References
- Acerbi, C., and B. Székely. 2014. Back-testing expected shortfall. Risk Magazine 27 (11):76–81.
- Artzner, P., F. Delbaen, J. M. Eber, and D. Heath. 1999. Coherent measures of risk. Mathematical Finance 9 (3):203–28. doi:10.1111/1467-9965.00068
- Balbás, A., B. Balbás, and R. Balbás. 2017. Differential equations connecting VaR and CVaR. Journal of Computational and Applied Mathematics 326:247–67. doi:10.1016/j.cam.2017.05.037
- Barczy, M., F. K. Nedényi, and L. Sütő, L. 2022. Probability equivalent level of Value at Risk and higher-order expected shortfalls. Insurance: Mathematics and Economics 108:107–28. doi:10.1016/j.insmatheco.2022.11.004
- Basel Committee on Banking Supervision. 2019. Minimum capital requirements for market risk. February 2019. Basel: Bank for International Settlements. BIS online publication No. bcbs457.
- Behan, D. F., S. H. Cox, Y. Lin, J. Pai, H. W. Pedersen, and M. Yi. 2010. Obesity and its relation to mortality and morbidity costs. Society of Actuaries, 59.
- Bellman, R., and K. L. Cooke. 1963. Differential-difference equations. New York: Academic Press.
- Berezansky, L., and E. Braverman. 2011. On nonoscillation of advanced differential equations with several terms. Abstract and Applied Analysis 2011:637142. doi:10.1155/2011/637142
- Chambers, C. P. 2009. An axiomatization of quantiles on the domain of distribution functions. Mathematical Finance 19 (2):335–42. doi:10.1111/j.1467-9965.2009.00369.x
- Cont, R., R. Deguest, and G. Scandolo. 2010. Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance 10 (6):593–606. doi:10.1080/14697681003685597
- Embrechts, P., and M. Hofert. 2013. A note on generalized inverses. Mathematical Methods of Operations Research 77 (3):423–32. doi:10.1007/s00186-013-0436-7
- Embrechts, P., M. Hofert, and R. Wang. 2016. Bernoulli and tail-dependence compatibility. Annals of Applied Probability 26 (3):1636–58.
- Embrechts, P., H. Liu, and R. Wang. 2018. Quantile-based risk sharing. Operations Research 66 (4):936–49. doi:10.1287/opre.2017.1716
- Embrechts, P., A. McNeil, and D. Straumann. 2002. Correlation and dependence in risk management: properties and pitfalls. Risk Management: Value at Risk and Beyond 1:176–223.
- Embrechts, P., G. Puccetti, L. Ruschendorf, R. Wang, and A. Beleraj. 2014. An academic response to Basel 3.5. Risks 2 (1):25–48. doi:10.3390/risks2010025
- Emmer, S., M. Kratz, and D. Tasche. 2015. What is the best risk measure in practice? A comparison of standard measures. Journal of Risk 18 (2):31–60. doi:10.21314/JOR.2015.318
- Fiori, A. M., and E. Rosazza Gianin. 2023. Generalized PELVE and applications to risk measures. European Actuarial Journal 13 (1):307–39. doi:10.1007/s13385-022-00320-6
- Frees, E. W. 2009. Regression modeling with actuarial and financial applications. Cambridge: Cambridge University Press.
- Gneiting, T. 2011. Making and evaluating point forecasts. Journal of the American Statistical Association 106 (494):746–62. doi:10.1198/jasa.2011.r10138
- Kou, S., and X. Peng. 2016. On the measurement of economic tail risk. Operations Research 64 (5):1056–72. doi:10.1287/opre.2016.1539
- Krause, D., M. Scherer, J. Schwinn, and R. Werner. 2018. Membership testing for Bernoulli and tail-dependence matrices. Journal of Multivariate Analysis 168:240–60. doi:10.1016/j.jmva.2018.07.014
- Li, H., and R. Wang. 2022. PELVE: Probability equivalent level of VaR and ES. Journal of Econometrics 234 (1):353–70. doi:10.1016/j.jeconom.2021.12.012
- Liu, F., and R. Wang. 2021. A theory for measures of tail risk. Mathematics of Operations Research 46 (3):1109–28. doi:10.1287/moor.2020.1072
- McNeil, A. J., R. Frey, and P. Embrechts. 2015. Quantitative risk management: Concepts, techniques and tools. Revised Edition. Princeton, NJ: Princeton University Press.
- Rudin, W. 1987. Real and Complex Analysis. International Series in Pure and Applied Mathematics, 3rd ed. New York: McGraw-Hill.
- Siewert, C. E., and E. E. Burniston. 1973. Exact analytical solutions of zez = a. Journal of Mathematical Analysis and Applications 43 (3):626–32. doi:10.1016/0022-247X(73)90281-3
- Wang, R., and R. Zitikis. 2021. An axiomatic foundation for the expected shortfall. Management Science 67:1413–29. doi:10.1287/mnsc.2020.3617