Journal of Evolutionary Economics

, Volume 23, Issue 3, pp 609–639 | Cite as

Time-varying beta: a boundedly rational equilibrium approach

Regular Article


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.


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

JEL Classification

G12 D84 



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.


  1. Abel A (2002) An exploration of the effects of pessimism and doubt on asset returns. J Econ Dyn Control 26:1075–1092CrossRefGoogle Scholar
  2. Adrian T, Franzoni F (2005) Learning about beta: time-varying factor loadings, expected returns, and the conditional CAPM. Working paper, Federal Reserve Bank of New YorkGoogle Scholar
  3. Alfarano S, Lux T, Wagner F (2005) Estimation of agent-based models: the case of an asymmetric herding model. Comput Econ 26:19–49CrossRefGoogle Scholar
  4. Anderson P, Arrow K, Pines D (1988) The economiy as an evolving complex system II. Addison-WelseyGoogle Scholar
  5. Ang A, Chen J (2007) CAPM over the long run: 1926–2001. J Empir Finance 14:1–40CrossRefGoogle Scholar
  6. Anufriev M, Bottazzi G, Pancotto F (2006) Equilibria, stability and asymptotic dominance in a speculative market with heterogeneous traders. J Econ Dyn Control 30(9–10):1787–1835CrossRefGoogle Scholar
  7. Anufriev M, Dindo P (2010) Wealth-driven selection in a financial market with heterogeneous agents. J Econ Behav Organ 73:327–358CrossRefGoogle Scholar
  8. Arthur W, Holland J, LeBaron B, Palmer R, Tayler P (1997) Asset pricing under endogeneous expectations in an artificial stock market. Econ Notes 26(2):297–330Google Scholar
  9. Basak S (2000) A model of dynamic equilibrium asset pricing with heterogeneous beliefs and extraneous beliefs. J Econ Dyn Control 24:63–95CrossRefGoogle Scholar
  10. Bernanke BS, Gertler M (1989) Agency costs, net worth, and business fluctuations. Am Econ Rev 79:14–31Google Scholar
  11. Bernanke BS, Gertler M, Gilchrist S (1999) Handbook of macroeconomics. In: Taylor JB, Woodford M (eds) The financial accelerator in a quantitative business cycle framework. Elsevier, pp 1341–1393Google Scholar
  12. Blume L, Easley D (1992) Evolution and market behavior. J Econ Theory 58:9–40CrossRefGoogle Scholar
  13. Böhm V, Chiarella C (2005) Mean variance preferences, expectations formation, and the dynamics of random asset prices. Math Financ 15:61–97CrossRefGoogle Scholar
  14. Böhm V, Wenzelburger J (2005) On the performance of efficient portfolios. J Econ Dyn Control 29:721–740CrossRefGoogle Scholar
  15. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31:307–327CrossRefGoogle Scholar
  16. Bollerslev T (1990) Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Rev Econ Stat 72(3):498–505CrossRefGoogle Scholar
  17. Bollerslev T, Engle R, Wooldridge J (1988) A capital asset pricing model with time varying covariances. J Polit Econ 96:116–131CrossRefGoogle Scholar
  18. Bos T, Newbold P (1984) An empirical investigation of the possibility of systematic stochastic risk in the market model. Journal of Business 57:35–41CrossRefGoogle Scholar
  19. Bottazzi G, Dindo P (2010) Evolution and market behavior with endogeneous investment rules. LEM and CAFED Working Paper 2010/20, Scuola Superiore Sant’Anna, Pisa, ItalyGoogle Scholar
  20. Braun P, Nelson D, Sunier A (1990) Good news, bad news, volatility and betas. J Finance 50:1575–1603CrossRefGoogle Scholar
  21. Brock W (1993) Pathways to randomness in the economy: emergent non-linearity and chaos in economics and finance. Estud Econ 8:3–55Google Scholar
  22. Brock W, Hommes C (1997) A rational route to randomness. Econometrica 65:1059–1095CrossRefGoogle Scholar
  23. Brock W, Hommes C (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J Econ Dyn Control 22:1235–1274CrossRefGoogle Scholar
  24. Campbell J, Vuolteenaho T (2004) Bad beta, good beta. Am Econ Rev 94(5):1249–1275CrossRefGoogle Scholar
  25. Carlstrom CT, Fuerst TS (1998) Agency costs and business cycles. Econ Theory 12:583–597CrossRefGoogle Scholar
  26. Chiarella C, Dieci R, Gardini L (2005) The dynamic interaction of speculation and diversification. Appl Math Financ 12(1):17–52CrossRefGoogle Scholar
