In the field of 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 ρ that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.
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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:[1]
That is, the risk of holding no assets is zero.
That is, if portfolio always has better values than portfolio under all scenarios then the risk of should be less than the risk of .[2]
Indeed, the risk of two portfolios together cannot get any worse than adding the two risks separately: this is the diversification principle.
Loosely speaking, if you double your portfolio then you double your risk.
The value is just adding cash to your portfolio , which acts like an insurance: the risk of is less than the risk of , and the difference is exactly the added cash . In particular, if then .
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:[3]
It is well known that value at risk is not, in general, 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 (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%. However if we held a portfolio that consisted of 50% of each bond by value then the 95% VaR is 35% since the probability of at least one of the bonds defaulting is 7.84% which exceeds 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) is a coherent risk measure, even though it is derived from Value at Risk which is not.
The tail value at risk (or tail conditional expectation) is a coherent risk measure only when the underlying distribution is continuous.
The entropic risk measure is a convex risk measure which is not coherent. It is related to the exponential utility.
The superhedging price is a coherent risk measure.
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.[4]
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:[5]
If instead of the sublinear property,R is convex, then R is a set-valued convex risk measure.
An acceptance set is convex (coherent) if and only if the corresponding risk measure is convex (coherent). As defined below it can be shown that and .
A convex risk measure can be represented as
such that is a penalty function.
A risk measure is coherent if and only if it can be represented as
such that .[6]
If for every X (where is the essential infimum) is a deviation risk measure, then there is a one-to-one relationship between D and an expectation-bounded coherent risk measure where for any
is expectation bounded if for any nonconstant X and for any constant X.[7]