Skip to main content

Examining heterogeneity in implied equity risk premium using penalized splines

Abstract

Financial data exhibit complex structures and relations and it is therefore not always possible or expedient to find a suitable parametric functional form to adequately describe the data. To overcome this problem, nonparametric techniques can be used to extract the functional process directly from the data without any a priori specification of the functional shape. We take advantage of this flexibility and use a penalized spline approach to model, over time, the implied equity risk premiums of companies that belong to a local stock exchange index. In finance and macroeconomic research it is common practice to use simple averaging techniques to aggregate the single values, thus obtaining an overview of the stock market of a country or particular groups defined by stock-specific characteristics. The objective is to obtain common patterns or dependencies from individual characteristics. A precondition here is a substantial heterogeneity of the individual stocks, because otherwise one constituent can represent the whole index and the required diversification effect fails. Hence, in this paper we explore if and how this assumption is justified. The examined stock indices are the Dow Jones Industrial Index and the German DAX 30. It turns out that the constituents of both indices show very stock-specific behaviors of their equity risk premium over time. Thus the application of these indices in, e.g., macroeconomic research seems adequate.

This is a preview of subscription content, access via your institution.

References

  • Aguilera, A.M., Ocana, F.A., Valderrama, M.J.: Stochastic modelling for evolution of stock-prices by means of functional principal component analysis. Appl. Stoch. Models Bus. Ind. 15(4), 227–234 (1999)

    Article  MATH  Google Scholar 

  • Akaike, H.: Information theory and an extension of the maximum likelihood principle. Breakth. Stat. 1, 610–624 (1973)

    Google Scholar 

  • Akritas, M., Politis, D.: Recent Advances and Trends in Nonparametric Statistics. North-Holland, Amsterdam (2003)

    MATH  Google Scholar 

  • Burnham, K., Anderson, D.R.: Model Selection and Multimodel Inference. Springer, Berlin (2002)

    MATH  Google Scholar 

  • Cheong, C., Lee, W.W., Yahaya, N.A.: Wavelet-based temporal cluster analysis on stock time series. In: ICOQSIA, pp. 6–8 (2005)

  • Claus, J., Thomas, J.: Equity premium as low as three percent? Empirical evidence from analysts earnings forecasts for domestic and international stock markets. J. Finance 56, 1629–1665 (2001)

    Article  Google Scholar 

  • Cornell, B.: The Equity Risk Premium: The Long-Run Future of the Stock Market. Wiley, New York (1999)

    Google Scholar 

  • Coull, B.A., Ruppert, D., Wand, M.P.: Simple incorporation of interactions into additive models. Biometrics 57, 539–545 (2001)

    Article  MathSciNet  Google Scholar 

  • deBoor, C.: A Practical Guide to Splines. Springer, Berlin (2001)

    Google Scholar 

  • Durban, M., Currie, I.: A note on p-spline additive models with correlated errors. Comput. Stat. 18, 263–292 (2003)

    MathSciNet  Google Scholar 

  • Durbán, M., Harezlak, J., Wand, M., Carroll, R.: Simple fitting of subject specific curves for longitudinal data. Stat. Med. 24, 1153–1162 (2005)

    Article  MathSciNet  Google Scholar 

  • Edward, E.O., Bell, P.W.: The Theory of Measurement of Business Income. University of California Press, Berkeley (1961)

    Google Scholar 

  • Eilers, P., Marx, B.: Flexible smoothing using B-splines and penalized likelihood. Stat. Sci. 11, 89–121 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Gebhardt, W.R., Lee, C.M.C., Swaminathan, B.: Toward an implied cost of capital. J. Account. Res. 39(1), 135–176 (2001)

    Article  Google Scholar 

  • Green, P.J., Silverman, B.W.: Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman & Hall, London (1994)

    MATH  Google Scholar 

  • Härdle, W., Tsybakov, A.: Local polynomial estimators of the volatility function in nonparametric autoregression. J. Econom. 81, 223–242 (1997)

    Article  MATH  Google Scholar 

  • Härdle, W., Müller, M., Sperlich, S., Werwatz, A.: Nonparametric and Semiparametric Models. Springer, Berlin (2004)

    MATH  Google Scholar 

  • Harris, R., Marston, F.: The market risk premium: Expectational estimates using analysts forecasts. J. Appl. Finance 11, 6–16 (1999)

    Google Scholar 

  • Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models. Chapman & Hall, London (1990)

    MATH  Google Scholar 

  • Ibbotson, R.G., Chen, P.: Long-run stock returns: participating in the real economy. Financ. Anal. J. 59(1), 88–98 (2003)

    Article  Google Scholar 

  • Ingrassia, S., Cerioli, A., Corbellini, A.: Some issues on clustering of functional data. In: Schader, M., Gaul, W., Vichi, M. (eds.) Between Data Science and Applied Data Analysis, pp. 49–56. Springer, Berlin (2003)

