C. Acerbi and D. Tasche, On the coherence of expected shortfall, Journal of Banking & Finance, vol.26, issue.7, pp.1487-1503, 2002.

J. Akahori, Some Formulae for a New Type of Path-Dependent Option, Ann. Appl. Probab, vol.5, issue.2, pp.383-388, 1995.

A. Alfons and M. Templ, Estimation of Social Exclusion Indicators from Complex Surveys: The R Package laeken, Journal of Statistical Software, vol.54, issue.15, pp.1-25, 2013.

C. Anderson and K. Turkman, The joint limiting distribution of sums and maxima of stationary sequences, Journal of Applied Probability, vol.28, issue.1, pp.33-44, 1991.

D. Ardia, K. Bluteau, K. Boudt, L. Catania, and D. Trottier, Markov-switching GARCH models in R: The MSGARCH package, Journal of Statistical Software, 2016.

J. Arismendi, Multivariate truncated moments, Journal of Multivariate Analysis, vol.117, pp.41-75, 2013.

P. Artzner, F. Delbaen, J. Eber, and D. Heath, Coherent measures of risks, Mathematical Finance, vol.9, pp.203-228, 1999.

P. Artzner, F. Delbaen, J. Eber, and D. Heath, Thinking coherently, Risk, vol.10, pp.68-71, 1997.

P. Athanasoglou, I. Daniilidis, and M. Delis, Bank procyclicality and output: Issues and policies, Journal of Economics and Business, vol.72, pp.58-83, 2014.

A. Aue, I. Berkes, and L. Horváth, Strong approximation for the sums of squares of augmented GARCH sequences, Bernoulli, vol.12, issue.4, pp.583-608, 2006.

A. Aue, S. Hörmann, L. Horváth, and M. Reimherr, Break detection in the covariance structure of multivariate time series models, The Annals of Statistics, vol.37, issue.6B, pp.4046-4087, 2009.

R. Bahadur, A note on quantiles in large samples, The Annals of Mathematical Statistics, vol.37, issue.3, pp.577-580, 1966.

G. Barone-adessi, F. Burgoin, G. , and K. , VaR without correlations for nonlinear portfolios, Risk, vol.11, pp.100-104, 1998.

G. Barone-adessi, K. Giannopoulos, and L. Vosper, VaR without correlations for nonlinear portfolios, Journal of Future Markets, vol.19, pp.583-602, 1999.

, Basel Committee on Banking Supervision. Amendment to the capital accord to incorporate market risks. Bank for International Settlements, 1996.

, Basel Committee on Banking Supervision. Minimum capital requirements for market risk, 2019.

F. Bec and C. Gollier, Term Structure and Cyclicity of Value-at-Risk: Consequences for the Solvency Capital Requirement, 2009.

F. Bellini and E. Di-bernardino, Risk management with expectiles, The European Journal of Finance, vol.23, issue.6, pp.487-506, 2017.

F. Bellini, B. Klar, A. Müller, and E. R. Gianin, Generalized quantiles as risk measures, Insurance: Mathematics and Economics, vol.54, pp.41-48, 2014.

A. Bera, A. Galvao, L. Wang, X. , and Z. , A new characterization of the normal distribution and test for normality, Econometric Theory, vol.32, issue.5, pp.1216-1252, 2016.

I. Berkes, S. Hörmann, and L. Horváth, The functional central limit theorem for a family of GARCH observations with applications, Statistics & Probability Letters, vol.78, issue.16, pp.2725-2730, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00595953

P. Billingsley, Convergence of probability measures. 1st, 1968.

T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, vol.31, issue.3, pp.307-327, 1986.

T. Bollerslev, Glossary to arch (garch, CREATES Research Paper, vol.49, 2008.

C. Bos and P. Janus, A Quantile-based Realized Measure of Variation: New Tests for Outlying Observations in Financial Data, 2013.