  27. Chiarella C, Dieci R, He X (2007) Heterogeneous expectations and speculative behaviour in a dynamic multi-asset framework. J Econ Behav Organ 62:402–427CrossRefGoogle Scholar
  28. Chiarella C, Dieci R, He X (2009) Heterogeneity, market mechanisms and asset price dynamics. In: Hens T, Schenk-Hoppe KR (eds) Handbook of financial markets: dynamics and evolution Elsevier, pp 277–344Google Scholar
  29. Chiarella C, Dieci R, He X (2010) A framework for CAPM with heterogeneous beliefs. In: Bischi G-I, Chiarella C, Gardini L (eds) Nonlinear dynamics in economics, finance and social sciences: essays in honour of John Barkley Rosser Jr., Springer, pp 353–369Google Scholar
  30. Chiarella C, Dieci R, He X (2011) Do heterogeneous beliefs diversify market risk?. Eur J Financ 17(3):241–258CrossRefGoogle Scholar
  31. Chiarella C, He X (2001) Asset price and wealth dynamics under Heterogeneous expectations. Quantitative Finance 1:509–526CrossRefGoogle Scholar
  32. Chiarella C, He X (2002) Heterogeneous beliefs, risk and learning in a simple asset pricing model. Comput Econ 19:95–132CrossRefGoogle Scholar
  33. Chiarella C, He X (2003) Dynamics of beliefs and learning under a l-processes – the heterogeneous case. J Econ Dyn Control 27:503–531CrossRefGoogle Scholar
  34. Chiarella C, He X, Hommes C (2006) A dynamic analysis of moving average rules. J Econ Dyn Control 30:1729–1753CrossRefGoogle Scholar
  35. Chiarella C, He X, Zheng M (2011) An analysis of the effect of noise in a heterogeneous agent financial market model. J Econ Dyn Control 35:148–162CrossRefGoogle Scholar
  36. Chordia T, Shivakumar L (2002) Momentum, business cycle, and time-varying expected returns. J Finance 57:985–1019CrossRefGoogle Scholar
  37. Collins D, Ledolter J, Rayburn J (1987) Some further evidence on the stochastic properties of systematic risk. Journal of Business 60(3):425–448CrossRefGoogle Scholar
  38. Day R, Huang W (1990) Bulls, bears and market sheep. J Econ Behav Organ 14:299–329CrossRefGoogle Scholar
  39. Detemple J, Murthy S (1994) Intertemporal asset pricing with heterogeneous beliefs. J Econ Theory 62:294–320CrossRefGoogle Scholar
  40. Dybvig P, Ross S (1985) Differential information and performance measurement using a security market line. J Finance 40:383–400CrossRefGoogle Scholar
  41. Engle R (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica 50:987–1008CrossRefGoogle Scholar
  42. Fabozzi F, Francis J (1978) Beta as a random coefficient. J Financ Quant Anal 13(1):101–106CrossRefGoogle Scholar
  43. Fama E, French K (1993) Common risk factors in the returns on stocks and bonds. J Financ Econ 33:3–56CrossRefGoogle Scholar
  44. Fama E, French K (2006) The value premium and the CAPM. J Finance 61(5):2163–2185CrossRefGoogle Scholar
  45. Farmer J, Gillemot L, Lillo F, Mike S, Sen A (2004) What really causes large price changes. Quantitative Finance 4:383–397CrossRefGoogle Scholar
  46. Ferson WE, Kandel S, Stambaugh RF (1999) Tests of asset pricing with time varying expected risk premiums and market betas. J Finance 42:201–220CrossRefGoogle Scholar
  47. Ferson WE, Siegel AF (1998) Stochastic discount factor bounds with conditioning information. Working paper, University ofWashingtonGoogle Scholar
  48. Ferson W, Harvey C (1991) The variation of economic risk premiums. J Polit Econ 99:385–415CrossRefGoogle Scholar
  49. Ferson W, Harvey C (1999) Conditioning variables and the cross-section of stock returns. J Finance 54(4):1325–1360CrossRefGoogle Scholar
  50. Gaunersdorfer A, Hommes C (2007) A nonlinear structural model for volatility clustering. In: Teyssiere G, Kirman A (eds) Long memory in economics. Springer, Berlin/Heidelberge, pp 265–288CrossRefGoogle Scholar
  51. Ghysels E (1998) On stable factor structures in the pricing of risk: Do time varying betas help or hurt. J Finance 53:549–574CrossRefGoogle Scholar
  52. Grundy BD, Martin JS (2001) Understanding the nature of the risks and source of the rewards to momentum investing. Rev Financ Stud 14:29–78CrossRefGoogle Scholar
  53. Hamilton J (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2):357–384CrossRefGoogle Scholar
  54. Hamilton J (1990) Analysis of time series subject to changes in regime. J Econom 45:39–70CrossRefGoogle Scholar
