Scenario based approach to value-at-risk optimization using mixed-integer programming

Question

For a discrete set of scenarios, minimising value at risk can be formulated as a mixed integer linear programming problem. If each scenario has equal probability then this can be written as

\begin{align} &\text{minimize} &\gamma\\ &\text{subject to} &(-r^{s}){'}X &\leq \gamma + M\cdot Y_{s} &&\text{$s = 1,\dots,S$} \tag1\\ &&\frac{1}{S}\sum_{s=1}^{S} Y_{s} &\leq \alpha \tag2\\ &&Y_{s} &\in \{0,1\} &&\text{$s = 1,\dots,S$} \\ &&\sum_{i=1}^{n}x_{i} &= 1 \end{align}

where $\alpha$ is the confidence level say $0.05$, $M$ is a big constant, $r$ is the return on assets, $x_{i}$ is the percentage in asset $i$, and $S$ is the number of scenarios.

If we assume that scenarios do not have same probabilities then constraint $(1)$ can be formulated as: $(-r^{s}\cdot P_{s}){'}X \leq \gamma + M\cdot Y_{s}$ where $P_{s}$ is the probability of scenario $s$. But I am struggling with redefining constraint $(2)$.

How can this constraint/problem be formulated if scenarios have different probabilities?

Answer 1

How about \begin{align}\min&\quad\gamma\\\text{s.t.}&\quad(-r^s)^\top X\leq \gamma + M Y_s \qquad s=1,\ldots,S\\&\quad\sum_{s=1}^SP_sY_s \leq \alpha\\&\quad \sum_{i=1}^nx_i=1\\&\quad Y_s\in\{0,1\}\end{align}