  Focusing on gains & losses relative to a risk-free benchmark instead of
terminal wealth, we consider an asset allocation problem to maximize
time-consistently a mean-risk reward function with a general risk measure which
is i) law-invariant, ii) cash- or shift-invariant, and iii) positively
homogeneous, and possibly plugged into a general function. Examples include
(relative) Value at Risk, coherent risk measures, variance, and generalized
deviation risk measures. We model the market via a generalized version of the
multi-dimensional Black-Scholes model using $\alpha$-stable L\'evy processes
and give supplementary results for the classical Black-Scholes model. The
optimal solution to this problem is a Nash subgame equilibrium given by the
solution of an extended Hamilton-Jacobi-Bellman equation. Moreover, we show
that the optimal solution is deterministic under appropriate assumptions.
