estimateFrontierByRisk

Estimate optimal portfolios with targeted portfolio risks

Syntax

[pwgt,pbuy,psell] = estimateFrontierByRisk(obj,TargetRisk)

[pwgt,pbuy,psell] = estimateFrontierByRisk(___,Name,Value)

Description

[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.

[pwgt,pbuy,psell] = estimateFrontierByRisk(___,Name,Value) adds name-optional name-value pair arguments for Portfolio or PortfolioMAD objects.

Examples

Obtain Portfolios with Targeted Portfolio Risks for a Portfolio Object

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

Use Portfolio Object to Solve for Portfolio Weights to Achieve Min/Max Objective

To obtain efficient portfolios for a Portfolio object that has targeted portfolio risks, estimateFrontierByRisk accepts one or more target portfolio risks and obtains efficient portfolios with the specified risks.

Load the stock price data and set portfolio constraints so that the total weight of the portfolio must equal 1 and the maximum allocation to for any one stock is 50%, while the minimum is 5%.

T = readtable('dowPortfolio.xlsx');

	Dates	DJI	AA	AIG	AXP	BA	C	CAT	DD	DIS	GE	GM	HD	HON	HPQ	IBM	INTC	JNJ	JPM	KO	MCD	MMM	MO	MRK	MSFT	PFE	PG	T	UTX	VZ	WMT	XOM
1	1/3/2006	1.0847e+04	28.7200	68.4100	51.5300	68.6300	45.2600	55.8600	40.6800	24.1800	33.6000	17.8200	39.7900	36.1400	28.3500	80.1300	24.5700	59.0800	37.7800	38.9800	32.7200	75.9300	52.2700	30.7300	26.1900	22.1600	56.3800	22.7000	54.9400	26.7900	44.9000	56.6400
2	1/4/2006	1.0880e+04	28.8900	68.5100	51.0300	69.3400	44.4200	57.2900	40.4600	23.7700	33.5600	18.3000	39.0500	35.9900	29.1800	80.0300	24.9000	59.9900	37.5600	38.9100	33.0100	75.5400	52.6500	31.0800	26.3200	22.8800	56.4800	22.8700	54.6100	27.5800	44.9900	56.7400
3	1/5/2006	1.0882e+04	29.1200	68.6000	51.5700	68.5300	44.6500	57.2900	40.3800	24.1900	33.4700	19.3400	38.6700	35.9700	28.9700	80.5600	25.2500	59.7400	37.6700	39.1000	33.0500	74.8500	52.5200	31.1300	26.3400	22.9000	56.3000	22.9200	54.4100	27.9000	44.3800	56.4500
4	1/6/2006	1.0959e+04	29.0200	68.8900	51.7500	67.5700	44.6500	58.4300	40.5500	24.5200	33.7000	19.6100	38.9600	36.5300	29.8000	82.9600	25.2800	60.0100	37.9400	39.4700	33.2500	75.4700	52.9500	31.0800	26.2600	23.1600	56.2400	23.2100	54.5800	28.0100	44.5600	57.5700
5	1/9/2006	1.1012e+04	29.3700	68.5700	53.0400	67.0100	44.4300	59.4900	40.3200	24.7800	33.6100	21.1200	39.3800	36.2300	30.1700	81.7600	25.4400	60.3800	38.5500	39.6600	33.8800	75.8400	53.1100	31.5800	26.2100	23.1600	56.6700	23.3000	55.2000	28.1200	44.4000	57.5400
6	1/10/2006	1.1012e+04	28.4400	69.1800	52.8800	67.3300	44.5700	59.2500	40.2000	25.0900	33.4300	20.7900	40.3300	36.1700	30.3300	82.1000	25.1000	60.4900	38.6100	39.7000	33.9100	75.3700	53.0400	31.2700	26.3500	22.7700	56.4500	23.1600	55.2400	28.2400	44.5400	57.9900
7	1/11/2006	1.1043e+04	28.0500	69.6000	52.5900	68.3000	44.9800	59.2800	38.8700	25.3300	33.6600	20.6100	41.4400	36.1900	30.8800	82.1900	25.1200	59.9100	38.5800	39.7200	34.5000	75.2200	53.3100	31.3900	26.6300	23.0600	56.6500	23.3400	54.4100	28.5800	45.2300	58.3800
8	1/12/2006	1.0962e+04	27.6800	69.0400	52.6000	67.9000	45.0200	60.1300	38.0200	25.4100	33.2500	19.7600	41.0500	35.7700	30.5700	81.6100	24.9600	59.6300	37.8700	39.5000	33.9600	74.5700	53.2300	31.4100	26.4800	22.9000	56.0200	23.2400	53.9000	28.6900	44.4300	57.7700
9	1/13/2006	1.0960e+04	27.8100	68.8400	52.5000	67.7000	44.9200	60.2400	37.8600	25.4700	33.3500	19.2000	40.4300	35.8500	31.4300	81.2200	24.7800	59.2600	37.8400	39.3700	33.6500	74.3800	53.2900	31.4000	26.5300	22.9900	56.4900	23.2700	54.1000	28.7500	44.1000	59.0600
10	1/17/2006	1.0896e+04	27.9700	67.8400	52.0300	66.9300	44.4700	60.8500	37.7500	25.1500	33.2000	18.6800	40.1100	35.5600	31.2000	81.0500	24.5200	58.7400	37.6400	39.1100	33.7700	73.9900	52.8500	31.1600	26.3400	22.6300	56.2500	23.1300	54.4100	28.1200	43.6600	59.6100
11	1/18/2006	1.0855e+04	27.8100	67.4200	51.8400	66.5800	44.4100	60.0400	37.5400	24.9700	33.0800	18.9800	40.4200	35.7100	31.2100	81.8300	21.7200	59.6100	37.2400	38.9100	34.1600	74.0700	52.8900	30.9900	26.1800	22.3600	56.5400	23.1100	54.0800	27.8300	43.8800	58.7800
12	1/19/2006	1.0881e+04	28.3300	66.9200	51.6600	66.4200	44.0200	60.6600	37.6900	26.0000	32.9500	19.1000	39.8300	35.8800	31.7700	81.1400	21.5300	59.6000	37.0300	39.0100	34.3600	73.8100	52.6800	31.2600	26.3600	23.2700	56.3900	23.1800	54.4100	28.0700	44.4900	59.5700
13	1/20/2006	1.0667e+04	27.6700	65.5500	50.4900	64.7900	41.9500	58.9800	37.3400	25.4900	31.7000	18.9000	38.7600	34.5700	31.2800	79.4500	20.9100	58.2800	36.0700	38.2100	35.0100	72.2100	52.1800	31.2000	25.7700	23.0300	55.7400	23.0100	53.4600	27.6400	43.7100	58.6300
14	1/23/2006	1.0689e+04	28.0300	65.4600	50.5300	65.3000	42.2400	59.2300	37.3700	25.2900	31.6300	20.6000	38.2900	34.7800	30.8800	79.5000	20.5200	58.6600	36.2800	38.6000	34.8600	72.6500	52.0900	30.8000	25.7100	23.1900	55.5500	22.7700	52.9400	27.6900	43.9500	59.2800
15	1/24/2006	1.0712e+04	28.1600	65.3900	51.8900	65.9200	42.2500	59.5300	37.0900	25.7600	31.3200	21.7300	39.0300	35.2500	30.9100	78.9500	20.4500	56.9000	36.1300	38.9800	35.0000	71.2100	51.7400	30.7600	25.6400	22.9100	55.7700	22.9600	54.8600	27.6000	44.4100	59.0500
16	1/25/2006	1.0710e+04	28.5700	64.6700	51.9700	65.1900	42.4500	60.2300	37.0500	25.2100	31.1300	22.4800	38.5700	34.7900	31.6400	79.0100	20.3800	56.0800	36.4800	39.2100	34.3200	70.0600	51.4900	31.1400	25.7600	23.1400	56.3500	23.4700	55.4000	28.0300	44.6500	58.3200
⋮

