Optimal portfolios with expected loss constraints and shortfall risk optimal martingale measures

Summary We study reward over penalty for risk ratios E[u(V)]/E[ρ(V)], V ∈ V, where V ⊆ L1(P) describes a linear space of attainable returns in an arbitrage-free market, u is concave and ρ ≥ 0 is convex. It turns out that maximizing such reward over penalty ratios is essentially equivalent to maximizing the ratio α(V) := E[V]/E[V−] or the expected profit over expected loss ratio E[V+]/E[V−]. The lowest upper bound α– := supV ∈ Vα(V) can be determined by solving an appropriate dual problem over the set of bounded equivalent martingale measures for V. This observation leads to the definition of shortfall risk optimal equivalent martingale measures.