Coherent risk measures alone are ineffective in constraining portfolio losses

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