Academic journal article Fuzzy Economic Review

Analysis of Beta Coefficients in the Brazilian Stock Market Using Fuzzy Linear Regression Methodology

Academic journal article Fuzzy Economic Review

Analysis of Beta Coefficients in the Brazilian Stock Market Using Fuzzy Linear Regression Methodology

Article excerpt

(ProQuest: ... denotes formulae omitted.)


The Capital Asset Pricing Model (CAPM) sets the expected return on any asset as a positive linear function of its systematic risk measured by means of the beta coefficient (β). This concept emphasizes the importance of systematic risk as a measure of non-diversifiable risk, the only risk that is remunerated in financial markets.

The basic underlying notion of this model is that every asset is affected by the market's general movements, assuming that the market factor is a systematic force. Other effects are assumed to be specific or unique to an individual asset and they diversify in a portfolio.

The beta coefficient is the ordinary least squares (OLS) estimator of the return on asset j on the portfolio return over a period of time. This estimation, besides using historical returns, requires other practical assumptions. Each assumption can significantly affect results.

An essential requirement for using beta to obtain the future risk of a financial asset is that it has predictive power. Since future values are calculated from past data, they must be stable over time so that the estimation is correct and precise. Therefore, the more stable a value is over time, the more useful it will be. Although beta is an indicator of risk, its value is not unique and its result will depend on the hypothesis and data that are used. Many authors have studied beta's historical evolution, and analyzed its capacity to make predictions from empirical and theoretical points of view.

The first decision that must be considered when calculating betas is the length of the sample period. A longer period provides more data, but the company itself could have changed its risk characteristics. Various studies, Levy (1971), Blume (1971), Altman et al. (1974), Eubank and Zumwalt (1981), Armitage and Brzeszczynski (2009), KiHoon and Satchell (2014), analyze the relation between the length of the estimation period and beta stationarity. They find that the prediction ability of betas (and consequently their stationarity) increases with the length of the period. However, this increase decreases in more diversified portfolios.

A conceptual problem arises when we try to determine the return on an asset. Financial theory does not specify if returns should be considered on a daily, weekly or even monthly basis. The calculation of betas will depend on which price is considered: closing price, average daily price, and so on.

Beta assets vary from one period to another because, in the first place, the risk measured by the beta coefficient of a value can vary over time. In the second place, each period's beta is calculated with a random error which increases as the coefficient goodness and the prediction power decrease. If we consider a portfolio, random errors committed in the calculation of individual betas will tend to cancel each other out, so a portfolio beta is more stable than a single beta value.

Two studies, Levy (1971) and Blume (1971), analyze the seasonality of betas of individual securities and portfolios. They observe that, whereas betas of portfolios with a high number of securities provide a considerable amount of information about future betas, the betas of individual securities provide much less. This result suggests that a portfolio's beta is more stable than a single security's beta. The same direct relationship between the portfolio size and the beta stationarity has been observed in various studies: Altman et al. (1974), Eubank and Zumwalt (1981), Tole (1981), Iglesias Antelo (1999).

In Jagannathan and Wang (1996) a conditional CAPM is specified, on the basis that the beta and expected returns vary over time. The results are better than those of the static model. Similarly, Mergner and Bullab (2008) uses 6 different techniques to make a study of 18 sectors in Europe, and shows that variable betas estimate the profitability of the sector, explained in terms of market movements, more efficiently than OLS. …

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