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.
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.