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CAPM with fuzzy returns and hypothesis testing

Author

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  • A. Mbairadjim Moussa

    (LAMETA - Laboratoire Montpelliérain d'Économie Théorique et Appliquée - UM1 - Université Montpellier 1 - UPVM - Université Paul-Valéry - Montpellier 3 - INRA - Institut National de la Recherche Agronomique - Montpellier SupAgro - Centre international d'études supérieures en sciences agronomiques - UM - Université de Montpellier - CNRS - Centre National de la Recherche Scientifique - Montpellier SupAgro - Institut national d’études supérieures agronomiques de Montpellier)

  • J. Sadefo Kamdem

    (LAMETA - Laboratoire Montpelliérain d'Économie Théorique et Appliquée - UM1 - Université Montpellier 1 - UPVM - Université Paul-Valéry - Montpellier 3 - INRA - Institut National de la Recherche Agronomique - Montpellier SupAgro - Centre international d'études supérieures en sciences agronomiques - UM - Université de Montpellier - CNRS - Centre National de la Recherche Scientifique - Montpellier SupAgro - Institut national d’études supérieures agronomiques de Montpellier)

  • A.F. Shapiro

    (Penn State - Pennsylvania State University [State College, PA] - Penn State System)

  • M. Terraza

    (LAMETA - Laboratoire Montpelliérain d'Économie Théorique et Appliquée - UM1 - Université Montpellier 1 - UPVM - Université Paul-Valéry - Montpellier 3 - INRA - Institut National de la Recherche Agronomique - Montpellier SupAgro - Centre international d'études supérieures en sciences agronomiques - UM - Université de Montpellier - CNRS - Centre National de la Recherche Scientifique - Montpellier SupAgro - Institut national d’études supérieures agronomiques de Montpellier)

Abstract

Over the last four decades, several estimation issues of the beta have been discussed extensively in many articles. An emerging consensus is that the betas are time-dependent and their estimates are impacted by the return interval and the length of the estimation period. These findings lead to the prominence of the practical implementation of the Capital Asset Pricing Model. Our goal in this paper is two-fold. After studying the impact of the return interval on the beta estimates, we analyze the sample size effects on the preceding estimation. Working in the framework of fuzzy set theory, we first associate the returns based on closing prices with the intraperiod volatility for the representation by the means of a fuzzy random variable in order to incorporate the effect of the interval period over which the returns are measured in the analysis. Next, we use these fuzzy returns to estimate the beta via fuzzy least square method in order to deal efficiently with outliers in returns, often caused by structural breaks and regime switches in the asset prices. A bootstrap test is carried out to investigate whether there is a linear relationship between the market portfolio fuzzy return and the given asset fuzzy return. Finally, the empirical results on French stocks suggest that our beta estimates seem to be more stable than the ordinary least square (OLS) estimates when the return intervals and the sample size change.

Suggested Citation

  • A. Mbairadjim Moussa & J. Sadefo Kamdem & A.F. Shapiro & M. Terraza, 2014. "CAPM with fuzzy returns and hypothesis testing," Post-Print hal-02901727, HAL.
    • Handle: RePEc:hal:journl:hal-02901727
    DOI: 10.1016/j.insmatheco.2013.11.001
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    Cited by:

    1. Dobrislav Dobrev∗ & Travis D. Nesmith & Dong Hwan Oh, 2017. "Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors," JRFM, MDPI, vol. 10(1), pages 1-14, February.
    2. Jules Sadefo Kamdem & Babel Raïssa Guemdjo Kamdem & Carlos Ougouyandjou, 2021. "S-ARMA Model and Wold Decomposition for Covariance Stationary Interval-Valued Time Series Processes," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 17(01), pages 191-213, March.
    3. Chikashi Tsuji, 2017. "A Non-linear Estimation of the Capital Asset Pricing Model: The Case of Japanese Automobile Industry Firms," Applied Finance and Accounting, Redfame publishing, vol. 3(2), pages 20-26, August.
    4. Alexey Shvedov, 2016. "Estimating the means and the covariances of fuzzy random variables," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 42, pages 121-138.
    5. Babel Raïssa Guemdjo Kamdem & Jules Sadefo-Kamdem & Carlos Ougouyandjou, 2020. "On Random Extended Intervals and their ARMA Processes," Working Papers hal-03169516, HAL.
    6. Pedro Antonio Martín-Cervantes & María del Carmen Valls Martínez, 2023. "Unraveling the relationship between betas and ESG scores through the Random Forests methodology," Risk Management, Palgrave Macmillan, vol. 25(3), pages 1-29, September.
    7. Brychykova, A., 2019. "Capital Asset Pricing Model Using Fuzzy Data and Application for the Russian Stock Market," Journal of the New Economic Association, New Economic Association, vol. 43(3), pages 58-77.
    8. Xingyu Dai & Roy Cerqueti & Qunwei Wang & Ling Xiao, 2025. "Volatility forecasting: a new GARCH-type model for fuzzy sets-valued time series," Annals of Operations Research, Springer, vol. 348(1), pages 735-775, May.
    9. Alfred Mbairadjim Moussa & Jules Sadefo Kamdem, 2022. "A fuzzy multifactor asset pricing model," Annals of Operations Research, Springer, vol. 313(2), pages 1221-1241, June.
    10. Leandro Maciel & Fernando Gomide & Rosangela Ballini, 2016. "Evolving Fuzzy-GARCH Approach for Financial Volatility Modeling and Forecasting," Computational Economics, Springer;Society for Computational Economics, vol. 48(3), pages 379-398, October.

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