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Concept# Risk measure

Summary

In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement.
A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable is . A risk measure should have certain properties:
Normalized
Translative
Monotone
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.
A set-valued risk measure is a function , where is a -dimensional Lp space, , 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
Value at risk
Expected shortfall
Superposed risk measures
Entropic value at risk
Drawdown
Tail conditional expectation
Entropic risk measure
Superhedging price
Expectile
Variance (or standard deviation) is not a risk measure in the above sense. This can be seen since it has neither the translation property nor monotonicity. That is, for all , and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure. To avoid any confusion, note that deviation risk measures, such as variance and standard deviation are sometimes called risk measures in different fields.
There is a one-to-one correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that and .
If is a (scalar) risk measure then is an acceptance set.
If is a set-valued risk measure then is an acceptance set.

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Distortion risk measure

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Coherent risk measure

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Expected shortfall

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