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| | | Nice to satisfy you, I am Marvella Shryock. Puerto Rico is exactly where he's been residing for many years and he will never move. One of the issues she loves most is to read comics and she'll be starting something else along with it. My day occupation is a meter reader.<br><br>Also visit my site: [http://www.eddysadventurestore.nl/nieuws/eliminate-candida-one-these-tips www.eddysadventurestore.nl] |
| {{Merge|Coherent risk measure|Distortion function|date=January 2014|section= General Framework of Wang Transform}}
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| {{Technical|date=August 2013}}
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| In the field of [[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 <math>\varrho</math> that satisfies properties of [[monotonicity]], [[Sub-additive|sub-additivity]], [[homogeneity (statistics)|homogeneity]], and [[translational invariance]].
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| ==Properties==
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| Consider a random outcome <math> X</math> viewed as an element of a linear space <math> \mathcal{L}</math> of measurable functions, defined on an appropriate probability space. A [[functional (mathematics)|functional]] <math>\varrho : \mathcal{L}</math> → <math>\R \cup \{+\infty\}</math> is said to be coherent risk measure for <math> \mathcal{L}</math> if it satisfies the following properties:<ref name="Artzner">{{cite doi|10.1111/1467-9965.00068}}</ref>
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| ; Normalized
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| : <math>\varrho(0) = 0</math>
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| That is, the risk of holding no assets is zero.
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| ; Monotonicity
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| : <math>\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} \;\mathrm{and}\; Z_1 \leq Z_2 \; \mathrm{a.s.} ,\; \mathrm{then} \; \varrho(Z_1) \geq \varrho(Z_2)</math>
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| That is, if portfolio <math>Z_2</math> always has better values than portfolio <math>Z_1</math> under [[almost surely|almost all]] scenarios then the risk of <math>Z_2</math> should be less than the risk of <math>Z_1</math>.<ref>{{cite journal|last=Wilmott|first=P.|year=2006|title=Quantitative Finance|publisher=Wiley|edition=2|volume=1|page=342}}</ref> E.g. If <math>Z_1</math> is an in the money call option (or otherwise) on a stock, and <math>Z_2</math> is also an in the money call option with a lower strike price.
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| ; Sub-additivity
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| : <math>\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} ,\; \mathrm{then}\; \varrho(Z_1 + Z_2) \leq \varrho(Z_1) + \varrho(Z_2)</math>
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| Indeed, the risk of two portfolios together cannot get any worse than adding the two risks separately: this is the [[Diversification (finance)|diversification]] principle.
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| ; Positive homogeneity
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| : <math>\mathrm{If}\; \alpha \ge 0 \; \mathrm{and} \; Z \in \mathcal{L} ,\; \mathrm{then} \; \varrho(\alpha Z) = \alpha \varrho(Z)</math>
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| Loosely speaking, if you double your portfolio then you double your risk.
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| ; Translation invariance
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| If <math> A</math> is a deterministic portfolio with guaranteed return <math> a</math> and <math> Z \in \mathcal{L}</math> then
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| : <math>\varrho(Z + A) = \varrho(Z) - a </math>
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| The portofolio <math> A</math> is just adding cash <math>a</math> to your portfolio <math>Z</math>. In particular, if <math>a=\varrho(Z)</math> then <math>\varrho(Z+A)=0</math>.
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| ===Convex risk measures===
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| The notion of coherence has been subsequently relaxed. Indeed, the notions of Sub-additivity and Positive Homogeneity can be replaced by the notion of [[convex function|convexity]]:<ref>{{cite journal|last=Föllmer|first=H.|last2=Schied|first2=A.|year=2002|title=Convex measures of risk and trading constraints|journal=Finance and Stochastics|volume=6|issue=4|pages=429–447}}</ref>
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| ; Convexity
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| : <math>If \ Z_1,Z_2 \in \mathcal{L}\text{ and }\lambda \in [0,1] \text{ then }\varrho(\lambda Z_1 + (1-\lambda) Z_2) \leq \lambda \varrho(Z_1) + (1-\lambda) \varrho(Z_2)</math>
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| == General Framework of Wang Transform ==
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| ;Wang transform of the decumulative distribution function
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| A Wang transform of the decumulative distribution function is an increasing function <math> g \colon [0,1] \rightarrow [0,1]</math> where <math> g(0)=0</math> and <math> g(1)=1</math>. <ref name="Wang">{{cite journal|last=Wang|first=Shuan|year=1996|title=Premium Calculation by Transforming the Layer Premium Density|journal=ASTIN Bulletin|volume=26|issue=1|pages=71–92}}</ref> This function is called ''distortion function'' or Wang transform function.
