This paper presents a general theory that works out the relation between coherent risk measures, valuation bounds, and certain classes of portfolio optimization problems. It is economically general in the sense that it works for any cash stream spaces, be it in dynamic trading settings, one-step models, or even deterministic cash streams. It is mathematically general in the sense that, the core results are established for (possibly infinite-dimensional) linear spaces. The valuation theory presented seems to fill a gap between arbitrage valuation on the one hand and single agent utility maximization or full-fledged equilibrium theory on the other hand. Coherent valuation bounds strike a balance in that the bounds can be sharp enough to be useful in the practice of pricing and still be generic, i.e., somewhat independent of personal preferences, in the way many coherent risk measures are somewhat generic.