  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
$\rho$-arbitrage for a risk measure $\rho$. We show how to determine
analytically whether such $\rho$-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.
