# American Institute of Mathematical Sciences

June  2021, 6(2): 99-116. doi: 10.3934/puqr.2021005

## Dual representation of expectile-based expected shortfall and its properties

 1 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China 2 School of Mathematical Sciences & Shanghai Advanced Institute of Finance (CAFR), Shanghai Jiao Tong University, Shanghai 200030, China

Received  November 18, 2020 Accepted  April 19, 2021 Published  June 2021

Fund Project: This research is supported by National Science Foundation of China (Grant No. 11971310, 11671257); “Assessment of Risk and Uncertainty in Finance” (Grant No. AF0710020) from Shanghai Jiao Tong University.

An expectile can be considered a generalization of a quantile. While expected shortfall is a quantile-based risk measure, we study its counterpart—the expectile-based expected shortfall—where expectile takes the place of a quantile. We provide its dual representation in terms of a Bochner integral. Among other properties, we show that it is bounded from below in terms of the convex combination of expected shortfalls, and also from above by the smallest law invariant, coherent, and comonotonic risk measures, for which we give the explicit formulation of the corresponding distortion function. As a benchmark to the industry standard expected shortfall, we further provide its comparative asymptotic behavior in terms of extreme value distributions. Based on these results, we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.

Citation: Mekonnen Tadese, Samuel Drapeau. Dual representation of expectile-based expected shortfall and its properties. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 99-116. doi: 10.3934/puqr.2021005
##### References:
 [1] Acerbi, C. and Tasche, D., On the coherence of expected shortfall, Journal of Banking & Finance, 2002, 26(7): 1487-1503. [2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203-228. doi: 10.1111/1467-9965.00068. [3] Bellini, F. and Bignozzi, V., On elicitable risk measures, Quantitative Finance, 2015, 15(5): 725-733. doi: 10.1080/14697688.2014.946955. [4] Bellini, F. and Di Bernardino, E., Risk management with expectiles, The European Journal of Finance, 2017, 23(6): 487-506. doi: 10.1080/1351847X.2015.1052150. [5] Bellini, F., Klar, B., Müller, A. and Rosazza Gianin, E., Generalized quantiles as risk measures, Insurance: Mathematics and Economics, 2014, 54: 41-48. doi: 10.1016/j.insmatheco.2013.10.015. [6] Biagini, S. and Frittelli, M., On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures, In: Optimality and Risk-Modern Trends in Mathematical Finance, Springer Berlin Heidelberg, 2009: 1-28. [7] Chen, J. M., On exactitude in financial regulation: Value-at-risk, expected shortfall, and expectiles, Risks, 2018, 6(2): 61. doi: 10.3390/risks6020061. [8] Cheridito, P. and Li, T., Dual characterization of properties of risk measures on Orlicz hearts, Mathematics and Financial Economics, 2008, 2(1): 29-55. doi: 10.1007/s11579-008-0013-7. [9] Daouia, A., Girard, S. and Stupfler, G., Tail expectile process and risk assessment, Preprint Hal-01744505, 2019. [10] de Haan, L. and Ferreira, A., Extreme Value Theory: An Introduction, Springer, 2006. [11] Delbaen, F., A remark on the structure of expectiles, Preprint arXiv: 1307.5881, 2013. [12] Delbaen, F., Bellini, F., Bignozzi, V. and Ziegel, J. F., Risk measures with the CxLS property, Finance and Stochastics, 2016, 20(2): 433-453. doi: 10.1007/s00780-015-0279-6. [13] 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): 31-60. doi: 10.21314/JOR.2015.318. [14] Föllmer, H. and Schied, A., Stochastic Finance (4 edition), De Gruyter, Berlin, Boston, 2016. [15] Gneiting, T., Making and evaluating point forecasts, Journal of the American Statistical Association, 2011, 106(494): 746-762. doi: 10.1198/jasa.2011.r10138. [16] Hille, E. and Phillips, R. S., Functional Analysis and Semi-groups, American Mathematical Society, 1957. [17] Hua, L. and Joe, H., Second order regular variation and conditional tail expectation of multiple risks, Insurance: Mathematics and Economics, 2011, 49(3): 537-546. doi: 10.1016/j.insmatheco.2011.08.013. [18] Jones, M., Expectiles and M-quantiles are quantiles, Statistics & Probability Letters, 1994, 20(2): 149-153. [19] Kaina, M. and Rüschendorf, L., On convex risk measures on Lp-spaces, Mathematical Methods of Operations Research, 2009, 69(3): 475-495. doi: 10.1007/s00186-008-0248-3. [20] Koenker, R., When are expectiles percentiles? Econometric Theory, 1993, 9(3): 526-527. doi: 10.1017/S0266466600007921. [21] Lv, W., Mao, T. and Hu, T., Properties of second-order regular variation and expansions for risk concentration, Probability in the Engineering and Informational Sciences, 2012, 26(4): 535-559. doi: 10.1017/S0269964812000174. [22] Mao, T. and Hu, T., Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks, Insurance: Mathematics and Economics, 2012, 51(2): 333-343. doi: 10.1016/j.insmatheco.2012.06.003. [23] Mao, T., Ng, K. W. and Hu, T., Asymptotic expansions of generalized quantiles and expectiles for extreme risks, Probability in the Engineering and Informational Sciences, 2015, 29(3): 309-327. doi: 10.1017/S0269964815000017. [24] McNeil, A. J., Frey, R. and Embrechts, P., Quantitative risk management: Concepts, techniques and tools: Revised edition, Princeton University Press, 2015. [25] Newey, W. K. and Powell, J. L., Asymmetric least squares estimation and testing, Econometrica, 1987, 55(4): 819-847. doi: 10.2307/1911031. [26] Shapiro, A., On Kusuoka representation of law invariant risk measures, Mathematics of Operations Research, 2013, 38(1): 142-152. doi: 10.1287/moor.1120.0563. [27] Tadese, M. and Drapeau, S., Relative bound and asymptotic comparison of expectile with respect to expected shortfall, Insurance: Mathematics and Economics, 2020, 93: 387-399. doi: 10.1016/j.insmatheco.2020.06.006. [28] Tang, Q. and Yang, F., On the Haezendonck–Goovaerts risk measure for extreme risks, Insurance: Mathematics and Economics, 2012, 50(1): 217-227. doi: 10.1016/j.insmatheco.2011.11.007. [29] Tasche, D., Expected shortfall and beyond, Journal of Banking & Finance, 2002, 26(7): 1519-1533. [30] Taylor, J. W., Estimating value at risk and expected shortfall using expectiles, Journal of Financial Econometrics, 2008, 6(2): 231-252. [31] Weber, S., Distribution-invariant risk measures, information, and dynamic consistency, Mathematical Finance, 2006, 16(2): 419-441. doi: 10.1111/j.1467-9965.2006.00277.x. [32] Ziegel, J. F., Coherence and elicitability, Mathematical Finance, 2016, 26(4): 901-918. doi: 10.1111/mafi.12080.

