What is a “coherent” risk measure?

What is a coherent risk measure, and why do we care? Can you give a simple example of a coherent risk measure as opposed to a non-coherent one, and the problems that a coherent measure addresses in portfolio choice?

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I'm just providing a global answer to the question, as I think it can be interesting for some beginners in quant finance.

The properties given by TheBridge:

Normalize

$\rho (\emptyset)=0$

This means you have no risk in taking no position.

$\rho(A_1+A_2) \leq \rho(A_1)+\rho(A_2)$

Having a position in two different can only decrease the risk of the portfolio (diversification)

Positive homogeneity

$\rho(\lambda A) = \lambda \rho(A)$

Doubling a position in an asset A doubles your risk.

And finally,

Translation invariance

$\rho(A + x) = \rho(A)-x$

That is, adding cash to a portfolio only diminishes the risk.

So a risk-measure is said to be coherent if and only if it has all these properties.

Note that this is just a convention, but it is motivated by the fact that all these properties are the ones an investor expects to hold for a risk measure.

Finally, notice that neither VaR nor Var are coherent risk measures, wherease the Expected Shortfall is.

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there are 4 defining properties of coherent risk measures

you can find them here as well as examples for coherent and counterexamples of those kind of risk measures

Regards

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Thanks. But I still don't get it: who defines these properties, where do they come from, is there any theoretical basis for imposing them, and why? –  Dimitris Feb 1 '11 at 15:57
I think these are just formal ways of describing informal "common-sense" ideas about risk. In the Wikipedia article, each axiom has a short sentence that descries the motivation -- such as "the risk of two portfolios together cannot get any worse than adding the two risks separately". –  Curt Hagenlocher Feb 2 '11 at 15:06

Coherent risk measures were created to address the problem that extant risk measures, like VaR, did not: namely that a risk measure should reward diversification.

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I don't think that we should care if a risk measure is coherent.

The reason that VaR is not coherent is because it need not be sub-additive. I'm willing to stand corrected, but I doubt that VaR is very far from sub-additive in practical situations. And I don't see a great deal of harm if it were. I have several problems with VaR but non-coherent is not among them.

The homogeneity condition is wrong. I call this the Amaranth condition -- it turns out that being all of one side of a market is risky.

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I'm not sure I get what you don't "like" in the homogeneity. Could you please explain a bit further? –  SRKX Sep 27 '11 at 16:36
The homogeneity condition claims that it is only 100 times more risky to own all of one side of a market than to share it equally with 99 others. I find that hard to believe. –  Patrick Burns Sep 27 '11 at 20:06
I agree with you about homogeneity for large relative positions is not sensible, but it is worth noting that VaR doesn't address that, at least in implementations I have seen. –  Brian B Oct 25 '11 at 18:56