Measures of Residual Risk with Connections to Regression, Risk Tracking, Surrogate Models, and Ambiguity

Measures of residual risk are developed as extension of measures of risk. They view a random variable of interest in concert with an auxiliary random vector that helps to manage, predict, and mitigate the risk in the original variable. Residual risk can be exemplified as a quantification of the improved situation faced by a hedging investor compared to that of a single-asset investor, but the notion reaches further with deep connections emerging with forecasting and generalized regression. We establish the fundamental properties in this framework and show that measures of residual risk along with generalized regression can play central roles in the development of risk-tuned approximations of random variables, in tracking of statistics, and in estimation of the risk of conditional random variables. The paper ends with dual expressions for measures of residual risk, which lead to further insights and a new class of distributionally robust optimization models.