show all references

1When the measure μ is finite, most literature defines strong measurability in terms of finite-valued functions. In this particular situation, this definition of measurability is equivalent to the current definition of measurability, see Hille and Phillips [16], for instance.

2We say $L_1$ and $L_2$ in $L^1$ are comonotone, if $( L_1(\omega)-L_1(\omega'))( L_2(\omega)-L_2(\omega'))\geq 0$ for all $(\omega, \omega')\in \Omega\times \Omega$ .

3We say $F_L$ is attracted by an extreme value distribution function $H$ and denoted by $MDA(H)$ if there exist constants $c_n>0$ and $d_n\in \mathbb{R}$ for each $n$ in $\mathbb{N}$ such that　　　　　　　　　　　　　　　　　　　　 $\lim_{n\nearrow \infty} F_L^n(c_n x+d_n) = H(x).$

4A measurable function $r: \mathbb{R} \to \mathbb{R}$ is said to be slowly varying if $\lim_{t\nearrow \infty}\frac{r(tx)}{r(t)} = 1$ for each $x$ in $\mathbb{R}$ .

5 ${\cal{W}}$ is a function such that $xe^x = y$ if and only if $x = {\cal{W}}(y)$ .

##### References:
 [1] Acerbi, C. and Tasche, D., On the coherence of expected shortfall, Journal of Banking & Finance, 2002, 26(7): 1487-1503. [2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3): 203-228. doi: 10.1111/1467-9965.00068. [3] Bellini, F. and Bignozzi, V., On elicitable risk measures, Quantitative Finance, 2015, 15(5): 725-733. doi: 10.1080/14697688.2014.946955. [4] Bellini, F. and Di Bernardino, E., Risk management with expectiles, The European Journal of Finance, 2017, 23(6): 487-506. doi: 10.1080/1351847X.2015.1052150. [5] Bellini, F., Klar, B., Müller, A. and Rosazza Gianin, E., Generalized quantiles as risk measures, Insurance: Mathematics and Economics, 2014, 54: 41-48. doi: 10.1016/j.insmatheco.2013.10.015. [6] Biagini, S. and Frittelli, M., On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures, In: Optimality and Risk-Modern Trends in Mathematical Finance, Springer Berlin Heidelberg, 2009: 1-28. [7] Chen, J. M., On exactitude in financial regulation: Value-at-risk, expected shortfall, and expectiles, Risks, 2018, 6(2): 61. doi: 10.3390/risks6020061. [8] Cheridito, P. and Li, T., Dual characterization of properties of risk measures on Orlicz hearts, Mathematics and Financial Economics, 2008, 2(1): 29-55. doi: 10.1007/s11579-008-0013-7. [9] Daouia, A., Girard, S. and Stupfler, G., Tail expectile process and risk assessment, Preprint Hal-01744505, 2019. [10] de Haan, L. and Ferreira, A., Extreme Value Theory: An Introduction, Springer, 2006. [11] Delbaen, F., A remark on the structure of expectiles, Preprint arXiv: 1307.5881, 2013. [12] Delbaen, F., Bellini, F., Bignozzi, V. and Ziegel, J. F., Risk measures with the CxLS property, Finance and Stochastics, 2016, 20(2): 433-453. doi: 10.1007/s00780-015-0279-6. [13] 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): 31-60. doi: 10.21314/JOR.2015.318. [14] Föllmer, H. and Schied, A., Stochastic Finance (4 edition), De Gruyter, Berlin, Boston, 2016. [15] Gneiting, T., Making and evaluating point forecasts, Journal of the American Statistical Association, 2011, 106(494): 746-762. doi: 10.1198/jasa.2011.r10138. [16] Hille, E. and Phillips, R. S., Functional Analysis and Semi-groups, American Mathematical Society, 1957. [17] Hua, L. and