Big Swings in the Data and Perceived Changes in the Risk Premia

Martin Sola; Fabio Spagnolo; Francisco Terfi

doi:10.1515/snde-2024-0118

Article

Big Swings in the Data and Perceived Changes in the Risk Premia

Martin Sola , Fabio Spagnolo and Francisco Terfi

Published/Copyright: July 22, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Studies in Nonlinear Dynamics & Econometrics

Abstract

Stock markets experience periods where stocks or market returns are consistently higher than their mean and other periods where the individual stocks and markets’ volatility fluctuates from high to low. Since these periods do not necessarily coincide, a related question is whether periods where individual stock markets are higher than their mean, usually identified as αs different from zero in the conditional regressions, disappear once the researcher accounts for changing states of the economy. In this spirit, we develop and estimate a state-dependent version of the CAPM pricing model that accounts for considerable swings in the data. We use U.S. financial data to assess the model’s validity and find support for a state-dependent version of the CAPM for the data under consideration. We show how important it is to consider changes in stock and market returns and changes in their variance-covariances, and that, when not accounting for changes in market conditions, may spuriously yield significant α values. We stress that to assess changes in the risk premium, we should not only focus on βs but also allow for changes in the market premium; otherwise, changes in risk premia may be over- or underestimated. In addition, the classification between investment opportunities may be mistaken for a single regime model, even when rolling regressions are used.

Keywords: non-diversifiable risk premium; Markov chain; structural breaks

JEL Classification: G00; G12; E44; C32

Corresponding author: Martin Sola, Department of Economics, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350 (C1428BCW), Ciudad de Buenos Aires, Argentina, E-mail: msola@utdt.edu

Acknowledgements

The authors thank Martin Gonzalez-Rozada, Constantino Hevia, Demian Pouzo, and Zacharias Psaradakis for helpful comments and discussions. The views expressed in this paper are solely the responsibility of the authors.

Appendix

Table 7:

Joint estimation of Equation (1).

Apple
μ 0 j = 0.05 μ 0 m = 0.068 ( 0.0649 ) ( 0.018 ) μ 1 j = 0.114 μ 1 m = 0.022 ( 0.033 ) ( 0.007 ) p 00 j = 0.858 p 00 m = 0.749 ( 0.108 ) ( 0.12 ) p 11 j = 0.976 p 11 m = 0.837 ( 0.031 ) ( 0.1 )
Ω 1 = 0.024 0.001 ( 0.008 ) ( 0.001 ) 0.001 0.001 ( 0.001 ) ( 0.001 ) Ω 2 = 0.01 0.004 ( 0.004 ) ( 0.002 ) 0.004 0.002 ( 0.002 ) ( 0.001 ) Ω 3 = 0.055 0.024 ( 0.014 ) ( 0.011 ) 0.024 0.015 ( 0.011 ) ( 0.006 ) Ω 4 = 0.018 − 0.015 ( 0.007 ) ( 0.007 ) − 0.015 0.015 ( 0.001 ) ( 0.003 )
Log-likelihood: 270.69		Akaike: −6.685
Citigroup
μ 0 j = − 0.363 μ 0 m = 0.019 ( 0.051 ) ( 0.020 ) μ 1 j = 0.003 μ 1 m = − 0.983 ( 0.048 ) ( 0.019 ) p 00 j = 0.985 p 00 m = 0.922 ( 0.011 ) ( 0.051 ) p 11 j = 0.819 p 11 m = 0.928 ( 0.147 ) ( 0.067 )
Ω 1 = 0.706 0.072 ( 0.309 ) ( 0.051 ) 0.072 0.010 ( 0.051 ) ( 0.008 ) Ω 2 = 0.065 0.022 ( 0.016 ) ( 0.006 ) 0.022 0.012 ( 0.006 ) ( 0.003 ) Ω 3 = 0.010 0.0004 ( 0.004 ) ( 0.002 ) 0.0004 0.001 ( 0.002 ) ( 0.0003 ) Ω 4 = 0.008 0.002 ( 0.001 ) ( 0.001 ) 0.002 0.002 ( 0.001 ) ( 0.0005 )
Log-likelihood: 279.279		Akaike: −6.914
Thermo Fisher Scientific
μ 0 j = 0.056 μ 0 m = − 0.002 ( 0.026 ) ( 0.034 ) μ 1 j = 0.066 μ 1 m = 0.026 ( 0.034 ) ( 0.023 ) p 00 j = 0.941 p 00 m = 0.825 ( 0.026 ) ( 0.147 ) p 11 j = 0.933 p 11 m = 0.705 ( 0.037 ) ( 0.293 )
Ω 1 = 0.0581 0.030 ( 0.0415 ) ( 0.019 ) 0.030 0.023 ( 0.019 ) ( 0.008 ) Ω 2 = 0.007 0.006 ( 0.005 ) ( 0.005 ) 0.006 0.006 ( 0.005 ) ( 0.005 ) Ω 3 = 0.011 0.008 ( 0.006 ) ( 0.005 ) 0.008 0.011 ( 0.005 ) ( 0.005 ) Ω 4 = 0.008 0.002 ( 0.002 ) ( 0.001 ) 0.002 0.001 ( 0.001 ) ( 0.0003 )
Log-likelihood: 312.778		Akaike: −7.807
Amazon
μ 0 j = 0.107 μ 0 m = 0.028 ( 0.033 ) ( 0.004 ) μ 1 j = 0.020 μ 1 m = 0.014 ( 0.028 ) ( 0.016 ) p 00 j = 0.919 p 00 m = 0.917 ( 0.043 ) ( 0.050 ) p 11 j = 0.952 p 11 m = 0.793 ( 0.026 ) ( 0.114 )
Ω 1 = 0.022 0.011 ( 0.011 ) ( 0.006 ) 0.011 0.006 ( 0.006 ) ( 0.003 ) Ω 2 = 0.017 0.001 ( 0.004 ) ( 0.0009 ) 0.001 0.0003 ( 0.004 ) ( 0.0001 ) Ω 3 = 0.034 0.009 ( 0.010 ) ( 0.005 ) 0.009 0.014 ( 0.005 ) ( 0.003 ) Ω 4 = 0.044 0.002 ( 0.013 ) ( 0.001 ) 0.002 0.0014 ( 0.001 ) ( 0.001 )
Log-likelihood: 259.099		Akaike: −6.376

