  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.

Portfolio Management;Risk Management