CVaR Convexity Depends on Loss and Gain Sign Conventions

Summary

The document addresses whether Conditional Value at Risk (CVaR) is convex or concave and what that means for optimization. The response explains that the answer depends on the quantity used as the function’s input and the sign convention: CVaR expressed as a function of losses is convex, while reversing the sign or expressing risk in terms of gains can yield a concave formulation. This distinction helps reconcile apparently conflicting textbook statements.

The response also connects convexity to coherence, noting that coherent risk measures satisfy convexity, unlike Value at Risk in general. A second answer cautions against inferring that concavity prevents global optimization: convex and concave objectives have global optima when optimized in the appropriate direction, while non-convexity is the more relevant concern for that guarantee. The exchange is conceptual rather than a proof or worked portfolio-optimization example, so conventions and objective direction should be checked in any specific formulation.

Key ideas

• CVaR as a function of portfolio losses is convex under the convention described.
• Using gains or reversing the sign can make the corresponding function concave.
• The sign convention explains why sources may assign different curvature to CVaR.
• Convexity is part of the coherence properties that distinguish CVaR from VaR in general.
• Both convex and concave objectives can have global optima when maximized or minimized appropriately.

Tags

# CVaR is concave risk measure or convex?


# 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?

## Answer by Sanjay (score 3)

https://quant.stackexchange.com/a/47065

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.

## Answer by Dhruv Mahajan (score 2)

https://quant.stackexchange.com/a/53920

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.

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