J. Boudoukh, M. Richardson, and R. Whitelaw, The best of both worlds, Risk, vol.11, issue.5, pp.64-67, 1998.

F. Boussama, Ergodicité, mélange et estimation dans les modeles GARCH, 1998.

M. Bräutigam, M. Dacorogna, and M. Kratz, Predicting risk with risk measures: an empirical study, p.1803, 2018.

M. Bräutigam, M. Dacorogna, and M. Kratz, Pro-Cyclicality of Traditional Risk Measurements: Quantifying and Highlighting Factors at its Source, 2019.

M. Bräutigam and M. Kratz, Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH(p,q) processes, 2019.

M. Bräutigam and M. Kratz, On the Dependence between Functions of Quantile and Dispersion Estimators, 2019.

M. Bräutigam and M. Kratz, On the Dependence between Quantiles and Dispersion Estimators, ESSEC Working Paper, p.1807, 2018.

M. Bräutigam and M. Kratz, The Impact of the Choice of Risk and Dispersion Measure on Procyclicality, 2020.

M. Brautigam, Online Appendix: Pro-cyclicality of Risk Measurements -Empirical Quantification and Theoretical Confirmation

L. Breiman and . Probability, classics in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.7, 1992.

M. Carrasco and X. Chen, Mixing and moment properties of various GARCH and stochastic volatility models, Econometric Theory, vol.18, issue.1, pp.17-39, 2002.

J. Chen, On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles, Risks, vol.6, issue.2, pp.1-29, 2018.

S. Chen, Nonparametric estimation of expected shortfall, Journal of Financial Econometrics, vol.6, issue.1, pp.87-107, 2008.

T. Chow and J. Teugels, The sum and the maximum of iid random variables, Proceedings of the 2nd Prague Symposium on Asymptotic Statistics, pp.81-92, 1978.

P. F. Christoffersen, Evaluating interval forecasts, International Economic Review, vol.39, pp.841-862, 1998.

M. Dacorogna, R. Gencay, U. Muller, O. Pictet, and R. Olsen, An introduction to High-Frequency Finance, 2001.

D. B. Dahl, D. Scott, C. Roosen, A. Magnusson, J. Swinton et al., Export Tables to LaTeX or HTML. R package version 1.8-4, 2019.

J. Daníelsson, The Emperor has no clothes: Limits to risk modelling, Journal of Banking and Finance, vol.26, pp.1273-1296, 2002.

A. Dasgupta and L. Haff, Asymptotic values and expansions for the correlation between different measures of spread, Journal of Statistical Planning and Inference, vol.136, issue.7, pp.2197-2212, 2006.

H. David, Early sample measures of variability, Statistical Science, pp.368-377, 1998.

J. Davidson, Stochastic limit theory: An introduction for econometricians, 1994.

G. De-rossi and A. Harvey, Quantiles, expectiles and splines, Journal of Econometrics, vol.152, issue.2, pp.179-185, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00553823

Z. Ding, C. Granger, and R. Engle, A long memory property of stock market returns and a new model, Journal of Empirical Finance, vol.1, issue.1, pp.83-106, 1993.

J. Duan, Augmented GARCH (p, q) process and its diffusion limit, Journal of Econometrics, vol.79, issue.1, pp.97-127, 1997.

D. Duttweiler, The mean-square error of Bahadur's order-statistic approximation, The Annals of Statistics, pp.446-453, 1973.

M. Ekström, A general central limit theorem for strong mixing sequences, Statistics & Probability Letters, vol.94, pp.236-238, 2014.

P. Embrechts and G. Samorodnitsky, Sample Quantiles of heavy tailed stochastic processes, Stochastic Processes and their Applications, vol.59, pp.217-233, 1995.

S. Emmer, M. Kratz, and D. Tasche, What is the best risk measure in practice? A comparison of standard risk measures, Journal of Risk, vol.18, issue.2, pp.31-60, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00921283

R. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica: Journal of the Econometric Society, vol.50, issue.4, pp.987-1007, 1982.

R. Engle and T. Bollerslev, Modelling the persistence of conditional variances, Econometric Reviews, vol.5, issue.1, pp.1-50, 1986.

