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

Operations Research Asked on January 1, 2022

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 + Mcdot 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:
$$(-r^{s}cdot P_{s}){‘}X leq gamma + Mcdot Y_{s}$$ 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?

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