# How to calculate minimum variance portfolio with constraints?

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gsourop on 7 Feb 2023
Commented: Sargondjani on 7 Feb 2023
Dear everyone,
I would like to calculate the weights of a minimum variance portfolio with leverage and short sale constraints. I have found the code below, which winsorizes the weights to 1, namely there the investor cannot leverage her wealth. However, I need to adjust it, so that the investor can leverage up to 2 times her portfolio and short sale up to 1 time her portfolio, in other word the weights (denoted as w) should follow the rule -1<=w<=2 .
% Portfolio problem
prob = optimproblem('ObjectiveSense','minimize');
% Variables
% Portfolio weights
x = optimvar('x',nAssets,1,'LowerBound',0); % x >= 0 (long-only portfolio)
% Objective
% min x'*Sigma*x (Variance)
prob.Objective = x'*Sigma*x;
% Constraints
% Sum of weights equal to 1 (fully-invested)
prob.Constraints.sumToTau = sum(x) == 1;
% Solve problem
sol = solve(prob);
w = sol.x;
Sargondjani on 7 Feb 2023
I dont know how you modelled borrowing. Is it simply modelled as a risk free asset?
Anyway, you would have to set the lower bound for that specific asset to -2*Value of portfolio. Then you need to set the lower bound for the sum weights of all other asset to -1 (ie. you can short the risky portfolio once).

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