Time consistency

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 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 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 {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. A dynamic risk measure ( ρ t ) t = 0 T {displaystyle left( ho _{t} ight)_{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,Yin L^{0}({mathcal {F}}_{T})} and t ∈ { 0 , 1 , . . . , T − 1 } : ρ t + 1 ( X ) ≥ ρ t + 1 ( Y ) {displaystyle tin {0,1,...,T-1}: ho _{t+1}(X)geq ho _{t+1}(Y)} implies ρ t ( X ) ≥ ρ t ( Y ) {displaystyle ho _{t}(X)geq ho _{t}(Y)} .