# estimateFrontierByRisk

Estimate optimal portfolios with targeted portfolio risks

## Description

example

[pwgt,pbuy,psell] = estimateFrontierByRisk(obj,TargetRisk) estimates optimal portfolios with targeted portfolio risks 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

## Examples

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To obtain efficient portfolios that have targeted portfolio risks, the estimateFrontierByRisk function accepts one or more target portfolio risks and obtains efficient portfolios with the specified risks. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio risks of 12%, 14%, and 16%. This example uses the default 'direct' method to estimate the optimal portfolios with targeted portfolio risks.

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 = estimateFrontierByRisk(p, [0.12, 0.14, 0.16]);

display(pwgt);
pwgt = 4×3

0.3984    0.2659    0.1416
0.3064    0.3791    0.4474
0.0882    0.1010    0.1131
0.2071    0.2540    0.2979

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 estimateFrontierByRisk to estimate optimal portfolios with targeted portfolio risks.

[pwgt, pbuy, psell] = estimateFrontierByRisk(p,[0.0324241, 0.0694534 ])
pwgt = 3×2

0.0000    0.5000
0.6907    0.0000
0.3093    0.5000

0.0000    0.5000
0.6907    0.0000
0.3093    0.5000

psell = 3×2

0     0
0     0
0     0

The estimateFrontierByRisk 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

To obtain efficient portfolios that have targeted portfolio risks, the estimateFrontierByRisk function accepts one or more target portfolio risks and obtains efficient portfolios with the specified risks. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio risks of 12%, 14%, and 16%. This example uses the default'direct' method to estimate the optimal portfolios with targeted portfolio risks. The 'direct' method uses fmincon to solve the optimization problem that maximizes portfolio return, subject to the target risk as the quadratic nonlinear constraint. setSolver specifies the solverType and SolverOptions for fmincon.

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);

p = setSolver(p, 'fmincon', 'Display', 'off', 'Algorithm', 'sqp', ...
'ConstraintTolerance', 1.0e-8, 'OptimalityTolerance', 1.0e-8, 'StepTolerance', 1.0e-8);

pwgt = estimateFrontierByRisk(p, [0.12, 0.14, 0.16]);

display(pwgt);
pwgt = 4×3

0.3984    0.2659    0.1416
0.3064    0.3791    0.4474
0.0882    0.1010    0.1131
0.2071    0.2540    0.2979

To obtain efficient portfolios that have targeted portfolio risks, the estimateFrontierByRisk function accepts one or more target portfolio risks and obtains efficient portfolios with the specified risks. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio risks of 12%, 20%, and 30%.

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 ];

rng(11);

p = PortfolioCVaR;
p = simulateNormalScenariosByMoments(p, m, C, 2000);
p = setDefaultConstraints(p);
p = setProbabilityLevel(p, 0.95);

pwgt = estimateFrontierByRisk(p, [0.12, 0.20, 0.30]);

display(pwgt);
pwgt = 4×3

0.5363    0.1387         0
0.2655    0.4991    0.3830
0.0570    0.1239    0.1461
0.1412    0.2382    0.4709

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.

To obtain efficient portfolios that have targeted portfolio risks, the estimateFrontierByRisk function accepts one or more target portfolio risks and obtains efficient portfolios with the specified risks. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio risks of 12%, 20%, and 25%. This example uses the default 'direct' method to estimate the optimal portfolios with targeted portfolio risks.

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 ];

rng(11);

p = simulateNormalScenariosByMoments(p, m, C, 2000);
p = setDefaultConstraints(p);

pwgt = estimateFrontierByRisk(p, [0.12, 0.20, 0.25]);

display(pwgt);
pwgt = 4×3

0.1610         0         0
0.4784    0.2137    0.0047
0.1116    0.1384    0.1200
0.2490    0.6480    0.8753

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.

To obtain efficient portfolios that have targeted portfolio risks, the estimateFrontierByRisk function accepts one or more target portfolio risks and obtains efficient portfolios with the specified risks. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio risks of 12%, 20%, and 25%. This example uses the default 'direct' method to estimate the optimal portfolios with targeted portfolio risks. The 'direct' method uses fmincon to solve the optimization problem that maximizes portfolio return, subject to the target risk as the quadratic nonlinear constraint. setSolver specifies the solverType and SolverOptions for fmincon.

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 ];

rng(11);

p = simulateNormalScenariosByMoments(p, m, C, 2000);
p = setDefaultConstraints(p);

p = setSolver(p, 'fmincon', 'Display', 'off', 'Algorithm', 'sqp', ...
'ConstraintTolerance', 1.0e-8, 'OptimalityTolerance', 1.0e-8, 'StepTolerance', 1.0e-8);
plotFrontier(p);

pwgt = estimateFrontierByRisk(p, [0.12 0.20, 0.25]);

display(pwgt);
pwgt = 4×3

0.1613         0         0
0.4777    0.2139    0.0037
0.1118    0.1381    0.1214
0.2492    0.6480    0.8749

## 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

Note

If no initial portfolio is specified in obj.InitPort, it is assumed to be 0 so that pbuy = max(0,pwgt) and psell = max(0,-pwgt). For more information on setting an initial portfolio, see setInitPort.

Data Types: object

Target values for portfolio risk, specified as a NumPorts vector.

Note

If any TargetRisk values are outside the range of risks for efficient portfolios, the target risk is replaced with the minimum or maximum efficient portfolio risk, depending on whether the target risk is below or above the range of efficient portfolio risks.

Data Types: double

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Method to estimate frontier by risk for Portfolio or PortfolioMAD objects, specified as the comma-separated pair consisting of 'Method' and a character vector with one of the following values:

Data Types: char

## Output Arguments

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Optimal portfolios on the efficient frontier with specified target returns from TargetRisk, 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 on 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 optimal portfolios with targeted portfolio risks.