# estimateFrontierLimits

Estimate optimal portfolios at endpoints of efficient frontier

## Description

example

[pwgt,pbuy,psell] = estimateFrontierLimits(obj) estimates optimal portfolios at endpoints of efficient frontier for Portfolio, PortfolioCVaR, or PortfolioMAD objects. For details on the respective workflows when using these different objects, see Portfolio Object Workflow, PortfolioCVaR Object Workflow, and PortfolioMAD Object Workflow.

example

[pwgt,pbuy,psell] = estimateFrontierLimits(obj,Choice) estimates optimal portfolios at endpoints of efficient frontier with an additional option specified for the Choice argument.

## Examples

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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 = 3×2

0.3000    0.3000
0.7000         0
0    0.7000

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 = 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.

## Input Arguments

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Object for portfolio, specified using Portfolio, PortfolioCVaR, or PortfolioMAD object. For more information on creating a portfolio object, see

Data Types: object

(Optional) Indicator which portfolios to obtain at the extreme ends of the efficient frontier, specified as a character vector with values 'Both' or "Both", 'Min' or "Min", or 'Max' or "Max". The options for a Choice action are:

• [] — Compute both minimum-risk and maximum-return portfolios.

• 'Both' or "Both" — Compute both minimum-risk and maximum-return portfolios.

• 'Min' or "Min" — Compute minimum-risk portfolio only.

• 'Max' or "Max" — Compute maximum-return portfolio only.

Data Types: char | string

## Output Arguments

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Optimal portfolios at the endpoints of the efficient frontier TargetReturn, returned as a NumAssets-by-NumPorts matrix. pwgt is returned for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).

Purchases relative to an initial portfolio for optimal portfolios at the endpoints of the efficient frontier, returned as NumAssets-by-NumPorts matrix.

Note

If no initial portfolio is specified in obj.InitPort, that value is assumed to be 0 such that pbuy = max(0, pwgt) and psell = max(0, -pwgt).

pbuy is returned for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).

Sales relative to an initial portfolio for optimal portfolios on the efficient frontier, returned as a NumAssets-by-NumPorts matrix.

Note

If no initial portfolio is specified in obj.InitPort, that value is assumed to be 0 such that pbuy = max(0, pwgt) and psell = max(0, -pwgt).

psell is returned for Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).

## Tips

You can also use dot notation to estimate the optimal portfolios at the endpoints of the efficient frontier.