Coherent risk measure

In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function that satisfies properties of monotonicity, sub additivity, homogeneity, and translational invariance.

Consider a random outcome viewed as an element of a linear space of measurable functions, defined on an appropriate probability space. A functional → is said to be coherent risk measure for if it satisfies the following properties:

That is, the risk when holding no assets is zero.

That is, if portfolio always has better values than portfolio under almost all scenarios then the risk of should be less than the risk of E. g. If is an in the money call option (or otherwise) on a stock, and is also an in the money call option with a lower strike price. In financial risk management, monotonicity implies a portfolio with greater future returns has less risk.

Indeed, the risk of two portfolios together cannot get any worse than adding the two risks separately: this is the diversification principle. In financial risk management, sub additivity implies diversification is beneficial. The sub additivity principle is sometimes also seen as problematic.

Loosely speaking, if you double your portfolio then you double your risk. In financial risk management, positive homogeneity implies the risk of a position is proportional to its size.

It is well known that value at risk is not a coherent risk measure as it does not respect the sub additivity property. An immediate consequence is that value at risk might discourage diversification.

The entropic value at risk is a coherent risk measure.

The tail value at risk (or tail conditional expectation) is a coherent risk measure only when the underlying distribution is continuous. The Wang transform function (distortion function) for the tail value at risk is The concavity of proves the coherence of this risk measure in the case of continuous distribution.

The PH risk measure (or Proportional Hazard Risk measure) transforms the hazard rates using a coefficient
The Wang transform function (distortion function) for the PH risk measure is The concavity of if proves the coherence of this risk measure.

g entropic risk measures are a class of information theoretic coherent risk measures that involve some important cases such as CVaR and EVaR.