  We consider market players with tail-risk-seeking behaviour as exemplified by
the S-shaped utility introduced by Kahneman and Tversky. We argue that risk
measures such as value at risk (VaR) and expected shortfall (ES) are
ineffective in constraining such players. We show that, in many standard market
models, product design aimed at utility maximization is not constrained at all
by VaR or ES bounds: the maximized utility corresponding to the optimal payoff
is the same with or without ES constraints. By contrast we show that, in
reasonable markets, risk management constraints based on a second more
conventional concave utility function can reduce the maximum S-shaped utility
that can be achieved by the investor, even if the constraining utility function
is only rather modestly concave. It follows that product designs leading to
unbounded S-shaped utilities will lead to unbounded negative expected
constraining utilities when measured with such conventional utility functions.
To prove these latter results we solve a general problem of optimizing an
investor expected utility under risk management constraints where both investor
and risk manager have conventional concave utility functions, but the investor
has limited liability. We illustrate our results throughout with the example of
the Black--Scholes option market. These results are particularly important
given the historical role of VaR and that ES was endorsed by the Basel
committee in 2012--2013.

Risk Management