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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 t {\displaystyle t} although it is certain that A is riskier than B at time t + 1 {\displaystyle t+1} . As the name suggests a time inconsistent risk measure can lead to inconsistent behavior in financial risk management.

Mathematical definition

A dynamic risk measure ( ρ t ) t = 0 T {\displaystyle \left(\rho _{t}\right)_{t=0}^{T}} on L 0 ( F T ) {\displaystyle L^{0}({\mathcal {F}}_{T})} is time consistent if ∀ X , Y ∈ L 0 ( F T ) {\displaystyle \forall X,Y\in L^{0}({\mathcal {F}}_{T})} and t ∈ { 0 , 1 , . . . , T − 1 } : ρ t + 1 ( X ) ≥ ρ t + 1 ( Y ) {\displaystyle t\in \{0,1,...,T-1\}:\rho _{t+1}(X)\geq \rho _{t+1}(Y)} implies ρ t ( X ) ≥ ρ t ( Y ) {\displaystyle \rho _{t}(X)\geq \rho _{t}(Y)} .¹

Equivalent definitions

Equality For all t ∈ { 0 , 1 , . . . , T − 1 } : ρ t + 1 ( X ) = ρ t + 1 ( Y ) ⇒ ρ t ( X ) = ρ t ( Y ) {\displaystyle t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)} Recursive For all t ∈ { 0 , 1 , . . . , T − 1 } : ρ t ( X ) = ρ t ( − ρ t + 1 ( X ) ) {\displaystyle t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))} Acceptance Set For all t ∈ { 0 , 1 , . . . , T − 1 } : A t = A t , t + 1 + A t + 1 {\displaystyle t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}} where A t {\displaystyle A_{t}} is the time t {\displaystyle t} acceptance set and A t , t + 1 = A t ∩ L p ( F t + 1 ) {\displaystyle A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})} ² Cocycle condition (for convex risk measures) For all t ∈ { 0 , 1 , . . . , T − 1 } : α t ( Q ) = α t , t + 1 ( Q ) + E Q [ α t + 1 ( Q ) ∣ F t ] {\displaystyle 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}]} where α t ( Q ) = * ⁡ e s s s u p X ∈ A t E Q [ − X ∣ F t ] {\displaystyle \alpha _{t}(Q)=\operatorname {*} {esssup}_{X\in A_{t}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]} is the minimal penalty function (where A t {\displaystyle A_{t}} is an acceptance set and * ⁡ e s s s u p {\displaystyle \operatorname {*} {esssup}} denotes the essential supremum) at time t {\displaystyle t} and α t , t + 1 ( Q ) = * ⁡ e s s s u p X ∈ A t , t + 1 E Q [ − X ∣ F t ] {\displaystyle \alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]} .³

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:

• ρ T − 1 c o m := ρ T − 1 {\displaystyle \rho _{T-1}^{com}:=\rho _{T-1}}
• ∀ t < T − 1 : ρ t c o m := ρ t ( − ρ t + 1 c o m ) {\displaystyle \forall t<T-1:\rho _{t}^{com}:=\rho _{t}(-\rho _{t+1}^{com})} ⁴

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 α t {\displaystyle \alpha _{t}} at time t is defined by

ρ t ( X ) = ess sup Q ∈ Q E Q [ − X | F t ] {\displaystyle \rho _{t}(X)={\text{ess}}\sup _{Q\in {\mathcal {Q}}}E^{Q}[-X|{\mathcal {F}}_{t}]}

such that Q = { Q ∈ M 1 : E [ d Q d P | F j ] ≤ α j − 1 E [ d Q d P | F j − 1 ] ∀ j = 1 , . . . , T } {\displaystyle {\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\}} .⁵

Dynamic superhedging price

The dynamic superhedging price is a time consistent risk measure.⁶

Dynamic entropic risk

The dynamic entropic risk measure is a time consistent risk measure if the risk aversion parameter is constant.⁷

Continuous time

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

ρ g ( X ) := E g [ − X ] {\displaystyle \rho _{g}(X):=\mathbb {E} ^{g}[-X]}

for a sublinear choice of function g {\displaystyle g} where E g {\displaystyle \mathbb {E} ^{g}} denotes a g-expectation. If the function g {\displaystyle g} is convex, then the corresponding risk measure is convex.⁸

References

1. 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) https://web.archive.org/web/20121019142204/http://www.princeton.edu/~dito/papers/timeincVaR_Oct08.pdf ↩

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) https://web.archive.org/web/20110902182345/http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf ↩

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] http://www.math.hu-berlin.de/~penner/Foellmer_Penner.pdf ↩

4. 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) https://web.archive.org/web/20121019142204/http://www.princeton.edu/~dito/papers/timeincVaR_Oct08.pdf ↩

5. 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. https://web.archive.org/web/20110719042954/http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf ↩

6. 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) https://web.archive.org/web/20110719042923/http://wws.mathematik.hu-berlin.de/~penner/penner.pdf ↩

7. 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) https://web.archive.org/web/20110719042923/http://wws.mathematik.hu-berlin.de/~penner/penner.pdf ↩

8. Rosazza Gianin, E. (2006). "Risk measures via g-expectations". Insurance: Mathematics and Economics. 39: 19–65. doi:10.1016/j.insmatheco.2006.01.002. /wiki/Doi_(identifier) ↩