Empirical study of value‐at‐risk and expected shortfall models with heavy tails

Emerald - Tập 7 Số 2 - Trang 117-135 - 2006
Fotios C.Harmantzis1, LinyanMiao1, YifanChien1
1Stevens Institute of Technology, Hoboken, New Jersey, USA

Tóm tắt

PurposeThis paper aims to test empirically the performance of different models in measuring VaR and ES in the presence of heavy tails in returns using historical data.Design/methodology/approachDaily returns of popular indices (S&P500, DAX, CAC, Nikkei, TSE, and FTSE) and currencies (US dollar vs Euro, Yen, Pound, and Canadian dollar) for over ten years are modeled with empirical (or historical), Gaussian, Generalized Pareto (peak over threshold (POT) technique of extreme value theory (EVT)) and Stable Paretian distribution (both symmetric and non‐symmetric). Experimentation on different factors that affect modeling, e.g. rolling window size and confidence level, has been conducted.FindingsIn estimating VaR, the results show that models that capture rare events can predict risk more accurately than non‐fat‐tailed models. For ES estimation, the historical model (as expected) and POT method are proved to give more accurate estimations. Gaussian model underestimates ES, while Stable Paretian framework overestimates ES.Practical implicationsResearch findings are useful to investors and the way they perceive market risk, risk managers and the way they measure risk and calibrate their models, e.g. shortcomings of VaR, and regulators in central banks.Originality/valueA comparative, thorough empirical study on a number of financial time series (currencies, indices) that aims to reveal the pros and cons of Gaussian versus fat‐tailed models and Stable Paretian versus EVT, in estimating two popular risk measures (VaR and ES), in the presence of extreme events. The effects of model assumptions on different parameters have also been studied in the paper.

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Tài liệu tham khảo

Acerbi, C. and Tasche, D. (2002), “On the coherence of expected shortfall”, Journal of Banking and Finance, Vol. 26, pp. 1487‐503.

Angelidis, T., Benos, A. and Degiannakis, S. (2004), “The use of GARCH model in VAR estimation”, Statistical Methodology, Vol. 1 No. 2, pp. 105‐28.

Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999), “Coherent measures of risk”, Math of Finance, Vol. 9 No. 3, pp. 203‐28.

Bali, T.G. (2003), “An extreme value approach to estimating volatility and value at risk”, Journal of Business, Vol. 76 No. 1, pp. 83‐108.

Basak, S. and Shapiro, A. (2001), “Value‐at‐risk based risk management: optimal policies and asset prices”, Review of Financial Studies, Vol. 14 No. 2, pp. 371‐405.

Berkowitz, J. and O'Brien, J.M. (2002), “How accurate are value‐at‐risk models at commercial banks?”, Journal of Finance, Vol. 57, pp. 1093‐111.

Chambers, J.M., Mallows, C.L. and Stuck, B.W. (1976), “A method for simulating stable random variables”, Journal of the American Statistical Association, Vol. 71, pp. 340‐4.

Coles, S. (2001), An Introduction to Statistical Modeling of Extreme Values, Spring Series in Statistics, Springer‐Verlag, London.

Coronado, M. (2000), “ Extreme value theory (EVT) for risk managers: pitfalls and opportunities in the use of EVT in measuring VaR”, Proceedings of the V Spanish and III Italian‐Spanish Conference on Actuarial and Financial Mathematics, Madrid, October.

Crouhy, M., Galai, D. and Mark, R. (2001), Risk Management, McGraw‐Hill, New York, NY.

da Silva, A.L.C. and Vaz de Melo Mendes, B. (2003), “Value‐at‐risk and extreme returns in Asian stock markets”, International Journal of Business, Vol. 8 No. 1, pp. 17‐40.

Danielsson, J. and de Vries, C. (1997), “Tail index and quantile estimation with very high frequency data”, Journal of Empirical Finance, Vol. 4, pp. 241‐57.

Delbaen, F. (2001), “Coherent risk measures”, lecture notes, Scuola Nomale di Pisa, Pisa.

Diebold, F.X., Schuermann, T. and Stroughair, J. (1998), “Pitfalls and opportunities in the use of extreme value theory in risk management”, The Journal of Risk Finance, Vol. 1, pp. 30‐6.

Duffie, D. and Pan, J. (1997), “An overview of value‐at‐risk”, Journal of Derivatives, Vol. 7, pp. 7‐49.

Efron, B. and Tibshirani, J. (1993), An Introduction to the Bootstrap, Chapman & Hall, London.

Embrechts, P. (2000), Extremes and Integrated Risk Management, published in association with UBS Warburg and Risk Books, London.

Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997), Modelling Extremal Events for Insurance and Finance, Springer Verlag, Berlin.

Embrechts, P., Resnick, S. and Samorodnitsky, G. (1999), “Extreme value theory as a risk management tool”, North American Actuarial Journal, Vol. 3, pp. 30‐41.

Fernandez, V. (2003), “Extreme value theory and value at risk”, Revista de Analisis Economico, Vol. 18 No. 1, pp. 57‐85.

Föellmer, H. and Schied, A. (2002), “Robust preferences and convex risk measures”, Advances in Finance and Stochastics, pp. 39‐56.

Gençay, R. and Selçuk, F. (2004), “Extreme value theory and value‐at‐risk: relative performance in emerging markets”, International Journal of Forecasting, Vol. 20, pp. 287‐303.

