How to calculate minimum variance portfolio with constraints?

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]

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).

Answer 1

[Alejandra Pena-Ordieres]

Hello,

If you want to use a risk-free asset to leverage the portfolio, you can use the Portfolio object in MATLAB to achieve what you want by following this example Leverage in Portfolio Optimization with a Risk-Free Asset. In your case, to leverage up to the full value of your portfolio, you'd need to set your upper budget to 2.

% Leverage lower and upper bound
leverageLB = 0; % No leverage
leverageUB = 1; % Up to 100% leverage
p = setBudget(p,1+leverageLB,1+leverageUB);

Using the problem-based formulation that you show in your question, you'd need to modify the costraints of the optimization problem as follows

% Constraints
% No risk-free asset allocation allowed.
prob.Constraints.NoRiskFreeAllocation = sum(x) >= 1+leverageLB;
% Allow up to 100% leverage
prob.Constraints.LeverageUpperBound = sum(x) <= 1+leverageUB; 

If you want to allow a positive allocation of the risk-free asset, you'd have to decrese the lower budget to a value less than 1.

If you want to short-sell assets to leverage the portfolio, you'd need to use the setBounds method of the Portfolio object

% Allow assets to short-sell
p = setBounds(p,-1,2);
% Fully-invested portfolio
p = setBudget(1,1);

Using the problem-based formulation, you'd need to modify the lower and upper bounds of the variables

% Variables
% Portfolio weights
x = optimvar('x',nAssets,1,'LowerBound',-1,'UpperBound',2);