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:
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:
- Translative in M
- Value at risk
- Expected shortfall
- Entropic value at risk
- Tail conditional expectation
- Entropic risk measure
- Superhedging price
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.
Relation to acceptance set
Risk measure to acceptance set
- If is a (scalar) risk measure then is an acceptance set.
- If is a set-valued risk measure then is an acceptance set.
Acceptance set to risk measure
- If is an acceptance set (in 1-d) then defines a (scalar) risk measure.
- If is an acceptance set then is a set-valued risk measure.
Relation with deviation risk measure
is called expectation bounded if it satisfies for any nonconstant X and for any constant X.
- Coherent risk measure
- Dynamic risk measure
- Managerial risk accounting
- Risk management
- Risk metric - the abstract concept that a risk measure quantifies
- RiskMetrics - a model for risk management
- Spectral risk measure
- Distortion risk measure
- Value at risk
- Conditional value-at-risk
- Entropic value at risk
- Risk return ratio
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