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?
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:
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)
$\rho(\lambda A) = \lambda \rho(A)$
Doubling a position in an asset A doubles your risk.
$\rho(A + x) = \rho(A)-x$
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.
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.
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.