Reverse stress testing in skew-elliptical models

Authors:
Jonathan von Schroeder, Thorsten Dickhaus and Taras Bodnar

Journal:
Theor. Probability and Math. Statist. **109** (2023), 101-127

MSC (2020):
Primary 62E15; Secondary 62P05

DOI:
https://doi.org/10.1090/tpms/1199

Published electronically:
October 3, 2023

MathSciNet review:
4652996

Full-text PDF

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References |
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Additional Information

Abstract: Stylized facts about financial data comprise skewed and heavy-tailed (log-)returns. Therefore, we revisit previous results on reverse stress testing under elliptical models, and we extend them to the broader class of skew-elliptical models. In the elliptical case, an explicit formula for the solution is provided. In the skew-elliptical case, we characterize the solution in terms of an easy-to-implement numerical optimization problem. As specific examples, we investigate the classes of skew-normal and skew-t models in detail. Since the solutions depend on population parameters, which are often unknown in practice, we also tackle the statistical task of estimating these parameters and provide confidence regions for the most likely scenarios.

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References
- C. Adcock and A. Azzalini,
*A selective overview of skew-elliptical and related distributions and of their applications*, Symmetry **12** (2020), no. 1, 118.
- C. Adcock, M. Eling, and N. Loperfido,
*Skewed distributions in finance and actuarial science: a review*, The European Journal of Finance **21** (2015), no. 13-14, 1253â€“1281.
- C. J. Adcock,
*Asset pricing and portfolio selection based on the multivariate extended skew-Student-$t$ distribution*, Ann. Oper. Res. **176** (2010), 221â€“234. MR **2607165**
- â€”,
*Meanâ€“varianceâ€“skewness efficient surfaces, Steinâ€™s lemma and the multivariate extended skew-Student distribution*, European J. Oper. Res. **234** (2014), no. 2, 392â€“401. MR **3144728**
- R. B. Arellano-Valle, M. D. Branco, and M. G. Genton,
*A unified view on skewed distributions arising from selections*, Canad. J. Statist. **34** (2006), no. 4, 581â€“601. MR **2347047**
- A. Azzalini,
*The skew-normal and related families*, Cambridge University Press, Cambridge, 2014. MR **3468021**
- â€”,
*The R package sn: The skew-normal and related distributions such as the skew-$t$*, 2019, R package version 1.5-4.
- A. Azzalini and G. Regoli,
*Some properties of skew-symmetric distributions*, Ann. Inst. Statist. Math. **64** (2012), no. 4, 857â€“879. MR **2927774**
- Basel Committee on Banking Supervision,
*Principles for sound stress testing practices and supervision*, Bank for International Settlements, Basel, 2009, ISBN: 92-9131-784-5.
- S. Bhattacharyya and P. J. Bickel,
*Adaptive estimation in elliptical distributions with extensions to high dimensions*, Preprint, available from http://sites.science.oregonstate.edu/~bhattash/Research_files/mixture_elliptic.pdf, 2014.
- T. Bodnar,
*An exact test on structural changes in the weights of the global minimum variance portfolio*, Quant. Finance **9** (2009), no. 3, 363â€“370. MR **2510189**
- T. Bodnar, S. Dmytriv, N. Parolya, and W. Schmid,
*Tests for the weights of the global minimum variance portfolio in a high-dimensional setting*, IEEE Trans. Signal Process. **67** (2019), no. 17, 4479â€“4493. MR **3999772**
- T. Bodnar and A. K. Gupta,
*Robustness of the inference procedures for the global minimum variance portfolio weights in a skew-normal model*, The European Journal of Finance **21** (2015), no. 13-14, 1176â€“1194.
- T. Bodnar, S. Mazur, and N. Parolya,
*Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions*, Scand. J. Stat. **46** (2019), no. 2, 636â€“660. MR **3948571**
- W. Breymann, A. Dias, and P. Embrechts,
*Dependence structures for multivariate high-frequency data in finance*, Quant. Finance **3** (2003), no. 1, 1â€“14. MR **1972372**
- P. Burman and W. Polonik,
*Multivariate mode hunting: data analytic tools with measures of significance*, J. Multivariate Anal. **100** (2009), no. 6, 1198â€“1218. MR **2508381**
- A. Capitanio,
*On the canonical form of scale mixtures of skew-normal distributions*, Statistica **80** (2020), no. 2, 145â€“160.
- J. Chen and Y. Huang,
*Finite-sample properties of the adjusted empirical likelihood*, J. Nonparametr. Stat. **25** (2013), no. 1, 147â€“159. MR **3039975**
- J. T. Chen, A. K. Gupta, and C. G. Troskie,