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| The ''dual distortion function'' is <math>\tilde{g}(x) = 1 - g(1-x)</math>.<ref name="PropertiesDRM">{{cite doi|10.1007/s11009-008-9089-z}}</ref><ref name="Wirch">{{cite web|title=Distortion Risk Measures: Coherence and Stochastic Dominance|author=Julia L. Wirch|author2=Mary R. Hardy|url=http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|format=pdf|accessdate=March 10, 2012}}</ref>
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| Given a [[probability space]] <math>(\Omega,\mathcal{F},\mathbb{P})</math>, then for any [[random variable]] <math>X</math> and any distortion function <math>g</math> we can define a new [[probability measure]] <math>\mathbb{Q}</math> such that for any <math>A \in \mathcal{F}</math> it follows that
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| <math>\mathbb{Q}(A) = g(\mathbb{P}(X \in A)).</math> <ref name="PropertiesDRM"/>
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| ;Actuarial premium principle
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| For any increasing concave Wang transform function, we could define a corresponding premium principle :<ref name="Wang"/>
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| <math> \varrho(X)=\int_0^{+\infty}g\left(\bar{F}_X(x)\right) dx</math>
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| ;Coherent risk measure
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| A coherent risk measure could be defined by a Wang transform of the decumulative distribution function <math>g</math> if on only if <math>g</math> is concave.<ref name="Wang"/>
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| ==Examples of risk measure==
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| === Value at risk ===
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| It is well known that [[value at risk]] '''is not''', in general, a coherent risk measure as it does not respect the sub-additivity property. An immediate consequence is that [[value at risk]] might discourage diversification.<ref name="Artzner"/>
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| [[Value at risk]] is, however, coherent, under the assumption of [[Elliptical distribution|elliptically distributed]] losses (e.g. [[normally distributed]]) when the portfolio value is a linear function of the asset prices. However, in this case the value at risk becomes equivalent to a mean-variance approach where the risk of a portfolio is measured by the variance of the portfolio's return.
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| The Wang transform function (distortion function) for the Value at Risk is <math> g(x)=\mathbf{1}_{x\geq 1-\alpha}</math>. The non-concavity of <math> g</math> proves the non coherence of this risk measure.
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| ;Illustration
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| As a simple example to demonstrate the non-coherence of value-at-risk consider looking at the VaR of a portfolio at 95% confidence over the next year of two default-able zero coupon bonds that mature in 1 years time denominated in our numeraire currency.
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| Assume the following:
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| * The current yield on the two bonds is 0%
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| * The two bonds are from different issuers
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| * Each bonds has a 4% [[probability of default]]ing over the next year
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| * The event of default in either bond is independent of the other
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| * Upon default the bonds have a recovery rate of 30%
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| Under these conditions the 95% VaR for holding either of the bonds is 0 since the probability of default is less than 5%. However if we held a portfolio that consisted of 50% of each bond by value then the 95% VaR is 35% since the probability of at least one of the bonds defaulting is 7.84% which exceeds 5%. This violates the sub-additivity property showing that VaR is not a coherent risk measure.
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| ===Average value at risk===
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| The average value at risk (sometimes called [[expected shortfall]] or conditional value-at-risk) is a coherent risk measure, even though it is derived from Value at Risk which is not.
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| ===Entropic value at risk===
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| The [[entropic value at risk]] is a coherent risk measure.<ref name=Ahmadi2>{{cite journal|last=Ahmadi-Javid|first=Amir|title=Entropic value-at-risk: A new coherent risk measure|journal=Journal of Optimization Theory and Applications|year=2012|volume=155|pages=1105–1123|doi=10.1007/s10957-011-9968-2|issue=3}}</ref>
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| ===Tail value at risk===
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| The [[tail value at risk]] (or tail conditional expectation) is a coherent risk measure only when the underlying distribution is [[continuous distribution|continuous]].
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| The Wang transform function (distortion function) for the [[tail value at risk]] is <math> g(x)=\min(\frac{x}{\alpha},1)</math>. The concavity of <math> g</math> proves the coherence of this risk measure in the case of continuous distribution.
