, Volume 17, Issue 4, pp 557–583 | Cite as

The dynamic power law model

  • Bryan KellyEmail author


I propose a new measure of common, time-varying tail risk for large cross sections of stock returns. Stock return tails are described by a power law in which the power law exponent is allowed to transition smoothly through time as a function of recent data. It is motivated by asset pricing theory and is estimable via quasi-maximum likelihood. Estimates indicate substantial time variation in stock return tails, and that the risk of extreme returns rises in weak economic conditions.


Power law Finance Asset prices Crash Tail risk 


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  1. Bansal, R., Yaron, A.: Risks for the long run: A potential resolution of asset pricing puzzles. J. Financ., 1481–1509 (2004)Google Scholar
  2. Barro, R.: Rare disasters and asset markets in the twentieth century. Q. J. Econ. 121 (3), 823–866 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  3. Berk, R.H.: Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Stat. 37 (1), 51–58 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  4. Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econ. 31 (3), 307– 327 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  5. Bollerslev, T.: A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev. Econ. Stat., 542–547 (1987)Google Scholar
  6. Bollerslev, T., Tauchen, G., Zhou, H.: Expected stock returns and variance risk premia. Rev. Financ. Stud. 22 (11), 4463–4492 (2009)CrossRefGoogle Scholar
  7. Campbell, J., Shiller, R.: The dividend-price ratio and expectations of future dividends and discount factors. Rev. Financ. Stud. 1 (3), 195–228 (1988)CrossRefGoogle Scholar
  8. Drechsler, I., Yaron, A.: What’s vol got to do with it? Rev. Financ. Stud. 24 (1), 1–45 (2011)CrossRefGoogle Scholar
  9. Drost, F.C., Werker, B.J.: Closing the GARCH gap: Continuous time GARCH modeling. J. Econ. 74 (1), 31–57 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  10. Dupuis, D.: Exceedances over high thresholds: A guide to threshold selection. Extremes 1 (3), 251–261 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  11. Embrechts, P., Kluppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer Verlag (1997)Google Scholar
  12. Engle, R.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: J. Econ. Soc., 987–1007 (1982)Google Scholar
  13. Epstein, L., Zin, S.: Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57 (4), 937–969 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  14. Eraker, B., Shaliastovich, I.: An equilibrium guide to designing affine pricing models. Math. Financ. 18 (4), 519–543 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. Fama, E.: Mandelbrot and the stable Paretian hypothesis. J. Bus. 36 (4), 420–429 (1963)CrossRefGoogle Scholar
  16. Fama, E., French, K.: Common risk factors in the returns on stocks and bonds. J. Financ. Econ. 33 (1), 3–56 (1993)CrossRefzbMATHGoogle Scholar
  17. Gabaix, X.: Variable rare disasters: An exactly solved framework for ten puzzles in macro-finance. Q. J. Econ. 127 (2), 645–700 (2012)CrossRefGoogle Scholar
  18. Gabaix, X., Gopikrishnan, P., Plerou, V., Stanley, H.: Institutional investors and stock market volatility. Q. J. Econ. 121 (2), 461–504 (2006)CrossRefzbMATHGoogle Scholar
  19. Gardes, L., Girard, S.: A moving window approach for nonparametric estimation of the conditional tail index. J. Multivar. Anal. 99 (10), 2368–2388 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. Gardes, L., Girard, S.: Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels. Extremes 13 (2), 177–204 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  21. Gardes, L., Stupfler, G.: Estimation of the conditional tail index using a smoothed local Hill estimator. Extremes, 1–31 (2013)Google Scholar
  22. Gnedenko, B., Kolmogorov, A.: Limit distributions for sums of independent random variables. Addison-Wesley (1968)Google Scholar
  23. Goegebeur, Y., Guillou, A., Schorgen, A.: Nonparametric regression estimation of conditional tails: the random covariate case. Stat., 1–24 (2013)Google Scholar
  24. Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica: J. Econ. Soc., 1029–1054 (1982)Google Scholar
  25. Hayashi, F.: Econometrics. Princeton University Press (2011)Google Scholar
  26. Hill, B.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)CrossRefzbMATHGoogle Scholar
  27. Jansen, D.W., De Vries, C.G.: On the frequency of large stock returns: Putting booms and busts into perspective. Rev. Econ. Stat., 18–24 (1991)Google Scholar
  28. Kelly, B., Jiang, H.: Tail Risk and Asset Prices, Chicago Booth Working Paper (2013)Google Scholar
  29. Krasker, W.: The “peso problem” in testing the efficiency of forward exchange markets. J. Monet. Econ. 6 (2), 269–276 (1980)CrossRefGoogle Scholar
  30. Kuester, K., Mittnik, S., Paolella, M.S.: Value-at-risk prediction: A comparison of alternative strategies. J. Finan. Econ. 4 (1), 53–89 (2006)Google Scholar
  31. Mandelbrot, B.: The variation of certain speculative prices. J. Bus. 36 (4), 394 (1963)CrossRefGoogle Scholar
  32. Matthys, G., Beirlant, J.: Adaptive threshold selection in tail index estimation. Extremes Integr. Risk Manag., 37–49 (2000)Google Scholar
  33. McNeil, A.J., Frey, R.: Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. J. empir. finan. 7 (3), 271–300 (2000)CrossRefGoogle Scholar
  34. Mittnik, S., Paolella, M.S., Rachev, S.T.: Diagnosing and treating the fat tails in financial returns data. J. empir. finan. 7 (3), 389–416 (2000)CrossRefGoogle Scholar
  35. Nelson, D.B.: Stationarity and persistence in the GARCH (1, 1) model. Econ. theory 6 (03), 318–334 (1990)CrossRefGoogle Scholar
  36. Newey, W., West, K.: A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  37. Newey, W.K., McFadden, D.: Large sample estimation and hypothesis testing. Handb. Econ. 4, 2111–2245 (1994)MathSciNetGoogle Scholar
  38. Neyman, J., Scott, E.L.: Consistent estimates based on partially consistent observations. Econometrica: J. Econ. Soc., 1–32 (1948)Google Scholar
  39. Rietz, T.: The equity risk premium: A solution. J. Monet. Econ. 22 (1), 117–131 (1988)CrossRefGoogle Scholar
  40. Wachter, J.A.: Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility? J. Financ. 68 (3), 987–1035 (2013)CrossRefGoogle Scholar
  41. White, H.: Cambridge University Press, Estimation, Inference and Specification Analysis, Econometric Society Monographs (1996)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA

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