The Influence of Sample Size on the Dynamics of Beta Factors

Table of contents

List of Abbreviations

List of Tables and Figures

1 Introduction
1.1 Problem Definition and Objectives
1.2 Course of the Investigation

2 Conceptual Framework
2.1 The Capital Asset Pricing Model
2.2 Determining Beta Factors through Least Squares Regression

3 Beta Factor Regression
3.1 Influence of the Sample Size on the Beta Regression
3.1.1 The Commerzbank Stock
3.1.2 The Sample Size of 250 Days
3.1.3 Beta Regressions with Samples Sizes up to one Month
3.1.4 Beta Regressions with Sample Sizes between three Months and less than one Year
3.1.5 Beta Regressions with Sample Sizes of two Years and above
3.2 Optimal Sample Size

4 Conclusion

References

Appendixes

List of Abbreviations

illustration not visible in this excerpt

List of Tables and Figures

Figure 1 Daily performance of the Commerzbank and the DAX since 2003

Figure 2 Daily stock and index return in comparison

Figure 3 Scatterplot of the daily return

Figure 4 Rolling beta with a sample size of 250 daily returns

Figure 5 Rolling beta with a sample size of five daily returns

Figure 6 Rolling beta with a sample size of 21 daily returns

Figure 7 Rolling beta with a sample size of 63 daily returns

Figure 8 Rolling beta with a sample size of 125 daily returns

Figure 9 Rolling beta with a sample size of 188 daily returns

Figure 10 Rolling beta with a sample size of 500 daily returns

Figure 11 Rolling beta with a sample size of 1,000 daily returns

1 Introduction

1.1 Problem Definition and Objectives

The capital asset pricing model (CAPM) was introduced by William Sharpe, John Lint-ner, and Jan Mossin in the 1960s on the basis of Harry Markowitz achievements in the field of portfolio theory. Since then, the CAPM has been one of the most widely used models for evaluating the fair price of assets in a portfolio. A major element of the CAPM is the beta factor. The beta factor measures how the expected return of a stock or a portfolio correlates with the return of the whole market.

The calculation of beta factors is a difficult task, as past stock return provides a basis for the risk measurement of future developments. Fluctuation within the market and indi-vidual stocks depends on several factors including business or branch news, both positive and negative. Obviously the fluctuation of a stock does affect beta factors. As the value of beta is decisive for the portfolio selection process, it is necessary to provide the CAPM with a beta that represents the best possible estimate of correlation with the market. On account of this, the calculation of beta factors is very complex as betas will vary over time. In addition, sample size variation can cause change within the beta. Nevertheless, the importance of the beta calculation is a factor often ignored. The aim of this seminar paper is to show the influence of the sample size on the beta factor. Furthermore, it shall attempt to define the determinants of an ideal sample size.

1.2 Course of the Investigation

This task will be completed in two separate sections. The first section will deal with im-portant theoretical concepts that are crucial for the understanding and utilization of the findings of this paper. The conceptual framework includes an illustration of the CAPM. Furthermore, it will present the theoretical and mathematical derivation of beta factor calculation through ordinary least squares (OLS) regressions. This will prepare the read-er for the second section of this paper which is to discuss the influence of sample size on the beta factor. The return of the Commerzbank serves as an example for the beta calculation. In order to receive an impression of the influence of the sample size on the beta factor, we will perform the calculation with a variety of different sample sizes. The beta regressions and the graphs of rolling betas are calculated with the statistical soft-ware R. After the analysis of the calculated betas, it will be necessary to define the main determinants that influence beta factors. Based on those findings, the sample sizes which are most suitable to calculate a useful beta value will be identified. Finally, the outcomes of the analysis will be summarized in the conclusion.

2 Conceptual Framework

2.1 The Capital Asset Pricing Model

The capital asset pricing model was introduced in literature by William Sharpe, John Lintner, and Jan Mossin in the mid 1960s. The CAPM relies on the portfolio selection process drawn by Harry Markowitz. According to this process, all investors are rational and mean-variance optimizers (Bodie, Kane, & Marcus, 2003, p.264). In other words, all investors aim to place their money in the most efficient portfolios. As investors tend to have an aversion to risk, greater risk is only accepted by a higher return. Due to this and the fact that all investors have access to the same information and analyze assets in the same way, they are all holding the same portfolio of risky assets. The portfolio that comprises all assets available is known as the market portfolio (M). Each investor holds a certain percentage of M, with the same weight of individual assets. With regard to the risk aversion, each investor blends his investment with risky and risk-free assets. An in-vestor who is risk averse holds a larger percentage of risk-free assets than one who is willing to take higher risks and vice versa. (Bodie et al., 2003, pp. 263-266)

The stocks in a portfolio must have a greater estimated return than the risk-free asset. An investor would prefer lesser risk over higher risk if there was not difference in re-turn. As a result, the risk of an individual asset requires an appropriate return. Thus, in-vestors request a risk premium in form of higher expected return for holding risky stocks in their portfolios (Allen, Brealey, & Myers, 2006, p. 188).

