Backtesting Extreme Value Theory models of expected shortfall

  1. Alfonso Novales 1
  2. Laura Garcia-Jorcano 2
  1. 1 Universidad Complutense de Madrid
    info

    Universidad Complutense de Madrid

    Madrid, España

    ROR 02p0gd045

  2. 2 Universidad de Castilla-La Mancha
    info

    Universidad de Castilla-La Mancha

    Ciudad Real, España

    ROR https://ror.org/05r78ng12

Revista:
Documentos de Trabajo (ICAE)

ISSN: 2341-2356

Año de publicación: 2019

Número: 24

Páginas: 1-51

Tipo: Documento de Trabajo

DIALNET GOOGLE SCHOLAR lock_openDocta Complutense editor

Otras publicaciones en: Documentos de Trabajo (ICAE)

Resumen

We use stock market data to analyze the quality of alternative models and proceduresfor fore- casting expected shortfall (ES) at different significance levels. We compute ES forecasts from conditional models applied to the full distribution of returns as well asfrom modelsthat focus on tail events using extreme value theory (EVT). We also apply the semiparametric filtered historical simulation (FHS) approach to ES forecasting to obtain 10-day ES forecasts. At the 10-day hori- zon we also combine FHS with EVT. The performance of the different models is assessed using six differentES backtests recentlyproposedintheliterature.Ourresultssuggestthatconditional EVT-basedmodelsproducemore accurate 1-day and 10-day ES forecasts than do non-EVT based models. Under either approach, asymmetric probability distributions for return innovations tend to produce better forecasts. Incorporating EVT in parametric or semiparametric approaches also improves ES forecasting performance. These qualitative results are alsovalidforthe recent crisis period, even though all models then underestimate the level of risk. FHS narrows the range of numerical forecasts obtained from alternative models, thereby reducing model risk. Combining EVT and FHS seemsto be best approach forobtainingaccurateESforecasts.

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Información de financiación

The authors gratefully acknowledge financial support from the grants ECO2015-67305-P, Prometeo II /2013/015, Programa de Ayudas a la Investigacion from Banco de España, and Programa de Financiación de Universidad Complutense de Madrid - Santander Universidades.

