Value at risk and efficiency under dependence and heavy-tailedness: models with common shocks

Springer Science and Business Media LLC - Tập 7 - Trang 285-318 - 2010
Rustam Ibragimov1, Johan Walden2
1Department of Economics, Harvard University, Cambridge, (USA)
2Haas School of Business, University of California at Berkeley, Berkeley, USA

Tóm tắt

This paper presents an analysis of diversification and portfolio value at risk for heavy-tailed dependent risks in models with multiple common shocks. We show that, in the framework of value at risk comparisons, diversification is optimal for moderately heavy-tailed dependent risks with common shocks and finite first moments, provided that the model is balanced, i.e., that all the risks are available for portfolio formation. However, diversification is inferior in balanced extremely heavy-tailed risk models with common factors. Finally, in several unbalanced dependent models, diversification is optimal, even though there is extreme heavy-tailedness in common shocks or in idiosyncratic parts of the risks. Analogues of the obtained results further hold for efficiency comparisons of linear estimators in random effects models with dependent and heavy-tailed observations.

Tài liệu tham khảo

An M.Y.: Logconcavity versus logconvexity: a complete characterization. J Econ Theor 80, 350–369 (1998) Andrews, D.W.K.: Cross-section regression with common shocks. Cowles Foundation Discussion Paper 1428. Available at http://cowles.econ.yale.edu/P/cd/d14a/d1428.pdf (2003) Andrews D.W.K.: Cross-section regression with common shocks. Econometrica 73(5), 1551–1585 (2005) Axtell R.L.: Zipf distribution of U.S. firm sizes. Science 293, 1818–1820 (2001) Bagnoli M., Bergstrom T.: Log-concave probability and its applications. Econ Theor 26, 445–469 (2005) Bai J.: Panel data models with interactive fixed effects. Econometrica 77, 1229–1279 (2009) Birkes D., Seely J., Azzam A.-M.: An efficient estimator of the mean in a two-stage nested model. Technometrics 23, 143–148 (1981) Birnbaum Z.W.: On random variables with comparable peakedness. Ann Math Stat 19, 76–81 (1948) Bouchard J.-P., Potters M.: Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management, 2nd edn. Cambridge University Press, Cambridge (2004) Cochran W.G.: The combination of estimates from different experiments. Biometrics 10, 101–129 (1954) de la Peña V.H., Ibragimov R., Sharakhmetov S.: On sharp Burkholder-Rosenthal-type inequalities for infinite-degree U-statistics. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 38, 973–990 (2002) de la Peña V.H., Ibragimov R., Sharakhmetov S.: On extremal distributions and sharp Lp-bounds for sums of multilinear forms. Ann. Probab. 31, 630–675 (2003) Dharmadhikari S.W., Joag-Dev K.: Unimodality, Convexity and Applications. Academic Press, Boston (1988) El-Bassiouni M.Y., Abdelhafez M.E.M.: Interval estimation of the mean in a two-stage nested model. J Stat Comput Simul 67, 333–350 (2000) Embrechts P., Klüppelberg C., Mikosch T.: Modelling Extremal Events for Insurance and Finance. Springer, New York (1997) Embrechts P., McNeil A.J., Frey R.: Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press, Princeton (2005) Fang K.T., Kotz S., Ng K.W.: Symmetric Multivariate and Related Distributions, Monographs on Statistics and Applied Probability vol. 36. Chapman and Hall Ltd., London (1990) Gabaix X.: Zipf’s law for cities: an explanation. Quart J Econ 114, 739–767 (1999) Gabaix X.: Power laws in economics and finance. Ann Rev Econ 1, 255–293 (2009) Gabaix X., Gopikrishnan P., Plerou V., Stanley H.E.: Institutional investors and stock market volatility. Quart J Econ 121, 461–504 (2006) Ibragimov, R.: New Majorization Theory in Economics and Martingale Convergence Results in Econometrics. Ph.D. dissertation, Yale University (2005) Ibragimov R.: Efficiency of linear estimators under heavy-tailedness: convolutions of α-symmetric distributions. Econ Theor 23, 501–517 (2007) Ibragimov, R. Heavy tailed densities. In: Durlauf, S.N., Blume, L.E. (eds.) ‘The New Palgrave Dictionary of Economics Online’. Palgrave Macmillan. http://www.dictionaryofeconomics.com/article?id=pde2008H000191 (2009a) Ibragimov R.: Portfolio diversification and value at risk under thick-tailedness. Quant Finance 9, 565–580 (2009b) Ibragimov R., Jaffee D., Walden J.: Non-diversification traps in markets for catastrophic risk. Rev Financ Stud 22, 959–993 (2009) Ibragimov R., Walden J.: The limits of diversification when losses may be large. J Bank Finance 31, 2551–2569 (2007) Ibragimov R., Walden J.: Portfolio diversification under local and moderate deviations from power laws. Insur Math Econ 42, 594–599 (2008) Jansen D.W., de Vries C.G.: On the frequency of large stock returns: putting booms and busts into perspective. Rev. Econ. Stat. 73, 18–32 (1991) Kaas R., Goovaerts M., Tang Q.: Some useful counterexamples regarding comonotonicity. Belg Actuar Bull 4, 1–4 (2004) Karlin S.: Total Positivity, vol. I. Stanford University Press, Stanford (1968) Koch G.G.: A general approach to the estimation of variance components. Technometrics 9, 93–118 (1967a) Koch G.G.: A procedure to estimate the population mean in random effects models. Technometrics 9, 577–585 (1967b) Lhabitant F.S.: Equally weighted index (HFRX). In: Gregoriou, G.N. (eds) Encyclopedia of Alternative Investments, pp. 166–167. Chapman and Hall, London (2008) Loretan M., Phillips P.C.B.: Testing the covariance stationarity of heavy-tailed time series. J Empir Finance 1, 211–248 (1994) Low L.: An application of majorization to comparison of variances. Technometrics 12, 141–145 (1970) Marshall A.W., Olkin I.: Inequalities: Theory of Majorization and its Applications. Academic Press, New York (1979) McCulloch J.H.: Measuring tail thickness to estimate the stable index alpha: a critique. J Bus Econ Stat 15, 74–81 (1997) Moscone F., Tosetti E.: A review and comparison of tests of cross-sectional independence in panels. J Econ Surv 23, 528–561 (2009) Nešlehová J., Embrechts P., Chavez-Demoulin V.: Infinite mean models and the LDA for operational risk. J Oper Risk 1, 3–25 (2006) Rachev S.T., Menn C., Fabozzi F.J.: Fat-tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing. Wiley, Hoboken, NJ (2005) Rachev S.T., Mittnik S.: Stable Paretian Models in Finance. Wiley, New York (2000) Shaked M., Shanthikumar J.G.: Stochastic Orders. Springer, New York (2007) Silverberg G., Verspagen B.: The size distribution of innovations revisited: An application of extreme value statistics to citation and value measures of patent significance. J Econom 139, 318–339 (2007) Weiler H., Culpin D.: Variance of weighted means. Technometrics 12, 757–773 (1970) Zolotarev V.M.: One-Dimensional Stable Distributions. American Mathematical Society, Providence (1986)