Risk Measures
Value at Risk
If \(Y\) is a loss function, and \(\beta \in(0,1)\), we define,
\[
\operatorname{VaR}_\beta(Y):=\min \{\gamma: \mathbb{P}(Y \geq \gamma) \leq 1-\beta\}
\]
Conditional Value at Risk (CVaR)
\[
\operatorname{CVaR}_\beta(Y):=\mathbb{E}\left(Y \mid Y \geq \operatorname{VaR}_\beta(Y)\right)
\]
As an Optimization Problem
View VaR and CVaR as the solution to an optimization problem,
\[
\operatorname{CVaR}_\beta(Y)=\min _\gamma\left(\gamma+\frac{1}{1-\beta} \mathbb{E}[\max (Y-\gamma, 0)]\right)
\]
The optimizer here \(\bar \gamma\) is \(\operatorname{VaR}_\beta(Y)\).
Please see the coding example in the next post how we formulate the problem in practice. Here the \(\max\) function is not linear, and we do need another variable say \(z=\max(Y-\gamma,0)\), then \(z\ge0\) and \(z \ge Y-\gamma\), which becomes linear constraints.