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.
of measurable functions, defined on an appropriate probability space.
if it satisfies the following properties:[1] That is, the risk when holding no assets is zero.
is also an in the money call option with a lower strike price.
In financial risk management, sub-additivity implies diversification is beneficial.
In financial risk management, translation invariance implies that the addition of a sure amount of capital reduces the risk by the same amount.
Indeed, the notions of Sub-additivity and Positive Homogeneity can be replaced by the notion of convexity:[5] 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.
[1] Value at risk is, however, coherent, under the assumption of elliptically distributed losses (e.g. normally distributed) when the portfolio value is a linear function of the asset prices.
However, in this case the value at risk becomes equivalent to a mean-variance approach where the risk of a portfolio is measured by the variance of the portfolio's return.
As a simple example to demonstrate the non-coherence of value-at-risk consider looking at the VaR of a portfolio at 95% confidence over the next year of two default-able zero coupon bonds that mature in 1 years time denominated in our numeraire currency.
Assume the following: Under these conditions the 95% VaR for holding either of the bonds is 0 since the probability of default is less than 5%.
This violates the sub-additivity property showing that VaR is not a coherent risk measure.
The average value at risk (sometimes called expected shortfall or conditional value-at-risk or
The domain can be extended for more general Orlitz Hearts from the more typical Lp spaces.
[7] The tail value at risk (or tail conditional expectation) is a coherent risk measure only when the underlying distribution is continuous.
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
g-entropic risk measures are a class of information-theoretic coherent risk measures that involve some important cases such as CVaR and EVaR.
The coherence of this risk measure is a consequence of the concavity of
The superhedging price is a coherent risk measure.
of the assets, then a set of portfolios is the proper way to depict risk.
Set-valued risk measures are useful for markets with transaction costs.
[8] A set-valued coherent risk measure is a function
[11] For any increasing concave Wang transform function, we could define a corresponding premium principle :[10]
A coherent risk measure could be defined by a Wang transform of the cumulative distribution function
A lower semi-continuous convex risk measure
is the set of probability measures absolutely continuous with respect to P (the "real world" probability measure), i.e.
[6] A lower semi-continuous risk measure is coherent if and only if it can be represented as such that
