Time consistency is a large question that deals with time and actions in time. The concept means keeping something consistent with time, and it is much broader question than commonly understood. Keeping something consistent with time means not only consistency but controlling the effects of the passage of time as well. Thus the concept deals with control over time.
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Time consistency and information economics
Time consistency is a key concept of information economics. Time consistency in information economics is tied to information theory's key finding that the information value of something is the greater the less of it is known in advance.
Time consistency and financial risk
Time consistency is also 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 more risky than portfolio (B) at some time in the future, then it is guaranteed to be more risky 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
Mathematical definition
A dynamic risk measure
Equivalent definitions
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:
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
such that
Dynamic superhedging price
The dynamic superhedging price is a time consistent risk measure.
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:
for a sublinear choice of function