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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.

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Notes

  1. For example, for book-to-market portfolios (constructed based on a book-to-market trading strategy that goes long the highest decile portfolio of stocks sorted on book-to-market ratios (value stocks) and short the lowest decile portfolio of book-to-market ratio stocks (growth stocks)), betas of the highest decile of book-to-market stocks reached over 2.5 during the 1940s and fell to − 0.5 at the end of 2001; see for example Kothari et al. (1995), Campbell and Vuolteenaho (2004) and Adrian and Franzoni (2005).

  2. More recently, Harvey (2001) used instrumental variables to estimate betas and showed that the estimates are very sensitive to the choice of instruments used to proxy for time-variation in the conditional betas.

  3. By questioning the conventional wisdom that there exists strong evidence of a book-to market effect (that stocks with high book-to-market ratios have higher average returns than what the CAPM predicts), Ang and Chen (2007) developed a methodology for consistently estimating time-varying betas in a conditional CAPM and found that a single-factor model performs substantially better in explaining the book-to-market premium.

  4. For example, Jegadeesh and Titman (2001) and Grundy and Martin (2001) attribute the momentum profits to stock-specific returns, and Chordia and Shivakumar (2002) attribute the profits to cross-sectional dispersion in expected returns or persistence in expected returns.

  5. See for example Anderson et al. (1988), Brock (1993), Arthur et al. (1997), Tesfatsion and Judd (2006).

  6. See, for example, Lintner (1969), Williams (1977), Huang and Litzenberger (1988), Detemple and Murthy (1994), Zapatero (1998), Basak (2000) and Abel (2002).

  7. See, for example, Day and Huang (1990), Kirman (1992), Farmer et al. (2004), Lux (2004), Chiarella et al. (2006), Alfarano et al. (2005), Gaunersdorfer and Hommes (2007), and He and Li (2007). We refer the reader to Tesfatsion and Judd (2006) and Hens and Schenk-Hoppe (2009), in particular, Hommes (2006), LeBaron (2006) and Chiarella et al. (2009) for surveys of recent literature on HAMs.

  8. Recent studies with many risky assets include Wenzelburger (2004), Westerhoff (2004), Böhm and Chiarella (2005), Böhm and Wenzelburger (2005), Chiarella et al. (2005, 2007), Westerhoff and Dieci (2006) and Horst and Wenzelburger (2008), showing that complex price dynamics may also result within a multi-asset market framework with heterogeneous beliefs. Chiarella et al. (2007) show that diversification does not always have a stabilizing role, but may act as a further source of instability in the financial market. Wenzelburger (2004) introduces a reference portfolio and Böhm and Wenzelburger (2005) show that the returns realized with an efficient portfolio do not necessarily outperform those of non-efficient portfolios. By allowing social interaction among consumers, Horst and Wenzelburger (2008) show that asset prices may behave in a non-ergodic manner.

  9. The current paper is not related to the time-varing consumer tastes literature. The risk tolerance in market equilibrium can be used to measure consumer tastes. The model we develop is based on CARA utility function and constant market fractions of different types of investors. Hence the risk tolerance in market equilibrium is a constant. By assuming either a CRRA utility function or the evolution of market fractions, the risk tolerance in market equilibrium can be time-varying, a more complicated case which we leave for future research.

  10. Schwert (1989) conducts one of the first detailed studies of the effects of the business cycle on stock returns. More recent efforts in the capital structure literature that attempt to link the corporate investment and leverage policies of firms includes Hennessy and Whited (2007) who introduce firms’ exposure to systematic risk, thus allowing for business cycle variation in financing, investment and default decisions.

  11. In Bernanke and Gertler (1989), Carlstrom and Fuerst (1998), and Bernanke et al. (1999), the heterogeneity stems from idiosyncratic productivity and borrowers’ idiosyncratic ability to repay their loans since with no cross-sectional heterogeneity of borrowers’ ability to repay, there is no risk at all from the point of view of lenders, and hence no financial friction.

  12. One of the implications of the CARA utility function is that the investor’s optimal demand is independent of his/her wealth, which simplifies the analysis significantly. Instead, a CRRA utility function would imply that the optimal decision depends on investor’s wealth and hence the market equilibrium involves both prices and the investor’s wealth shares, see for example Chiarella and He (2001). In this case, the characterization of market equilibrium becomes more complicated and wealth-driven market selection becomes an important issue, see for example Blume and Easley (1992). When investors have heterogeneous beliefs or demand functions, we refer the reader to Anufriev et al. (2006), Anufriev and Dindo (2010) and Bottazzi and Dindo (2010).

  13. In this paper, in order to make clear the various relationships we focus on the case of constant market fractions, rather than time varying dynamics of the market fractions driven by some fitness function as in Brock and Hommes (1998). Thus study we leave to a future paper.

  14. The concept of a consensus belief was initially developed in Lintner (1969) and Rubinstein (1974) where sufficient conditions under which equilibrium returns are determined as if there exists an representative investor whose endowment, beliefs, and tastes are a composite of the actual individuals are given. Here we follow the construction in Chiarella et al. (2010).

