Risk Measures

Value at Risk

If $Y$ is a loss function, and $β \in (0, 1)$ , we define,

{VaR}_{β} (Y) := min {γ : P (Y \geq γ) \leq 1 - β}

Conditional Value at Risk (CVaR)

{CVaR}_{β} (Y) := E (Y ∣ Y \geq {VaR}_{β} (Y))

As an Optimization Problem

View VaR and CVaR as the solution to an optimization problem,

{CVaR}_{β} (Y) = min_{γ} (γ + \frac{1}{1 - β} E [max (Y - γ, 0)])

The optimizer here $\bar{γ}$ is ${VaR}_{β} (Y)$ .

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 $z = max (Y - γ, 0)$ , then $z \geq 0$ and $z \geq Y - γ$ , which becomes linear constraints.