I would like to optimize a portfolio allocation (maximizing the exposure or the expected return), but with VaR or CVaR contraints. (some parts of my portfolio cannot exceed a certain VaR)
How can I achieve that? Is there a way to turn the problem into a linear programming problem? or to approximate the results?
Any links or ideas are welcome.
Answer
In my experience, a VaR or CVaR portfolio optimization problem is usually best specified as minimizing the VaR or CVaR and then using a constraint for the expected return. As noted by Alexey, it is much better to use CVaR than VaR. The main benefit of a CVaR optimization is that it can be implemented as a linear programming problem. Another option I have tried is the technique in this paper:
http://www.math.uwaterloo.ca/~tfcolema/articles/bank_article.pdf
Another option is the two-step heuristic where one first finds the mean-variance efficient frontier and then you could calculate whatever are the relevant portfolio statistics on only the mean-variance efficient portfolios. In this way you could exclude portfolios that have too high a VaR or CVaR (or mixed CVaR deviation) for your consideration.
However, as you say you are particularly concerned about the VaR or CVaR of certain parts of your portfolio. As noted above, VaR constraints for different different groups of assets would require non-linear constraints. However, CVaR constraints for different assets could be calculated using linear constraints (though it would also be possible to implement a relatively slower methodology using non-linear constraints). For guidance on how to implement this as a linear constraint, it might help to follow
http://www.soa.org/library/proceedings/arch/2008/arch-2008-iss1-cox-lin.aspx
with the only difference that you would want to calculate the CVaR over the relevant groups of securities.
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