Given portfolio p, the estimateFrontierLimits function obtains the endpoint portfolios.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
      0.00408 0.0289 0.0204 0.0119;
      0.00192 0.0204 0.0576 0.0336;
      0 0.0119 0.0336 0.1225 ];
 
p = Portfolio;
p = setAssetMoments(p, m, C);
p = setDefaultConstraints(p);
pwgt = estimateFrontierLimits(p);

disp(pwgt);

    0.8891         0
    0.0369         0
    0.0404         0
    0.0336    1.0000

When any one, or any combination of the constraints from 'Conditional' BoundType, MinNumAssets, and MaxNumAssets are active, the portfolio problem is formulated as mixed integer programming problem and the MINLP solver is used.

Create a Portfolio object for three assets.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ];
AssetCovar = [ 0.00324625 0.00022983 0.00420395;
               0.00022983 0.00049937 0.00019247;
               0.00420395 0.00019247 0.00764097 ];  
p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar);
p = setDefaultConstraints(p);           

Use setBounds with semicontinuous constraints to set xi = 0 or 0.02 <= xi <= 0.5 for all i = 1,...NumAssets.

p = setBounds(p, 0.02, 0.7,'BoundType', 'Conditional', 'NumAssets', 3);                    

When working with a Portfolio object, the setMinMaxNumAssets function enables you to set up the limits on the number of assets invested (as known as cardinality) constraints. This sets the total number of allocated assets satisfying the Bound constraints that are between MinNumAssets and MaxNumAssets. By setting MinNumAssets = MaxNumAssets = 2, only two of the three assets are invested in the portfolio.

p = setMinMaxNumAssets(p, 2, 2);  

Use estimateFrontierLimits to estimate the optimal portfolios at endpoints of the efficient frontier.

[pwgt, pbuy, psell] = estimateFrontierLimits(p,'Both')

pwgt = 3×2

    0.3000    0.3000
    0.7000         0
         0    0.7000

pbuy = 3×2

    0.3000    0.3000
    0.7000         0
         0    0.7000

psell = 3×2

     0     0
     0     0
     0     0

The estimateFrontierLimits function uses the MINLP solver to solve this problem. Use the setSolverMINLP function to configure the SolverType and options.

p.solverTypeMINLP

ans = 
'OuterApproximation'

p.solverOptionsMINLP

ans = struct with fields:
                           MaxIterations: 1000
                    AbsoluteGapTolerance: 1.0000e-07
                    RelativeGapTolerance: 1.0000e-05
                  NonlinearScalingFactor: 1000
                  ObjectiveScalingFactor: 1000
                                 Display: 'off'
                           CutGeneration: 'basic'
                MaxIterationsInactiveCut: 30
                      ActiveCutTolerance: 1.0000e-07
                  IntMasterSolverOptions: [1x1 optim.options.Intlinprog]
    NumIterationsEarlyIntegerConvergence: 30

Given portfolio p, the estimateFrontierLimits function obtains the endpoint portfolios.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
    0.00408 0.0289 0.0204 0.0119;
    0.00192 0.0204 0.0576 0.0336;
    0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;

rng(11);

AssetScenarios = mvnrnd(m, C, 20000);

p = PortfolioCVaR;
p = setScenarios(p, AssetScenarios);
p = setDefaultConstraints(p);
p = setProbabilityLevel(p, 0.95);


pwgt = estimateFrontierLimits(p);

disp(pwgt);

    0.8451         0
    0.0613         0
    0.0451         0
    0.0485    1.0000

The function rng($$seed$$) resets the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.

Given portfolio p, the estimateFrontierLimits function obtains the endpoint portfolios.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
    0.00408 0.0289 0.0204 0.0119;
    0.00192 0.0204 0.0576 0.0336;
    0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;

rng(11);

AssetScenarios = mvnrnd(m, C, 20000);

p = PortfolioMAD;
p = setScenarios(p, AssetScenarios);
p = setDefaultConstraints(p);

pwgt = estimateFrontierLimits(p);

disp(pwgt);

    0.8817         0
    0.0431         0
    0.0387         0
    0.0366    1.0000

The function rng($$seed$$) resets the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.