    Google Scholar 

  • Jiang, X., Lee, B.: An empirical test of the accounting-based residual income model and the traditional dividend discount model. J. Bus. 78, 1465–1504 (2005)

    Article  Google Scholar 

  • Jobert, A., Platania, A., Rogers, L.C.G.: A Bayesian solution to the equity premium puzzle. Preprint, available at: http://www.statslab.cam.ac.uk/~chris

  • Krivobokova, T., Kauermann, G.: A note on penalized spline smoothing with correlated errors. J. Am. Stat. Assoc. (2007)

  • Lee, C.: Accounting-based valuation: Impact on business practices and research. Account. Horiz. 13(4), 413–425 (1999)

    Google Scholar 

  • Lee, C., Myers, J., Swaminathan, B.: What is the intrinsic value of the Dow? J. Finance 54, 1693–1741 (1999)

    Article  Google Scholar 

  • Lessard, D.R.: International portfolio diversification: a multivariate analysis for a group of Latin American countries. J. Finance 28(3), 619–633 (1973)

    Article  Google Scholar 

  • Linton, O., Nielsen, J., Tangaard, C., Mammen, E.: Estimating the yield curve by kernel smoothing methods. J. Econom. 105(1), 185–223 (2001)

    Article  MATH  Google Scholar 

  • Masry, E., Tjostheim, D.: Nonparametric estimation and identification of nonlinear ARCH time series: strong convergence and asymptotic normality. Econom. Theory 11, 258–289 (1995)

    Article  MathSciNet  Google Scholar 

  • McCulloch, C.E., Searle, S.R.: Generalized, Linear, and Mixed Models. Wiley, New York (2000)

    Google Scholar 

  • Mehra, R., Prescott, E.C.: The equity premium: a puzzle. J. Monet. Econ. 15(2), 145–61 (1985)

    Article  Google Scholar 

  • Meric, I., Meric, G.: Co-movements of European equity markets before and after the 1987 crash. Multinatl. Finance J. 1, 137–152 (1997)

    Google Scholar 

  • Ngo, L., Wand, M.: Smoothing with mixed model software. J. Stat. Softw. 9(1), 2–3 (2004)

    Google Scholar 

  • Nychka, D., Cummins, D.: Comment on: Eilers, P., Marx, B., Flexible smoothing with B-splines and penaltie. Stat. Sci. 11, 89–121 (1996)

    Article  Google Scholar 

  • Ohlson, J.A.: Earnings, book-values and dividends in equity valuation. Contemp. Account. Res. 11(2), 661–687 (1995)

    Article  Google Scholar 

  • Opsomer, J., Wang, Y., Yang, Y.: Nonparametric regression with correlated errors. Stat. Sci. 16, 134–153 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • O’Sullivan, F.: A statistical perspective on illposed inverse problems. Stat. Sci. 1, 505–527 (1986)

    MathSciNet  Google Scholar 

  • Pagan, A., Ullah, A.: Nonparametric Econometrics. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  • Pinheiro, J., Bates, D.: Mixed-Effects Models in S and Splus. Springer, New York (2002)

    Google Scholar 

  • Robinson, G.K.: That BLUP is a good thing: the estimation of random effects. Stat. Sci. 6, 15–51 (1991)

    Article  MATH  Google Scholar 

  • Ruppert, D.: Statistics and Finance. Springer, New York (2004)

    MATH  Google Scholar 

  • Ruppert, D., Wand, M., Carroll, R.: Semiparametric Modelling. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  • Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Finance 19, 425–442 (1964)

    Article  Google Scholar 

  • Siegel, J.: The equity premium: stock and bond returns since 1802. Financ. Anal. J. 48(1), 28–46 (1992)

    Article  Google Scholar 

  • Stanton, R.: A nonparametric model of term structure dynamics and the market price of interest rate risk. J. Finance 52(5), 1973–2002 (1997)

    Article  Google Scholar 

  • Wager, C., Vaida, F., Kauermann, G.: Model selection for P-spline smoothing using Akaike information criteria. Aust. N. Z. J. Stat. 48(4), 417–428 (2006)

    Article  MathSciNet  Google Scholar 

  • Wand, M.: Smoothing and mixed models. Comput. Stat. 18, 223–249 (2003)

    MATH  Google Scholar 

  • Wang, Y.: Smoothing spline models with correlated random errors. J. Am. Stat. Assoc. 93, 341–348 (1998)

    Article  MATH  Google Scholar 

  • Williams, J.B.: The Theory of Investment Value. Harvard University Press, Cambridge (1938)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael Wegener.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wegener, M., Kauermann, G. Examining heterogeneity in implied equity risk premium using penalized splines. AStA 92, 35–56 (2008). https://doi.org/10.1007/s10182-007-0052-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10182-007-0052-z

Keywords