only differ in the allocation of weights for the stocks due to other estimated covariance matrix by the use of the historical or adjusted beta. The adjusted betas are adjusted accordingly to the equations as displayed in table 3. The first forecasted period does not have a portfolio based on adjusted betas since the adjusted beta requires two prior periods in order to estimate a linear relationship between betas. The findings for the first two forecasted periods suggests that the historical beta is a poor forecaster, the average standard deviation of the first two periods are on average roughly seven times higher than the standard deviation of the actual minimum variance portfolio. These results are most likely biased due to invalid data for the first periods, as argued in the data section. With the emphasis on the latter periods, the historical portfolio yields on average a standard deviation which is 2.6 times higher than the standard deviation of the actual minimum variance portfolio. All portfolios yield a positive Sharpe ratio which implies that all portfolios have an positive return.
18
Table 6: Comparison of the standard deviation and the Sharpe ratio of the estimations based on the historical beta and the Blume adjusted beta. Actual are the statistics of the actual Minimum Variance Portfolio of the estimated period where the betas of the stocks for that period are known. All portfolios are based on the market model and uses portfolios of ten equally weighted stocks. The difference between the performance of the portfolios is only subjected to the allocation of weights to the stocks in the portfolios.
Table 6 displays the results for the ten-stocks portfolio. Noteworthy is the poor performance of the historical portfolio in terms of realized returns. The portfolios based on the historical betas yields in general a negative Sharpe ratio and this implies that these portfolios realized a negative return over the forecasted period. A negative Sharpe ratio implies that in order to minimize the risk of the portfolio, one should accept a negative return. It should thus be better to invest in the risk-free asset and receive a positive return than to invest in riskier assets which will reduce the wealth of the investor. The portfolios that uses the adjusted beta technique outperforms the historical portfolio again in terms of replicating the actual minimum variance portfolio. Furthermore the adjusted portfolio provides a better trade-off between risk and return than the portfolio based on historical betas.
A comparison with table 5 shows that the standard deviation of nearly all portfolios formed with the grouping procedure are higher than the portfolios formed with individual stocks. Although both procedures do not suffer from any restrictions, a higher standard deviation for the portfolios formed with the grouping procedure is a logical consequence of this procedure. There are more diversifying opportunities at the individual stock level than with a ten-stock portfolio basis. Grouping stocks reduces the diversification possibilities by reducing the covariance matrix and furthermore the individual stocks are less restrictive due to the composition of the portfolios. Although the ten-stock portfolio can get any weight to minimize the standard deviation of the forecast portfolio, the weights within the ten-stock portfolios are not able to change and this in turn results in a sub-optimal distribution for the minimum variance portfolio.
In terms of estimating power, grouping stocks results in better estimates than the individual stocks for all periods. On average the standard deviation of the historical portfolios is about 4.4 times higher than the standard deviation of the actual minimum variance portfolio for individual stocks. In
Forecast period Historical Adjusted Actual
St.Dev Sharpe St.Dev Sharpe St.Dev Sharpe
1986/1990 6.613% 0.215 3.753% 0.278
1991/1995 5.856% -0.087 9.235% 0.008 2.429% 0.047
1996/2000 2.063% -0.205 9.150% 0.033 1.824% -0.281
2001/2005 2.263% 0.276 1.954% 0.354 1.583% 0.518
19 contrast, the standard deviation of the historical method for the ten-stock portfolios are on average only 1.88 times higher than the standard deviation of the actual portfolio. A possible explanation is that grouping betas reduces the number of extreme betas, which reduces the effect of estimation errors. This in turn results to more stable betas for the portfolios, since the extreme betas have lower power in the constructed portfolio than what they should have in the individual stock framework. Adjusting the betas yields a beneficial effect and increases the accuracy of the estimated minimum variance portfolio.
5.2 Multi index model
Table 7: The regression coefficients of the Blume adjustment method for the Fama-French three factor model. The model which is regressed is 𝜷𝒕= 𝒂 + 𝒃 ∗ 𝜷𝒕−𝟏 and only the regression coefficients are given. All regressions are based on individual stocks, yet the ten-stock portfolio yields similar results. All regression coefficients are significant at the 1% level.
