Weighted V@R and its Properties

摘要

The paper deals with the study of a coherent risk measure, which we call Weighted V@R. It is a risk measure of the form $rho_mu(X)=intlimits_{[0,1]}hbox{TV}@hbox{R}_{uplambda}(hbox{X}) mu(hbox{d}uplambda),$ where μ is a probability measure on [0,1] and TV@R stands for Tail V@R. After investigating some basic properties of this risk measure, we apply the obtained results to the financial problems of pricing, optimization, and capital allocation. It turns out that, under some regularity conditions on μ, Weighted V@R possesses some nice properties that are not shared by Tail V@R. To put it briefly, Weighted V@R is “smoother” than Tail V@R. This allows one to say that Weighted V@R is one of the most important classes (or maybe the most important class) of coherent risk measures.