When Coherent Risk Constraints Fail to Limit Tail-Risk Seeking

Summary

The paper examines whether coherent risk measures can control investors who have limited liability or seek excessive tail risk. It introduces risk-measure-specific statistical arbitrage, termed rho-arbitrage, and argues that when such opportunities exist, constraints based on coherent measures may fail to curb those behaviours. The authors provide analytical tests for the existence of these portfolios in complete markets and in the Markowitz model, then study numerical examples of incomplete markets.

For expected shortfall constraints, the results depend strongly on the probability model chosen by the risk manager; the paper finds that ineffectiveness is possible in realistic settings. Since value at risk constraints are weaker than expected shortfall constraints, the analysis also bears on VaR-based controls. The authors contrast these findings with expected utility constraints, which they report remain effective in any arbitrage-free market. The conclusions therefore depend on both the constraint and the assumed market probability model, and do not establish that every coherent risk limit will fail in every market.

Key ideas

• Risk-measure-specific statistical arbitrage can undermine coherent risk constraints.
• The paper gives analytical conditions for identifying such opportunities in complete markets and the Markowitz model.
• Expected shortfall constraints can be ineffective in some realistic incomplete markets.
• The assessment depends heavily on the probability model selected by the risk manager.
• The paper reports that reasonable expected utility constraints remain effective in arbitrage-free markets.

Tags

# The ineffectiveness of coherent risk measures


# The ineffectiveness of coherent risk measures









We show that coherent risk measures are ineffective in curbing the behaviour of investors with limited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term $ρ$-arbitrage for a risk measure $ρ$. We show how to determine analytically whether such $ρ$-arbitrage portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether expected shortfall constraints are ineffective in these markets. We find that the answer depends heavily upon the probability model selected by the risk manager but that it is certainly possible for expected shortfall constraints to be ineffective in realistic markets. Since value at risk constraints are weaker than expected shortfall constraints, our results can be applied to value at risk. By contrast, we show that reasonable expected utility constraints are effective in any arbitrage-free market.

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