# Mean-CVAR Optimization

I’m curious if anyone has any familiarity with Mean-CVAR optimization. I’m curious how people do this in practice. I saw some matlab code on wilmott based strictly on the paper from below, but I also discovered you do it with different optimizations. Meucci refers to mean-CVAR optimization in many of his papers, but I couldn’t find any code on his site that shows how he does it. Two big questions: 1) do people include the risk-free asset when they do this. In MVO including a risk-free asset in the covariance matrix makes it impossible to invert (if you’re doing the returns with risk-free values). This need not be the case for mean-cvar 2) why isn’t the optimization performed in terms of CVAR squared to make it more comparable to MVO? I understand that CVAR is convex which is the reason why it is used instead of VAR, but imagine you have a risky asset that you calculated the CVAR for an a risk-free asset with 0 CVAR and such that the combined CVAR is w*CVAR where w is the weight for the risky asset and 1-w is the weight for the risk-free asset. If you optimize max(w*u-lambda*w*CVAR), then choosing w doesn’t make sense. However, if you optimize max(w*u-lambda*w^2*CVAR^2), you would choose w=u/(lambda*CVAR^2). The classic paper is from Rockafellar and Uryasev (I found here: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.132.1960&rep=rep1&type=pdf) I also ran across another paper from Krokhmal & et al (here: http://www.engineering.uiowa.edu/~krokhmal/pdf/cvar.pdf) that shows how the efficient frontier in mean-CVAR (just like in MVO) is the same regardless of whether you use an optimization that minimizes cvar subject to the mean return being greater than an amount, an optimization that minimizes the negative of the return with the CVAR below a threshold, and optimizing with the CVAR minus a risk coefficient times the return.

i’m also interested in this.