Relative bound and asymptotic comparison of expectile with respect to expected shortfall. (English) Zbl 1448.62064
Summary: Expectile bears some interesting properties in comparison to the industry wide expected shortfall in terms of assessment of tail risk. We study the relationship between expectile and expected shortfall using duality results and the link to optimized certainty equivalent. Lower and upper bounds of expectile are derived in terms of expected shortfall as well as a characterization of expectile in terms of expected shortfall. Further, we study the asymptotic behavior of expectile with respect to expected shortfall as the confidence level goes to 1 in terms of extreme value distributions. We use concentration inequalities to illustrate that the estimation of value at risk requires larger sample size than expected shortfall and expectile for heavy tail distributions when \(\alpha\) is close to 1. Illustrating the formulation of expectile in terms of expected shortfall, we also provide explicit or semi-explicit expressions of expectile and some simulation results for some classical distributions.
MSC:
| 62G32 | Statistics of extreme values; tail inference |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 91G05 | Actuarial mathematics |
Software:
QRMReferences:
| [1] | Acerbi, C.; Tasche, D., On the coherence of expected shortfall, J. Bank. Financ., 26, 7, 1487-1503 (2002) |
| [2] | Armenti, Y.; Crépey, S.; Drapeau, S.; Papapantoleon, A., Multivariate shortfall risk allocation and systemic risk, SIAM J. Financ. Math., 9, 1, 90-126 (2018) · Zbl 1408.91236 |
| [3] | Artzner, P.; Delbaen, F.; Eber, J.-M.; Heath, D., Coherent measures of risk, Math. Finance, 9, 3, 203-228 (1999) · Zbl 0980.91042 |
| [4] | Asimit, A. V.; Furman, E.; Tang, Q.; Vernic, R., Asymptotics for risk capital allocations based on conditional tail expectation, Insurance Math. Econom., 49, 3, 310-324 (2011) · Zbl 1228.91029 |
| [5] | Bartl, D.; Tangpi, L., Non-asymptotic rates for the estimation of risk measures (2020), ArXiV |
| [6] | Bellini, F.; Bignozzi, V., On elicitable risk measures, Quant. Finance, 15, 5, 725-733 (2015) · Zbl 1395.91506 |
| [7] | Bellini, F.; Di Bernardino, E., Risk management with expectiles, Eur. J. Finance, 23, 6, 487-506 (2017) |
| [8] | Bellini, F.; Klar, B.; Müller, A.; Rosazza Gianin, E., Generalized quantiles as risk measures, Insurance Math. Econom., 54, 41-48 (2014) · Zbl 1303.91089 |
| [9] | Ben-Tal, A.; Teboulle, M., An old-new concept of convex risk measures: The optimized certainity equivalent, Math. Finance, 17, 3, 449-476 (2007) · Zbl 1186.91116 |
| [10] | Brazauskas, V.; Jones, B. L.; Puri, M. L.; Zitikis, R., Estimating conditional tail expectation with actuarial applications in view, J. Statist. Plann. Inference, 138, 11, 3590-3604 (2008) · Zbl 1152.62027 |
| [11] | Chen, J. M., On exactitude in financial regulation: Value-at-risk, expected shortfall, and expectiles, Risks, 6, 2, 61 (2018) |
| [12] | Daouia, A.; Girard, S.; Stupfler, G., Estimation of tail risk based on extreme expectiles, J. R. Stat. Soc. Ser. B Stat. Methodol., 80, 2, 263-292 (2018) · Zbl 1383.62235 |
| [13] | Delbaen, F., A remark on the structure of expectiles (2013), Preprint ArXiV |
| [14] | Delbaen, F.; Bellini, F.; Bignozzi, V.; Ziegel, J. F., Risk measures with the CxLS property, Finance Stoch., 20, 2, 433-453 (2016) · Zbl 1376.91173 |
| [15] | Emmer, S.; Kratz, M.; Tasche, D., What is the best risk measure in practice? A comparison of standard measures, J. Risk, 18, 2, 31-60 (2015) |
| [16] | Föllmer, H.; Schied, A., Convex measures of risk and trading constraints, Finance Stoch., 6, 4, 429-447 (2002) · Zbl 1041.91039 |
