The capital asset pricing model (CAPM) was introduced by William Sharpe, John Lintner, and Jan Mossin in the 1960s on the basis of Harry Markowitz’s achievements in the field of portfolio theory. Since then, the CAPM has been one of the most widely used models for evaluating the price of portfolio assets. 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.
(...) 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 complex as betas will vary over time. In addition, sample size variation can cause change within the beta.
(...) 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.
Table of Contents
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
Research Objectives and Core Themes
The primary objective of this seminar paper is to investigate the influence of varying sample sizes on the calculation of beta factors. By utilizing the Commerzbank stock as a case study, the author examines how different temporal windows for return data affect the stability and reliability of beta estimates within the Capital Asset Pricing Model (CAPM) framework.
- Theoretical foundations of the Capital Asset Pricing Model (CAPM).
- Mathematical derivation of beta factors using Ordinary Least Squares (OLS) regression.
- Comparative analysis of rolling beta performance across various time-based sample sizes.
- Evaluation of statistical indicators like R-squared and standard error in beta reliability.
- Identification of determinants for selecting an optimal sample size in risk assessment.
Excerpt from the Book
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 generally 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 regression 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.
Summary of Chapters
1 Introduction: This chapter outlines the problem definition regarding beta factor variability and introduces the scope of the investigation using the Commerzbank stock.
2 Conceptual Framework: This section provides the theoretical background of the CAPM and the mathematical derivation of beta factors using the Gauss-Markov theorem and OLS regression.
3 Beta Factor Regression: This chapter presents a detailed empirical analysis of how different sample sizes, ranging from one week to four years, impact the stability and accuracy of beta coefficients.
4 Conclusion: This chapter synthesizes the findings, confirming that beta is a time-varying factor and recommending a sample size between six months and two years for optimal risk measurement.
Keywords
Beta Factor, CAPM, Sample Size, Commerzbank, Ordinary Least Squares, OLS, Rolling Beta, Financial Risk, Stock Return, Statistical Stability, Standard Error, R-squared, Portfolio Theory, Market Volatility, Econometrics
Frequently Asked Questions
What is the fundamental focus of this research?
The paper examines how changing the size of the historical return sample influences the calculation of beta factors and, consequently, the assessment of stock market risk.
Which specific themes are addressed in this paper?
The core themes include the validity of the CAPM, the statistical mechanics of OLS regression, the impact of market volatility on beta, and the practical selection of sample sizes for risk management.
What is the primary research goal?
The goal is to determine how sample size variation affects beta stability and to identify the optimal sample length for creating a reliable risk measurement tool.
Which scientific methodology is applied?
The author uses a comparative empirical analysis, performing OLS regressions with various sample sizes on Commerzbank stock data, and evaluating the outcomes through rolling beta graphs and statistical error measures.
What is covered in the main section of the paper?
The main section evaluates rolling betas across different temporal intervals, starting from weekly data up to four-year samples, assessing their fluctuation and responsiveness to market events.
Which keywords define this work?
Key terms include Beta Factor, CAPM, Rolling Beta, Sample Size, OLS, and Financial Volatility.
How does the Commerzbank stock example illustrate the author's argument?
Commerzbank serves as a volatile case study that highlights how aggressive stock movements, particularly during the US Subprime Crisis, can distort beta values if the chosen sample size is too short or too long.
What conclusion does the author reach regarding the optimal sample size?
The author concludes that a sample size of between six months and two years is most appropriate, as it balances the need for stability with the necessity of reflecting current market dynamics.
Why are extremely large sample sizes considered potentially misleading?
Extremely large samples may incorporate outdated information and force the beta into a narrow range, which may fail to capture recent, significant shifts in a stock's risk profile.
- Quote paper
- Kevin Rink (Author), 2008, The Influence of Sample Size on the Dynamics of Beta Factors, Munich, GRIN Verlag, https://www.grin.com/document/147114