Setup a portfolio of ten stocks.

symbol = T.Properties.VariableNames(3:12)';
dailyReturn = tick2ret(T{:,3:12});
p = Portfolio('AssetList',symbol,'RiskFreeRate',0.01/252);

Estimate the mean and variance of portfolio based on daily returns using estimateAssetMoments.

p = estimateAssetMoments(p,dailyReturn);

Set the total weight of portfolio to 1 where the LowerBudget = 1 and the UpperBudget = 1.

p = setBudget(p,1,1)

p = 
  Portfolio with properties:

                       BuyCost: []
                      SellCost: []
                  RiskFreeRate: 3.9683e-05
                     AssetMean: [10×1 double]
                    AssetCovar: [10×10 double]
                 TrackingError: []
                  TrackingPort: []
                      Turnover: []
                   BuyTurnover: []
                  SellTurnover: []
                          Name: []
                     NumAssets: 10
                     AssetList: {'AA'  'AIG'  'AXP'  'BA'  'C'  'CAT'  'DD'  'DIS'  'GE'  'GM'}
                      InitPort: []
                   AInequality: []
                   bInequality: []
                     AEquality: []
                     bEquality: []
                    LowerBound: []
                    UpperBound: []
                   LowerBudget: 1
                   UpperBudget: 1
                   GroupMatrix: []
                    LowerGroup: []
                    UpperGroup: []
                        GroupA: []
                        GroupB: []
                    LowerRatio: []
                    UpperRatio: []
                  MinNumAssets: []
                  MaxNumAssets: []
    ConditionalBudgetThreshold: []
        ConditionalUpperBudget: []
                     BoundType: []

Set the maximum allocation for any one security to 50%, and set the minimum for any one security to 5%.

p.LowerBound = repmat(0.05,[10,1]);
p.UpperBound = repmat(0.5,[10,1]);

Calculate the portfolio weights for the maximum Sharpe ratio using estimateMaxSharpeRatio.

w1 = estimateMaxSharpeRatio(p)

w1 = 10×1

    0.0500
    0.0500
    0.0500
    0.0500
    0.0500
    0.0500
    0.0500
    0.4357
    0.0500
    0.1643

Estimate moments of portfolio returns using estimatePortMoments.