  15. It follows from \(I\theta_a^{-1}=I\sum_{h=1}^H n_h \theta_h^{-1}=\sum_{i=1}^I \theta_i^{-1}\) that the average risk aversion is a harmonic mean of the risk aversion of H agents types weighted by their market fractions. In the case when the market fractions evolve over time, the average risk aversion becomes time-varying. In the case when all agents have the same risk aversion coefficient, it also corresponds to the average risk aversion; this is what we assume for the numerical analysis in Section 4.

  16. Based on the assumptions, agents within the same group h have the same optimal portfolio ζ h,t , but they may differ in terms of their wealth (and optimal consumption).

  17. The chartists may or may not have information on the fundamental values, however in this paper we assume that they form their expectations based on historical returns.

  18. When the fundamentalists are engaged some learning processes, the parameter α would be time-varying. This realistic feature of α would make the analysis more complicated and we leave this task as future research.

  19. Under no arbitrage argument, whether a normally distributed dividend yield leads to a random walk fundamental price process is a subtle issue not discussed here.

  20. Note that this simplification is not realistic, in particular when the fundamentalists learn from market data. However we maintain this simplification in order bring out more clearly the impact of the chartists on the market equilibrium and time-varying beta.

  21. Given that the returns contain both capital gain and dividend yield components, the trend followers also take dividends into account implicitly. However, since the dividends are paid only once/twice a year, the trend followers are more concerned about the capital gain rather than the dividend yield.

  22. Compared to the fundamentalists, the trend followers take advantage of market excess volatility when extrapolating their expected return from historical returns, they also update their variance/covariance estimate consistently, which is reflected by the time-varying component in \({\boldsymbol{\Omega}}_{c,t}\).

  23. The connections between the fundamental price process and the demand noise process will be pointed out in the next section.

  24. Note also from Eq. 3.8 that conditional variances of returns are not constant, which is reflected in the assumption of the martingale fundamental price process. One way to make this consistent with the constant covariance of the returns is to assume that the conditional variance of the demand shock depend on the fundamental price.

  25. In some cases, this indirect approach to deal with the stochastic model may not fully characterise the nature of the stochastic model and we refer the reader to Chiarella et al. (2011) for a recent discussion.

  26. Among the key parameters of the model, a decrease of either the market fraction of the fundamentalists, n f , or the speed of the mean reversion, α, has a similar effect to a reduction of δ. The effect of λ is less clear. It is associated with the second moment, hence it does not affect the local stability of the steady state. However, an increase in λ seems to be associated with more and more irregular fluctuations once the steady state is destabilized. A general discussion on the impact of various parameters will be given in Section 5.

  27. Assuming heterogeneous risk aversion does not imply substantial differences with respect the main message of this paper (though this may not be the case in general, see for example Chiarella and He (2002)), except that fundamental prices become affected by the distribution of agent-types (i.e. by parameter n f ).

  28. If δ is the annualised chartist parameter, the average memory length of chartist weighted average return (3.2) is given by \(\mathcal{ \ell }:=\sum_{s=0}^{\infty }s(1-\delta )\delta ^{s}=\delta /(1-\delta )\) years. If K is a different time frequency (so that expectations are updated K times per year), the chartist parameter is converted according to δ (K) = Kδ/[1 + (K − 1)δ], which doesn’t affect the original average memory length, as can easily be checked.

  29. For interpretation of the references to color in figures, the reader is referred to the web version of this article.

  30. As can be argued from Fig. 1, and reported by related studies (Chiarella et al. 2005, 2007), in such multi-asset models the attractor developing from the local bifurcation may be entirely located, initially, in a lower dimensional ‘manifold’ of the phase space. As a consequence, we may observe systematic fluctuations of prices and returns of some assets, while other assets still remain at their steady state levels.

  31. For different selections of risk and return parameters, it is possible to observe qualitatively similar situations where asset 2, or asset 3 is the ‘first’ to be destabilized. Numerical simulation reveals that this may depend in a quite complicated way on risk-return tradeoffs.

  32. Note that in the simulations we also assume \({\bf{p}}_{0}={\bf{p}}^{\ast }\), and therefore the deterministic fundamental price p  ∗  represents the unconditional mean of the price process.

  33. Note that a common practice in empirical work involving ‘rolling’ betas, is represented by rolling OLS estimation over 60 months (5 years), though different choices can be found in the literature. In particular, the use of monthly returns appears to reduce the impact of transaction costs. Such issues are not considered within our stylized model. Therefore, we may use higher frequencies and time windows of different length, to illustrate the time-varying nature of the betas, and to emphasize the impact of model parameters on their dynamic patterns. Note also that a general feature of the rolling betas is a slow time variation, not necessarily mean-reverting to some fixed level, but not monotonic either.

  34. The relatively small amplitude of fluctuations of the price and return on the market portfolio contrasts with the large fluctuations of asset returns in the post-shock phase, which reveals that agents are actively trading in a way to exploit emerging correlations among the risky assets, to reduce portfolio risk.

