Coherent Risk Measures and von Neumann–Morgenstern Utility Axioms

# Does a coherent risk measure satisfy the four axioms of von Neumann–Morgenstern?


# Does a coherent risk measure satisfy the four axioms of von Neumann–Morgenstern?












What is the relationship between the axioms of Artzner et al (1999) for coherent risk measures and the axioms of von Neumann-Morgenstern (1944) for the expected utility theory?

## Answer by Ram Ahluwalia (score 7, accepted)

https://quant.stackexchange.com/a/3825

The von Neumann-Morgenstern utility axioms are normative criteria for rational choice. In contrast, he Artzner/Uryasev axioms are normative criteria that some argue must hold for any measure that aims to measure portfolio risk.

What they have in common is simply that they are normative criteria. The substance of the axioms are different, however, since they cover different domains.

In the case of utility theory, the axioms represent constraints on preferences that hold for rational agents. An investor that did not hold one of the axioms could be deprived of their wealth thru a series of gambles or choices. (The easiest way to see this is to imagine gaming an individual whose preferences are not transitive.)

Coherent risk measures are normative claims that some argue must be satisfied by a risk measure. For example, most of us would quite readily agree to the axiom of "sub-additivity" which states that the risk of two portfolios combined cannot have greater risk than the two risks separately (by virtue of diversification). Some measures such as Value-at-Risk do not satisfy this property, while others such as conditional value-at-risk do. At worst, the portfolios are perfectly correlated so there is no diversification benefit. So Artzner/Uryasev would argue that any risk measure (such as Value-at-Risk) that can show an increase in portfolio risk when two assets are combined is incoherent.

There are other axioms that govern related fields such as the axioms of euclidean geometry or the three axioms of probability. In all cases the axioms i) create a foundation for developing theories that are functions of the axioms (such as the pythagorean theorem, conditional probability, or utility of money theories), ii) the axioms allow us to test that our beliefs conform to normative expectations which has applied, practical value.

Digression: Interestingly, in each case there are scenarios where one axiom is relaxed and new branches of science are opened up. For example, eliminating the requirement that "parallel lines never intersect" opens up the idea of Riemannian geometry. You can argue behavioral finance and bounded rationality involve relaxing one or more of the VNM utility axioms.

For completness, the von Neumann-Morgenstern utility axioms are completeness, transitivity, continuity, and independence.

- Completeness implies that for every pair of choices, the agent prefers A over B, B over A, or is indifferent

- Transitivity implies that if an investor prefers A>B, and B>C, then the investor prefers A>C

- Continuity requires that investor's preferences are continuous

- Independence assumes that preferences (say between a gamble A and gamble B) hold independently even if a third independent gamble is introduced. Note this one is somewhat controversial -- see the Ellsberg Paradox to see why

The Coherent Risk Measures are:

- Normalized -- holding no assets entails zero risk

- Monotonicity -- if in any scenario Portfolio A results in less loss than Portfolio B, then Portfolio A is less risky than Portfolio B

- Sub-additivity -- portfolio risk cannot increase when adding two risks separately

- Positive homogeneity -- doubling your position size doubles your risk

- Translation invariance -- anytime you add cash to a portfolio you reduce portfolio risk

See also this post for some more insight on the Coherent Risk measure properties.

## Answer by Alexandre Street (score 2)

https://quant.stackexchange.com/a/3844

See the paper "On the conditional value at risk Probability dependent utility function" by Alexandre Street, on Theory and Decision, 2010. It shows that the well know CVAR fails in the independence axiom but it also provides good insights for that. The CVAR (redefined for revenues and not for losses - see the above paper) is convex in the probability set. This is the main reason why this measure does not follow the independence axiom.

If you don't have access to that journal, you can find a manuscript on the authors page.