A portfolio of independent, but not identically distributed, returns is optimized under the variance risk measure, in the high-dimensional limit where the number N of the different assets in the portfolio and the sample size T are assumed large with their ratio r=N/T kept finite, with a ban on...

The optimization of a large random portfolio under the Expected Shortfall risk measure with an ℓ₂ 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...

The contour map of estimation error of Expected Shortfall (ES) is constructed. It allows one to quantitatively determine the sample size (the length of the time series) required by the optimization under ES of large institutional portfolios for a given size of the portfolio, at a given...

Investors who optimize their portfolios under any of the coherent risk measures are naturally led to regularized portfolio optimization when they take into account the impact their trades make on the market. We show here that the impact function determines which regularizer is used. We also show...