A novel nonlinear programming approach for estimating CAPM beta of an asset using fuzzy regression
Highlights
► A novel nonlinear programming approach for estimating CAPM beta. ► A novel fuzzy regression method with fuzzy output. ► Proposals based on both of possibility concepts and central tendency for fuzzy regression. ► Constructing membership functions by means of probability density function.
Introduction
The evaluation of risky assets is one of the major research tasks in finance. In the development of portfolio theory, Markowitz (1952) defined risk in terms of a well-known statistical measure known as the variance. Specifically, Markowitz quantified risk as the variance about the expected return of an asset. Although the total risk of an asset can be measured by its variance, this risk measure can be divided into two general types of risk: systematic risk and unsystematic risk. Sharpe (1964) defined systematic risk as the portion of an asset’s variability that can be attributed to a common factor, and the portion of an asset’s variability that can be diversified away as unsystematic risk (Fabozzi, 1999). In context of measuring the systematic risk, there are couples of Capital Asset Pricing Models (CAPM) in the literature; the most popular one is the Sharpe–Lintner Black mean–variance CAPM. According to this model, the typical measure of asset riskiness is the beta coefficient that is known as systematic risk measure compares the variability of an asset’s historical returns to the market as a whole. That is, beta measures the expected change of an asset for every percentage change in the benchmark index (Clarfeld & Bernstein, 1997). While making investment decisions, investors are concerned only with the systematic risk, because the unsystematic risk is diversified away by a well-balanced portfolio. For this reason, β is only concern that investors have when they value securities (Maximiliano, 2001).
The beta coefficient is mainly used for two purposes. The first involves the ranking of assets and portfolios with respect to systematic risk by practitioners. The second deals with testing CAPM and mean–variance efficiency. Beta is generally estimated by using the standard market model, which is expressed as the following linear regression model:where Rit is the realized return on asset i over interval t; Rmt is the realized return on the market index over interval t; αi is the constant term for asset i; βi is the sensitivity of asset i returns to the market index returns measured as the covariance between the asset return and the market portfolio return (cov(Ri, Rm)/var(Rm)). According to Eq. (1), the market return Rmt and the asset return Rit correspond to the input and the output respectively.
Due to estimation procedure of least square method (LSM), there is a clash between the financial and the statistical theory. Because, even one observed data may cause a large effect over LSM estimates in the regression models. However, the omitting such data can completely change the model structure. That is, LSM procedure is highly sensitive to extreme observations, so that estimations of LSM might not robust in the periods with wide market fluctuations. In here, the sensitivity of beta can be traced to a combination of two factors (Shalit and Yitzhaki, 2002):
- •
Incompatibility between standard statistical methods and financial theory. In particular, the LSM regression estimator is based on a quadratic weighting scheme that tends to contravene the assumptions of risk aversion.
- •
Probability distribution of market returns with ‘‘fat” tails; that is the data do not follow the normal distribution.
Accordingly, the factors mentioned above make beta coefficient sensitive to market fluctuations, therefore LSM is inappropriate to estimate beta coefficient. In the regression analysis, the researchers encounter two types of extreme observations known as outlier and leverage. The outlier has more influence on the regression line than the leverage has. To decrease influences on beta coefficient of extreme observations, or to remove those from data set, Robust Regression Techniques (RRT) are used by analysts. The most known RRT’s are Bi-square, Huber, Hampel, Andrews and Tukey M methods. For all that, the omitted observations might be valuable for investors, since they might carry substantial information about the state of nature of the financial markets. Besides, it is common to encounter such on extreme observations during periods with high volatility in the financial markets. In such cases, investors can make risky investment decisions to boost their profits as well.
The other shortcoming of RRT’s is that they don’t take into account the possibilistic imprecision included by data. In such situations, the fuzzy regression approaches are powerful alternatives against traditional techniques due to using the fuzzy numbers for imprecision of interest. Besides, intuitively some plausible semantic description of imprecise properties might be made easily by the fuzzy theory. The fuzzy numbers are able to improve modeling of problems where the output and the inputs (numerical and continuous) are affected by imprecision. In addition, because of problems such as shortness of observation number and non-standard normal distribution of residuals, the probabilistic assumptions considered for estimations of LSM and RRT’s might be invalid. In such cases, the fuzzy regression approaches ensures the reasonable results for problems without any probabilistic assumption as well.
