Editing Value at risk (section)

===VaR, CVaR, RVaR and EVaR===

The VaR is not a [[coherent risk measure]] since it violates the sub-additivity property, which is

:<math>\mathrm{If}\; X,Y \in \mathbf{L} ,\; \mathrm{then}\; \rho(X + Y) \leq \rho(X) + \rho(Y).</math>

However, it can be bounded by coherent risk measures like [[Conditional Value-at-Risk]] (CVaR) or [[entropic value at risk]] (EVaR). 
CVaR is defined by  average of VaR values for confidence levels between 0 and {{mvar|&alpha;}}.

However VaR, unlike CVaR, has the property of being a [[robust statistics|robust statistic]].
A related class of risk measures is the 'Range Value at Risk' (RVaR), which is a robust version of CVaR.<ref name="RVaR">{{cite journal|last1=Cont|first1=Rama|last2=Deguest|first2=Romain|last3=Giacomo|first3=Giacomo|title=Robustness and Sensitivity Analysis of Risk Measurement Procedures|journal=Quantitative Finance|volume=10|year=2010|issue=6|pages=593–606|doi=10.1080/14697681003685597|s2cid=158678050|url=https://hal.archives-ouvertes.fr/hal-00413729/file/robustriskarxiv.pdf}}</ref>

For <math> X\in \mathbf{L}_{M^+} </math> (with <math>\mathbf{L}_{M^+} </math> the set of all [[Borel measure|Borel]] [[measurable function]]s whose [[moment-generating function]] exists for all positive real values) we have

:<math>\text{VaR}_{1-\alpha}(X)\leq \text{RVaR}_{\alpha,\beta}(X) \leq \text{CVaR}_{1-\alpha}(X)\leq\text{EVaR}_{1-\alpha}(X),</math>

where

:<math>
\begin{align}
&\text{VaR}_{1-\alpha}(X):=\inf_{t\in\mathbf{R}}\{t:\text{Pr}(X\leq t)\geq 1-\alpha\},\\
&\text{CVaR}_{1-\alpha}(X) := \frac{1}{\alpha}\int_0^{\alpha} \text{VaR}_{1-\gamma}(X)d\gamma,\\
&\text{RVaR}_{\alpha,\beta}(X) := \frac{1}{\beta-\alpha}\int_{\alpha}^{\beta} \text{VaR}_{1-\gamma}(X)d\gamma,\\
&\text{EVaR}_{1-\alpha}(X):=\inf_{z>0}\{z^{-1}\ln(M_X(z)/\alpha)\},
\end{align}
</math>

in which <math> M_X(z) </math> is the moment-generating function of {{mvar|X}} at {{mvar|z}}. In the above equations the variable {{mvar|X}} denotes the financial loss, rather than wealth as is typically the case.