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 ⊆ L 1 ( 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 – := sup V ∈ V α( 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.