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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.

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