| [17] | Föllmer, H.; Schied, A., Stochastic Finance (2016), De Gruyter: De Gruyter Berlin, Boston · Zbl 1343.91001 |
| [18] | Fournier, N.; Guillin, A., On the rate of convergence in Wasserstein distance of the empirical measure, Probab. Theory Related Fields, 162, 3, 707-738 (2015) · Zbl 1325.60042 |
| [19] | Gao, F.; Shaochen, W., Asymptotic behavior of the empirical conditional value-at-risk, Insurance Math. Econom., 49, 3, 345-352 (2011) · Zbl 1228.91035 |
| [20] | Gneiting, T., Making and evaluating point forecasts, J. Amer. Statist. Assoc., 106, 494, 746-762 (2011) · Zbl 1232.62028 |
| [21] | Guo, S.; Xu, H., Distributionally robust shortfall risk optimization model and its approximation, Math. Program., 174, 1, 473-498 (2019) · Zbl 1421.90097 |
| [22] | de Haan, L.; Ferreira, A., Extreme Value Theory: An Introduction (2006), Springer · Zbl 1101.62002 |
| [23] | Holzmann, H.; Klar, B., Expectile asymptotics, Electron. J. Stat., 10, 2, 2355-2371 (2016) · Zbl 1397.62075 |
| [24] | Hua, L.; Joe, H., Second order regular variation and conditional tail expectation of multiple risks, Insurance Math. Econom., 49, 3, 537-546 (2011) · Zbl 1228.91039 |
| [25] | Kalkbrener, M., An axiomatic approach to capital allocation, Math. Finance, 15, 3, 425-437 (2005) · Zbl 1102.91049 |
| [26] | Koenker, R., When are expectiles percentiles?, Econom. Theory, 9, 3, 526-527 (1993) |
| [27] | Kolla, R. K.; A., P. L.; Bhat, S. P.; Jagannathan, K. P., Concentration bounds for empirical conditional value-at-risk: The unbounded case, Oper. Res. Lett., 17, 1, 16-20 (2019) · Zbl 1476.91220 |
| [28] | Lv, W.; Mao, T.; Hu, T., Properties of second-order regular variation and expansions for risk concentration, Probab. Engrg. Inform. Sci., 26, 4, 535-559 (2012) · Zbl 1261.62043 |
| [29] | Major, P., On the invariance principle for sums of independent identically distributed random variables, J. Multivariate Anal., 8, 4, 487-517 (1978) · Zbl 0408.60028 |
| [30] | Mao, T.; Hu, T., Second-order properties of the Haezendonck-Goovaerts risk measure for extreme risks, Insurance Math. Econom., 51, 2, 333-343 (2012) · Zbl 1284.91256 |
| [31] | Mao, T.; Ng, K. W.; Hu, T., Asymptotic expansions of generalized quantiles and expectiles for extreme risks, Probab. Engrg. Inform. Sci., 29, 3, 309-327 (2015) · Zbl 1373.62216 |
| [32] | Mao, T.; Yang, F., Risk concentration based on expectiles for extreme risks under FGM copula, Insurance Math. Econom., 64, 429-439 (2015) · Zbl 1348.91175 |
| [33] | McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative risk management: Concepts, techniques and tools: Revised edition (2015), Princeton University Press · Zbl 1337.91003 |
| [34] | Newey, W. K.; Powell, J. L., Asymmetric least squares estimation and testing, Econometrica, 55, 4, 819-847 (1987) · Zbl 0625.62047 |
| [35] | Shapiro, A., On Kusuoka representation of law invariant risk measures, Math. Oper. Res., 38, 1, 142-152 (2013) · Zbl 1291.91125 |
| [36] | Tang, Q.; Yang, F., On the Haezendonck-Goovaerts risk measure for extreme risks, Insurance Math. Econom., 50, 1, 217-227 (2012) · Zbl 1239.91084 |
| [37] | Tasche, D., Expected shortfall and beyond, J. Bank. Financ., 26, 7, 1519-1533 (2002) · Zbl 1119.62106 |
| [38] | Tasche, D., Capital allocation to business units and sub-portfolios: the Euler principle (2008), Preprint ArXiV |
| [39] | Taylor, J. W., Estimating value at risk and expected shortfall using expectiles, J. Financ. Econ., 6, 2, 231-252 (2008) |
| [40] | Weber, S., Distribution-invariant risk measures, information, and dynamic consistency, Math. Finance, 16, 2, 419-441 (2006) · Zbl 1145.91037 |
| [41] | Ziegel, J. F., Coherence and elicitability, Math. Finance, 26, 4, 901-918 (2016) · Zbl 1390.91336 |
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