$L_p$ regularized portfolio optimization

TL;DR

This paper explores how different $L_p$ regularizers influence the stability of portfolio optimization under coherent risk measures, highlighting the stabilizing effects of $L_1$ and $L_p$ norms for $p>1$ especially in the context of Expected Shortfall.

Contribution

It establishes a link between impact functions and regularizers, showing how $L_p$ norms affect estimation error sensitivity in portfolio optimization.

Findings

$L_p$ regularizers with $p>1$ reduce estimation error sensitivity.

$L_1$ regularizer shifts or eliminates instability depending on implementation.

Impact functions determine the choice of regularizer in market impact-aware optimization.

Abstract

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 that any regularizer based on the norm $L_p$ with $p>1$ makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with $p<1$ do not. The $L_1$ norm represents a border case: its "soft" implementation does not remove the instability, but rather shifts its locus, whereas its "hard" implementation (equivalent to a ban on short selling) eliminates it. We demonstrate these effects on the important special case of Expected Shortfall (ES) that is on its way to becoming the next global regulatory market risk measure.