Table 7 provides an overview of the regression coefficients of the Fama-French betas. The market risk premium shows a similar mean-reversing regression tendency as the beta for the market model. The beta corresponding with the size factor also provides a relative stable coefficient for the former beta but the constant term of the regressions fluctuates over time. Furthermore I expect that mean of the SMB beta should be approximately zero, yet the regression tendencies concentrates the stock for all higher than the expected mean of zero. This suggests that although the SMB factor does provide a moderate regression tendency it cannot be classified as a mean-reversing tendency of the betas or the assumption that the mean of the SMB and HML betas are zero is incorrect. Finally the regression coefficients of the betas of the value factor do not show any sign of stable tendency towards a certain range of beta values. Based on the stable regression tendency of the MRP factor and the somewhat regression tendency of the SMB factor the adjustment can yield a better estimate for future betas in comparison with the historical betas. The question remains if the regressions indeed provide better estimates of the future betas and in what extent the adjustment of the HML beta, following the
Regression tendency implied between periods
MRP factor SMB factor HML factor
20 regression coefficients in table 7, distorts the accuracy of the estimations. Table 8 provides the mean square errors for the betas in the Fama-French model.
Table 8: The mean square error of the historical beta and adjusted beta with respect to the actual beta of the forecasted period for both the individual stocks and the ten-stock portfolios. The mean square error is calculated using the following formula: 𝑴𝑺𝑬 = 𝑵𝟏∑(𝜷𝑭− 𝜷𝑨)𝟐 where 𝜷𝑭 and 𝜷𝑨 are the forecasted beta and the actual beta for a given period and N is the number of betas included in the sample. The aggregated mean square error is taken over the betas for the MRP, SMB and HML indices for a given forecasted period.
The MSE displayed in the table 8 is the aggregated mean square error for all betas in the Fama-French model for a given forecasted period. Adding more indices does not necessary have to increase the aggregated mean square error dependable on the forecasting power of the historical and the adjusted beta. Table 8 shows that the mean square error for the Fama-French betas are substantially higher than the mean square error of the market model. Dissecting the aggregated mean square error of the Fama-French betas into separate MSEs provide some valuable insights about the forecasting performance of the separate betas.
The forecasting errors of the beta for the market risk premium is approximately the same as the mean square error of the beta in the market model. Both the betas corresponding with the SMB and HML factor are poor forecasters of future betas. On average the mean square error of the SMB is near one, while the HML factor has an even higher mean square error. This results in the high aggregated mean square error for the Fama-French model. Adjusting the betas with the regressions found in table 7 increases the aggregated mean square error of the estimations which implies that Blume’s adjustment method negatively influences the forecasting power of the betas. Again the betas corresponding with the SMB and HML factors are responsible for the poor estimations. The poor performance of the HML factor for the adjusted beta is in line with the findings of the regression coefficients and supports the observation that the beta for the value factor does not have a stable character but appears to move
Forecasted period
Mean Square Error individual stocks
Mean Square Error 10-stock portfolio
Historical beta Adjusted beta Historical beta Adjusted beta
1991/1995 2.165 2.200 1.655 1.724
1996/2000 2.065 1.950 1.458 1.618
2001/2005 0.672 1.239 0.248 0.962
21 randomly though time. The poor performance of the SMB beta is a bit surprising, since the regressions coefficients do show a moderate regression tendency.
In line with the findings for the market model, the mean square error of the Fama-French model decreases when ten-stock portfolios are used instead of individual stocks. The decrease in the mean square error is less dramatic than the reduction in the MSE for the market model. Grouping the stocks has a beneficial effect on the accuracy of the historical SMB beta, while the HML beta remains unaffected. The adjusted betas have again a lower forecasting power than the historical betas.
Table 9: Comparison of the standard deviation and the Sharpe ratio of the estimations based on the historical method and the Blume adjusted method. Actual are the statistics of the actual minimum variance portfolio of the forecasted period. All portfolios are based on the Fama-French multi index model and uses individual stocks. The difference between the performance of the portfolios is only subjected to the allocation of the stocks in the portfolio.
The results of the minimum variance portfolios based on the historical betas and the adjusted betas for the Fama-French model are displayed in table 9. Table 9 supports the findings in table 7 and table 8 that the portfolio based on the historical beta is a better proxy for the actual minimum variance portfolio than the portfolio formed with the adjusted betas and thus that Blume’s adjustment technique does not increase the accuracy for the betas of the SMB and HML factors. Only the forecasted period 2001-2005 yield nearly the same results for the standard deviation of the historical and adjusted portfolio. A comparison of the standard deviations of the single stock approaches for the market model and the Fama-French model suggests that both models yield approximately the same standard deviations for all periods both in the portfolio based on historical weights and the actual minimum variance portfolio. While the standard deviations between both models are nearly the same, there is a difference between the Sharpe ratios. The market model achieves for most periods higher Sharpe ratio than the Fama-French model which implies that the market model provides a better trade-off between risk and return.
