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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?

One Answer

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}

Answered by k88074 on January 1, 2022

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