Local stability and differentiability of the Mean-Conditional Value at Risk model defined on the mixed-integer loss functions

In this paper, we study local stability of the mean-risk model with Conditional Value at Risk measure where the mixed-integer value function appears as a loss variable. This model has been recently introduced and studied in Schulz and Tiedemann [16]. First, we generalize the qualitative results for the case with random technology matrix. We employ the contamination techniques to quantify a possible effect of changes in the underlying probability distribution on the optimal value. We use the generalized qualitative results to express the explicit formula for the directional derivative of the local optimal value function with respect to the underlying probability measure. The derivative is used to construct the bounds. Similarly, we can approximate the behavior of the local optimal value function with respect to the changes of the risk-aversion parameter which determines our aversion to risk.