An extremely volatile stock is considered greater risk when compared to a stock that does not move as strong. The beta factor measures the movement of a stock and the whole market together. The beta of the market is defined as one. A stock that moves stronger than the market has a beta greater than one, and a stock that does not move as strong as the market has a beta factor less than one. For instance, if market return de-clines by one percent then the return of a stock with a beta coefficient of 1.6 will de-crease by 1.6 percent, according to the theory of the CAPM. Mathematically the beta can be calculated as the covariance of the return of an individual stock i and the return of the market divided by the variance of the market:

illustration not visible in this excerpt (Bodie et al., 2003, p. 265). (1)

The following shows why the beta of the market portfolio must always equal to one. With the identity [illustration not visible in this excerpt]it can be seen that [illustration not visible in this excerpt].

The linear relationship between a stock’s return and the return of the market is presented in the CAPM formula

illustration not visible in this excerpt, (2)

where [illustration not visible in this excerpt] denotes the return and [illustration not visible in this excerpt]the risk-free asset (Allen et al., 2006, pp. 188-189). The formulas (1) and (2) are the core ideas of the CAPM, which are used for the beta calculation and the determination of fair stock return, respectively.

2.2 Determining Beta Factors through Least Squares Regression

The previous section demonstrated that the beta factor is useful for the pricing of a stock. In general, beta factors are calculated through the method of least squares with the very last returns of the stock. Investigating the sample size of the returns and its im-pact on the beta factor will be the major focus of this paper. First, it is necessary to briefly introduce the standard method for calculating a beta factor. The estimation of the beta factor can be derived from the Gauss-Markov theorem. The multi-level derivation will be illustrated here through its basic steps.

According to Gauss and Markov, in a linear model the least squares estimator is a best linear unbiased estimator (BLUE), if the errors, [illustration not visible in this excerpt], with [illustration not visible in this excerpt], (a) have an expect-ed value of zero, [illustration not visible in this excerpt], (b) the variance of all errors is equal, [illustration not visible in this excerpt], and (c) the errors do not correlate, [illustration not visible in this excerpt], with [illustration not visible in this excerpt] (Murray, 2006, p. 91). Addition-ally, the dependent variable is defined as [illustration not visible in this excerpt], where [illustration not visible in this excerpt] and [illustration not visible in this excerpt]^[1] are the intercept and the slope of the function, respectively, and [illustration not visible in this excerpt] denotes the independent variable. Now the coefficients of the linear function can be estimated through a linear estimator, which is a linear combination of the dependent variable:

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That in turn is the slope of the linear model, with [illustration not visible in this excerpt] as real constants. [illustration not visible in this excerpt] is an unbiased estimator for [illustration not visible in this excerpt] under the preconditions

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Minimizing the variance ofillustration not visible in this excerpt,illustration not visible in this excerpt, in accordance with the preconditions (4) makes [illustration not visible in this excerpt] a BLUE estimator for [illustration not visible in this excerpt], with the constants given through

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[illustration not visible in this excerpt] denotes the mean of the independent variable (Murray, 2006, pp. 125-127). Through this, the BLUE estimator can be derived as

[illustration not visible in this excerpt](Murray, 2006, p. 129). (6)

Finally, this is the formula for the method of ordinary least squares, where we yield the beta factor through the linear regression. The numerator in (6) equals the covariance,illustration not visible in this excerpt, and the denominator equals the variance, [illustration not visible in this excerpt]. This is equivalent to the CAPM definition of the beta, with [illustration not visible in this excerpt].

The quality of the regression outcome is an important factor for the reliability of the beta factor. Thus, it is essential to revert to statistical methods to measure the quality of the regression. The coefficient of determination, R-squared, measures how much the model fits the empirical data. An R-squared of zero means that there is no relation be-tween the model and the data. On the other hand, a value of one indicates that the model fits the data perfectly. That is the case, if the regression line goes through all data points (Altman & Bland, 2005). The standard error is a statistical measure that measures the quality of the obtained mean value through the regression in comparison to the real pop-ulation mean (Altman & Bland, 2005). Thus, a higher sample size leads to a lower stan-dard error.

