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'{{ distinguish2| [[deviation risk measure]]s, e.g. [[standard deviation]] }} 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 [[downside risk|risks]] taken by [[financial institutions]], such as banks and insurance companies, acceptable to the [[regulator (economics)|regulator]]. In recent years attention has turned towards [[coherent risk measure|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 <math>X</math> is <math>\rho[X]</math>. A risk measure <math>\rho: \mathcal{L} \to \mathbb{R} \cup \{+\infty\}</math> should have certain properties:<ref>{{cite journal|last=Artzner|first=Philippe|last2=Delbaen|first2=Freddy|last3=Eber|first3=Jean-Marc|last4=Heath|first4=David|year=1999|title=Coherent Measures of Risk|journal=Mathematical Finance|volume=9|issue=3|pages=203–228|url=http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf|format=pdf|accessdate=February 3, 2011}}</ref> ; Normalized : <math>\rho(0) = 0</math> ; Translative : <math>\mathrm{If}\; a \in \mathbb{R} \; \mathrm{and} \; Z \in \mathcal{L} ,\;\mathrm{then}\; \rho(Z + a) = \rho(Z) - a</math> ; Monotone : <math>\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} \;\mathrm{and}\; Z_1 \leq Z_2 ,\; \mathrm{then} \; \rho(Z_1) \geq \rho(Z_2)</math> ==Set-valued== In a situation with <math>\mathbb{R}^d</math>-valued portfolios such that risk can be measured in <math>m \leq d</math> 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 cost]]s.<ref>{{cite journal|last=Jouini|first=Elyes|last2=Meddeb|first2=Moncef|last3=Touzi|first3=Nizar|year=2004|title=Vector–valued coherent risk measures|journal=Finance and Stochastics|volume=8|issue=4|pages=531–552}}</ref> ===Mathematically=== A set-valued risk measure is a function <math>R: L_d^p \rightarrow \mathbb{F}_M</math>, where <math>L_d^p</math> is a <math>d</math>-dimensional [[Lp space]], <math>\mathbb{F}_M = \{D \subseteq M: D = cl (D + K_M)\}</math>, and <math>K_M = K \cap M</math> where <math>K</math> is a constant [[solvency cone]] and <math>M</math> is the set of portfolios of the <math>m</math> reference assets. <math>R</math> must have the following properties:<ref>{{cite doi|10.1137/080743494}}</ref> ; Normalized : <math>K_M \subseteq R(0) \; \mathrm{and} \; R(0) \cap -\mathrm{int}K_M = \emptyset</math> ; Translative in M : <math>\forall X \in L_d^p, \forall u \in M: R(X + u1) = R(X) - u</math> ; Monotone : <math>\forall X_2 - X_1 \in L_d^p(K) \Rightarrow R(X_2) \supseteq R(X_1)</math> == 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 <math>Var(X + a) = Var(X) \neq Var(X) - a</math> for all <math>a \in \mathbb{R}</math>, and a simple counterexample for monotonicity can be found. The standard deviation is a [[deviation risk measure]]. ==Relation to Acceptance Set== There is a [[bijection|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> ===Risk Measure to Acceptance Set=== * If <math>\rho</math> is a (scalar) risk measure then <math>A_{\rho} = \{X \in L^p: \rho(X) \leq 0\}</math> is an acceptance set. * If <math>R</math> is a set-valued risk measure then <math>A_R = \{X \in L^p_d: 0 \in R(X)\}</math> is an acceptance set. ===Acceptance Set to Risk Measure=== * If <math>A</math> is an acceptance set (in 1-d) then <math>\rho_A(X) = \inf\{u \in \mathbb{R}: X + u1 \in A\}</math> defines a (scalar) risk measure. * If <math>A</math> is an acceptance set then <math>R_A(X) = \{u \in M: X + u1 \in A\}</math> is a set-valued risk measure. ==Relation with deviation risk measure== There is a [[bijection|one-to-one]] relationship between a [[deviation risk measure]] ''D'' and an expectation-bounded risk measure <math>\rho</math> where for any <math>X \in \mathcal{L}^2</math> * <math>D(X) = \rho(X - \mathbb{E}[X])</math> * <math>\rho(X) = D(X) - \mathbb{E}[X]</math>. <math>\rho</math> is called expectation bounded if it satisfies <math>\rho(X) > \mathbb{E}[-X]</math> for any nonconstant ''X'' and <math>\rho(X) = \mathbb{E}[-X]</math> for any constant ''X''.<ref name="Rockafellar">{{cite journal|title=Deviation Measures in Risk Analysis and Optimization|first1=Tyrrell|last1=Rockafellar|first2=Stanislav|last2=Uryasev|first3=Michael|last3=Zabarankin|year=2002|url=http://www.ise.ufl.edu/uryasev/Deviation_measures_wp.pdf|format=pdf|accessdate=October 13, 2011}}</ref> == 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]] * [[Value at risk]] * [[Conditional value-at-risk]] * [[Entropic Value at Risk]] hfuq hfbbnvcnjeswjhiute6uio ghkjryutlkil sdfibyt5r 2yyrityi9nm ryueu43u4u7uyewgihekmfkhrdnjjg jedueiuehejthr4eherwjtejrterhknghs//oeiohfhfhfhfhr oaszbgfti u5iu pt ipoieptw djfdjf dhbhhdhfsmdmmvifjeo jfjfkfke fsohzbalopqmcaixdsaq kdodvctqohdncaohf jdnncfaoowygfbdjospjd gobfg fs wfte bghfo jkjd ffshhht bwes nbuv wtx cqa jhr jfhg vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb ==References== {{Reflist}} ==Further reading== *{{cite book | last = Crouhy | authorlink = | first = Michel | coauthors = D. Galai, and R. Mark | title = Risk Management | publisher = [[McGraw-Hill]] | year = 2001 | location = | pages = 752 pages | url = | doi = | id = ISBN 0-07-135731-9 }} *{{cite book | last = Kevin | first = Dowd | authorlink = | coauthors = | title = Measuring Market Risk | edition = 2nd | publisher = [[John Wiley & Sons]] | year = 2005 | location = | pages = 410 pages | url = | doi = | id = ISBN 0-470-01303-6 }} * {{cite book|first1=Hans|last1=Foellmer|first2=Alexander|last2=Schied|title=Stochastic Finance|Publisher=[[Walter de Gruyter]]|year=2004|isbn=311-0183463|series=de Gruyter Series in Mathematics|volume=27|location=Berlin|pages=xi+459|mr=2169807}} * {{cite book|first1=Alexander|last1=Shapiro|first2=Darinka|last2=Dentcheva|last3=Ruszczyński|first3=Andrzej|authorlink3=Andrzej Piotr Ruszczyński|title=Lectures on stochastic programming. Modeling and theory|publisher=[[Society for Industrial and Applied Mathematics]]|year=2009|isbn=978-0898716870|series=MPS/SIAM Series on Optimization|volume=9|location=Philadelphia|pages=xvi+436|mr=2562798}} [[Category:Actuarial science]] [[Category:Mathematical finance]] [[Category:Financial risk]]'