CAPM and APT like models with risk measures

The paper deals with optimal portfolio choice problems when risk levels are given by coherent risk measures, expectation bounded risk measures or general deviations. Both static and dynamic pricing models may be involved. Unbounded problems are characterized by new notions such as compatibility and strong compatibility between pricing rules and risk measures. Surprisingly, it is pointed out that the lack of bounded optimal risk and/or return levels arises in practice for very important pricing models (for instance, the Black and Scholes model) and risk measures (V aR, CV aR, absolute deviation and downside semi-deviation, etc.). Bounded problems will present a Market Price of Risk and generate a pair of benchmarks. From these benchmarks we will introduce APT and CAPM like analyses, in the sense that the level of correlation between every available security and some economic factors will expalin the security expected return. On the contray, the risk level non correlated with these factors will have no influence on any return, despite we are dealing with very general risk functions that are beyond the standard deviation.Research partially supported by “RD_Sistemas SA”, “Comunidad Autónoma de Madrid” (Spain), Grant s−0505/tic/000230, and “MEyC” (Spain), Grant SEJ2006−15401−C0