Risk premium: insights over the threshold

Abstract

The aim of this article is 2-fold: first to test the adequacy of Pareto distributions to describe the tail of financial returns in emerging and developed markets, and second to study the possible correlation between stock market indices observed returns and return's extreme distributional characteristics measured by Value at Risk and Expected Shortfall. We test the empirical model using daily data from 41 countries, in the period from 1995 to 2005. The findings support the adequacy of Pareto distributions and the use of a log linear regression estimation of their parameters, as an alternative for the usually employed Hill's estimator. We also report a significant relationship between extreme distributional characteristics and observed returns, especially for developed countries.

Acknowledgements

The views expressed in this work are those of the authors and do not reflect those of the Banco Central do Brasil or its members. We would like to thank an unknown reviewer, Javier Perote, Alfonso Novales and seminar participants at XIII Foro de Finanzas. This research was funded by MEC Grants BEC2002-0279 and SEJ2005-05485. The usual disclaimers apply.

Notes

¹ Extreme value theory provides a natural approach to VAR and ES estimation and there is already a considerable literature on the subject. See Danielsson and de Vries (1997), Embrechts et al. (1997) and Diebold et al. (1998).

² Nawrocki (1999) presents a review of such literature.

³ See, e.g. Embrechts et al. (1997, p. 348).

⁴ We used Newey–West procedure in order to generate a covariance matrix that is consistent in the presence of both heteroskedasticity and autocorrelation of unknown form. Some recent articles like Cho and Engle (1999) Andersen et al. (2003) and Hwang and Salmon (2006) use this procedure.

⁵ ‘The Application of Basel II to Trading Activities and the Treatment of Double Default Effects’ (BIS, 2005) suggests the use of a one-to-ten days horizon in the measurement of VAR.

⁶ Barry et al. (2002) found that mean returns for small firms exceed mean returns for large firms. Fama and French (1995) proposed size as a factor for their pricing model. This effect has been reversed in recent periods (Al-Rjoub et al. 2005).

⁷ Considering a significance level of 0.10.

⁸ See French and Poterba (1991) or Shapiro (1999) for a review of the home bias effect.

⁹ We also tried to improve the model introducing in the regression the size of each country's market. A three-factor model was fitted where two of them are beta and the natural logarithm of the market capitalization. The coefficient for size was negative in almost all regressions suggesting that small markets get a premium. However, due to colinearity problems results are of dubious statistical significance. Full results are available on request.