Time consistency (finance)

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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

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Time consistency is a property in financial risk related to dynamic risk measures. The purpose of the time the 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   although it is certain that A is riskier than B at time  . As the name suggests a time inconsistent risk measure can lead to inconsistent behavior in financial risk management.

Mathematical definition

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A dynamic risk measure   on   is time consistent if   and   implies  .[1]

Equivalent definitions

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Equality
For all  
Recursive
For all  
Acceptance Set
For all   where   is the time   acceptance set and  [2]
Cocycle condition (for convex risk measures)
For all   where   is the minimal penalty function (where   is an acceptance set and   denotes the essential supremum) at time   and  .[3]

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:

  •  
  •  [1]

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.

Time consistent alternative

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The time consistent alternative to the dynamic average value at risk with parameter   at time t is defined by

 

such that  .[4]

Dynamic superhedging price

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The dynamic superhedging price is a time consistent risk measure.[5]

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.[5]

Continuous time

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In continuous time, a time consistent coherent risk measure can be given by:

 

for a sublinear choice of function   where   denotes a g-expectation. If the function   is convex, then the corresponding risk measure is convex.[6]

References

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  1. ^ a b Cheridito, Patrick; Stadje, Mitja (October 2008). "Time-inconsistency of VaR and time-consistent alternatives" (PDF). Archived from the original (PDF) on October 19, 2012. Retrieved November 29, 2010. {{cite journal}}: Cite journal requires |journal= (help)
  2. ^ Acciaio, Beatrice; Penner, Irina (February 22, 2010). "Dynamic risk measures" (PDF). Archived from the original (PDF) on September 2, 2011. Retrieved July 22, 2010. {{cite journal}}: Cite journal requires |journal= (help)
  3. ^ Föllmer, Hans; Penner, Irina (2006). "Convex risk measures and the dynamics of their penalty functions" (PDF). Statistics and Decisions. 24 (1): 61–96. Retrieved June 17, 2012.[permanent dead link]
  4. ^ Cheridito, Patrick; Kupper, Michael (May 2010). "Composition of time-consistent dynamic monetary risk measures in discrete time" (PDF). International Journal of Theoretical and Applied Finance. Archived from the original (PDF) on July 19, 2011. Retrieved February 4, 2011.
  5. ^ a b Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (PDF). Archived from the original (PDF) on July 19, 2011. Retrieved February 3, 2011. {{cite journal}}: Cite journal requires |journal= (help)
  6. ^ Rosazza Gianin, E. (2006). "Risk measures via g-expectations". Insurance: Mathematics and Economics. 39: 19–65. doi:10.1016/j.insmatheco.2006.01.002.