Sensitivity analysis in applications with deviation, risk, regret, and error measures

The envelope formula is obtained for optimization problems with positively homogeneous convex functionals defined on a space of random variables. Those problems include linear regression with general error measures and optimal portfolio selection with the objective function being either a general deviation measure or a coherent risk measure subject to a constraint on the expected rate of return. The obtained results are believed to be novel even for Markowitz’s mean-variance portfolio selection but are far more general and include explicit envelope relationships for the rates of return of portfolios that minimize lower semivariance, mean absolute deviation, deviation measures of L^p-type and semi-L^p type, and conditional value-at-risk. In each case, the envelope theorem yields explicit estimates for the absolute value of the difference between deviation/risk of optimal portfolios with the unperturbed and perturbed asset probability distributions in terms of a norm of the perturbation.