R. Engle and S. Manganelli, CAViaR: Conditional autoregressive value at risk by regression quantiles, Journal of Business & Economic Statistics, vol.22, issue.4, pp.367-381, 2004.

R. Engle and V. Ng, Measuring and testing the impact of news on volatility, The Journal of Finance, vol.48, issue.5, pp.1749-1778, 1993.

E. Banking and A. , Results from the 2018 market risk benchmarking exercise, 2019.

, European Union, EC (Solvency II)". Official Journal of European Union, vol.138, 2009.

. Falk and R. Reiss, Independence of order statistics, The Annals of Probability, pp.854-862, 1988.

M. Falk, Asymptotic independence of median and MAD, Statistics & Probability Letters, vol.34, issue.4, pp.341-345, 1997.

T. Ferguson, Asymptotic joint distribution of sample mean and a sample quantile, 1999.

R. Fisher, On the probable error of a coefficient of correlation deduced from a small sample, Metron, vol.1, pp.3-32, 1921.

C. Francq, L. Horvath, and J. Zakoïan, Merits and drawbacks of variance targeting in GARCH models, Journal of Financial Econometrics, vol.9, issue.4, pp.619-656, 2011.

C. Francq and J. Zakoian, GARCH models: structure, statistical inference and financial applications, 2019.

J. Geweke, Modeling the persistence of conditional variances: a comment, Econometric Reviews, vol.5, pp.57-61, 1986.

A. Ghalanos and . Rugarch, Univariate GARCH models. R package version 1.4-1, 2019.

J. Ghosh, A new proof of the Bahadur representation of quantiles and an application, The Annals of Mathematical Statistics, pp.1957-1961, 1971.

L. Glosten, R. Jagannathan, R. , and D. , On the relation between the expected value and the volatility of the nominal excess return on stocks, The Journal of Finance, vol.48, issue.5, pp.1779-1801, 1993.

S. Gorard, Revisiting a 90-year-old debate: the advantages of the mean deviation, British Journal of Educational Studies, vol.53, issue.4, pp.417-430, 2005.

P. Hall and A. Welsh, Limit theorems for the median deviation, Annals of the Institute of Statistical Mathematics, vol.37, issue.1, pp.27-36, 1985.

F. Hampel, The influence curve and its role in robust estimation, Journal of the American Statistical Association, vol.69, issue.346, pp.383-393, 1974.

M. Higgins and A. Bera, A class of nonlinear ARCH models, International Economic Review, vol.33, issue.1, pp.137-158, 1992.

H. Ho, T. Lin, H. Chen, and W. Wang, Some results on the truncated multivariate t distribution, Journal of Statistical Planning and Inference, vol.142, issue.1, pp.25-40, 2012.

H. Ho and T. Hsing, On the asymptotic expansion of the empirical process of long-memory moving averages, The Annals of Statistics, vol.24, issue.3, pp.992-1024, 1996.

M. Hofert, K. Hornik, A. J. Mcneil, and . Qrmtools, Tools for Quantitative Risk Management. R package version 0.0-10, 2018.

S. Hörmann, Augmented GARCH sequences: Dependence structure and asymptotics, Bernoulli, vol.14, issue.2, pp.543-561, 2008.

T. Hsing, A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables, The Annals of Probability, pp.938-947, 1995.

J. Hull and A. White, Incorporating volatility updating into the historical simulation method for value-at-risk, Journal of Risk, vol.1, issue.1, pp.5-19, 1998.

I. A. Ibragimov, Some limit theorems for stationary processes, Theory of Probability & Its Applications, vol.7, pp.349-382, 1962.

A. Jamalizadeh, M. Khosravi, and N. Balakrishnan, Recurrence relations for distributions, Computational Statistics & Data Analysis, vol.53, issue.4, pp.847-852, 2009.

A. Jamalizadeh, R. Pourmousa, and N. Balakrishnan, Truncated and limited skew-normal and skew-t distributions: properties and an illustration, Communications in Statistics-Theory and Methods, vol.38, pp.2653-2668, 2009.