[risk1, ret1] = estimatePortMoments(p,w1)

risk1 = 
0.0090

ret1 = 
0.0013

Calculate portfolio weights given a target risk.

TargetRisk = 0.01

TargetRisk = 
0.0100

Use estimateFrontierByRisk to estimate optimal portfolios with targeted portfolio risks.

w3 = estimateFrontierByRisk(p,TargetRisk)

w3 = 10×1

    0.0500
    0.0500
    0.0500
    0.0500
    0.0500
    0.0500
    0.0500
    0.3419
    0.0500
    0.2581

Estimate moments of portfolio returns using estimatePortMoments, where risk3 is estimates for standard deviations of portfolio returns for each portfolio and ret3 is estimates for means of portfolio returns for each portfolio.

[risk3,ret3] = estimatePortMoments(p,w3)

risk3 = 
0.0100

ret3 = 
0.0013

Obtain Portfolios with Targeted Portfolio Risks for a Portfolio Object with BoundType, MinNumAsset, and MaxNumAsset Constraints

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

pbuy = 3×2

    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
                    IntMainSolverOptions: [1×1 optim.options.Intlinprog]
    NumIterationsEarlyIntegerConvergence: 30
                     ExtendedFormulation: 0
                            NumInnerCuts: 10
                     NumInitialOuterCuts: 10

Obtain Portfolios with Targeted Portfolio Risks for a Portfolio Object Using the Direct Method and Solver Options

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', ...
        'SpecifyObjectiveGradient', true, 'SpecifyConstraintGradient', true, ...
        '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

Obtain Portfolios with Targeted Portfolio Risks for a PortfolioCVaR Object

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.

Obtain Portfolios with Targeted Portfolio Risks for a PortfolioMAD Object

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

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

display(pwgt);

pwgt = 4×3

    0.1611         0         0
    0.4774    0.2137    0.0047
    0.1126    0.1384    0.1200
    0.2488    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.

Obtain Portfolios with Targeted Portfolio Risks for a PortfolioMAD Object Using the Direct Method and Solver Options

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

p = setSolver(p, 'fmincon', 'Display', 'off', 'Algorithm', 'sqp', ...
        'SpecifyObjectiveGradient', true, 'SpecifyConstraintGradient', true, ...
        '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    0.0000    0.0000
    0.4777    0.2139    0.0037
    0.1118    0.1381    0.1214
    0.2492    0.6480    0.8749

Input Arguments

obj — Object for portfolio

object

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

• Portfolio

• PortfolioCVaR

• PortfolioMAD

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

TargetRisk — Target values for portfolio risk

vector

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.

Example: [pwgt,pbuy,psell] = estimateFrontierByRisk(p,‘method’,‘direct’)

Method — Method to estimate frontier by risk for Portfolio or PortfolioMAD objects

'direct' (default) | character vector with value 'direct' or 'iterative'

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:

• 'direct' — Construct one optimization problem to maximize the portfolio return with target risk as the nonlinear constraint, and solve it directly using fmincon, instead of iteratively exploring the efficient frontier. For an example of using the 'direct' option, see Obtain Portfolios with Targeted Portfolio Risks for a Portfolio Object Using the Direct Method and Solver Options and Obtain Portfolios with Targeted Portfolio Risks for a PortfolioMAD Object Using the Direct Method and Solver Options.

• 'iterative' — One-dimensional optimization using fminbnd to find the portfolio return between min and max return on the frontier that minimizes the difference between the actual risk and target risk. Then the portfolio weights are obtained by solving a frontier by return problem. Consequently, the 'iterative' method is slower than the 'direct'.

Data Types: char

Output Arguments

pwgt — Optimal portfolios on efficient frontier with specified target risks

matrix

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).

pbuy — Purchases relative to initial portfolio for optimal portfolios on efficient frontier

matrix

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).

psell — Sales relative to initial portfolio for optimal portfolios on efficient frontier

matrix

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.

[pwgt,pbuy,psell] = obj.estimateFrontierByRisk(TargetRisk);

or

[pwgt,pbuy,psell] = obj.estimateFrontierByRisk(TargetRisk,Name,Value);

Version History

Introduced in R2011a