Risk measure: Difference between revisions
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==Relation to Acceptance Set== |
==Relation to Acceptance Set== |
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There is a [[one-to-one]] correspondence between an [[acceptance set]] and a corresponding risk measure. As defined below it can be shown that |
There is a [[one-to-one]] correspondence between an [[acceptance set]] and a corresponding risk measure. As defined below it can be shown that <math>R_{A_R}(X) = R(X)</math> and <math>A_{R_A} = A</math>.<ref>{{cite doi|10.1007/s11579-011-0047-0}}</ref> |
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===Risk Measure to Acceptance Set=== |
===Risk Measure to Acceptance Set=== |
Revision as of 16:25, 20 April 2012
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.
Mathematically
A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents the risk at hand. The common notation for a risk measure associated with a random variable is . A risk measure should have certain properties:[1]
- Normalized
- Translative
- Monotone
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.[2]
Mathematically
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:[3]
- Normalized
- Translative in M
- Monotone
Examples
Well known risk measures
- Value at risk
- Expected shortfall
- Tail conditional expectation
- Entropic risk measure
- Superhedging price
- ...
Variance
Variance (or standard deviation) is not a risk measure. This can be seen since it has neither the translation property or monotonicity. That is for all , and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure.
Relation to Acceptance Set
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 .[4]
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
There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure where for any
- .
is called expectation bounded if it satisfies for any nonconstant X and for any constant X.[5]
See also
- 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
References
- ^ Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk" (pdf). Mathematical Finance. 9 (3): 203–228. Retrieved February 3, 2011.
- ^ Jouini, Elyes; Meddeb, Moncef; Touzi, Nizar (2004). "Vector–valued coherent risk measures". Finance and Stochastics. 8 (4): 531–552.
- ^ Hamel, Andreas; Heyde, Frank (December 11, 2008). "Duality for Set-Valued Risk Measures" (pdf). Retrieved July 22, 2010.
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instead. - ^ Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization" (pdf). Retrieved October 13, 2011.
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Further reading
- Crouhy, Michel (2001). Risk Management. McGraw-Hill. pp. 752 pages. ISBN 0-07-135731-9.
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suggested) (help) - Kevin, Dowd (2005). Measuring Market Risk (2nd ed.). John Wiley & Sons. pp. 410 pages. ISBN 0-470-01303-6.
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