# Time-varying beta: a boundedly rational equilibrium approach

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## Abstract

The conditional CAPM with time-varying betas has been widely used to explain the cross-section of asset returns. However, most of the literature on time-varying beta is motivated by econometric estimation using various latent risk factors rather than explicit modelling of the stochastic behaviour of betas through agents’ behaviour, such as momentum trading. Misspecification of beta risk and the lack of any theoretical guidance on how to specify risk factors based on the representative agent economy appear empirically challenging. In this paper, we set up a dynamic equilibrium model of a financial market with boundedly rational and heterogeneous agents within the mean-variance framework of repeated one-period optimisation and develop an explicit dynamic behaviour CAPM relation between the expected equilibrium returns and time-varying betas. By incorporating the two most commonly used types of investors, fundamentalists and chartists, into the model, we show that there is a systematic change in the market portfolio, risk-return relationships, and time varying betas when investors change their behaviour, such as the chartists acting as momentum traders. In particular, we demonstrate the stochastic nature of time-varying betas. We also show that the commonly used rolling window estimates of time-varying betas may not be consistent with the ex-ante betas implied by the equilibrium model. The results provide a number of insights into an understanding of time-varying beta.

## Keywords

Equilibrium asset prices CAPM Time-varying betas Heterogeneous expectations Fundamentalism Momentum traders## JEL Classification

G12 D84## Notes

### Acknowledgements

We would like to thank Alan Kirman and Cars Hommes for helpful comments as well as conference participants at WEHIA 2006 (Bologna), COMPLEXITY 2006 (Aix-en-Provence), CEF 2006 (Cyprus), MDEF08 (Urbino), and the 2009 Workshop on Evolution and Market Behavior in Economics and Finance (Pisa) for helpful comments and suggestions. In particular we would like to thank the editors of this special issue, Giulio Bottazzi and Pietro Dindo, and three referees for their helpful comments and valuable suggestions which have significantly improved the paper. The usual caveat applies. Financial support for Chiarella and He from the Australian Research Council (ARC) under Discovery Grant (DP0773776) is gratefully acknowledged. Dieci acknowledges support from MIUR under the project PRIN-2004137559.

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