Gençay, R., Selçuk, F. and Ulugülyağci, A. (2003), “High volatility, thick tails and extreme value theory in value‐at‐risk estimation”, Insurance: Mathematics and Economics, Vol. 33, pp. 337‐56.

Gilli, M. and Këllezi, E. (2003), “An application of extreme value theory for measuring risk”, Department of Econometrics, University of Geneva and FAME, Geneva.

Gupta, A. and Liang, B. (2004), “Do hedge funds have enough capital? A value‐at‐risk approach”, EFA 2003 Annual Conference Paper No. 376, Journal of Financial Economics, Vol. 77 No. 1, pp. 219‐53.

Hull, J. and White, A. (1998), “Value at risk when daily changes in market variables are not normally distributed”, Journal of Derivatives, Vol. 5, pp. 9‐19.

Janicki, A. and Weron, A. (1994), Simulation and Chaotic Behavior of Stable Stochastic Processes, Marcel Dekker, New York, NY.

Jorion, P. (1997), Value‐at‐Risk: The New Benchmark for Controlling Market Risk, McGraw‐Hill, Chicago.

Longin, F.M. (2000), “From value at risk to stress testing: the extreme value approach”, Journal of Banking and Finance, Vol. 24 No. 7, pp. 1097‐130.

McNeil, A.J. (1997), “Estimating the tails of loss severity distributions using extreme value theory”, Theory ASTIN Bulletin, Vol. 27 No. 1, pp. 1117‐37.

McNeil, A.J. (1998), “Calculating quantile risk measures for financial return series using extreme value theory”, Department of Mathematics, ETH E‐Collection, Swiss Federal Technical University, Zurich.

McNeil, A.J. (1999), “Extreme value theory for risk managers”, Internal Modeling CAD II, Risk Books, London, pp. 93‐113.

McNeil, A.J. and Frey, R. (2000), “Estimation of tail‐related risk measures for heteroscedastic Financial Time Series: an extreme value approach”, Journal of Empirical Finance, Vol. 7, pp. 271‐300.

Mandelbrot, B. (1963), “The variation of certain speculative prices”, Journal of Business, Vol. 36, pp. 394‐419.

Mandelbrot, B. (1997), Fractals and Scaling in Finance, Springer‐Verlag, Berlin.

Meyfredi, J‐C. (2005), “Is there a gain to explicitly modelling extremes? A risk management analysis”, working paper, Edhec‐Risk and Asset Management Research Centre, Nice.

Mittnik, S., Rachev, S.T., Doganoglu, T. and Chenyao, D. (1999), “Maximum likelihood estimation of Stable Paretian models”, Mathematical and Computer Modelling, Vol. 29, pp. 275‐93.

Neftci, S. (2000), “Value at risk calculations, extreme events, and tail estimation”, The Journal of Derivatives, Spring, pp. 1‐15.

Rachev, S.T. and Mittnik, S. (2000), Stable Paretian Models in Finance, Wiley, New York, NY.

Reiss, R.D. and Thomas, M. (2001), Statistical Analysis of Extreme Values: From Insurance, Finance, Hydrology and Other Fields, Birkhäuser Verlag, Basel.

Samorodnitsky, G. and Taqqu, M.S. (1994), Stable Non‐Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman & Hall, New York, NY and London.

Van Der Vaart, A.W. (1998), Asymptotic Statistics, Cambridge University Press, Cambridge.

Wagner, N. and Marsh, T.A. (2000), On Adaptive Tail Index Estimation for Financial Return Models, Institute for Business and Economic Research, University of California, Berkeley, CA.

Yamai, Y. and Yoshiba, T. (2001), “ Comparative analyses of expected shortfall and value‐at‐risk (2): expected utility maximization and tail risk”, IMES Discussion Paper Series 2001‐E‐14, Institute for Monetary and Economic Studies, Tokyo.

Yamai, Y. and Yoshiba, T. (2002a), “Comparative analyses of expected shortfall and value‐at‐risk: their estimation error, decomposition, and optimization”, Monetary and Economic Studies, pp. 87‐122.

Yamai, Y. and Yoshiba, T. (2002b), “Comparative analyses of expected shortfall and value‐at‐risk (3): their validity under market stress”, Monetary and Economic Studies, pp. 181‐237.

Zolotarev, V.M. (1986), “One‐dimensional stable distributions”, American Mathematical Society Translations of Mathematics Monographs, Vol. 65, American Mathematical Society, Providence, RI (translation of the original 1983 Russian edition).

Bredin, D. and Hyde, S. (2001), “FOREX Risk: Measurement and evaluation using value‐at‐risk”, Journal of Business Finance and Accounting, Vol. 31, pp. 1389‐417.

Chavez‐Demoulin, V., Davison, A.C. and McNeil, A.J. (2005), “A point process approach to value‐at‐risk estimation”, Quantitative Finance, Vol. 5.

Harmantzis, F., Chien, Y. and Miao, L. (2005), “Empirical comparative study for expected shortfall and value‐at‐risk”, paper presented at INFORMS Applied Probability Conference, Ottawa.

Harmantzis, F., Marinelli, C. and Rachev, S.T. (2004), “A comparative study of some univariate models for value‐at‐risk”, paper presented at INFORMS Applied Probability Conference, Beijing.

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Emerald

Tập 7 Số 2

117-135

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