The fuzzy regression analysis was first introduced by Tanaka, Uejima, and Asai (1982), who established his idea on the basis of the possibility theory. He modeled the procedure of parameter estimation as linear programming problem, where the inputs and output are crisp and fuzzy number respectively. Since the membership functions (msf) of fuzzy sets are often described as possibility distributions, this approach is usually called possibilistic regression analysis (Tanaka and Lee, 1998, Tanaka and Watada, 1988) where the fuzzy coefficients are non-interactive; that is, coefficients are determined independently from each other. Models in which coefficients are used with interactive possibility distributions were first proposed by Tanaka and Ishibuchi, 1991, Tanaka et al., 1995 where msf’s are quadratic, and possibility distributions are exponential. A modified form of possibilistic regression was proposed by Savic and Pedrycz (1991) while possibilistic regression for fuzzy input–output data was studied by Sakawa and Yano (1992). Sakawa and Yano introduced the multi-objective fuzzy linear regression model with fuzzy input-output-parameters.
Another approach of fuzzy regression is the fuzzy least squares, which is based on the notation of distance between the estimated and the observed outputs, and goodness-of-fit. Celmins, 1987a, Celmins, 1987b dealt quadratic msf’s based on the least squares fitting with indicators of discord, data spread dilator etc., and Diamond, 1987, Diamond, 1988 proposed models for the least squares fitting for crisp inputs and fuzzy output, for fuzzy inputs-output where the distances of fuzzy numbers are defined to measure the best fit for models (Tanaka & Lee, 1998).
Recently, even there are many approaches in the literature, the fuzzy regression models can be roughly classified into three categories by conditions of inputs and output as follows:
- •
Both input and output data are non-fuzzy number.
- •
Input data is non-fuzzy number but output data is fuzzy number.
- •
Input and output data are both fuzzy number.
The motivation of this study is that when investors make a decision about their investment strategies, they are interested in an asset return itself rather than market return. Therefore, in this study, it is assumed that the returns of market (input) and asset (output) are crisp and fuzzy number respectively, since the asset return depends on market. Because of these characteristic of returns, the second case that is the fuzzy regression with crisp input and fuzzy output is appropriate to estimate beta coefficient. Besides, the fuzzy regression with crisp inputs and fuzzy output can be categorized into three alternative groups as well:
- •
Proposals based on the use of possibility concepts (Peters, 1994, Redden and Woodall, 1994, Tanaka and Lee, 1998, Tanaka and Watada, 1988, Özelkan and Duckstein, 2000).
- •
Proposals based on the minimization of central values, mainly through the use of the least squares method (Celmins, 1987a, Celmins, 1987b, Diamond, 1987, Diamond, 1988, Kao and Chyu, 2003).
- •
Proposals based on both of possibility concepts and central tendency (Tanaka and Lee, 1998, Donoso et al., 2006, Kocadagli, 2009, Kocadagli, 2011).
In order to construct msf of any asset return, it is first assumed that an asset return includes imprecision, and its probability distribution is known, or it can be measured. Thus, msf of the asset return can be constructed by Civanlar and Trussell (1985) approach, which is based on the consistency principle. To reconcile the minimization of the estimated deviations of the central tendency with the minimization of the estimated deviations in the spreads of asset return’s msf, the quadratic objective function is proposed. Besides, to solve the problem of determining h-cut level in essence of the traditional fuzzy regression techniques, the constraint system is constituted by means of the membership degrees of the observed asset returns and the estimated asset returns. Thus, the novel non-linear programming approach with constraints, which provides the consistent solutions for h-cut level problem in the existence of extreme observations, is improved.
In the following sections, the theoretical structure of the proposed fuzzy regression approach is introduced in details. For implementation, the beta coefficients of assets of ISCTR, PETKIM and THYAO traded in Istanbul Stock Exchange 100 (ISE100) are evaluated by the proposed approach and traditional techniques, and then the results of the analysis are discussed in the application section.
Section snippets
Constructing the membership function of output
According to Civanlar and Trussell (1985), in order to define a reasonable msf of a fuzzy set, any probability density function (pdf) defining feature of the elements of fuzzy set can be used. These requirements are quantitatively described below:
- •
E{μ(x)|x is distributed according to the underlying pdf} ⩾ c where the confidence level c should be close to unity.
- •
0 ⩽ μ(x) ⩽ 1.
- •
∫ μ2(x)dx should be minimized. This condition is required to obtain a selective msf, that is, the size of the set should be as
Applications
The aim of applications in this section is to show how to cope with clash between statistical and financial theory that appear in the existence of extreme observations. In analysis, the sample data sets of assets of ISCTR, PETKIM and THYAO traded in Istanbul Stock Exchange 100 (ISE100) are used. The sample data includes returns of ISE100 index and an asset for forty opening and closing sessions in January 2010. In Figures, the axes x and y of scatter plots correspond to the market and the asset
Results and discussion
After the outliers are determined by Cook’s distance criteria, the beta coefficients of assets are evaluated by the proposed approach, LSM and RRT’s, and then the estimation results are summarized in Tables and Figures. According to these results, although some outliers have the low Cook’s distance, RRT’s are not sensitive to these observations that might contain substantial information. As seen in Fig. 5, although the regression line of LSM is sensitive to outliers, its statistical assumptions
Conclusions
In the existence of extreme observations, there is a clash between statistical and financial theory. Therefore, to overcome this incompatibility, the novel non-linear programming model is improved by means of the fuzzy regression. This novel approach not only overcomes a clash between statistical and financial theory but also provides a solution to h-cut level problem in the fuzzy regression approaches. Thus, the extreme observations carried significant information for investors can be utilized
References (32)
Last squares fitting to fuzzy vector data
Fuzzy Sets and Systems
(1987)Multidimensional last squares fitting of fuzzy regression models
Mathematical Modeling
(1987)Fuzzy least squares
Information Science
(1988)- et al.
