In the last teen years many new risk functions have been introduced(coherent risk measures, expectation bounded risk measures, generalizeddeviations, etc.) and many actuarial and/or financial problems have beenrevisited by using them. The use of new risk functions is well justified by therapid development and evolution of the financial markets and the growingpresence of skewness and kurtosis, among many other reasons, but thepractical final result of many problems may critically depend on the concreterisk function we are drawing on. This paper deals with optimizationproblems involving risk functions and proposes several risk level upperbounds that apply regardless of the considered function. In particular bothcapital requirements and usual central moments and dispersions are boundedfrom above. The methodology is general enough and applies for perfect orimperfect financial markets, static or dynamic models, pricing or hedgingissues, portfolio choice problems, optimal reinsurance problems, etc.