{{about|the property which maintains consistency in the measurement of financial risk over time|the property in game theory|dynamic inconsistency}}
'''Time consistency''' in the context of finance is the property of not having mutually contradictory evaluations of risk at different points in time. This property implies that if investment A is considered riskier than B at some future time, then A will also be considered riskier than B at every prior time.
==Time consistency and financial risk==
Time consistency is a property in financial risk related to dynamic risk measures. The purpose of the time-consistent property is to categorize the risk measures which satisfy the condition that if portfolio (A) is riskier than portfolio (B) at some time in the future, then it is guaranteed to be riskier 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.
{{Technical|date=February 2018}}
===Mathematical definition=== 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|last1=Cheridito|first1=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|access-date=November 29, 2010|archive-url=https://web.archive.org/web/20121019142204/http://www.princeton.edu/~dito/papers/timeincVaR_Oct08.pdf|archive-date=October 19, 2012|url-status=dead|df=mdy-all}}</ref>
====Equivalent definitions==== ; Equality : 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>
; Recursive : For all <math>t \in \{0,1,...,T-1\}: \rho_t(X) = \rho_t(-\rho_{t+1}(X))</math>
; Acceptance Set : 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|last1=Acciaio|first1=Beatrice|last2=Penner|first2=Irina|date=February 22, 2010|title=Dynamic risk measures|url=http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf|access-date=July 22, 2010|url-status=dead|archive-url=https://web.archive.org/web/20110902182345/http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf|archive-date=September 2, 2011}}</ref>
; Cocycle condition (for convex risk measures) : 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 (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|last1=Föllmer|first1=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|access-date=June 17, 2012}}{{Dead link|date=July 2018 |bot=InternetArchiveBot |fix-attempted=no }}</ref>
===Construction=== 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: * <math>\rho^{com}_{T-1} := \rho_{T-1}</math> * <math>\forall t < T-1: \rho^{com}_t := \rho_t(-\rho^{com}_{t+1})</math><ref name="composition" />
===Examples===
====Value at risk and average value at risk==== Both dynamic value at risk and dynamic average value at risk are not a time consistent risk measures.
====Time consistent alternative==== The time consistent alternative to the dynamic average value at risk with parameter <math>\alpha_t</math> at time ''t'' is defined by : <math>\rho_t(X) = \text{ess}\sup_{Q \in \mathcal{Q}} E^Q[-X|\mathcal{F}_t]</math> 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|access-date=February 4, 2011|url-status=dead|archive-url=https://web.archive.org/web/20110719042954/http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf|archive-date=July 19, 2011}}</ref>
====Dynamic superhedging price==== 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|access-date=February 3, 2011|url-status=dead|archive-url=https://web.archive.org/web/20110719042923/http://wws.mathematik.hu-berlin.de/~penner/penner.pdf|archive-date=July 19, 2011}}</ref>
====Dynamic entropic risk==== The dynamic entropic risk measure is a time consistent risk measure if the risk aversion parameter is constant.<ref name="penner_thesis" />
==== Continuous time ==== In continuous time, a time consistent coherent risk measure can be given by: : <math>\rho_g(X) := \mathbb{E}^g[-X]</math> 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, then the corresponding risk measure is convex.<ref>{{Cite journal | last1 = Rosazza Gianin | first1 = E. | doi = 10.1016/j.insmatheco.2006.01.002 | title = Risk measures via g-expectations | journal = Insurance: Mathematics and Economics | volume = 39 | pages = 19–65 | year = 2006 }}</ref>
==References== {{Reflist}}
Category:Financial risk modeling Category:Mathematical finance Category:Financial economics