Time consistency (finance) explained

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 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

t

although it is certain that A is riskier than B at time

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

\left(\rhot\right)

T
t=0
on
0(l{F}
L
T)
is time consistent if

\forallX,Y\in

0(l{F}
L
T)
and

t\in\{0,1,...,T-1\}:\rhot+1(X)\geq\rhot+1(Y)

implies

\rhot(X)\geq\rhot(Y)

.[1]

Equivalent definitions

Equality
  • For all

    t\in\{0,1,...,T-1\}:\rhot+1(X)=\rhot+1(Y)\rhot(X)=\rhot(Y)

    Recursive
  • For all

    t\in\{0,1,...,T-1\}:\rhot(X)=\rhot(-\rhot+1(X))

    Acceptance Set
  • For all

    t\in\{0,1,...,T-1\}:At=At,t+1+At+1

    where

    At

    is the time

    t

    acceptance set and

    At,t+1=At\cap

    p(l{F}
    L
    t+1

    )

    [2]
    Cocycle condition (for convex risk measures)
  • For all

    t\in\{0,1,...,T-1\}:\alphat(Q)=\alphat,t+1(Q)+EQ[\alphat+1(Q)\midl{F}t]

    where

    \alphat(Q)=\operatorname*{ess

    sup}
    X\inAt

    EQ[-X\midl{F}t]

    is the minimal penalty function (where

    At

    is an acceptance set and

    \operatorname*{esssup}

    denotes the essential supremum) at time

    t

    and

    \alphat,t+1(Q)=\operatorname*{ess

    sup}
    X\inAt,t+1

    EQ[-X\midl{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:

    com
    \rho
    T-1

    :=\rhoT-1

    \forallt<T-1:

    com
    \rho
    t

    :=

    com
    \rho
    t+1

    )

    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

    \alphat

    at time t is defined by

    \rhot(X)=ess\supQ

    } E^Q[-X|\mathcal{F}_t]such that

    l{Q}=\left\{Q\inl{M}1:E\left[

    dQ
    dP

    |l{F}j\right]\leq\alphaj-1E\left[

    dQ
    dP

    |l{F}j-1\right]\forallj=1,...,T\right\}

    .[4]

    Dynamic superhedging price

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

    Dynamic entropic risk

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

    Continuous time

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

    \rhog(X):=Eg[-X]

    for a sublinear choice of function

    g

    where

    Eg

    denotes a g-expectation. If the function

    g

    is convex, then the corresponding risk measure is convex.[6]

    Notes and References

    1. Cheridito. Patrick. Stadje. Mitja. October 2008. Time-inconsistency of VaR and time-consistent alternatives. November 29, 2010. https://web.archive.org/web/20121019142204/http://www.princeton.edu/~dito/papers/timeincVaR_Oct08.pdf. October 19, 2012. dead. mdy-all.
    2. Acciaio. Beatrice. Penner. Irina. February 22, 2010. Dynamic risk measures. July 22, 2010. dead. https://web.archive.org/web/20110902182345/http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf. September 2, 2011.
    3. Föllmer. Hans. Penner. Irina. Convex risk measures and the dynamics of their penalty functions. Statistics and Decisions. 24. 1. 2006. 61–96. June 17, 2012.
    4. Patrick. Cheridito. Michael. Kupper. Composition of time-consistent dynamic monetary risk measures in discrete time. International Journal of Theoretical and Applied Finance. May 2010. February 4, 2011. dead. https://web.archive.org/web/20110719042954/http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf. July 19, 2011.
    5. Penner. Irina. 2007. Dynamic convex risk measures: time consistency, prudence, and sustainability. February 3, 2011. dead. https://web.archive.org/web/20110719042923/http://wws.mathematik.hu-berlin.de/~penner/penner.pdf. July 19, 2011.
    6. Rosazza Gianin . E. . 10.1016/j.insmatheco.2006.01.002 . Risk measures via g-expectations . Insurance: Mathematics and Economics . 39 . 19–65 . 2006 .