在$\ell_1$约束下优化预期短缺--一种分析方法
Optimizing Expected Shortfall under an $\ell_1$ constraint -- an analytic approach
摘要：
预期损失是当前金融监管机构对市场风险的度量，是指高于高分位数的平均亏损。它的估计和优化对样本波动是高度不稳定的，并且在临界比率$r=N/T$以上变得不可能，其中$N$是投资组合中不同资产的数量，并且$T$是可用时间序列的长度。临界比率取决于置信水平$α$，这意味着我们在$α-r$平面上有一行临界点。估计ES的大的波动可以通过应用正则化器来衰减。在本文中，我们分析计算ES下的$\ell_1$正则化副本的方法借用随机系统的统计物理。禁止卖空，即一个约束，使所有的投资组合权重非负，是一个特殊情况下的非对称$\ell_1$正则化。结果给出了正则化ES的样本外和样本内估计、估计误差、最优投资组合权重的分布和正则化器从投资组合中消除的资产密度。结果表明，没有短期约束作为一个高波动性截止，在这个意义上说，它设置的高波动性元素的权重为零的概率比那些低波动性项目。该截止值使纵横比$r=N/T$重新标准化，从而扩展了优化的可行性范围。我们发现，有一个非平凡的映射之间的正则化和非正则化的问题，对应于重整化的序参数。
英文摘要：
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 $α$, which means we have a line of critical points on the $α-r$ 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.
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