# Time consistency (finance)

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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.

Finance is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

Risk is the potential for uncontrolled loss of something of value. Values can be gained or lost when taking risk resulting from a given action or inaction, foreseen or unforeseen. Risk can also be defined as the intentional interaction with uncertainty. Uncertainty is a potential, unpredictable, and uncontrollable outcome; risk is an aspect of action taken in spite of uncertainty.

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

Financial risk is any of various types of risk associated with financing, including financial transactions that include company loans in risk of default. Often it is understood to include only downside risk, meaning the potential for financial loss and uncertainty about its extent.

In financial mathematics, a conditional risk measure is a random variable of the financial risk as if measured at some point in the future. A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra.

In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement.

### Mathematical definition

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

#### Equivalent definitions

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

• ${\displaystyle \rho _{T-1}^{com}:=\rho _{T-1}}$
• ${\displaystyle \forall 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.

Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses.

#### Time consistent alternative

The time consistent alternative to the dynamic average value at risk with parameter ${\displaystyle \alpha _{t}}$ at time t is defined by

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

such that ${\displaystyle {\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}}$. [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. [5]

#### Continuous time

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

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

for a sublinear choice of function ${\displaystyle g}$ where ${\displaystyle \mathbb {E} ^{g}}$ denotes a g-expectation. If the function ${\displaystyle g}$ is convex, then the corresponding risk measure is convex. [6]

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## 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 requires |journal= (help)
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 requires |journal= (help)
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.
4. 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.
5. 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 requires |journal= (help)
6. Rosazza Gianin, E. (2006). "Risk measures via g-expectations". Insurance: Mathematics and Economics. 39: 19–65. doi:10.1016/j.insmatheco.2006.01.002.