D. James, K. Hornik, and . Chron, Chronological Objects which Can Handle Dates and Times. R package version 2.3-53. S original by David James, R port by Kurt Hornik, 2018.

H. Kim, Moments of truncated Student-t distribution, Journal of the Korean Statistical Society, vol.37, issue.1, pp.81-87, 2008.

M. Kratz, Y. Lok, and A. Mcneil, Multinomial VaR Backtests: A simple implicit approach to backtesting expected shortfall, Journal of Banking and Finance, vol.88, pp.393-407, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01424279

W. Kruskal, Ordinal measures of association, Journal of the American Statistical Association, vol.53, issue.284, pp.814-861, 1958.

C. Kuan, J. Yeh, and Y. Hsu, Assessing value at risk with care, the conditional autoregressive expectile models, Journal of Econometrics, vol.150, issue.2, pp.261-270, 2009.

R. Kulik, Optimal rates in the Bahadur-Kiefer representation for GARCH sequences, 2006.

O. Lee, Functional central limit theorems for augmented GARCH (p, q) and FIGARCH processes, Journal of the Korean Statistical Society, vol.43, issue.3, pp.393-401, 2014.

E. Lehmann, Elements of Large-Sample Theory, 1999.

P. Lin, K. Wu, A. , and I. , Asymptotic joint distribution of sample quantiles and sample mean with applications, Communications in Statistics-Theory and Methods, vol.9, issue.1, pp.51-60, 1980.

R. Loynes, The variance and the range of i.I.D. random variables, Communications in Statistics -Theory and Methods, vol.19, issue.4, pp.1419-1432, 1990.

S. Mazumder and R. Serfling, Bahadur representations for the median absolute deviation and its modifications, Statistics & Probability Letters, vol.79, issue.16, pp.1774-1783, 2009.

T. Mikosch and C. St?ric?, Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process, The Annals of Statistics, vol.28, issue.5, pp.1427-1451, 2000.

A. Milhøj, A multiplicative parameterization of arch models, 1987.

R. Miura, A note on look-back options based on order statistics, Hitotsubashi J. Commerce Management, vol.27, pp.15-28, 1992.

J. Morgan, Introduction to riskmetrics, 1994.

D. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica: Journal of the Econometric Society, vol.59, issue.2, pp.347-370, 1991.

M. Neumann, A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics, ESAIM: Probability and Statistics, vol.17, pp.120-134, 2013.

W. Newey and J. Powell, Asymmetric least squares estimation and testing, Econometrica: Journal of the Econometric Society, pp.819-847, 1987.

J. P. Nolan, Stable Distributions -Models for Heavy Tailed Data, progress, 2018.

S. Pantula, Modeling the persistence of conditional variances: a comment, Econometric Reviews, vol.5, pp.79-97, 1986.

C. Pérignon and D. R. Smith, The level and quality of Value-at-Risk disclosure by commercial banks, Journal of Banking & Finance, vol.34, issue.2, pp.362-377, 2010.

. Pham-gia and T. Hung, The mean and median absolute deviations, Mathematical and Computer Modelling, vol.34, pp.921-936, 2001.

W. Philipp, A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables, The Annals of Probability, pp.319-350, 1977.

D. Politis, J. Romano, and M. Wolf, Subsampling for heteroskedastic time series, Journal of Econometrics, vol.81, issue.2, pp.281-317, 1997.

M. Pritsker, The hidden dangers of historical simulation, Journal of Banking & Finance, vol.30, issue.2, pp.561-582, 2006.

R. Pyke and . Spacings, Journal of the Royal Statistical Society. Series B (Methodological), pp.395-449, 1965.

M. Quagliariello, Does macroeconomy affect bank stability? A review of the empirical evidence, Journal of Banking Regulation, vol.9, issue.2, pp.102-115, 2008.

. R-core-team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, 2019.

. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, vol.26, issue.7, pp.1443-1471, 2002.

