Non-parametric and semi-parametric asset pricing

📝 Abstract

We find that the CAPM fails to explain the small firm effect even if its non-parametric form is used which allows time-varying risk and non-linearity in the pricing function. Furthermore, the linearity of the CAPM can be rejected, thus the widely used risk and performance measures, the beta and the alpha, are biased and inconsistent. We deduce semi-parametric measures which are non-constant under extreme market conditions in a single factor setting; on the other hand, they are not significantly different from the linear estimates of the Fama-French three-factor model. If we extend the single factor model with the Fama-French factors, the simple linear model is able to explain the US stock returns correctly.

💡 Analysis

We find that the CAPM fails to explain the small firm effect even if its non-parametric form is used which allows time-varying risk and non-linearity in the pricing function. Furthermore, the linearity of the CAPM can be rejected, thus the widely used risk and performance measures, the beta and the alpha, are biased and inconsistent. We deduce semi-parametric measures which are non-constant under extreme market conditions in a single factor setting; on the other hand, they are not significantly different from the linear estimates of the Fama-French three-factor model. If we extend the single factor model with the Fama-French factors, the simple linear model is able to explain the US stock returns correctly.

📄 Content

Non-Parametric and Semi-Parametric Asset Pricing

Péter Erdős Department of Finance, Budapest University of Technology and Economics Budapest, Műegyetem rkp. 9., 1111. HUNGARY erdos@finance.bme.hu Phone: +36 1 463 1083, Fax +36 1 463 2745

Mihály Ormos* Department of Finance, Budapest University of Technology and Economics Budapest, Műegyetem rkp. 9., 1111. HUNGARY ormos@finance.bme.hu Phone: +36 1 463 4220, Fax +36 1 463 2745

Dávid Zibriczky Department of Computer Science and Information Theory, Budapest University of Technology and Economics Budapest, Műegyetem rkp. 9., 1111. HUNGARY zibriczky@finance.bme.hu Phone: +36 70 259 9901, Fax +36 1 463 2745

*corresponding author This paper is appearing in Economic Modelling. Please cite this article as: Erdős P, Ormos M, Zibriczky D (2011) Non-parametric and semi-parametric asset pricing. Economic Modelling 28:(3) pp. 1150-1162., DOI: 10.1016/j.econmod.2010.12.008 This is the pre-print version of our accepted paper before typesetting.

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Non-Parametric and Semi-Parametric Asset Pricing

Abstract We find that the CAPM fails to explain the small firm effect even if its non-parametric form is used which allows time-varying risk and non-linearity in the pricing function. Furthermore, the linearity of the CAPM can be rejected, thus the widely used risk and performance measures, the beta and the alpha, are biased and inconsistent. We deduce semi-parametric measures which are non-constant under extreme market conditions in a single factor setting; on the other hand, they are not significantly different from the linear estimates of the Fama-French three-factor model. If we extend the single factor model with the Fama-French factors, the simple linear model is able to explain the US stock returns correctly.

Keywords: Asset pricing; Kernel regression; Risk measures; Semi-parametric models; Non- parametric models JEL classification numbers: C14; C51; G12

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1. Introduction The Capital Asset Pricing Model (CAPM, Sharpe, 1964; Lintner, 1965; Mossin, 1966) is the most broadly applied equilibrium model in financial literature and among practitioners. However, despite its success, there are many studies denying the validity of the CAPM. Among others, Banz (1981), Basu (1983), Bhandari (1988) and Fama and French (1995) provide evidence against the CAPM. Researchers try to override the problem of the CAPM by alternative models which have fallen in three classes: (1) Multifactor extensions of the linear model such as the Merton’s ICAPM (1973) or Ross’ (1976) APT; (2) conditional versions of the linear model allowing time-variation in the market risk; (3) non-linear asset pricing models. The Fama-French three-factor-model (Fama and French, 1996) extends the single factor model adding two risk factors, one for the small firm effect and one for the ‘relative distress’. Carhart (1997) involves the momentum factor which is also priced besides the three Fama-French factors. Keim and Stambaugh (1986) and Breen et al. (1989) show that conditional betas are not constant. While Fama and French (1989), Chen (1991), and Ferson and Harvey (1991) prove that betas vary over the business cycles. Ferson (1989), Ferson and Harvey (1991, 1993), Ferson and Korajczyk (1995), and Jaganathan and Wang (1996), among others, provide further evidence that the market risks vary from time to time. Jagannathan and Wang (1996) formalize a conditional CAPM which exhibits some empirical evidence that betas are time-varying. However, they argue that firm size effect is not significant in this setting. Zhang (2006) finds that the conditional international CAPM with exchange risk provides the least pricing error. However, Stapleton and Subrahmanyam (1983) verify a linear relationship between risk and return in the case of the CAPM; there are many studies that contradict this result. Barone Adesi (1985) proposes a quadratic three-moment CAPM. Bansal and Viswanathan (1993) show that a non-linear, two-

3 factor model, extending the market risk factor with the one-period yield in the next period, outperforms the CAPM. Bansal et al. (1993) shows that a non-linear arbitrage-pricing model is superior over the linear conditional, and the linear unconditional models, for pricing international equities, bonds and forward currency contracts. Chapman (1997) argues that non-linear pricing kernel in CCAPM performs better than the standard CAPM. Dittmar (2002) argues that a cubic pricing kernel induces much less pricing error than a linear one. Asgharian and Karlsson (2008) confirm the pricing ability of the non-linear model suggested by Dittmar (2002) on international equity, allowing time varying risk prices. Akdeniz et al. (2002) elaborate a new non-linear approach that allows time-varying betas and is called threshold CAPM. The model allows beta to change when the threshold variable hits a certa

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