Conditional risk–return relationship in a time-varying beta model

Abstract

We investigate the asymmetric risk–return relationship in a time-varying beta CAPM. A state space model is established and estimated by the Adaptive Least Squares with Kalman foundations proposed by McCulloch. Using S&P 500 daily data from 1987:11–2003:12, we find a positive risk–return relationship in the up market (positive market excess returns) and a negative relationship in the down market (negative market excess returns). This supports the argument of Pettengill et al., who use a constant beta model. However, our model outperforms theirs by eliminating the unexplained returns and improving the accuracy of the estimated risk price.

Keywords:

• Capital asset pricing
• Time varying parameter models
• Asymmetric risk-return relationship

Acknowledgements

We are grateful to the editors, the referees, Glenn Pettengill, Ajay Samant, Sri Sundaram, and Mark Wheeler for helpful comments. Naturally, all remaining errors are ours.

Notes

†Many studies apply the PSM methodology to other markets. These include: Fletcher (1997, 2000) and Hung et al. (2004) for the UK market; Isakov (1999) for the Swiss market; Lam (2001) and Ho et al. (2006) for the Hong Kong market; Elsas et al. (2003) for the German market; Hodoshima et al. (2000) for the Japanese market; Faff (2001) for the Australian market; and Sandoval and Saens (2004) for four Latin American markets. The overwhelming preponderance of these studies supports the PSM conclusion.

†The purpose of the first step is to reduce the ‘errors-in-the-variables problem’, because the portfolio betas are supposed to be more precise estimates of true betas than the individual stock betas. The second step estimates the portfolio betas in a fresh, subsequent period in order to minimize the ‘regression problem’ that positive and negative sampling errors are bunched within portfolios. See Fama and MacBeth (1973) for a detailed discussion.

‡Petersen (2005) claims that the Fama–MacBeth methodology is superior to a pool time-series/cross-section estimation. In finance applications, residuals of a given year may be correlated across firms (cross-sectional dependence), which is called the time effect. He argues that the Fama–MacBeth methodology is designed to address the time effect and shows that the standard errors in the Fama–MacBeth methodology are unbiased in the presence of the time effect.

§Note that PSM regress security returns on excess market returns while we regress excess security returns on excess market returns. Thus, the intercept estimated by PSM ought to be equal to the risk-free rate. This may be another reason why they do not do a zero-null hypothesis test on the constant term.

†A constant covariance matrix would lead to a constant gain coefficient, which is not desirable. See McCulloch (2006) for details.

‡On the contrary, previous ALS studies arbitrarily set the initial values of the parameters. The ALSKF circumvents this problem and provides a simple but rigorous initialization.

§Fraser et al. (2004) and Galagedera and Faff (2004) also use industry-sorted portfolios to test the PSM model.

†We recognize the possibility that using larger and more mature firms in the S&P 500 may bias the results, and the fact that eliminating firms without complete data reduces the sample size. However, the current data selection criteria allow us to track the industrial classification codes for the stocks more easily and precisely. Using stocks with the same sample size also greatly simplifies the already-complicated Kalman filter estimations.

‡For example, Fraser et al. (2004) find that the betas estimated prior to October 1987 are different from the beta estimated immediately after the crash. They argue that adding the one single October 1987 observation to the estimation completely changed the forecast of the beta risk.

†As mentioned earlier, PSM have a single constant term and do not distinguish the intercepts in the up and down markets. Using our data to estimate such a single-constant setup shows an insignificant constant term. However, by theory the intercept should be zero no matter whether it is an up market or a down market. Therefore, our specification proves the advantage of our time-varying beta model over the PSM model.