Table 7 reports the maximum likelihood estimation of the joint equation model (1) for stock j and market m. The title of each panel indicates the stock used in the estimations. We report (in parenthesis) the standard error of the estimates.

Table 8:

Joint Estimation Equation (1) subject to the SD-CAPM Restrictions

Apple
μ 0 j = 0.071 μ 0 m = 0.008 ( 0.022 ) ( 0.007 ) μ 1 j = 0.026 μ 1 m = 0.029 ( 0.008 ) ( 0.006 ) p 00 j = 0.644 p 00 m = 0.859 ( 0.143 ) ( 0.062 ) p 11 j = 0.917 p 11 m = 0.912 ( 0.065 ) ( 0.042 )
Ω 1 = 0.045 0.015 ( 0.011 ) 0.015 0.014 ( 0.026 ) Ω 2 = 0.041 0.021 ( 0.015 ) 0.021 0.011 ( 0.015 ) Ω 3 = 0.013 0.002 ( 0.026 ) 0.002 0.001 ( 0.027 ) Ω 4 = 0.077 0.005 ( 0.028 ) 0.005 0.001 ( 0.028 )
Log-likelihood: 266.478 Akaike: −6.679
Citigroup
μ 0 j = 0.023 μ 0 m = 0.025 ( 0.010 ) ( 0.010 ) μ 1 j = 0.046 μ 1 m = 0.012 ( 0.030 ) ( 0.005 ) p 00 j = 0.928 p 00 m = 0.950 ( 0.020 ) ( 0.071 ) p 11 j = 0.945 p 11 m = 0.821 ( 0.068 ) ( 0.137 )
Ω 1 = 0.343 0.063 ( 0.111 ) 0.063 0.019 ( 0.042 ) Ω 2 = 0.014 0.0004 ( 0.039 ) 0.0004 0.0002 ( 0.008 ) Ω 3 = 0.062 0.022 ( 0.055 ) 0.022 0.012 ( 0.018 ) Ω 4 = 0.005 0.002 ( 0.005 ) 0.002 0.002 ( 0.010 )
Log-likelihood: 278.722 Akaike: −6.899
Thermo Fisher Scientific
μ 0 j = 0.036 μ 0 m = 0.098 ( 0.011 ) ( 0.002 ) μ 1 j = 0.006 μ 1 m = 0.010 ( 0.012 ) ( 0.005 ) p 00 j = 0.852 p 00 m = 0.867 ( 0.050 ) ( 0.128 ) p 11 j = 0.968 p 11 m = 0.768 ( 0.061 ) ( 0.240 )
Ω 1 = 0.032 0.019 ( 0.017 ) 0.019 0.019 ( 0.019 ) Ω 2 = 0.011 0.0001 ( 0.010 ) 0.0001 0.002 ( 0.006 ) Ω 3 = 0.007 0.002 ( 0.005 ) 0.002 0.002 ( 0.047 ) Ω 4 = 0.001 0.0003 ( 0.026 ) 0.003 0.009 ( 0.013 )
Log-likelihood: 311.154 Akaike: −7.764
Amazon
μ 0 j = 0.071 μ 0 m = 0.141 ( 0.022 ) ( 0.003 ) μ 1 j = 0.119 μ 1 m = 0.028 ( 0.009 ) ( 0.007 ) p 00 j = 0.751 p 00 m = 0.964 ( 0.261 ) ( 0.123 ) p 11 j = 0.876 p 11 m = 0.737 ( 0.187 ) ( 0.131 )
Ω 1 = 0.018 0.001 ( 0.032 ) 0.001 0.0002 ( 0.006 ) Ω 2 = 0.0001 0.00001 ( 0.011 ) 0.00001 0.00001 ( 0.005 ) Ω 3 = 0.031 0.003 ( 0.036 ) 0.003 0.003 ( 0.029 ) Ω 4 = 0.047 0.011 ( 0.028 ) 0.011 0.016 ( 0.019 )
Log-likelihood: 257.267 Akaike: −6.327

