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| {{about|the property which keeps a consistency in financial risk in time|the property in game theory|dynamic inconsistency}}
| | Hello! I am Jerilyn. I smile that I can join to the entire globe. I live in Norway, in the south region. I dream to check out the various nations, to get acquainted with intriguing people.<br><br>Feel free to visit my web blog - Hostgator Coupon, [http://currico.com.au/?option=com_k2&view=itemlist&task=user&id=100418 http://currico.com.au/?option=com_k2&view=itemlist&task=user&id=100418], |
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| '''Time consistency''' is a property in [[financial risk]] related to [[dynamic risk measure]]s. The purpose of the time consistent property is to categorize the [[risk measure]]s which satisfy the condition that if portfolio (A) is more risky than portfolio (B) at some time in the future, then it is guaranteed to be more risky at any time prior to that point. This is an important property since if it were not to hold then there is an event (with probability of occurring greater than 0) such that B is riskier than A at time <math>t</math> although it is certain that A is riskier than B at time <math>t+1</math>. As the name suggests a '''time inconsistent''' risk measure can lead to inconsistent behavior in [[financial risk management]].
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| ==Mathematical definition==
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| A dynamic risk measure <math>\left(\rho_t\right)_{t=0}^{T}</math> on <math>L^0(\mathcal{F}_T)</math> is time consistent if <math>\forall X, Y \in L^0(\mathcal{F}_T)</math> and <math>t \in \{0,1,...,T-1\}: \rho_{t+1}(X) \geq \rho_{t+1}(Y)</math> implies <math>\rho_t(X) \geq \rho_t(Y)</math>.<ref name="composition">{{cite journal|last=Cheridito|first=Patrick|last2=Stadje|first2=Mitja|date=October 2008|title=Time-inconsistency of VaR and time-consistent alternatives|url=http://www.princeton.edu/~dito/papers/timeincVaR_Oct08.pdf|accessdate=November 29, 2010|format=pdf}}</ref>
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| ===Equivalent definitions===
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| ; Equality
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| : For all <math>t \in \{0,1,...,T-1\}: \rho_{t+1}(X) = \rho_{t+1}(Y) \Rightarrow \rho_{t}(X) = \rho_{t}(Y)</math>
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| ; Recursive
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| : For all <math>t \in \{0,1,...,T-1\}: \rho_t(X) = \rho_t(-\rho_{t+1}(X))</math>
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| ; Acceptance Set
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| : For all <math>t \in \{0,1,...,T-1\}: A_t = A_{t,t+1} + A_{t+1}</math> where <math>A_t</math> is the time <math>t</math> [[acceptance set]] and <math>A_{t,t+1} = A_t \cap L^p(\mathcal{F}_{t+1})</math><ref>{{cite journal|last=Acciaio|first=Beatrice|last2=Penner|first2=Irina|date=February 22, 2010|title=Dynamic risk measures|url=http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf|accessdate=July 22, 2010|format=pdf}}</ref>
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| ; Cocycle condition (for [[convex risk measure]]s)
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| : For all <math>t \in \{0,1,...,T-1\}: \alpha_t(Q) = \alpha_{t,t+1}(Q) + \mathbb{E}^{Q}[\alpha_{t+1}(Q) \mid \mathcal{F}_t]</math> where <math>\alpha_t(Q) = \operatorname*{ess sup}_{X \in A_t} \mathbb{E}^{Q}[-X \mid \mathcal{F}_t]</math> is the minimal [[Penalty function (risk)|penalty function]] (where <math>A_t</math> is an acceptance set and <math>\operatorname*{ess sup}</math> denotes the [[essential supremum]]) at time <math>t</math> and <math>\alpha_{t,t+1}(Q) = \operatorname*{ess sup}_{X \in A_{t,t+1}} \mathbb{E}^{Q}[-X \mid \mathcal{F}_t]</math>.<ref>{{cite journal|last=Föllmer|first=Hans|last2=Penner|first2=Irina|title=Convex risk measures and the dynamics of their penalty functions|journal=Statistics and decisions|volume=24|issue=1|year=2006|pages=61–96|url=http://www.math.hu-berlin.de/~penner/Foellmer_Penner.pdf|format=pdf|accessdate=June 17, 2012}}</ref>
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| ==Construction==
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| Due to the recursive property it is simple to construct a time consistent risk measure. This is done by composing one-period measures over time. This would mean that:
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| * <math>\rho^{com}_{T-1} := \rho_{T-1}</math>
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| * <math>\forall t < T-1: \rho^{com}_t := \rho_t(-\rho^{com}_{t+1})</math><ref name="composition" />
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| ==Examples==
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| ===Value at risk and average value at risk===
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| Both dynamic [[value at risk]] and dynamic [[average value at risk]] are not a time consistent risk measures.
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| ====Time consistent alternative====
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| The time consistent alternative to the dynamic average value at risk with parameter <math>\alpha_t</math> at time ''t'' is defined by
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| : <math>\rho_t(X) = \text{ess}\sup_{Q \in \mathcal{Q}} E^Q[-X|\mathcal{F}_t]</math>
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| such that <math>\mathcal{Q} = \left\{Q \in \mathcal{M}_1: E\left[\frac{dQ}{dP}|\mathcal{F}_j\right] \leq \alpha_{j-1} E\left[\frac{dQ}{dP}|\mathcal{F}_{j-1}\right] \forall j = 1,...,T\right\}</math>.<ref>{{cite journal|first1=Patrick|last1=Cheridito|first2=Michael|last2=Kupper|title=Composition of time-consistent dynamic monetary risk measures in discrete time|journal=International Journal of Theoretical and Applied Finance|date=May 2010|url=http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf|format=pdf|accessdate=February 4, 2011}}</ref>
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| ===Dynamic superhedging price===
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| The dynamic [[superhedging price]] is a time consistent risk measure.<ref name="penner_thesis">{{cite journal|last=Penner|first=Irina|year=2007|title=Dynamic convex risk measures: time consistency, prudence, and sustainability|url=http://wws.mathematik.hu-berlin.de/~penner/penner.pdf|format=pdf|accessdate=February 3, 2011}}</ref>
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| ===Dynamic entropic risk===
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| The dynamic [[entropic risk measure]] is a time consistent risk measure if the [[risk aversion]] parameter is constant.<ref name="penner_thesis" />
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| == Continuous time ==
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| In continuous time, a time consistent coherent risk measure can be given by:
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| : <math>\rho_g(X) := \mathbb{E}^g[-X]</math>
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| for a [[sublinear]] choice of function <math>g</math> where <math>\mathbb{E}^g</math> denotes a [[g-expectation]]. If the function <math>g</math> is [[convex function|convex]], then the corresponding risk measure is convex.<ref>{{cite doi|10.1016/j.insmatheco.2006.01.002}}</ref>
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| ==References==
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| {{Reflist}}
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| [[Category:Financial risk]]
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| [[Category:Mathematical finance]]
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Hello! I am Jerilyn. I smile that I can join to the entire globe. I live in Norway, in the south region. I dream to check out the various nations, to get acquainted with intriguing people.
Feel free to visit my web blog - Hostgator Coupon, http://currico.com.au/?option=com_k2&view=itemlist&task=user&id=100418,