R. Rodriguez, Correlation, Encyclopedia of Statistical Sciences, pp.1375-1385, 1982.
URL : https://hal.archives-ouvertes.fr/hal-00008512

G. Roussas and D. Ioannides, Moment inequalities for mixing sequences of random variables, Stochastic Analysis and Applications, vol.5, issue.1, pp.60-120, 1987.

R. Team, RStudio: Integrated Development Environment for R. RStudio, 2016.

G. Schwert, Why does stock market volatility change over time?, The Journal of Finance, vol.44, issue.5, pp.1115-1153, 1989.

J. Segers, On the asymptotic distribution of the mean absolute deviation about the mean, 2014.

R. Serfling and S. Mazumder, Exponential probability inequality and convergence results for the median absolute deviation and its modifications, Statistics & Probability Letters, vol.79, issue.16, pp.1767-1773, 2009.

S. Taylor, Modelling financial time series, 1986.

A. Trapletti and K. Hornik, tseries: Time Series Analysis and Computational Finance. R package version 0.10-47, 2019.

A. Vaart and . Van-der, Asymptotic statistics, 1998.

M. Wendler, Bahadur representation for U-quantiles of dependent data, Journal of Multivariate Analysis, vol.102, issue.6, pp.1064-1079, 2011.

H. Wickham and J. Bryan, Read Excel Files

C. Withers and S. Nadarajah, Expansions for the joint distribution of the sample maximum and sample estimate, Sankhy?: The Indian Journal of Statistics, Series A, pp.109-123, 2008.

W. Wu, On the Bahadur representation of sample quantiles for dependent sequences, The Annals of Statistics, vol.33, pp.1934-1963, 2005.

D. Wuertz, T. Setz, and Y. Chalabi, fBasics: Rmetrics -Markets and Basic Statistics. R package version 3042.89, 2017.

D. Wuertz, T. Setz, Y. Chalabi, C. Boudt, P. Chausse et al., Rmetrics -Autoregressive Conditional Heteroskedastic Modelling, 2019.

Q. Yao and H. Tong, Asymmetric least squares regression estimation: a nonparametric approach, Journal of Nonparametric Statistics, vol.6, issue.2-3, pp.273-292, 1996.

J. Zakoian, Threshold heteroskedastic models, Journal of Economic Dynamics and Control, vol.18, issue.5, pp.931-955, 1994.

G. Zumbach, Correlations of the realized volatilities with the centered volatility increment, 2012.

G. Zumbach, The pitfalls in fitting GARCH (1, 1) processes, Advances in Quantitative Asset Management, pp.179-200, 2000.

G. Zumbach, Discrete time series, processes, and applications in finance, 2012.

, See the online-appendix in, vol.34

, We present for each index the plots of the logarithm of the SQP ratios against the annualized volatility (MAD)

C. Aus, . Deu, . Fra, . Gbr, . Ita et al., Real data FIGURE A.4: The logarithm of ES ratios as a function of annualized volatility (on real data) for ? = 95%. From left to right

, GARCH

. Figure-a, . Aus, . Can, . Deu, . Fra et al., The logarithm of ES ratios as a function of annualized volatility (on sample paths from GARCH(1,1) simulations with Gaussian innovations) for ? = 95%. From left to right, vol.5

, A.5.2 SQP ratios and annualized volatility (MAD)

, We see the mean of 1000 repetitions, and its corresponding empirical 2.5% and 97.5% quantiles. From left to right, top to bottom, the estimation methods are: Zumbach, fGARCH, rugarch

, Estimation Quality: Beta value FIGURE E.3: Parameter estimates of beta for sample 1 with the different methods

, Estimation Quality: Alpha value FIGURE E.5: Parameter estimates of alpha in sample 2 with the different methods

, We see the mean of 1000 repetitions, and its corresponding empirical 2.5% and 97.5% quantiles, Estimation Quality: Beta value FIGURE E, vol.6

E. , Overall summary On real data we observed the following (although not presented here, but only in the online-appendix

, ? In general all methods behave similarly (and sensitive) to changes in the data set