{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:56:39Z","timestamp":1760234199138,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,24]],"date-time":"2021-04-24T00:00:00Z","timestamp":1619222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio r=N\/T, where N is the number of different assets in the portfolio, and T is the length of the available time series. The critical ratio depends on the confidence level \u03b1, which means we have a line of critical points on the \u03b1\u2212r plane. The large fluctuations in the estimation of ES can be attenuated by the application of regularizers. In this paper, we calculate ES analytically under an \u21131 regularizer by the method of replicas borrowed from the statistical physics of random systems. The ban on short selling, i.e., a constraint rendering all the portfolio weights non-negative, is a special case of an asymmetric \u21131 regularizer. Results are presented for the out-of-sample and the in-sample estimator of the regularized ES, the estimation error, the distribution of the optimal portfolio weights, and the density of the assets eliminated from the portfolio by the regularizer. It is shown that the no-short constraint acts as a high volatility cutoff, in the sense that it sets the weights of the high volatility elements to zero with higher probability than those of the low volatility items. This cutoff renormalizes the aspect ratio r=N\/T, thereby extending the range of the feasibility of optimization. We find that there is a nontrivial mapping between the regularized and unregularized problems, corresponding to a renormalization of the order parameters.<\/jats:p>","DOI":"10.3390\/e23050523","type":"journal-article","created":{"date-parts":[[2021,4,24]],"date-time":"2021-04-24T21:49:20Z","timestamp":1619300960000},"page":"523","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimizing Expected Shortfall under an \u21131 Constraint\u2014An Analytic Approach"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5038-678X","authenticated-orcid":false,"given":"G\u00e1bor","family":"Papp","sequence":"first","affiliation":[{"name":"Institute for Physics, E\u00f6tv\u00f6s Lor\u00e1nd University, 1117 Budapest, Hungary"}]},{"given":"Imre","family":"Kondor","sequence":"additional","affiliation":[{"name":"Parmenides Foundation, 82049 Pullach, Germany"},{"name":"London Mathematical Laboratory, London W6 8RH, UK"},{"name":"Complexity Science Hub, Vienna 1080, Austria"}]},{"given":"Fabio","family":"Caccioli","sequence":"additional","affiliation":[{"name":"London Mathematical Laboratory, London W6 8RH, UK"},{"name":"Department of Computer Science, University College London, London WC1E 6BT, UK"},{"name":"Systemic Risk Centre, London School of Economics and Political Sciences, London WC2A 2AE, UK"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,24]]},"reference":[{"key":"ref_1","first-page":"77","article-title":"Portfolio selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. 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