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| ===Proportionnal Hazard (PH) risk measure ===
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| The PH risk measure (or Proportional Hazard Risk measure) transforms the hasard rates <math>\scriptstyle \left( \lambda(t) = \frac{f(t)}{\bar{F}(t)}\right)</math> using a coefficient <math> \xi</math>.
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| The Wang transform function (distortion function) for the PH risk measure is <math> g_{\alpha}(x) = x^{\xi} </math>. The concavity of <math> g</math> if <math>\scriptstyle \xi<\frac{1}{2}</math> proves the coherence of this risk measure.
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| [[File:Sample_of_Wang_transform_function_or_distortion_function.png|thumb|right|Sample of Wang transform function or distortion function]]
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| ===g-Entropic risk measures===
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| [[g-entropic risk measure]]s are a class of information-theoretic coherent risk measures that involve some important cases such as CVaR and EVaR.<ref name=Ahmadi2 />
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| ===The Wang risk measure===
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| The Wang risk measure is define by the following Wang transform function (distortion function) <math> g_{\alpha}(x)=\Phi\left[ \Phi^{-1}(x)-\Phi^{-1}(\alpha)\right]</math>. The coherence of this risk measure is a consequence of the concavity of <math> g</math>.
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| ===Entropic risk measure===
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| The [[entropic risk measure]] is a convex risk measure which is not coherent. It is related to the [[exponential utility]].
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| ===Superhedging price===
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| The [[superhedging price]] is a coherent risk measure.
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| ==Set-valued==
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| In a situation with <math>\mathbb{R}^d</math>-valued portfolios such that risk can be measured in <math>n \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>
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| ===Properties===
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| A set-valued coherent risk measure is a function <math>R: L_d^p \rightarrow \mathbb{F}_M</math>, where <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>
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| ; Normalized
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| : <math>K_M \subseteq R(0) \; \mathrm{and} \; R(0) \cap -\mathrm{int}K_M = \emptyset</math>
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| ; Translative in M
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| : <math>\forall X \in L_d^p, \forall u \in M: R(X + u1) = R(X) - u</math> | |
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| ; Monotone
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| : <math>\forall X_2 - X_1 \in L_d^p(K) \Rightarrow R(X_2) \supseteq R(X_1)</math>
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| ; Sublinear
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| ===Set-valued convex risk measure===
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| If instead of the sublinear property,''R'' is convex, then ''R'' is a set-valued convex risk measure.
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| ==Dual representation==
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| A [[lower semi-continuous]] convex risk measure <math>\varrho</math> can be represented as
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| : <math>\varrho(X) = \sup_{Q \in \mathcal{M}(P)} \{E^Q[-X] - \alpha(Q)\}</math> | |
| such that <math>\alpha</math> is a [[Penalty function (risk)|penalty function]] and <math>\mathcal{M}(P)</math> is the set of probability measures [[absolutely continuous]] with respect to ''P'' (the "real world" [[probability measure]]), i.e. <math>\mathcal{M}(P) = \{Q \ll P\}</math>.
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| A [[lower semi-continuous]] risk measure is coherent if and only if it can be represented as
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| : <math>\varrho(X) = \sup_{Q \in \mathcal{Q}} E^Q[-X]</math>
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| such that <math>\mathcal{Q} \subseteq \mathcal{M}(P)</math>.<ref>{{cite book|first1=Hans|last1=Föllmer|first2=Alexander|last2=Schied|title=Stochastic finance: an introduction in discrete time|publisher=Walter de Gruyter|year=2004|edition=2|isbn=978-3-11-018346-7}}</ref>
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| ==See also==
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| * [[Risk metric]] - the abstract concept that a risk measure quantifies
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| * [[RiskMetrics]] - a model for risk management
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| * [[Spectral risk measure]] - a subset of coherent risk measures
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| * [[Distortion risk measure]]
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| * [[Conditional value-at-risk]]
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| * [[Entropic value at risk]]
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| * [[Financial risk]]
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| ==References==
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| {{reflist|30em}}
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| ==External links==
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| * [http://www.princeton.edu/~dito/riskmeasures/CohConMon/ A list of important papers on coherent and convex risk measures]
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| [[Category:Actuarial science]]
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| [[Category:Mathematical finance]]
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| [[Category:Financial risk]]
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