Time consistency (finance)
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 formula_1 although it is certain that A is riskier than B at time formula_2. As the name suggests a time inconsistent risk measure can lead to inconsistent behavior in financial risk management. A dynamic risk measure formula_3 on formula_4 is time consistent if formula_5 and formula_6 implies formula_7. Equality
 For all formula_8
 Recursive
 For all formula_9
 Acceptance Set
 For all formula_10 where formula_11 is the time formula_1 acceptance set and formula_13
 Cocycle condition (for convex risk measures) For all formula_14 where formula_15 is the minimal penalty function (where formula_11 is an acceptance set and formula_17 denotes the essential supremum) at time formula_1 and formula_19. 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: Both dynamic value at risk and dynamic average value at risk are not a time consistent risk measures. The time consistent alternative to the dynamic average value at risk with parameter formula_22 at time "t" is defined by
 formula_23
such that formula_24. The dynamic superhedging price is a time consistent risk measure. The dynamic entropic risk measure is a time consistent risk measure if the risk aversion parameter is constant. In continuous time, a time consistent coherent risk measure can be given by:
 formula_25
for a sublinear choice of function formula_26 where formula_27 denotes a g-expectation. If the function formula_26 is convex, then the corresponding risk measure is convex.
Time consistency and financial risk
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.