Financiadores

Referencias bibliográficas

  • Aas, K. and Haff, I.H., The Generalized Hyperbolic Skew Student’s t-Distribution. Journal of Financial Econometrics, 2006, 4(2), 275-309.
  • Acerbi, C. and Szekely, B., Backtesting Expected Shortfall. Publication of MSCI. https://www.msci.com/www/research-paper/research-insight-backtesting/0128184734, 2014.
  • Acerbi, C. and Tasche, D., On the coherence of Expected Shortfall. Journal of Banking and Finance, 2002, 26, 1487-1503.
  • Alexander, C. and Sheedy, E, Developing a stress testing framework based on market risk models. Journal of Banking and Finance, 2008, 32, 2220-2236.
  • Artzner, P., Delbaen, F., Eber, J.M. and Heath, M., Coherent measures of risks. Mathematical Finance, 1999, 9, 203-228.
  • Azzalini, A. and Capitanio, A., Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2003, 65(2), 367-389.
  • Barone-Adesi, G. Bourgoin, F., and Giannopoulus, K., Don’t look back. Risk, 1998, 11, 100-103.
  • Barone-Adesi, G., Giannopoulus, K. and Vosper, L., VaR without correlations for portfolios of derivative securities. Journal of Futures Markets, 1999, 19, 583-602.
  • Barone-Adesi, G., Giannopoulos, K. and Vosper, L., Backtesting derivative portfolios with filtered historical simulation (FHS). European Financial Management, 2002, 8(1), 31-58.
  • Basel Committee on Banking Supervision, Standards: Minimum Capital requirements for market risk. Bank for International Settlements, 2016.
  • Berkowitz, J., Testing density forecasts, with applications to risk management. Review of Financial Studies, 2001, 14, 371-405.
  • Berkowitz, J., Christoffersen, P.F. and Pelletier, D., Evaluating Value-at-Risk models with desk-level data. Management Science, 2011, 57(12), 2213-2227.
  • Carver, L., Mooted VaR substitute cannot be backtested, says top quant. Risk, March, 2013, 8. Chan, K.F., and Gray, P., Using extreme value theory to measure value-at-risk for daily electricity spot prices. International Journal of Forecasting, 2006, 22(2), 283-300.
  • Chavez-Demoulin, V. and McGill, J., High-frequency financial data modeling using Hawkes processes. Journal of Banking and Finance, 2012, 36, 3415-3426.
  • Chen, J.M., Measuring market risk under the basel accords: VaR, stressed VaR, and Expected Shortfall. Aestimatio, The IEB International Journal of Finance, 2014, 8, 184-201.
  • Christoffersen, P.F., Evaluating interval forecasts. International Economic Review, 1998, 39, 841-862
  • Clift, S.S., Costanzino, N. and Curran, M., Empirical Performance of Backtesting Methods for Expected Shortfall. http://dx.doi.org/10.2139/ssrn.2618345, 2016.
  • Cont, R., Deguest, R. and Scandolo, G., Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance, 2010, 16(6), 593-291.
  • Costanzino, N. and Curran, M., Backtesting General Spectral Risk Measures with application to Expected Shortfall. http://dx.doi.org/10.2139/ssrn.2514403, 2015.
  • Crnkovic, C. and Drachman, J., Quality control. Risk, 1996, 9, 139-143. Daniels, H.E., Tail Probability Approximations. International Statistic Review, 1987, 55, 37-48.
  • Dan´ıelsson, J., Jorgensen, B.N., Mandira, S., Samorodnitsky, G. and de Vries, C.G., Subadditivity re-examined: the case for Value-at-Risk. www.riskresearch.org, 2005.
  • Dan´ıelsson, J., Jorgensen, B.N., Samorodnitsky, G., Sarma, M. and de Vries, C.G., Fat tails, VaR and subadditivity. Journal of Econometrics, 2013, 172(2), 283-291.
  • Dan´ıelsson, J. and de Vries, C.G., Value-at-Risk and extreme returns. Annales d’Economie et de Statistique, 2000, 239-270.
  • Davison, A. and Smith, R., Models for exceedances over high thresholds. Journal of the Royal Statistical Society. Series B, 1990, 393-442.
  • Degiannakis, S., Dent, P. and Floros, C., A Monte Carlo Simulation Approach to Forecasting Multi-period Value-at-Risk and Expected Shortfall Using the FIGARCH-skT Specification. The Manchester School, 2014, 82(1), 71-102.
  • Degiannakis, S., Floros, C. and Dent, P., Forecasting value-at-risk and expected shortfall using fractionally integrated models of conditional volatility: International evidence. International Review of Financial Analysis, 2013, 27, 21-33.
  • Diebold, F.X., Gunther, T.A. and Tay, A.S., Evaluating density forecasts with applications to financial risk management. International Economic Review, 1998, 39(4), 863-883.
