## 2020, issue 3, p. 43-58

Received 07.10.2020; Revised 12.10.2020; Accepted 23.10.2020

Published 27.10.2020; First Online 05.11.2020

https://doi.org/10.34229/2707-451X.20.3.5

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MSC 62E17, 62P05, 90C08, 91G70

A New Family of Expectiles and its Properties

Viktor Kuzmenko

V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv

Introduction. This paper considers a risk measure called expectile. Expectile is a characteristic of a random variable calculated using the asymmetric least square method. The level of asymmetry is defined by a parameter in the interval (0, 1). Expectile is used in financial applications, portfolio optimization problems, and other applications as well as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). But expectile has a set of advantageous properties. Expectile is both a coherent and elicitable risk measure that takes into account the whole distribution and assigns greater weight to the right tail.

The purpose of the paper. As a rule, expectile is compared with quantile (VaR). Our goal is to compare expectile with CVaR by introducing the same parameter – confidence level. To do this we first give a new representation of expectile using the weighted sum of mean and CVaR. Then we consider a new family of expectiles defined by two parameters. Such expectiles are compared with quantile and CVaR for different continuous and finite discrete distributions. Our next goal is to build a regular risk quadrangle where expectile is a risk function.

Results. We propose and substantiate two new expressions that define expectile. The first expression uses maximization by varying confidence level of CVaR and varying coefficient before CVaR. It is specified for continuous and finite discrete distributions. The second expression uses minimization of the new error function of the new expectile-based risk quadrangle. The use of two parameters in expectile definition changes the dependence of expectile on its confidence level and generates a new family of expectiles. Comparison of new expectiles with quantile and CVaR for a set of distributions shows that the proposed expectiles can be closer to the quantile than the standard expectile. We propose two variants for expectile linearization and show how to use them with a linear loss function.

Keywords: Expectile, EVaR, Quantile, Conditional Value-at-Risk, CVaR, Kusuoka representation, Fundamental Risk Quadrangle, Portfolio Safeguard package.

Cite as: Kuzmenko V. A New Family of Expectiles and its Properties. Cybernetics and Computer Technologies. 2020. 3. P. 43–58. https://doi.org/10.34229/2707-451X.20.3.5

References

1.     Newey W.K., Powell J.L. Asymmetric least squares estimation and testing. Econometrica. 1987. 55 (4). P. 819–847. https://doi.org/10.2307/1911031

2.     Yao Q., Tong H. Asymmetric least squares regression estimation: A nonparametric approach. Journal of Nonparametric Statistics. 1996. 6 (2–3). P. 273–292. https://doi.org/10.1080/10485259608832675

3.     Bellini F., Klar B., Müller A., Gianin E.R. Generalized quantiles as risk measures. Insurance: Mathematics and Economics. 2014. 54. P. 41–48. https://doi.org/10.1016/j.insmatheco.2013.10.015

4.     Rockafellar R.T., Uryasev S. Conditional Value-at-Risk for General Loss Distributions. Journal of Banking and Finance. 2002. 26 (7). P. 1443–1471. https://doi.org/10.1016/S0378-4266(02)00271-6

5.     Ziegel J.F. Coherence and Elicitability. Mathematical Finance. 2016. 26 (3). P. 901–918. https://doi.org/10.1111/mafi.12080

6.     Jakobsons E. Scenario aggregation method for portfolio expectile optimization. Statistics & Risk Modeling. 2016. 33 (1–2). P. 51–65. https://doi.org/10.1515/strm-2016-0008

7.     Colombo С. Portfolio Optimization with Expectiles. University of Milano, 2018. 157 p. https://doi.org/10.13140/RG.2.2.35097.67685

8.     Wagner A., Uryasev S. Portfolio Optimization with Expectile and Omega Functions. Risk Management (q-fin.RM). 2019. https://arxiv.org/abs/1910.14005

9.     Waltrup L. S., Sobotka F., Kneib T., Kauermann G. Expectile and quantile regression – David and Goliath? Statistical Modelling. 2015. 15 (5). P. 433–456. https://doi.org/10.1177/1471082X14561155

10.     Rockafellar R.T., Uryasev S. The Fundamental Risk Quadrangle in Risk Management, Optimization and Statistical Estimation. Surveys in Operations Research and Management Science. 2013. 18 (1). P. 33–53. https://doi.org/10.1016/j.sorms.2013.03.001

11.     Kuzmenko V., Golodnikov A., Uryasev S. CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles. J. Risk Financial Manag. 2019. 12 (3). P. 107. https://doi.org/10.3390/jrfm12030107

12.     Koenker R. Quantile Regression. Cambridge University Press, 2005. 349 p. https://doi.org/10.1017/CBO9780511754098

13.     AORDA Portfolio Safeguard. http://www.aorda.com/index.php/portfolio-safeguard/ (accessed 14.10.2020)

14.     Bellini F., Bernardino E.D. Risk management with Expectiles. The European Journal of Finance. 2017. 23 (6). P. 487–506. https://doi.org/10.1080/1351847X.2015.1052150

15.     Schnabel S.K., Eilers P.H.C. Optimal expectile smoothing. Computational Statistics and Data Analysis. 2009. 53 (12). P. 4168–4177. https://doi.org/10.1016/j.csda.2009.05.002

16.     Chen J.M. On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles. Risks. 2018. 6 (2). P. 61. https://doi.org/10.3390/risks6020061

17.     Kusuoka S. On law invariant coherent risk measures. In: Kusuoka S., Maruyama T. (eds). Advances in Mathematical Economics. 2001. 3. P. 83–95. https://doi.org/10.1007/978-4-431-67891-5_4

18.     Shapiro A. On Kusuoka Representation of Law Invariant Risk Measures. Mathematics of Operations Research. 2012. 38 (1). https://doi.org/10.1287/moor.1120.0563

19.     Pichler A., Shapiro A. Uniqueness of Kusuoka Representations. 2012. https://arxiv.org/abs/1210.7257v4

20.     Bellini F., Bignozzi V., Puccetti G. Conditional expectiles, time consistency and mixture convexity properties. Insurance: Mathematics and Economics. 2018. 82. P. 117–123. https://doi.org/10.1016/j.insmatheco.2018.07.001

21.     Jones M.C. Expectiles and M-quantiles are quantiles. Statistics & Probability Letters. 1994. 20 (2). P. 149–153. https://doi.org/10.1016/0167-7152(94)90031-0

22.     Weber S. Distribution-Invariant Risk Measures, Information, and Dynamic Consistency. Mathematical Finance. 2006. 16 (2). P. 419–441. https://doi.org/10.1111/j.1467-9965.2006.00277.x

23.     Gschöpf P. Measuring risk with expectile based expected shortfall estimates. Berlin: Humboldt-University, 2014. 61 p. http://dx.doi.org/10.18452/14215

24.     Delbaen F. A Remark on the Structure of Expectiles. 22 Jul 2013. 9 p. https://arxiv.org/abs/1307.5881

25.     Lan G., Zhou Z. Algorithms for stochastic optimization with function or expectation constraints. Comput Optim Appl. 2020. 76. (2). P. 461–498. https://doi.org/10.1007/s10589-020-00179-x

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