Estimating parameters of the CAPM under generalised asymmetric student-t distribution—The case of the Warsaw Stock Exchange sectoral indices
Vol 7, Issue 1, 2024
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Abstract
This paper analyses selected sub-indices listed on the Warsaw Stock Exchange (WSE) covering seven sectors: construction, IT, media, real estate, fuel, food, and telecommunications, from 3 January 2006 to 29 May 2020. We use daily, weekly, monthly, and quarterly data, resulting in 3600 daily, 751 weekly, 172 monthly, and 56 quarterly observations. The WIG index quotations were used to approximate the market portfolio and the Poland 10Y government bond yields for the risk-free rate. We have estimated the parameter β in CAPM regression using three different stochastic assumptions for the error term. The basic stochastic framework of the model utilises the generalised asymmetric student-t distribution (GAST). We have also estimated the parameter β based on the symmetric version of the GAST distribution and on the Gaussian one. These models can be treated as special cases of the basic framework. The estimates of the β parameter are influenced by the assumptions made about the error term. The data indicates that except for WIG-Paliwa, the Gaussian error term leads to larger β estimates than other non-Gaussian specifications. The inference about the shape parameters is not very certain, and the data does not strongly support the two-piece mechanism that enforces the asymmetry of the error term distribution. Furthermore, the estimates of the β parameter depend strongly on the frequency of the analysed series.
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1. De Forest E. On an Unsymmetrical Probability Curve. Analyst. 1882; 9: 135-142.
2. De Forest E. On an Unsymmetrical Probability Curve. Analyst. 1883; 10: 1-7.
3. Edgeworth FY. The law of error and the elimination of chance. The
4. London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1886; 21(131): 308-324. doi: 10.1080/14786448608627852
5. Pearson K. Asymmetrical Frequency Curves. Nature. 1893; 48: 615-616.
6. Pearson K. On Skew Probability Curves. Nature. 1895; 52: 317.
7. Genton MG. Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Chapman & Hall/CRC, Boca Raton; 2004.
8. Black F, Scholes M. The Pricing of Options and Corporate Liabilities. Journal of Political Economy. 1973; 81: 637-659.
9. Merton RC. Theory of Rational Option Pricing. Bell Journal of Economics. 1973; 4: 141-183.
10. Markovitz HM. Portfolio Selection. Journal of Finance. 1952; 7: 77-91.
11. Sharpe WF. Capital asset prices: a theory of market equilibrium under conditions of risk. The Journal of Finance. 1964; 19(3): 425-442. doi: 10.1111/j.1540-6261.1964.tb02865.x
12. Treynor JL. Toward a Theory of Market Value of Risky Assets. 1961; Unpublished Work.
13. Lintner J. Security Prices, Risk, and Maximal Gains from Diversification. The Journal of Finance. 1965; 20(4): 587. doi: 10.2307/2977249
14. Lintner J. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics. 1965; 47: 13-37.
15. Graham JR, Harvey CR. The Theory and Practice of Corporate Finance: Evidence from the Field. Journal of Financial Economics. 2001; 60: 187-243.
16. Altavilla C, Bochmann P, Ryck JD, et al. Measuring the Cost of Equity of Euro Area Banks. ECB Occasional. 2021. doi: 10.2139/ssrn.3797125
17. Kaplan PD, Peterson JD. Full-Information Industry Betas. Financial Management. 1998; 27(2): 85-93.
18. Fama EF, French KR. The Capital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives. 2004; 18: 25-46.
19. Andrei D, Cujean J, Wilson M. The Lost Capital Asset Pricing Model, The Review of Economic Studies. 2023; 90(6): 2703-2762. doi: 10.1093/restud/rdad013
20. Savor P, Wilson M. Asset pricing: A tale of two days. Journal of Financial Economics. 2014; 113(2): 171-201. doi: 10.1016/j.jfineco.2014.04.005
21. Black F. Capital Market Equilibrium with Restricted Borrowing. Journal of Business. 1972; 45(3): 444-455. doi: 10.1086/295472 doi
22. Fama EF, French K. Value Versus Growth: The International Evidence. Journal of Finance. 1998; 53(6): 1975-1999. doi: 10.1111/0022-1082.00080
23. Zhou XY, Yin G. Markowitz’s Mean-Variance Portfolio Selection with Regime Switching: A Continuous-Time Model. SIAM Journal on Control and Optimization. 2003; 42(4): 1466-1482. doi: 10.1137/s0363012902405583
24. Wei L. Empirical Study of CAPM in the 5G sector of Chinese stock market. In: Proceedings of the 2021 2nd International Conference on Big Data Economy and Information Management (BDEIM); 3-5 December 2021; Sanya, China.
25. Urom C, Chevallier J, Zhu B. A dynamic conditional regime-switching GARCH CAPM for energy and financial markets. Energy Economics. 2020; 85: 104577. doi: 10.1016/j.eneco.2019.104577
26. Adcock C. Capital Asset Pricing for UK Stocks under the Multivariate Skew-Normal Distribution. In: Genton MG (editor). Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Chapman & Hall/CRC, Boca Raton; 2004.
