{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:31:36Z","timestamp":1762522296230,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,6,27]],"date-time":"2019-06-27T00:00:00Z","timestamp":1561593600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MINECO\/FEDER, EU","award":["MTM2015-70840-P"],"award-info":[{"award-number":["MTM2015-70840-P"]}]},{"name":"MCIU\/AEI\/FEDER, EU","award":["PGC2018-098860-B-I00"],"award-info":[{"award-number":["PGC2018-098860-B-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper introduces a new family of the convex divergence-based risk measure by specifying     ( h , \u03d5 )    -divergence, corresponding with the dual representation. First, the sensitivity characteristics of the modified divergence risk measure with respect to profit and loss (P&amp;L) and the reference probability in the penalty term are discussed, in view of the certainty equivalent and robust statistics. Secondly, a similar sensitivity property of     ( h , \u03d5 )    -divergence risk measure with respect to P&amp;L is shown, and boundedness by the analytic risk measure is proved. Numerical studies designed for R\u00e9nyi- and Tsallis-divergence risk measure are provided. This new family integrates a wide spectrum of divergence risk measures and relates to divergence preferences.<\/jats:p>","DOI":"10.3390\/e21070634","type":"journal-article","created":{"date-parts":[[2019,6,27]],"date-time":"2019-06-27T11:26:18Z","timestamp":1561634778000},"page":"634","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Divergence-Based Risk Measures: A Discussion on Sensitivities and Extensions"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0544-3842","authenticated-orcid":false,"given":"Meng","family":"Xu","sequence":"first","affiliation":[{"name":"School of Economics, Sichuan University, Chengdu 610065, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6863-7847","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Angulo","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/1467-9965.00068","article-title":"Coherent measures of risk","volume":"9","author":"Artzner","year":"1999","journal-title":"Math. Financ."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"F\u00f6llmer, H., and Schied, A. (2002). Robust preferences and convex measures of risk. Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, Springer Berlin Heidelberg.","DOI":"10.1007\/978-3-662-04790-3_2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1473","DOI":"10.1016\/S0378-4266(02)00270-4","article-title":"Putting order in risk measures","volume":"26","author":"Frittelli","year":"2002","journal-title":"J. Bank Financ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1146\/annurev-financial-111914-042031","article-title":"The axiomatic approach to risk measures for capital determination","volume":"7","author":"Weber","year":"2015","journal-title":"Annu. Rev. Financ. Econ."},{"key":"ref_5","unstructured":"F\u00f6llmer, H., and Schied, A. (2019, June 24). Convex and Coherent Risk Measures. Available online: http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download?doi=10.1.1.335.3202&rep=rep1&type=pdf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1142\/S0219493711003334","article-title":"Entropic risk measures: coherence vs. convexity, model ambiguity and robust large deviations","volume":"11","author":"Knispel","year":"2011","journal-title":"Stoch. Dynam."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1287\/moor.1120.0559","article-title":"Entropy coherent and entropy convex measures of risk","volume":"38","author":"Laeven","year":"2013","journal-title":"Math. Oper. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1111\/j.1467-9965.2007.00311.x","article-title":"An old-new concept of convex risk measures: The optimized certainty equivalent","volume":"17","author":"Teboulle","year":"2007","journal-title":"Math. Financ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s10479-015-1801-0","article-title":"Certainty equivalent measures of risk","volume":"249","author":"Vinel","year":"2017","journal-title":"Ann. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s00186-008-0248-3","article-title":"On convex risk measures on Lp-spaces","volume":"69","author":"Kaina","year":"2009","journal-title":"Math. Oper. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1007\/s10957-011-9968-2","article-title":"Entropic Value-at-Risk: a new coherent risk measure","volume":"155","year":"2012","journal-title":"J. Optimiz. Theory App."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Pele, D., Lazar, E., and Dufour, A. (2017). Information entropy and measures of market risk. Entropy, 19.","DOI":"10.3390\/e19050226"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0167-6687(97)00031-0","article-title":"Axiomatic characterization of insurance prices","volume":"21","author":"Wang","year":"1997","journal-title":"Insur. Math. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Frittelli, M., and Gianin, E.R. (2005). Law invariant convex risk measures. Advanced Mathematical Economics, Springer Tokyo.","DOI":"10.1007\/4-431-27233-X_2"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhou, R., Liu, X., Yu, M., and Huang, K. (2017). Properties of risk measures of generalized entropy in portfolio selection. Entropy, 19.","DOI":"10.3390\/e19120657"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1111\/j.1468-0262.2006.00716.x","article-title":"Ambiguity aversion, robustness, and the variational representation of Preferences","volume":"74","author":"Maccheroni","year":"2006","journal-title":"Econometrica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"643","DOI":"10.2307\/1884324","article-title":"Risk, ambiguity, and the Savage axioms","volume":"75","author":"Ellsberg","year":"1961","journal-title":"Q. J. Econ."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Dupuis, P., and Ellis, R.S. (1997). A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons.","DOI":"10.1002\/9781118165904"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1287\/moor.12.2.224","article-title":"Penalty functions and duality in stochastic programming via \u03c6-divergence functionals","volume":"12","author":"Teboulle","year":"1987","journal-title":"Math. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Greselin, F., and Zitikis, R. (2018). From the classical Gini index of income inequality to a new Zenga-type relative measure of risk: A modeller\u2019s perspective. Econometrics, 6.","DOI":"10.3390\/econometrics6010004"},{"key":"ref_21","unstructured":"Maccheroni, F., Marinacci, M., and Rustichini, A. (2019, June 24). A Variational Formula for the Relative Gini Concentration Index. Available online: http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.561.6568."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.ejor.2017.06.059","article-title":"Risk analysis and decision theory: A bridge","volume":"264","author":"Borgonovo","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/S0927-5398(00)00011-6","article-title":"Sensitivity analysis of Values at Risk","volume":"7","author":"Gourieroux","year":"2000","journal-title":"J. Empir. Finance"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1111\/j.0960-1627.2004.00184.x","article-title":"Nonparametric estimation and sensitivity analysis of Expected Shortfall","volume":"14","author":"Scaillet","year":"2004","journal-title":"Math. Financ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1080\/14697681003685597","article-title":"Robustness and sensitivity analysis of risk measurement procedures","volume":"10","author":"Cont","year":"2010","journal-title":"Quant. Financ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1016\/j.ejor.2015.06.032","article-title":"Sensitivity analysis: a review of recent advances","volume":"248","author":"Borgonovo","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02926015","article-title":"Asymptotic behaviour and statistical applications of divergence measures in multinomial populations: a unified study","volume":"36","author":"Morales","year":"1995","journal-title":"Stat. Papers"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1515\/strm-2017-0027","article-title":"Optimal expected utility risk measures","volume":"35","author":"Geissel","year":"2017","journal-title":"Stat. Risk Model."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Kusuoka, S. (2001). On law invariant coherent risk measures. Advances in Mathematical Economics, Springer.","DOI":"10.1007\/978-4-431-67891-5"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10479-010-0747-5","article-title":"Kusuoka representation of higher order dual risk measures","volume":"181","author":"Dentcheva","year":"2010","journal-title":"Ann. Oper. Res."},{"key":"ref_31","first-page":"299","article-title":"Information-type measures of difference of probability distributions and indirect observations","volume":"2","year":"1967","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1006\/jeth.2001.2815","article-title":"Ambiguity made precise: A comparative eoundation","volume":"102","author":"Ghirardato","year":"2002","journal-title":"J. Econ. Theory"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6754","DOI":"10.1073\/pnas.1207805110","article-title":"Classical subjective expected utility","volume":"110","author":"Maccheroni","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1016\/j.ejor.2015.02.001","article-title":"Decision analysis under ambiguity","volume":"244","author":"Borgonovo","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Huber, P.J. (1981). Robust Statistics, John Wiley & Sons.","DOI":"10.1002\/0471725250"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"F\u00f6llmer, H., and Schied, A. (2016). Stochastic Finance: An Introduction in Discrete Time, De Gruyter. [4th ed.].","DOI":"10.1515\/9783110463453"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1016\/j.ejor.2017.11.051","article-title":"Capital allocation \u00e0 la Aumann-Shapley for non-differentiable risk measures","volume":"267","author":"Centrone","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"611","DOI":"10.2307\/1913073","article-title":"Values of markets with a continuum of traders","volume":"43","author":"Aumann","year":"1975","journal-title":"Econometrica"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.jmva.2011.06.005","article-title":"Qualitative and infinitesimal robustness of tail-dependent statistical functionals","volume":"103","author":"Schied","year":"2012","journal-title":"J. Multivariate. Anal."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s00780-013-0225-4","article-title":"Comparative and qualitative robustness for law-invariant risk measures","volume":"18","author":"Schied","year":"2014","journal-title":"Financ. Stoch."},{"key":"ref_41","first-page":"28","article-title":"New nonadditive measures of entropy for discrete probability distributions","volume":"10","author":"Sharma","year":"1975","journal-title":"Casp. J. Math. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"032003","DOI":"10.1088\/1751-8113\/45\/3\/032003","article-title":"A closed-form expression for the Sharma\u2013Mittal entropy of exponential families","volume":"45","author":"Nielsen","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.1111\/risa.13125","article-title":"Which parameters are important? Differential importance under uncertainty","volume":"38","author":"Borgonovo","year":"2018","journal-title":"Risk Anal."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1111\/risa.12434","article-title":"Sensitivity analysis using risk measures","volume":"36","author":"Tsanakas","year":"2016","journal-title":"Risk Anal."},{"key":"ref_45","unstructured":"Pichler, A., and Schlotter, R. (2019). Entropy based risk measures. Eur. J. Oper. Res."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Naudts, J. (2011). Generalised Thermostatistics, Springer.","DOI":"10.1007\/978-0-85729-355-8"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1111\/j.1467-9965.2009.00364.x","article-title":"Risk measures on Orlicz hearts","volume":"19","author":"Cheridito","year":"2009","journal-title":"Math. Financ."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/7\/634\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:01:41Z","timestamp":1760187701000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/7\/634"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,27]]},"references-count":47,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7]]}},"alternative-id":["e21070634"],"URL":"https:\/\/doi.org\/10.3390\/e21070634","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,6,27]]}}}