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.
Properties
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:Normalized
That is, the risk of holding no assets is zero.Monotonicity
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 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.
Sub-additivity
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.
Positive homogeneity
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.
Translation invariance
If is a deterministic portfolio with guaranteed return and thenThe portfolio is just adding cash to your portfolio. In particular, if then.
In financial risk management, translation invariance implies that the addition of a sure amount of capital reduces the risk by the same amount.
Convex risk measures
The notion of coherence has been subsequently relaxed. Indeed, the notions of Sub-additivity and Positive Homogeneity can be replaced by the notion of convexity:; Convexity
Examples of risk measure
Value at risk
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.Value at risk is, however, coherent, under the assumption of elliptically distributed losses 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.
The Wang transform function for the Value at Risk is. The non-concavity of proves the non coherence of this risk measure.
;Illustration
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:
- The current yield on the two bonds is 0%
- The two bonds are from different issuers
- Each bond has a 4% probability of defaulting over the next year
- The event of default in either bond is independent of the other
- Upon default the bonds have a recovery rate of 30%
Average value at risk
The average value at risk is a coherent risk measure, even though it is derived from Value at Risk which is not. The domain can be extended for more general Orlitz Hearts from the more typical Lp spaces.Entropic value at risk
The entropic value at risk is a coherent risk measure.Tail value at risk
The tail value at risk is a coherent risk measure only when the underlying distribution is continuous.The Wang transform function for the tail value at risk is. The concavity of proves the coherence of this risk measure in the case of continuous distribution.
Proportional Hazard (PH) risk measure
The PH risk measure transforms the hazard rates using a coefficient.The Wang transform function for the PH risk measure is. The concavity of if proves the coherence of this risk measure.
g-Entropic risk measures
s are a class of information-theoretic coherent risk measures that involve some important cases such as CVaR and EVaR.The Wang risk measure
The Wang risk measure is defined by the following Wang transform function . The coherence of this risk measure is a consequence of the concavity of.Entropic risk measure
The entropic risk measure is a convex risk measure which is not coherent. It is related to the exponential utility.Superhedging price
The superhedging price is a coherent risk measure.Set-valued
In a situation with -valued portfolios such that risk can be measured in 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.Properties
A set-valued coherent risk measure is a function, where and where is a constant solvency cone and is the set of portfolios of the reference assets. must have the following properties:; Normalized
; Translative in M
; Monotone
; Sublinear
General framework of Wang transform
;Wang transform of the cumulative distribution functionA Wang transform of the cumulative distribution function is an increasing function where and . This function is called distortion function or Wang transform function.
The dual distortion function is.
Given a probability space, then for any random variable and any distortion function we can define a new probability measure such that for any it follows that
;Actuarial premium principle
For any increasing concave Wang transform function, we could define a corresponding premium principle :
;Coherent risk measure
A coherent risk measure could be defined by a Wang transform of the cumulative distribution function if and only if is concave.
Set-valued convex risk measure
If instead of the sublinear property,R is convex, then R is a set-valued convex risk measure.Dual representation
A lower semi-continuous convex risk measure can be represented assuch that is a penalty function and is the set of probability measures absolutely continuous with respect to P, i.e.. The dual characterization is tied to spaces, Orlitz hearts, and their dual spaces.
A lower semi-continuous risk measure is coherent if and only if it can be represented as
such that.