Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics

Chun, So Yeon and Shapiro, Alexander and Uryasev, Stan (2011): Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics. Forthcoming in:

Abstract

We discuss linear regression approaches to estimation of law invariant conditional risk measures. Two estimation procedures are considered and compared; one is based on residual analysis of the standard least squares method and the other is in the spirit of the M-estimation approach used in robust statistics. In particular, Value-at-Risk and Average Value-at-Risk measures are discussed in details. Large sample statistical inference of the estimators is derived. Furthermore, finite sample properties of the proposed estimators are investigated and compared with theoretical derivations in an extensive Monte Carlo study. Empirical results on the real-data (different financial asset classes) are also provided to illustrate the performance of the estimators.

• Item Type: MPRA Paper
• Language: English
• Keywords: Value-at-Risk, Average Value-at-Risk, linear regression, least squares residuals, M-estimators, quantile regression, conditional risk measures, law invariant risk measures, statistical inference
• Subjects: C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; G - Financial Economics > G3 - Corporate Finance and Governance > G32 - Financing Policy ; Financial Risk and Risk Management ; Capital and Ownership Structure ; Value of Firms ; Goodwill; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General

References

Acerbi, C. 2002. Spectral measures of risk: a coherent representation of subjective risk a version. Journal of Banking & Finance 26, 1505-1518.

Artzner, P., F. Delbaen, J.-M. Eber, D. Heath. 1999. Coherent measures of risk. Mathematical Finance 9 203-228.

Berkowitz, J, M Pritsker, M Gibson, H Zhou. 2002. How accurate are value-at-risk models at commercial banks. Journal of Finance 57 1093-1111.

Bluhm, Christian, Ludger Overbeck, Christoph Wagner. 2002. An introduction to Credit Risk Modeling. 1st ed. Chapman and Hall/CRC.

Cai, Z., X. Wang. 2008. Nonparametric estimation of conditional var and expected shortfall. Journal of Econometrics 147(1) 120-130.

Chen, S. X., C. Y. Tang. 2005. Nonparametric inference of value-at-risk for dependent nancial returns. Journal of Financial Econometrics 3(2) 227-255.

Chernozhukov, V., L. Umantsev. 2001. Conditional value-at-risk: Aspects of modeling and estimation. Empirical Economics 26(1) 271-292.

Detlefsen, K., G. Scandolo. 2005. Conditional and dynamic con v ex risk measures. Finance and Stochastics 9(4) 539-561.

Duffie, D., K. J. Singleton. 2003. Credit Risk: Pricing, Measurement and Management. Princeton, Princeton University Press.

Engle, R. F., S Manganelli. 2004. Caviar: Conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics 22 367-381.

Fermanian, J.D., O. Scaillet. 2005. Sensitivity analysis of var and expected shortfall for portfolios under nettingagreements. Journal of Banking and Finance 29 927-958.

FitchRatings. 2006. Global credit derivatives survey. Frey, R., A. J. McNeil. 2002. Var and expected shortfall in portfolios of dependent credit risks: Conceptual and practical insights. Journal of Banking & Finance 26 (7) 1317-1334.

F�ollmer, H., A. Sc hied. 2011. Stochastic finance: An introduction in discrete time. Walter de Gruyter & Co., Berlin.

Frey , R., A. J. McNeil. 2002. Var and expected shortfall in portfolios of dependent credit risks: Conceptual and practical insights. Journal of Banking & Finance 26(7) 1317-1334.

Giesecke, K., S. Weber. 2008. Measuring the risk of large losses. Journal of Investment Management 6(4) 1-15.

Gneiting, T. 2009. Evaluating point forecasts. Institut f�ur Angewandte Mathematik, Universitpreprint, preprint.

Huber, P. J. 1981. Robust Statistics. Wiley, New York.

Jackson, Patricia, William Perraudin. 2000. Regulatory implications of credit risk modelling. Journal of Banking & Finance 24(1-2) 1{14.

Jorion, P. 2003. Financial Risk Manager Handbook. 2nd ed. Wiley, New York.

Koenker, R. 2005. Quantile Regression. Cambridge University Press, Cambridge, UK.

Leorato, S., F. Peracchi, A. V. Tanase. 2010. Asymptotically efficient estimation of the conditional expected shortfall. EIEF Working Papers Series 1013, Einaudi Institute for Economic and Finance (EIEF).

McNeil, A. J., R. Frey. 2000. Estimation of tail-related risk measures for heteroscedastic nancial time series: An extreme value approach. Journal of Empirical Finance 7 271-300.

O'Kane, D., S. Turnbull. 2003. Valuation of credit default swaps. Quantitative credit research quarterly, Lehman Brothers.

Peracchi, F., A. V. Tanase. 2008. On estimating the conditional expected shortfall. Applied Stochastic Models in Business and Industry 24 471-493.

Pflug, G.W., Ch., W. R�omisch. 2007. Modeling, Measuring and Managing Risk. World Scientic Publishing Co., London.

Rockafellar, R. T., S. Uryasev. 2002. Conditional value-at-risk for general loss distributions. Journal of Banking & Finance 26(7) 1443{1471.

Rockafellar, R. T., S. Uryasev, M. Zabarankin. 2008. Risk tuning with generalized linear regression. Mathematics of Operations Research 33 712-729.

Scaillet, O. 2004a. Nonparametric estimation and sensitivity analysis of expected shortfall. Mathematical Finance 14(1) 115-129.

Scaillet, O. 2004b. Nonparametric estimation of conditional expected shortfall. Insurance and Risk Management Journal 72639-660.

Shapiro, A. 1989. Asymptotic properties of statistical estimators in stochastic programming. Annals of Statistics 17 841-858.

Shapiro, A., D. Dentcheva, A. Ruszczynski. 2009. Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia.

Trindade, A., S. Uryasev, A. Shapiro, G. Zrazhevsky. 2007. Financial prediction with constrained tail risk. Journal of Banking & Finance 31 3524-3538.

Weber, S. 2006. Distribution-invariant risk measures, information, and dynamic consistency. Mathematical Finance 16 (2) 419-441.

Wozabal, D., N. Wozabal. 2009. Asymptotic consistency of risk functionals. Journal of Nonparametric Statistics 21 (8) 977-990.

Zhu, S., M. Fukushima. 2009. Worst-case conditional value-at-risk with application to robust portfolio management. Operations Research 57(5) 1155-1168.