Table 8 reports the maximum likelihood estimation of the joint equation model (1) for stock j and market m with the four restrictions imposed by the SD-CAPM. The title of each panel indicates the stock used in the estimations. We report (in parenthesis) the standard error of the estimates.

References

Abdymomunov, A., and J. Morley. 2011. “Time Variation of CAPM Betas Across Market Volatility Regimes.” Applied Financial Economics 21 (19): 1463–78. https://doi.org/10.1080/09603107.2011.577010.Search in Google Scholar

Ang, A., and J. Chen. 2007. “CAPM over the Long Run: 1926-2001.” Journal of Empirical Finance 14 (1): 1–40. https://doi.org/10.1016/j.jempfin.2005.12.001.Search in Google Scholar

Avramov, D., and T. Chordia. 2006. “Asset Pricing Models and Financial Market Anomalies.” The Review of Financial Studies 19 (3): 1001–40. https://doi.org/10.1093/rfs/hhj025.Search in Google Scholar

Bali, T., and R. Engle. 2010. “Resurrecting the Conditional CAPM with Dynamic Conditional Correlations.” Working paper, New York University Stern School of Business. Available at SSRN: https://ssrn.com/abstract=1364865 or https://doi.org/10.2139/ssrn.136486.Search in Google Scholar

Bauer, R., M. Cosemans, and P. C. Schotman. 2010. “Conditional Asset Pricing and Stock Market Anomalies in Europe.” European Financial Management 16 (2): 165–90. https://doi.org/10.1111/j.1468-036x.2008.00453.x.Search in Google Scholar

Bodurtha, J., and M. Nelson. 1991. “Testing the CAPM with Time-Varying Risks and Returns.” Journal of Finance 46 (4): 1485–505.10.1111/j.1540-6261.1991.tb04627.xSearch in Google Scholar

Bollerslev, T., R. F. Engle, and J. M. Wooldridge. 1988. “A Capital Asset Pricing Model with Time-Varying Covariances.” Journal of Political Economy 96 (1): 116–31. https://doi.org/10.1086/261527.Search in Google Scholar

Davies, R. B. 1977. “Hypothesis Testing when a Nuisance Parameter is Present Only Under the Alternative.” Biometrika 64 (2): 247–54. https://doi.org/10.2307/2335690.Search in Google Scholar

Davies, R. B. 1987. “Hypothesis Testing when a Nuisance Parameter is Present Only Under the Alternatives.” Biometrika 74 (1): 33–43. https://doi.org/10.2307/2336019.Search in Google Scholar

Fama, E. F., and K. R. French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1): 3–56. https://doi.org/10.1016/0304-405x.Search in Google Scholar

Fama, E. F., and K. R. French. 1992. “The Cross-Section of Expected Stock Returns.” Journal of Finance 47 (2): 427–65. https://doi.org/10.2307/2329112.Search in Google Scholar

Fama, E. F., and K. R. French. 1996. “Multifactor Explanations of Asset Pricing Anomalies.” The Journal of Finance 51 (1): 55–84. https://doi.org/10.2307/2329302.Search in Google Scholar

Ferson, W. E., and C. R. Harvey. 1999. “Conditioning Variables and the Cross Section of Stock Returns.” The Journal of Finance 54 (4): 1325–60. https://doi.org/10.1111/0022-1082.00148.Search in Google Scholar

Hall, S. G., D. K. Miles, and M. P. Taylor. 1989. “Modelling Asset Prices with Time-Varying Betas.” The Manchester School of Economic & Social Studies 57 (4): 340–56. https://doi.org/10.1111/j.1467-9957.1989.tb00821.x.Search in Google Scholar

Hamilton, J. D. 1994. Time Series Analysis. Princeton University Press.10.1515/9780691218632Search in Google Scholar

Hansen, B. 1992. “The Likelihood Ratio Test Under Nonstandard Conditions: Testing the Markov Switching Model of GNP.” Journal of Applied Econometrics 7 (S1): S61–82. https://doi.org/10.1002/jae.3950070506.Search in Google Scholar