Least-squares estimates in fuzzy regression analysis
European Journal of Operational Research
(2003) - et al.
Multi-objective fuzzy regression: a general framework
Computers and Operations Research
(2000) - et al.
Properties of certain fuzzy linear regression methods
Fuzzy Sets and Systems
(1994) - et al.
Multiobjective fuzz linear regression analysis for fuzzy input-output data
Fuzzy Sets and Systems
(1992) - et al.
Evulation of fuzzy linear regression models
Fuzzy Sets and Systems
(1991) - et al.
Possibilistic linear systems and their applications to the linear regresion model
Fuzzy Sets and Systems
(1988) - et al.
Exponential possibility regression analysis. linear systems by quadratic membership functions of fuzzy parameters
Fuzzy Sets and Systems
(1995)
Identification of possibilistic linear systems by quadratic membership functions of fuzzy parameters
Fuzzy Sets and Systems
Exponential possibility regression analysis by identification method of possibilistic coefficients
Fuzzy Sets and System
Insight of a fuzzy regression model
Fuzzy Sets and Systems
Constructing membership functions using statistical data
How to interpret measures of risk understanding risk in mutual fund selection
Journal of Accountancy
Cited by (21)
A robust support vector regression with exact predictors and fuzzy responses
2021, International Journal of Approximate ReasoningCitation Excerpt :The methods of fuzzy regression analysis have been suggested for linear and non-linear models. The methodologies of linear models can be classified as (1) possibilistic approaches (see for example [8–17,19,21]), (2) fuzzy least squares and fuzzy least absolutes methods (see for example [22–37]), and (3) machine learning techniques like evolutionary algorithms [38–43], SVMLs [44–47] and neural networks embedded in fuzzy regression analysis [43,48–52]. The first class of the methodologies attempts to minimize a linear/non-linear programming model by minimizing the total spread of its fuzzy parameters to support the observations at some specific levels.
A fuzzy additive regression model with exact predictors and fuzzy responses
2020, Applied Soft Computing JournalCitation Excerpt :The fuzzy regression analysis methods have been suggested for linear and nonlinear models. The linear modeling can be practiced through either (1) possibilistic approaches (see for example [10–21]), (2) fuzzy least-squares and fuzzy least-absolutes methods, or (3) evolutionary algorithms [22–29], support vector machines [30–33] and neural networks embedded in fuzzy regression analysis [34–41]. The first class of these methodologies is used to minimize a linear/nonlinear programming model by minimizing the total spread of its fuzzy parameters to support the observations at some specific levels.
Fuzzy spline univariate regression with exact predictors and fuzzy responses
2020, Journal of Computational and Applied MathematicsCitation Excerpt :In this regard, Chukhrova and Johannssen [1] presented a comprehensive and systematic review of the latest fuzzy regression analysis methodologies and their applications, up to 2019. These methodologies have been classified as (1) possibilistic approaches (see for example [2–14]), (2) fuzzy least squares and fuzzy least absolutes parametric/non-parametric methods., and (3) machine learning techniques. The first class aimed at minimizing a linear/non-linear programming model through the minimization of the total spread of its fuzzy parameters to support the observations at some specific level.
Fuzzy regression analysis: Systematic review and bibliography
2019, Applied Soft Computing JournalThe normalized interval regression model with outlier detection and its real-world application to house pricing problems
2015, Fuzzy Sets and SystemsCitation Excerpt :Based on Tanaka's possibilistic regression model, an approach with considering the central tendency was proposed [32]. The method was extended to the non-symmetrical case [6] and utilized in the CAPM beta estimation problem [17]. Further, the possibilistic regression model has been extended to the real time fuzzy regression analysis [24] and fuzzy autocorrelation models [37].
A novel portfolio selection model based on fuzzy goal programming with different importance and priorities
2015, Expert Systems with ApplicationsCitation Excerpt :Markowitz (1952) defined the risk as the variance that is basically measured as the expected value of the squared deviation from the expected return of an asset. Essentially, the total risk can be divided into two general types of risk: systematic risk and unsystematic risk (Kocadagli, 2013). Sharpe (1964) defined systematic risk as a portion of an asset’s variability that is caused by inherent uncertainty of benchmark market.