  55. Hansen L, Richard S (1987) The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models. Econometrica 55:587–613CrossRefGoogle Scholar
  56. Harvey C (2001) The specification of conditional expectations. J Empir Finance 8:573–638CrossRefGoogle Scholar
  57. He X, Li Y (2007) Power law behaviour, heterogeneity, and trend chasing. J Econ Dyn Control 31:3396–3426CrossRefGoogle Scholar
  58. Heckman J (2001) Micro data, heterogeneity, and evaluation of public policy: Nobel lecture. J Polit Econ 109:673–748CrossRefGoogle Scholar
  59. Hennessy CA, Whited TM (2007) How costly is external financing? evidence from a structural estimation. J Finance 52:1705–1745CrossRefGoogle Scholar
  60. Hens T, Schenk-Hoppe KR (2009) Handbook of financial markets: dynamics and evolution. Handbooks in Finance, ElsevierGoogle Scholar
  61. Hommes C (2006) Heterogeneous agent models in economics and finance. In: Tesfatsion L, Judd KL (eds) Agent-based computational economics. Handbook of Computational Economics, vol 2. North-Holland, pp 1109–1186Google Scholar
  62. Horst U, Wenzelburger J (2008) On non-ergodic asset prices. Econ Theory 34(2):207–234CrossRefGoogle Scholar
  63. Huang C-F, Litzenberger R (1988) Foundations for financial economics. Elsevier, North-HollandGoogle Scholar
  64. Jagannathan R, Wang Z (1996) The conditional CAPM and cross-section of expected returns. J Finance 51:3–53CrossRefGoogle Scholar
  65. Jegadeesh N, Titman S (1993) Returns to buying winners and selling losers: implications for stock market efficiency. J Finance 48:65–91CrossRefGoogle Scholar
  66. Jegadeesh N, Titman S (2001) Profitability of momentum strategies: an evaluation of alternative explanations. J Finance 56:699–720CrossRefGoogle Scholar
  67. Kirman A (1992) Whom or what does the representative agent represent?. J Econ Perspect 6:117–136Google Scholar
  68. Kothari S, Shanken J, Sloan R (1995) Another look at the cross-section of expected stock returns. J Finance 50(1):185–224CrossRefGoogle Scholar
  69. LeBaron B (2006) Agent-based computational finance. In: Tesfatsion L, Judd KL (eds) Agent-based computational economics. Handbook of computational economics, vol 2. North-Holland, pp 1187–1233Google Scholar
  70. Lewellen J, Nagel S (2006) The conditional CAPM does not explain asset-pricing anomalies. J Financ Econ 82(3):289–314CrossRefGoogle Scholar
  71. Lintner J (1969) The aggregation of investor’s diverse judgements and preferences in purely competitive security markets. J Financ Quant Anal 4:347–400CrossRefGoogle Scholar
  72. Lux T (2004) Financial power laws: empirical evidence, models and mechanisms. In: Cioffi C (ed) Power laws in the social sciences: discovering complexity and non-equilibrium in the social universe. Cambridge University PressGoogle Scholar
  73. Rubinstein M (1974) An aggregation theorem for securities markets. J Financ Econ 1:225–244CrossRefGoogle Scholar
  74. Rubinstein M (1975) Security market efficiency in an arrow-debreu economy. Am Econ Rev 65:812–824Google Scholar
  75. Schwert G (1989) Why does stock market volatility change over time?. J Finance 44:1115–1154CrossRefGoogle Scholar
  76. Tesfatsion L, Judd K (2006) Agent-based computational economics, vol 2. Handbook of Computational Economics, ElsevierGoogle Scholar
  77. Wang KQ (2003) Asset pricing with conditional information: a new test. J Finance 58(1):161–196CrossRefGoogle Scholar
  78. Wenzelburger J (2004) Learning to predict rationally when beliefs are heterogeneous. J Econ Dyn Control 28:2075–2104CrossRefGoogle Scholar
  79. Westerhoff F (2004) Multiasset market dynamics. Macroecon Dyn 8:591–616Google Scholar
  80. Westerhoff F, Dieci R (2006) The effectiveness of Keynes-Tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach. J Econ Dyn Control 30:293–322CrossRefGoogle Scholar
  81. Williams J (1977) Capital asset prices with heterogeneous beliefs. J Financ Econ 5:219–239CrossRefGoogle Scholar
  82. Zapatero F (1998) Effects of financial innovations on market volatility when beliefs are heterogeneous. J Econ Dyn Control 22:597–626CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.UTS Business SchoolUniversity of TechnologySydneyAustralia
  2. 2.Department of Mathematics for Economics and Social SciencesUniversity of BolognaBolognaItaly

Personalised recommendations