  Equivalent characterizations of multiportfolio time consistency are deduced
for closed convex and coherent set-valued risk measures on $L^p(\Omega,\mathcal
F, P; R^d)$ with image space in the power set of $L^p(\Omega,\mathcal
F_t,P;R^d)$. In the convex case, multiportfolio time consistency is equivalent
to a cocycle condition on the sum of minimal penalty functions. In the coherent
case, multiportfolio time consistency is equivalent to a generalized version of
stability of the dual variables. As examples, the set-valued entropic risk
measure with constant risk aversion coefficient is shown to satisfy the cocycle
condition for its minimal penalty functions, the set of superhedging portfolios
in markets with proportional transaction costs is shown to have the stability
property and in markets with convex transaction costs is shown to satisfy the
composed cocycle condition, and a multiportfolio time consistent version of the
set-valued average value at risk, the composed AV@R, is given and its dual
representation deduced.
