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This paper quantitatively examines a non-linear capital asset pricing model (CAPM) by using monthly stock returns of major automobile industry firms in Japan. Applying the maximum likelihood method, we derive the following interesting findings. (1) First, in the case where the distribution of stock returns has a fat-tail, our non-linear CAPM is highly effective. Because the parameters of our non-linear CAPM well capture fat-tailed return distributions, the non-linear model estimation derives reliable estimates of beta values. (2) Second, in the case where stock returns are normally distributed, our non-linear CAPM is also effective. Since the parameters of our non-linear CAPM also well capture normally distributed returns by adjusting its degrees of freedom parameter value, the non-linear model estimation similarly derives reliable beta estimates as those derived from the standard linear CAPM. (3) Finally, we further conduct the Wald tests based on the estimators from the standard CAPM and our non-linear CAPM, and we suggest that in the case where the distribution of stock returns has a fat-tail, the Wald test based on the estimators from our non-linear CAPM shall be more reliable than the Wald test based on the estimators from the standard linear CAPM.

A Non-linear Estimation of the Capital Asset Pricing Model: The Case of Japanese Automobile Industry Firms