[Paper Review] Feasibility of Portfolio Optimization under Coherent Risk Measures

Imre Kondor, Istvan Varga-Haszonits|ArXiv.org|Mar 15, 2008

Risk and Portfolio Optimization9 references4 citations

TL;DR

This paper demonstrates that portfolio optimization under coherent risk measures, including Expected Shortfall and Maximal Loss, can become unstable and yield unbounded (diverging to minus infinity) risk estimates when one asset dominates others in a finite sample. The instability arises due to the coherence axioms, which cause coherent risk measures to incorrectly signal extreme risk reduction when sampling fluctuations create a false dominance, undermining optimization feasibility in real-world, finite-data settings.

ABSTRACT

It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered on the special example of Expected Shortfall which is used here both as an illustration and as a prompt for generalization.

Motivation & Objective

• To investigate the feasibility of portfolio optimization under coherent risk measures in finite-sample settings.
• To determine whether the instability observed in Expected Shortfall (ES) is specific to ES or a general property of all coherent risk measures.
• To analyze the mathematical conditions under which coherent risk measures become unbounded during optimization.
• To assess the impact of sample fluctuations on the stability of coherent risk-based portfolio optimization.
• To evaluate whether constraints on portfolio weights can eliminate or merely mask this instability.

Proposed method

• The authors analyze the behavior of coherent risk measures under the condition that one asset dominates all others in a given finite sample.
• They use the axioms of coherent risk measures to derive a general condition under which the risk measure diverges to minus infinity when such dominance occurs.
• Theoretical analysis is applied to the Maximal Loss (Minimax) and Expected Shortfall (ES) as special cases to illustrate the instability.
• Phase transition analysis is conducted in the limit of large N (assets) and T (sample size), with fixed ratio N/T, to identify critical thresholds for feasibility.
• The replica method is applied to analytically derive the phase diagram for Expected Shortfall, confirming the existence of a critical N/T ratio.
• The study examines the role of portfolio constraints and shows they only mask, not eliminate, the instability caused by sample-driven dominance.

Experimental results

Research questions

• RQ1 Does the instability of Expected Shortfall in portfolio optimization extend to other coherent risk measures?
• RQ2 Under what conditions does a coherent risk measure become unbounded during portfolio optimization?
• RQ3 Is the dominance of a single asset in a finite sample sufficient to cause optimization failure under any coherent risk measure?
• RQ4 How does the critical ratio N/T (number of assets to sample size) affect the feasibility of optimization under coherent risk measures?
• RQ5 Can constraints on portfolio weights eliminate the instability caused by sample fluctuations in coherent risk measures?

Key findings

• The axioms of coherent risk measures imply that if one asset dominates all others in a finite sample, the risk measure diverges to minus infinity, rendering optimization infeasible.
• This instability is not limited to Expected Shortfall but is a general property of all coherent risk measures, as proven via the coherence axioms.
• For the Maximal Loss (Minimax) risk measure, the probability of optimization feasibility is less than one even when T > N, and it drops sharply as N/T exceeds 1/2 in the large-N, large-T limit.
• For Expected Shortfall, the critical N/T ratio at which optimization becomes unfeasible decreases with increasing confidence level α, forming a downward-sloping phase boundary.
• The instability arises from sample-to-sample fluctuations: a false dominance signal can occur even when no true dominance exists over infinite time horizons.
• Constraints on portfolio weights do not resolve the instability; they only shift the solution to the boundary, making results sensitive to random fluctuations rather than stable risk characteristics.