modern portfolio theory - 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?

Answer

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.

Sub-addiitivity

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

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.