The optimization of a large random portfolio under the Expected Shortfall risk measure with an \ell_2 regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number  of different assets in the portfolio is much less than the length  of the available time series, the regularizer plays a negligible role  , while in the (much more frequently occurring in practice) opposite limit, where the samples are comparable or even small compared to the number of different assets, the optimum is almost entirely determined by the regularizer. Our results clearly show that the transition region between these two extremes is relatively narrow, and it is only here that one can meaningfully speak of a trade-off between fluctuations and bias .
The optimization of a large random portfolio under the Expected Shortfall risk measure with an \ell_2 regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number N of different assets in the portfolio is much less than the length T of the available time series, the regularizer plays a negligible role even if its strength \eta is large , while in the opposite limit, where the size of samples is comparable to, or even smaller than the number of assets, the optimum is almost entirely determined by the regularizer. We construct the contour map of estimation error on the N/T vs. \eta plane and find that for a given value of the estimation error the gain in N/T due to the regularizer can reach a factor of about 4 for a sufficiently strong regularizer .
A
N
A
T
A
even if its strength \eta is large
D
(much more frequently occurring in practice)
R
samples are comparable or even small compared to
size of samples is comparable to, or even smaller than
D
different
R
Our results clearly show that the transition region between these two extremes is relatively narrow, and it is only here that one can meaningfully speak of a trade-off between fluctuations and bias
We construct the contour map of estimation error on the N/T vs. \eta plane and find that for a given value of the estimation error the gain in N/T due to the regularizer can reach a factor of about 4 for a sufficiently strong regularizer