*The distribution of stock returns when the market is up*, Comm. Statist. Theory Methods **32** (2003), no. 8, 1541â€“1558. MR **1996794**
- R. Cont,
*Empirical properties of asset returns: stylized facts and statistical issues*, Quant. Finance **1** (2001), no. 2, 223â€“236.
- A. DasGupta,
*Asymptotic theory of statistics and probability*, Springer Texts in Statistics, Springer, New York, 2008. MR **2664452**
- P. J. Dhrymes,
*Mathematics for econometrics*, fourth ed., Springer, New York, 2013. MR **3113324**
- T. Dickhaus,
*Simultaneous statistical inference with applications in the life sciences*, Springer, Heidelberg, 2014. MR **3184277**
- E. J. Elton, M. J. Gruber, S. J. Brown, and W. N. Goetzmann,
*Modern portfolio theory and investment analysis*, John Wiley & Sons, Hoboken, NJ, 2014.
- K. T. Fang, S. Kotz, and K. W. Ng,
*Symmetric multivariate and related distributions*, Monographs on Statistics and Applied Probability, vol. 36, Chapman and Hall, Ltd., London, 1990. MR **1071174**
- G. Giorgi, A. Guerraggio, and J. Thierfelder,
*Mathematics of optimization: smooth and nonsmooth case*, Elsevier Science B.V., Amsterdam, 2004. MR **2068816**
- P. Glasserman, C. Kang, and W. Kang,
*Stress scenario selection by empirical likelihood*, Quant. Finance **15** (2015), no. 1, 25â€“41. MR **3290600**
- J. L. Horowitz,
*Bootstrap methods in econometrics*, Annual Review of Economics **11** (2019), no. 1, 193â€“224.
- Z. Hu and R.-C. Yang,
*A new distribution-free approach to constructing the confidence region for multiple parameters*, PLOS ONE **8** (2013), no. 12, 1â€“13.
- Y. Kopeliovich, A. Novosyolov, D. Satchkov, and B. Schachter,
*Robust risk estimation and hedging: A reverse stress testing approach*, The Journal of Derivatives, no. 4, 10â€“25.
- Z. M. Landsman and E. A. Valdez,
*Tail conditional expectations for elliptical distributions*, N. Am. Actuar. J. **7** (2003), no. 4, 55â€“71. MR **2061237**
- H. Markowitz,
*Portfolio selection*, The Journal of Finance **7** (1952), 77â€“91. MR **103768**
- A. J. McNeil, R. Frey, and P. Embrechts,
*Quantitative risk management*, Princeton University Press, Princeton, NJ, 2005. MR **2175089**
- A. B. Owen,
*Empirical likelihood*, Chapman and Hall/CRC, New York, 2001.
- B. Pfaff and A. McNeil,
*Qrm: Provides r-language code to examine quantitative risk management concepts*, 2020, R package version 0.4-20.
- K. K Said, W. Ning, and Y. Tian,
*Likelihood procedure for testing changes in skew normal model with applications to stock returns*, Comm. Statist. Simulation Comput. **46** (2017), no. 9, 6790â€“6802. MR **3764939**
- T. Shushi,
*Skew-elliptical distributions with applications in risk theory*, Eur. Actuar. J. **7** (2017), no. 1, 277â€“296. MR **3661137**
- P. Thiuthad and N. Pal,
*Point estimation of the location parameter of a skew-normal distribution: some fixed sample and asymptotic results*, J. Stat. Theory Pract. **13** (2019), no. 2, Paper No. 37, 27. MR **3924748**
- P. Traccucci, L. Dumontier, G. Garchery, and B. Jacot,
*A triptych approach for reverse stress testing of complex portfolios*, Preprint, available from https://arxiv.org/abs/1906.11186, 2019.
- H. Wakaki,
*Discriminant analysis under elliptical populations*, Hiroshima Math. J. **24** (1994), no. 2, 257â€“298. MR **1284376**
- S. S. Wilks,
*The large-sample distribution of the likelihood ratio for testing composite hypotheses*, The Annals of Mathematical Statistics **9** (1938), no. 1, 60â€“62.

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Additional Information

**Jonathan von Schroeder**

Affiliation:
University of Bremen, Institute for Statistics, Bremen, Germany

Email:
j.von.schroeder@gmail.com

**Thorsten Dickhaus**

Affiliation:
University of Bremen, Institute for Statistics, Bremen, Germany

Email:
dickhaus@uni-bremen.de

**Taras Bodnar**

Affiliation:
Stockholm University, Department of Mathematics, Stockholm, Sweden

Email:
taras.bodnar@math.su.se

Keywords:
Bank regulation,
constrained optimization,
empirical likelihood,
most likely scenario,
parametric bootstrap,
risk management

Received by editor(s):
March 31, 2022

Accepted for publication:
October 28, 2022

Published electronically:
October 3, 2023

Additional Notes:
The first author was supported by the Deutsche Forschungsgemeinschaft (DFG, http://dx.doi.org/10.13039/501100001659) within the framework of RTG 2224, entitled â€ś$\pi ^3$: Parameter Identification â€“ Analysis, Algorithms, Applicationsâ€ť.

Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv