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Concept
Coherent risk measure
Summary
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.
If is a deterministic portfolio with guaranteed return and then
The 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.
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
It is well known that value at risk is not a coherent risk measure as it does not respect the sub-additivity property.
Official source
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