  35. Note that large market noise may considerably affect the dynamics and determine large shifts of the betas, even reversing their ordering. One reason is that under our parameter calibration (see Appendix) agents’ second moment beliefs Ω0 are somehow consistent with the base level of market noise (q = 0.005 in this example). In Fig. 5c, however, the increased market noise has no counterpart in Ω0, and this fact may produce large deviations of the betas away from their ‘steady state’ levels.

  36. Fundamentalist proportion n f has a more ambiguous effect, in general. For parameter n f ranging in the interval [0.2, 0.35], we typically observe more dispersed betas for intermediate values, and less dispersed betas at the extremes of the interval.

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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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Authors and Affiliations

  1. UTS Business School, University of Technology, PO Box 123 Broadway, Sydney, NSW, 2007, Australia

    Carl Chiarella & Xue-Zhong He

  2. Department of Mathematics for Economics and Social Sciences, University of Bologna, Viale Q. Filopanti 5, 40126, Bologna, Italy

    Roberto Dieci

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  1. Carl Chiarella
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Correspondence to Xue-Zhong He.

Appendix: some details on parameter calibration

Appendix: some details on parameter calibration

This Appendix provides some details about the parameter calibration in the numerical simulations of Sections 4 and 5. In particular, it points out the connections between the parameters characterizing demand noise (i.e. the fundamental price process), dividend yield noise and matrix Ω0 representing agents’ second-moment beliefs. For j = 1, 2, 3, we denote by

$$ \sigma_{\xi,j}^{2}:=Var_{t-1}\left(\xi_{j,t}-\xi_{j,t-1}\right) $$

the (constant) conditional variance of the noise process impacting on the demand of asset j. Since \(\mathbf{p}_{t}^{\ast}-\mathbf{p}_{t-1}^{\ast }={\boldsymbol{\epsilon}}_{t}=\mathbf{S}^{-1}({\boldsymbol{\xi}}_{t}-{\boldsymbol{\xi}} _{t-1})\), the conditional variance of the change in the fundamental price of asset j is thus given by

$$ Var_{t-1}(\epsilon_{j,t})=Var_{t-1}\left(p_{j,t}^{\ast}-p_{j,t-1}^{\ast}\right)=\frac {1}{s_{j}^{2}}\sigma_{\xi,j}^{2} $$

where s j is the fixed supply of asset j. The noise terms ϵ j,t (and ξ j,t  − ξ j,t − 1) are assumed to be i.i.d. normal in our numerical simulation. We also assume zero cross correlation between the demand shocks affecting different assets, as well as between their dividend yields. The standard deviation of the demand shock for asset j is assumed to be proportional to its average (i.e. ‘steady state’) demand, namely \(\sigma_{\xi,j}=qs_{j}p_{j}^{\ast}\) for q > 0, from which \( Var_{t-1}(\epsilon_{j,t})=q^{2}(p_{j}^{\ast})^{2}\). In the paper, the parameter q (and the standard deviation σ ξ,j ) represents an annualised parameter at the annualised time step. Due to the ‘random walk’ nature of demand noise (and fundamental prices used), the parameter q is converted to shorter time steps in the same way as the other standard deviation parameters, so that \(q/\sqrt{K}\) is the parameter rescaled to 1/K years (although we always use the same notation q irrespective of the time frequency).

Under the above parameter restrictions, the conditional variance of the return process in the ‘benchmark CAPM’ can be written as

$$ Var_{t-1}\left(r_{j,t}^{\ast}\right)=\frac{1}{\left(p_{j,t-1}^{\ast}\right)^{2}}Var_{t-1} \left(\epsilon_{j,t}\right)+Var_{t-1}\left(\rho_{j,t}\right)=\left( \frac{p_{j}^{\ast}} {p_{j,t-1}^{\ast}}\right) ^{2}q^{2}+\sigma_{\rho,j}^{2} $$

where \(\sigma_{\rho,j}^{2}=Var_{t-1}(\rho_{j,t})=Var(\rho_{j,t})\) is the variance of the i.i.d. dividend yield of asset j. This implies that the conditional variance of returns is not a constant in the simulated benchmark CAPM case, whereas for simplicity we have assumed constant second-moment beliefs in this case (matrix \(\Omega_{0}:=\emph{diag}(\sigma_{1}^{2} ,\sigma_{2}^{2},\sigma_{3}^{2})\)). For the sake of consistency, in order to keep the constant matrix Ω0 as close as possible to the actual variance/covariance return matrix in the ‘benchmark CAPM’ scenario, we set variance beliefs \(\sigma_{j}^{2}\) as \( \sigma_{j}^{2}=q^{2}+\sigma_{\rho,j}^{2}\) for j = 1, 2, 3.

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Chiarella, C., Dieci, R. & He, XZ. Time-varying beta: a boundedly rational equilibrium approach. J Evol Econ 23, 609–639 (2013). https://doi.org/10.1007/s00191-011-0233-5

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