Joe, H., Second order regular variation and conditional tail expectation of multiple risks, Insurance: Mathematics and Economics, 2011, 49(3): 537-546. doi: 10.1016/j.insmatheco.2011.08.013. [18] Jones, M., Expectiles and M-quantiles are quantiles, Statistics & Probability Letters, 1994, 20(2): 149-153. [19] Kaina, M. and Rüschendorf, L., On convex risk measures on Lp-spaces, Mathematical Methods of Operations Research, 2009, 69(3): 475-495. doi: 10.1007/s00186-008-0248-3. [20] Koenker, R., When are expectiles percentiles? Econometric Theory, 1993, 9(3): 526-527. doi: 10.1017/S0266466600007921. [21] Lv, W., Mao, T. and Hu, T., Properties of second-order regular variation and expansions for risk concentration, Probability in the Engineering and Informational Sciences, 2012, 26(4): 535-559. doi: 10.1017/S0269964812000174. [22] Mao, T. and Hu, T., Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks, Insurance: Mathematics and Economics, 2012, 51(2): 333-343. doi: 10.1016/j.insmatheco.2012.06.003. [23] Mao, T., Ng, K. W. and Hu, T., Asymptotic expansions of generalized quantiles and expectiles for extreme risks, Probability in the Engineering and Informational Sciences, 2015, 29(3): 309-327. doi: 10.1017/S0269964815000017. [24] McNeil, A. J., Frey, R. and Embrechts, P., Quantitative risk management: Concepts, techniques and tools: Revised edition, Princeton University Press, 2015. [25] Newey, W. K. and Powell, J. L., Asymmetric least squares estimation and testing, Econometrica, 1987, 55(4): 819-847. doi: 10.2307/1911031. [26] Shapiro, A., On Kusuoka representation of law invariant risk measures, Mathematics of Operations Research, 2013, 38(1): 142-152. doi: 10.1287/moor.1120.0563. [27] Tadese, M. and Drapeau, S., Relative bound and asymptotic comparison of expectile with respect to expected shortfall, Insurance: Mathematics and Economics, 2020, 93: 387-399. doi: 10.1016/j.insmatheco.2020.06.006. [28] Tang, Q. and Yang, F., On the Haezendonck–Goovaerts risk measure for extreme risks, Insurance: Mathematics and Economics, 2012, 50(1): 217-227. doi: 10.1016/j.insmatheco.2011.11.007. [29] Tasche, D., Expected shortfall and beyond, Journal of Banking & Finance, 2002, 26(7): 1519-1533. [30] Taylor, J. W., Estimating value at risk and expected shortfall using expectiles, Journal of Financial Econometrics, 2008, 6(2): 231-252. [31] Weber, S., Distribution-invariant risk measures, information, and dynamic consistency, Mathematical Finance, 2006, 16(2): 419-441. doi: 10.1111/j.1467-9965.2006.00277.x. [32] Ziegel, J. F., Coherence and elicitability, Mathematical Finance, 2016, 26(4): 901-918. doi: 10.1111/mafi.12080.
Graphs of $es_\alpha$ , $tce_\alpha$ , and $ES_\alpha$ for Pareto, standard Student $t$ , standard normal, and beta distributions.
Graph of $\gamma_\beta$ for $\alpha = 70 \%, \alpha = 80 \%$ , and $\alpha = 90 \%$ .
Graph of $\varphi$ and $\varphi_{\lambda,\beta,\delta}$ for $\alpha = 90 \%, \lambda = 20 \%$ , $\beta = 90 \%$ , and $\delta = 0 \%$ .
Graph of $ES_\alpha/es_\alpha$ (for Pareto, standard Student $t$ , and standard normal distributions) and $(\hat{x}-ES_\alpha)/(\hat{x}-es_\alpha)$ (for the beta distribution).
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