Forecast period
Historical Adjusted Actual
St.Dev Sharpe St.Dev Sharpe St.Dev Sharpe
1986/1990 6.230% 0.175 1.337% 0.696
1991/1995 4.829% -0.163 10.367% 0.037 0.138% -1.686
1996/2000 1.268% 0.278 9.217% 0.577 0.931% 1.261
2001/2005 2.424% 0.084 2.630% -0.049 0.838% 0.709
22
Table 10: Comparison of the standard deviation and the Sharpe ratio of the estimations based on the historical method and the Blume adjusted method. Actual are the statistics of the minimum variance portfolio of the forecasted period. All portfolios are based on the fama French multi index model and uses portfolios of ten stocks. The difference between the performance of the portfolios is only subjected to the allocation of the stocks in the portfolio.
Forecast period Historical Adjusted Actual
St.Dev Sharpe St.Dev Sharpe St.Dev Sharpe
1986/1990 6.662% 0.163 4.357% 0.281
1991/1995 5.937% -0.162 9.739% -0.013 3.224% -0.056
1996/2000 2.838% -0.183 3.167% -0.236 2.514% -0.145
2001/2005 2.550% 0.170 2.323% 0.207 1.870% 0.366
2006/2010 2.207% -0.206 1.410% -0.155 0.770% -0.194
Finally table 10 displays the results of the minimum variance portfolios when grouped stocks are used. In line with the findings for the market model and the reduced MSEs for the historical betas, grouping stocks does increase the accuracy of the forecasted portfolios. It appears that, in contrast with the findings of the MSEs in table 8, adjusting the betas does have a beneficial effect to approach the actual minimum variance portfolio. The standard deviations of all portfolios are again higher than with the single stock approach, most likely due to the similar reasons as for the market model.
5.3 Hypotheses
The first hypothesis states that Blume’s adjustment technique provides stable regression coefficients for the Fama-French model and the market model. This hypothesis is true for the market model yet it is only partially true for the Fama-French model. The market risk premium provides a stable regression tendency and the size factor shows a relative stable coefficient for the prior beta but does not provide a stable constant value. The regressions of the betas of the value factor do not show any stable regression coefficients and appears to be random through time. The use of portfolios of ten stocks does not alter the regression coefficients.
23 The last hypothesis states that portfolios constructed with the covariance matrix of adjusted betas yield a better estimate for the actual minimum variance portfolio. With historical betas there is no difference in the standard deviations of the portfolios between the market model and the Fama-French model. There is however a difference between the performance of the portfolios for the market model and the Fama-French model for the adjusted betas. For the market model adjusting the betas does increase the accuracy of the forecasted minimum variance portfolio, while for the Fama-French model adjusting the betas yield in a worse performance of the forecasted minimum variance portfolio for individual stocks. The use of portfolios of ten stocks does increase the forecast of the historical beta for both models but also increases the standard deviation of these portfolios. The use of grouped stocks does have a beneficial effect on the performance of the forecasted minimum variance portfolio when adjusted betas are used. The forecasted portfolio based on adjusted betas outperforms the forecasted portfolio of the historical betas when stocks are first sorted in portfolios.
Blume’s adjustment technique is, based on the findings of the hypotheses, not a proper technique to increase the forecasting ability for the Fama-French betas. It should be noted that this observation is based on only a few periods for the Japanese stock market with an incomplete dataset and further research is required to support or reject this observation. This paper does support the theory that Blume’s adjustment technique does increase the forecasting performance for the market model.
6. Conclusion
This paper finds that the market model is a better forecaster of future betas. It provides better estimates of the stock allocation compared with the Fama-French three factor model and is also a less demanding model. For the single index model the estimations can be improved by either grouping the portfolios or by the use of a beta adjustment technique. This paper finds evidence of the mean reversing property of betas corresponding with the market index, yet the other betas related with the size and value indices of the Fama and French model do not appear to be mean reversing and the regressions and findings suggests that especially the HML beta is non-stationary over time or at least subjected to high estimation errors. As a consequence adjusting the betas of the value and size factors in the multi index model yields worse results in comparison with the historical betas of this model.
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