3 Beta Factor Regression

This following section will provide a detailed analysis of the beta factor development for different sample sizes. Because a general analysis of beta factors would go beyond the scope of this work it is necessary to focus on a single stock. The Commerzbank is a stock with both high volatility and a rather high beta factor when compared to other stocks in the German stock index DAX. At the end of the period under observation, the Commerzbank disclosed the second highest beta of all companies listed in the DAX (Deutsche Börse, 2008). The DAX serves as the benchmark index in this seminar paper. The sample size comprises 1,422 daily returns of the Commerzbank and the DAX be-tween January 1, 2003, and July 31, 2008.

3.1 Influence of the Sample Size on the Beta Regression

3.1.1 The Commerzbank Stock

The daily close of the stock price and the index performance of the Commerzbank and the DAX, respectively, are shown in Figure 1. In both graphs, the first halves of the sample sizes, shown on the abscissa, have a comparable development. These samples represent the development of the index and the stock between January 2003 and approx-imately December 2005. Subsequently, the stock price shows a much stronger increase than the DAX. However, the stock also shows a larger downswing in summer 2006, es-pecially in the latest month. This development results from the huge impact of the US Subprime Crisis on the banking branch (BBC News, 2008). Taking the return of the stock and the index into account, both show heavy fluctuation in 2003. Nonetheless, the Commerzbank shows greater volatility within its daily return most of the time. This can be observed particularly in most recent times. As both graphs differ in their scaling, Fig-ure 2 gives a better direct comparison of both returns. Here the black bars indicate that the positive and negative returns of the bank are greater in the examined period than those of the index, which is shown in red. On top of that, the gap between the return grows slightly over the time. Based on these returns, we will analyze beta coefficients for different time periods.

3.1.2 The Sample Size of 250 Days

The most common procedure of calculating beta factors is by utilizing the most recent 250 daily returns of a stock. There is no mathematical or theoretical reason why so many institutions chose to calculate stock betas with a sample size of 250. Perhaps it is due to the fact that there are roughly 250 trading days per year. Nevertheless, the Deutsche Börse, for instance, provides only the Beta-250 (Deutsche Börse, 2005, pp. 7-9). This example implies that at least in Germany a sample of 250 trading days is gen-erally accepted. Because of this we perform the regression for this sample size first.

The regression for the most recent 250 trading days, which is the period form August 15, 2007, to July 31, 2008, yields a beta of 1.7191. The graphical regression for this beta is shown in Figure 3. The regression line is indicated in blue. The slope of the re-gression is the beta coefficient. With a standard error of 0.0918 percent, the regression provides a satisfactory value. The coefficient of determination, R-squared, is 0.5867. This figure is a rather weak value for the coefficient of determination.

With respect to the development of the beta, it is essential to plot the graph for the rolling beta, which is shown in Figure 4. It is noticeable that the sample comprises slightly less than 1,200 returns. This can be explained with the calculation process of the beta. There are at least 250 returns required to calculate the beta, which is why there cannot exist any rolling beta before the 1,172nd sample. In the following regressions with different sample sizes the scaling of the abscissa will vary accordingly.

Regarding the development of the rolling beta, it can be observed that the rolling beta shows an upward and downward movement comparable to stock price trends. The rolling beta has two major lows: one between the samples 700 and 850 with a beta of approximately 0.9, and the other at sample 250 with a value of circa 1. Whereas the beta fluctuates between approximately 1.1 and close to 1.4 within the first 1,000 samples it is rising tremendously in the most recent year. In this period, the beta rises from about 1.0 to over 1.7. Particularly since November 2007, the Commerzbank beta rose from 1.0846 to the previously mentioned value of 1.7191. The increase within nine months has a strong impact on the required expected return of the CAPM. This can be demonstrated with an example: If the risk free rate is [illustration not visible in this excerpt] percent, for instance, and the expected re-turn of the market accounts for 8 percent then the CAPM would require [illustration not visible in this excerpt] as an expected return for the Com-merzbank stock. In comparison to that, in November 2007 a stock return of just [illustration not visible in this excerpt] would be necessary, resulting in a dif-ference of 3.18 percent. Thus, this first exemplified calculation shows the impact on the beta, which in turn results in different CAPM pricing. Moreover, the rolling beta with the sample size of 250 shows how huge the changes of a beta can be within less than one year.

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^[1] This is not the CAPM beta. Here alpha and beta denote the coefficient in the linear regression model. Alpha and beta are commonly used as coefficients in regressions.