CVaR is concave risk measure or convex?

I see in pflug modeling and measuring risk book, CVaR is concave... But the other book definate cvar is convex... If assume cvar is concave, then cvar optimization problem give us a global optimal point?

CVaR is a convex function in the underlying portfolio (measured as for instance absolute value or profit). I won't get into proving anything so instead I am going to link the first result from Google search: https://pdfs.semanticscholar.org/a5df/128eed59668b525a743a4e7f3f0efe12f930.pdf

In fact, one of the reasons that we in general think of CVaR as a superior risk measure to VaR is the fact that CVaR is a coherent risk measure and VaR is not. Convexity needs to be satisfied in order for risk measure to be coherent.

• Thank you for your answer. Do you study the pflug "Modeling, Measuring and Managing Risk" book? Commented Aug 11, 2019 at 20:01
• Pflug prove the Cvar is concave... Commented Aug 11, 2019 at 20:02
• In "portfolio optimization with copula based extention conditional value at risk" paper, author said cvar is superaddative Commented Aug 11, 2019 at 20:05

It does not even matter if it’s concave or convex wrt global optimisation, both concave and convex functions have global optimal points albeit the only difference is maximum vs minimum which is easily incorporated with just a negative sign.

As for CVaR concave or convex can simply be a result of whether it’s defined on losses or gains, with positive or negative sign respectively, so in one case it’ll be convex, another concave. But it doesn’t matter in optimisation.

I think you’re confusing concave with non-convex.