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 $\alpha$, which means we have a line of critical points on the $\alpha -r$ 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 ${\ell }_1$ 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 ${\ell }_1$ 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.

中文翻译：

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ℓ1约束下的预期短缺优化—一种分析方法

预期缺口（ES）是高于较高分位数的平均损失，是当前的金融监管市场风险度量。它的估计和优化对于样本波动非常不稳定，超过临界比率就变得不可能$\mathrm{[R}=ñ/Ť$，其中N是投资组合中不同资产的数量，T是可用时间序列的长度。关键比率取决于置信度$\alpha$，这意味着我们在 $\alpha -\mathrm{[R}$飞机。ES的估计中的大波动可以通过应用调节器来减弱。在本文中，我们将在${\ell }_{\mathrm{1个}}$通过从随机系统的统计物理学中借来的复制品的方法来进行正则化。禁止卖空（即限制所有投资组合权重均为非负值的约束）是不对称的特殊情况${\ell }_{\mathrm{1个}}$正则化。给出了针对正则化ES的样本外和样本内估计量，估计误差，最优投资组合权重的分布以及正则化器从投资组合中消除的资产密度的结果。从无意义约束的意义上说，它比高波动性项目将高波动性元素的权重设置为零的可能性更高，因此它是高波动性的截止。该截止值使宽高比重新正常化$\mathrm{[R}=ñ/Ť$，从而扩大了优化的可行性范围。我们发现在正则化和非正则化问题之间存在非平凡的映射，对应于阶数参数的重新规范化。