%0 Journal Article %A Grechuk, Bogdan %A Zabarankin, Michael %D 2018 %T Sensitivity Analysis in Applications with Deviation, Risk, Regret, and Error Measures %U https://figshare.le.ac.uk/articles/journal_contribution/Sensitivity_Analysis_in_Applications_with_Deviation_Risk_Regret_and_Error_Measures/10226282 %2 https://figshare.le.ac.uk/ndownloader/files/18447350 %K IR content %X 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 ${\cal L}^p$-type and semi-${\cal L}^p$ 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. %I University of Leicester