  • Diebold, F.X., Schuermann, T. and Stroughair, J.D., Pitfalls and opportunities in the use of extreme value theory in risk management. The Journal of Risk Finance, 2000, 50, 264-272.
  • Ding, Z., Granger, C.W.J. and Engle, R.F., A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1993, 1, 83-106.
  • Du, Z. and Escanciano, J.C., Backtesting Expected Shortfall: Accounting for Tail Risk. Management Science, 2016, 63(4), 940-958.
  • Embrechts, P, Lambrigger, D. and Wuthrich, M., Multivariate extremes and the aggregation ¨ of dependence risks: examples and counter-examples. Extremes, 2009, 12(2), 107-127.
  • Embrechts, P., McNeil, A. and Straumann, D., Correlation and dependence in risk management: properties and pitfalls. Risk management: value at risk and beyond, 2002, 1, 176-223.
  • Embrechts, P., Ne ˘slehova, J. and W ´ uthrich, M.V., Additivity properties for Value-at-Risk ¨ under Archimedean dependence and heavy-tailedness. Insurance: Mathematics and Economics, 2009, 44(2), 164-169.
  • Emmer, S., Kratz, M. and Tasche, D., What is the best risk measure in practice? A comparison of standard measures. Journal of Risk, 2015, 18(2).
  • Engle R.F. and Manganelli, S., CAViaR: conditional autoregressive value at risk by regression quantiles. Journal of Business and Economic Statistics, 2004, 22, 367-381.
  • Ergen, I., Two-step methods in VaR prediction and the importance of fat tails. Quantitative Finance, 2015, 15(6), 1013-1030.
  • Ergün, A. and Jun, J., Time-varying higher-order conditional moments and forecasting in- ¨ traday VaR and expected shortfall. Quarterly Review of Economics and Finance, 2010, 50, 264-272.
  • Fernandez, C. and Steel, M., On Bayesian Modelling of Fat Tails and Skewness. Journal of the American Statistical Association, 1998, 93(441), 359-371.
  • Fissler, T., Ziegel, J.F. and Gneiting, T., Expected Shortfall is jointly elicitable with value at risk-implications for backtesting. arXiv preprint arXiv:1507.00244, 2015.
  • Garcia, R., Renault, E. and Tsafack, G., Proper conditioning for coherent VaR in portfolio ´ management. Management Science, 2007, 53(3), 483-494.
  • Garcia-Jorcano, L. and Novales, A., Volatility specifications versus probability distributions in VaR estimation. Manuscript, 2017.
  • Gerlach, R. and Chen, C. W., Bayesian expected shortfall forecasting incorporating the intraday range, Journal of Financial Econometrics, 2014, 14(1), 128-158.
  • Giot, P. and Laurent, S., Value-at-Risk for long and short trading positions. Journal of Applied Econometrics, 2003a, 18, 641-664.
  • Giot, P., and Laurent, S., Market risk in commodity markets: a VaR approach. Energy Economics, 2003b, 25, 435-457.
  • Gneiting T., Making and evaluation point forecasts. Journal of the American Statistical Association, 2011, 106, 746-762.
  • Gonzalo, J., and Olmo, J., Which extreme values are really extreme?. Journal of Financial Econometrics, 2004, 2(3), 349-369.
  • Graham, A. and Pal, J., Backtesting value-at-risk tail losses on a dynamic portfolio. ´ Journal of Risk Model Validation, 2014, 8(2), 59-96.
  • Hansen, B., Autorregressive conditional density estimation. International Economic Review, 1994, 35, 705-730.
  • Herrera, R., Energy risk management through self-exciting marked point process. Energy Economics, 2013, 38, 64-76.
  • Hill, B.M., A simple general approach to inference about the tail of a distribution. The Annals of Statistic, 1975, 3(5), 1163-1174.
  • Ibragimov, R., New Majorization Theory In Economics And Martingale Convergence Results In Econometrics. Ph.D. Thesis, Yale University, 2005.
  • Ibragimov, R., Portfolio diversification and value at risk under thicktailedness. Quantitative Finance, 2009, 9, 565-580.
  • Ibragimov, R. and Walden, J., The limits of diversification when losses may be large. Journal of banking and finance, 2007, 31(8), 2551-2569.
  • Jalal, A. and Rockinger, M., Predicting tail-related risk measures: The consequences of using GARCH filters for non-GARCH data. Journal of Empirical Finance, 2008, 15, 868-877.
  • Johnson, N.L., Systems of frequency curves generated by methods of translations. Biometrika, 1949, 36, 149-176.
  • Kerkhof, J. and Melenberg, B., Backtesting for risk-based regulatory capital. Journal of Banking and Finance, 2004, 28, 1845-1865.
