Coherent Risk Measures: Axioms, Diversification, and Practical Critiques
Summary
The document introduces coherence as a set of properties for portfolio risk measures: zero risk for an empty position, subadditivity, positive homogeneity, and translation invariance. These axioms express expectations such as diversification reducing measured risk, scaling a position scaling its risk, and adding cash lowering risk. It identifies Expected Shortfall as coherent and says VaR and variance are not, emphasizing subadditivity as VaR’s potential failure.
The answers also challenge whether coherence should be a decisive criterion. One argues that VaR may be close to subadditive in practice and questions whether violations cause much harm; another rejects homogeneity on the grounds that concentrated exposure can itself be risky. The document offers conceptual arguments rather than empirical evidence or a worked portfolio example, and its statements about which measures satisfy the axioms are not developed in detail. Readers should treat coherence as one framework for evaluating risk measures, alongside practical concerns about model use and portfolio choice.
Key ideas
- A coherent risk measure satisfies normalization, subadditivity, positive homogeneity, and translation invariance.
- Subadditivity formalizes the idea that combining positions should not increase measured risk beyond the sum of their separate risks.
- The document identifies Expected Shortfall as coherent and VaR and variance as non-coherent.
- The answers disagree about how important coherence is for practical risk management.
- A critique argues that positive homogeneity may not reflect risks from concentrated market exposure.
Tags
Full text
# What is a "coherent" risk measure? # 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 by SRKX (score 13, accepted) https://quant.stackexchange.com/a/2045 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. ## Answer by shabbychef (score 3) https://quant.stackexchange.com/a/108 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. ## Answer by Patrick Burns (score 3) https://quant.stackexchange.com/a/2047 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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