27. Azzalini A, Dalla Valle A. The Multivariate Skew-Normal Distribution. Biometrika. 1996; 83: 715-726.
28. Adcock CJ, Shutes K. Fat Tails and the Capital Asset Pricing Model. In: Dunis CL (editor). Advances in Quantitative Asset Management. Kluwer Academic Publishers; 2000.
29. Affleck-Graves J, McDonald B. Nonnormalities and tests of asset-pricing theories. Journal of Finance. 1989; 44: 889-908.
30. MacKinlay AC, Richardson MP. Using generalized method of moments to test mean variance efficiency. Journal of Finance. 1991; 46: 511-527.
31. Zhou G. Asset-Pricing Tests Under Alternative Distributions. The Journal of Finance. 1993; 48(5): 1927. doi: 10.2307/2329073
32. Harvey CR, Siddique A. Conditional Skewness in Asset Pricing Tests. Journal of Finance. 2000; 55: 1263-1295. doi: 10.1111/0022-1082.00247
33. Li L, Gan Q, Zhuo Z, et al. Testing the CAPM Theory Based on a New Model for Fama-French 25 Portfolio Returns. Theoretical Economics Letters. 2014; 04(08): 666-680. doi: 10.4236/tel.2014.48085
34. Theodossiou AK, Theodossiou P. Stock return outliers and beta estimation: The case of U.S. pharmaceutical companies. Journal of International Financial Markets, Institutions and Money. 2014; 30: 153-171. doi: 10.1016/j.intfin.2014.02.002
35. Bao T, Diks C, Li H. A generalized CAPM model with asymmetric power distributed errors with an application to portfolio construction. Economic Modelling. 2018; 68: 611-621. doi: 10.1016/j.econmod.2017.03.035
36. Harvey A, Lange RJ. Volatility Modeling with a Generalized t-Distribution. Journal of Time Series Analysis. 2017; 38(2): 175-190. doi: 10.1111/jtsa.12224
37. Li R, Nadarajah S. A review of Student’s t distribution and its generalizations. Empirical Economics. 2018; 58: 1461-1490. doi: 10.1007/s00181-018-1570-0
38. Zhu D, Zinde-Walsh V. Properties and estimation of asymmetric exponential power distribution. Journal of Econometrics. 2009; 148(1): 86-99. doi: 10.1016/j.jeconom.2008.09.038
39. Zhu D, Galbraith JW. A generalized asymmetric Student-t distribution with application to financial econometrics. 2010; Journal of Econometrics. 2010; 157(2): 297-305. doi: 10.1016/j.jeconom.2010.01.013
40. Fernández C, Steel MF. On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association. 1998; 93(441): 359-371. doi: 10.1080/01621459.1998.10474117
41. Theodossiou P. Financial data and the skewed generalized t distribution. Management Science. 1998; 44: 1650-1661.
42. Mazur B, Pipień M. Modelling Time Varying Asymmetry and Tail Behaviour in Long Series of Daily Financial Returns. Studies in Nonlinear Dynamics and Econometrics. 2018; 22(5).
43. Mazur B, Pipień M. A Family of Flexible Multivariate Distributions with Applications in Empirical Finance. Acta Physica Polonica A. 2020; 138: 65-73. doi: 10.12693/APhysPolA.138.65
44. Ferreira JTAS, Steel MFJ. A Constructive Representation of Univariate Skewed Distributions. Journal of the American Statistical Association. 2006; 101: 823-829.
45. Hansen B. Autoregressive Conditional Density Estimation. International Economic Review. 1994; 35(3): 705-730.
46. Arnold BC, Groeneveld RA. Measuring the Skewness with respect to the Mode. The American Statistician. 1995; 49(1): 34-38.
47. Jensen MC, Black F, Scholes MS. The Capital Asset Pricing Model: Some Empirical Tests. In: Jensen MC (editor). Studies in the Theory of Capital Markets. Praeger Publishers Inc.; 1972.
48. Chamberlain G. A characterization of the distributions that imply mean-variance utility functions. Journal of Economic Theory. 1983; 29: 185-201.
49. Gibbons MR, Ross SA, Shanken J. A Test of the Efficiency of a Given Portfolio. Econometrica. 1989; 57(5): 1121. doi: 10.2307/1913625
50. Shephard N. An Estimator for Predictive Regression: Reliable Inference for Financial Economics, manuscript. Available online: https://arxiv.org/abs/2008.06130 (accessed on 27 January 2024).
51. Gomola A, Pipień M. Maximum Likelihood Estimates of CAPM Beta’s for the Banking Sector Indices under Heavy Tailed and Asymmetric Distributions - the Case of NYSE and WSE. 2022; Unpublished Work.
DOI: https://doi.org/10.24294/fsj.v7i1.5632
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