  • Kourouma, L., Dupre, D., Sanfilippo, G. and Taramasco, O., Extreme value at risk and expected shortfall during financial crisis. http://dx.doi.org/10.2139/ssrn.1744091, 2011.
  • Kuester, K., Mittnik, S. and Paolella, M.S., Value-at-Risk prediction: a comparison of alternative strategies. Journal of Financial Econometrics, 2006, 4, 53-89.
  • Kupiec, P., Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 1995, 2, 174-184.
  • Lambert, P. and Laurent, S., Modelling Financial Time Series using GARCH-type models with a skewed student distribution for the innovations. Mimeo, Université de Liege , 2001.
  • Longin, F., The choice of the distribution of asset returns: how extreme value theory can help? Journal of Banking and Finance, 2005, 29, 1017-1035.
  • Lönnbark, C., Approximation methods for multiple period Value at Risk and Expected Short- ¨ fall prediction. Quantitative Finance, 2016, 16(6), 947-968.
  • Lugannani, R. and Rice, S.O., Saddlepoint approximation for the distribution of the sum of independent random variables. Advanced Applied Probability, 1980, 12, 475-490.
  • Mabrouk, S. and Saadi, S., Parametric value-at-risk analysis: evidence from stock indices. The Quartely Review of Economics and Fiannce, 2012, 52, 305-321.
  • Marinelli, C., D’Addona, S. and Rachev, S., A comparison of some univariate models for value-at-risk and expected shortfall. International Journal of Theoretical and Applied Finance, 2007, 10, 1043-1075.
  • McMillan, D.G. and Kambourodis, D., Are RiskMetrics forecasts good enough? Evidence from 31 stock markets. International Review of Financial Analysis, 2009, 18, 117-124.
  • McMillan, D.G. and Speight, A.E., Daily volatility forecasts: Reassessing the performance of GARCH models. Journal of Forecasting, 2004, 23(6), 449-460.
  • McNeil, A. and Frey, R., Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 2000, 7, 271-300.
  • Nelson, D.B., Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 1991, 59(2), 347-370.
  • Pascual, L., Romo, J. and Ruiz, E., Bootstrap prediction for returns and volatilities in garch models. Computational Statistics and Data Analysis, 2006, 50(9), 2293-2312.
  • Pickands, J., Statistical inference using extreme order statistics. The Annals of Statistics, 1975, 3, 119-131.
  • Righi, M.B., and Ceretta, P.S., A comparison of Expected Shortfall estimation models. Journal of Economics and Business, 2015, 78, 14-47.
  • Righi, M.B. and Ceretta, P.S., Individual and flexible Expected Shortfall backtesting. Journal of Risk Model Validation, 2013, 7(3), 3-20.
  • Rocco, M., Extreme value theory in finance: a survey. Journal of Economic Surveys, 2014, 28(1), 82-108.
  • Rosenblatt, M., Remarks on a multivariate transformation. Annals of Mathematical Statistics, 1952, 23, 470-472.
  • Ruiz, E. and Pascual, L., Bootstrapping financial time series. Journal of Economic Surveys, 2002, 16(3), 271-300
  • Smith, R., Estimating tails of probability distributions. The Annals of Statistics, 1987, 15, 1174-1207.
  • So, M.K. and Wong, C.M., Estimation of multiple period expected shortfall and median shortfall for risk management. Quantitative Finance, 2012, 12(5), 739-754.
  • Taylor, J.W., Using exponentially weighted quantile regression to estimate value at risk and expected shortfall. Journal of Financial Econometrics, 2007, 6(3), 382-406.
  • Theodossiou, P., Financial data and skewed generalized t distribution. Management Science, 1998, 44, 1650-1661.
  • Tolikas, K., Unexpected tails in risk measurement: Some international evidence. Journal of Banking and Finance, 2014, 40, 476-493.
  • Wong, W.K., Fan, G. and Zeng, Y., Capturing tail risks beyond VaR. Review of Pacific Basin Financial Markets and Polices, 2012, 15(03), 1250015.
  • Wong, W.K., Backtesting trading risk of commercial banks using expected shortfall. Journal of Banking and Finance, 2008, 3, 1404-1415.
  • Wong, W.K., Backtesting value-at-risk based on tail losses. Journal of Empirical Finance, 2010, 17(3), 526-538.
  • Yamai, Y. and Yoshiba, T., 2005. Value-at-risk versus expected shortfall: A practical perspective. Journal of Banking and Finance, 29(4), 997-1015.
  • Zhou, J., Extreme risk measures for REITs: A comparison among alternative methods. Applied Financial Economics, 2012, 22, 113-126.
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