Hansen, B. 1996. “Erratum: the Likelihood Ratio Test Under Nonstandard Conditions: Testing the Markov Switching Model of GNP.” Journal of Applied Econometrics 11 (2): 195–8.10.1002/(SICI)1099-1255(199603)11:2<195::AID-JAE375>3.0.CO;2-2Search in Google Scholar

He, Z., F. O’Connor, and J. Thijssen. 2018. “Is Gold a Sometimes Safe Haven or an Always Hedge for Equity Investors? A Markov-Switching CAPM Approach for US and UK Stock Indices.” International Review of Financial Analysis 60: 30–7. https://doi.org/10.1016/j.irfa.2018.08.010.Search in Google Scholar

Huang, H. C. 2000. “Tests of Regimes - Switching CAPM.” Applied Financial Economics 10 (5): 573–8. https://doi.org/10.1080/096031000416451.Search in Google Scholar

Huang, H. C. 2001. “Tests of CAPM with Nonstationary Beta.” International Journal of Finance and Economics 6 (3): 255–68. https://doi.org/10.1002/ijfe.146.Search in Google Scholar

Jagannathan, R., and Z. Wang. 1996. “The Conditional CAPM and the Cross-Section of Expected Returns.” Journal of Finance 51 (1): 3–53. https://doi.org/10.2307/2329301.Search in Google Scholar

Lettau, M., and S. Ludvigson. 2001. “Resurrecting the (C)CAPM: A Cross-Sectional Test when Risk Premia Are Time-Varying.” Journal of Political Economy 109 (6): 1238–87. https://doi.org/10.1086/323282.Search in Google Scholar

Lewellen, J., and S. Nagel. 2003. “The Conditional CAPM Does Not Explain Asset-Pricing Anomalies.” Journal of Financial Economics 63: 289–314. https://doi.org/10.1016/j.jfineco.2005.05.012.Search in Google Scholar

Lintner, J. 1965. “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.” Review of Economics and Statistics 47 (1): 13–37. https://doi.org/10.2307/1924119.Search in Google Scholar

Markowitz, H. 1952. “Portfolio Selection.” The Journal of Finance 7 (1): 77–91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x.Search in Google Scholar

Pouzo, D., Z. Psaradakis, and M. Sola. 2022. “Maximum Likelihood Estimation in Markov Regime-Switching Models with Covariate-Dependent Transition Probabilities.” Econometrica 90 (4): 1681–710. https://doi.org/10.3982/ecta17249.Search in Google Scholar

Psaradakis, Z., and M. Sola. 1998. “Finite-Sample Properties of the Maximum Likelihood Estimator in Autoregressive Models with Markov Switching.” Journal of Econometrics 86 (2): 369–86. https://doi.org/10.1016/s0304-4076/98/00010-4.Search in Google Scholar

Psaradakis, Z., and N. Spagnolo. 2003. “On the Determination of the Number of Regimes in Markov-Switching Autoregressive Models.” Journal of Time Series Analysis 24 (2): 237–52. https://doi.org/10.1111/1467-9892.00305.Search in Google Scholar

Ravn, M. O., and M. Sola. 1995. “Stylized Facts and Regime Changes: Are Prices Procyclical?” Journal of Monetary Economics 36 (3): 497–526. https://doi.org/10.1016/0304-3932/95/01231-1.Search in Google Scholar

Roll, R. 1977. “A Critique of the Asset Pricing Theory’s Tests Part I: on past and Potential Testability of the Theory.” Journal of Financial Economics 4 (2): 129–76. https://doi.org/10.1016/0304-405x/77/90009-5.Search in Google Scholar

Ross, S. A. 1976. “The Arbitrage Theory of Capital Asset Pricing.” Journal of Economic Theory 13 (3): 341–60. https://doi.org/10.1016/0022-0531/76/90046-6.Search in Google Scholar

Sharpe, W. F. 1964. “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.” The Journal of Finance 19 (3): 425–42. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x.Search in Google Scholar

Vendrame, V., C. Guermat, and J. Tucker. 2018. “A Conditional Regime Switching CAPM.” International Review of Financial Analysis 56: 1–11. https://doi.org/10.1016/j.irfa.2017.12.001.Search in Google Scholar

Vendrame, V., C. Guermat, and J. Tucker. 2023. “A Conditional Higher-Moment CAPM.” International Review of Financial Analysis 86: 102524. https://doi.org/10.1016/j.irfa.2023.102524.Search in Google Scholar

Received: 2024-10-29

Accepted: 2025-07-09

Published Online: 2025-07-22

You are currently not able to access this content.

https://doi.org/10.1515/snde-2024-0118

Keywords for this article

non-diversifiable risk premium; Markov chain; structural breaks