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financial risk evaluation: Value at Risk (VaR) and Conditional Value at Risk (CVaR). These refined formulations are developed with the intention of offering resilient and practically implementable tools to address the complexities inherent in economic risk analysis. Moreover, the newly established expressions are seamlessly integrated into the GARCH modeling framework, thereby enriching its predictive capabilities. In order to verify both the practical relevance and theoretical soundness of the presented methodology, it is systematically employed regarding the daily return series of a varied portfolio of stocks. The outcomes of the numerical experiments provide compelling evidence of the approach\u2019s reliability and effectiveness, emphasizing its suitability for advancing contemporary risk management strategies in financial environments.<\/jats:p>","DOI":"10.3390\/axioms14050391","type":"journal-article","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T09:58:22Z","timestamp":1747821502000},"page":"391","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Assessing Expected Shortfall in Risk Analysis Through Generalized Autoregressive Conditional Heteroskedasticity Modeling and the Application of the Gumbel Distribution"],"prefix":"10.3390","volume":"14","author":[{"given":"Bingjie","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yihui","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jia","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1024-3201","authenticated-orcid":false,"given":"Tao","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Albrecher, H., Binder, A., Lautscham, V., and Mayer, P. 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