Before revision:  We show that for a finite probability space the calculation of the values of AV@R reduces to linear vector optimization problems which can be solved using known algorithms. The formulation of AV@R as a linear vector optimization problem is an extension of the corresponding scalar result by Rockafellar and Uryasev .
After revision:  We show that for a finite probability space the calculation of the values of both version over finite probability spaces. Several examples illustrate various features of the theoretical constructions .