Conditional Bounds with LowerBound Defined as Empty or Zero

When using setBounds with the BoundType set to 'Conditional' and the LowerBound input argument is empty ([ ]) or 0, the Conditional bound is not effective and is equivalent to a Simple bound.

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, 'Budget', 1);
p = setBounds(p, 0, 0.5, 'BoundType', 'Conditional');  
p = setMinMaxNumAssets(p, 3, 3);
estimateFrontier(p, 10)

Warning: Conditional bounds with 'LowerBound' as zero are equivalent to simple bounds.
Consider either using strictly positive 'LowerBound' or 'simple' as the 'BoundType'
instead. 
> In internal.finance.PortfolioMixedInteger/checkBoundType (line 46)
  In Portfolio/checkarguments (line 204)
  In Portfolio/setBounds (line 80) 
Warning: The solution may have less than 'MinNumAssets' assets with nonzero weight. To
enforce 'MinNumAssets' requirement, set strictly positive lower conditional bounds. 
> In internal.finance.PortfolioMixedInteger/hasIntegerConstraints (line 44)
  In Portfolio/estimateFrontier (line 51) 

ans =

  Columns 1 through 8

    0.5000    0.3555    0.3011    0.3299    0.3585    0.3873    0.4160    0.4448
    0.5000    0.5000    0.4653    0.3987    0.3322    0.2655    0.1989    0.1323
    0.0000    0.1445    0.2335    0.2714    0.3093    0.3472    0.3850    0.4229

  Columns 9 through 10

    0.4735    0.5000
    0.0657         0
    0.4608    0.5000

In all the 10 optimal allocations, there are allocations (the first and last ones) that only have two assets, which is in conflict with the MinNumAssets constraint that three assets should be allocated. Also there are two warnings, which actually explain what happens. In this case, the 'Conditional' bound constraints are defined as xi = 0 or 0 <= xi <= 0.5, which are internally modeled as 0*vi<=xi<=0.5*vi, where vi is 0 or 1, where 0 indicates not allocated, and 1 indicates allocated. Here, vi=1, which still allows the asset to have a weight of 0. In other words, setting LowerBound as 0 or empty, doesn’t clearly define the minimum allocation for an allocated asset. Therefore, a 0 weighted asset is also considered as an allocated asset. To fix this warning, follow the instructions in the warning message, and set a LowerBound value that is strictly positive.

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, 'Budget', 1);
p = setBounds(p, 0.3, 0.5, 'BoundType', 'Conditional');  
p = setMinMaxNumAssets(p, 3, 3);
estimateFrontier(p, 10)

ans =

  Columns 1 through 8

    0.3000    0.3180    0.3353    0.3489    0.3580    0.3638    0.3694    0.3576
    0.4000    0.3820    0.3642    0.3479    0.3333    0.3199    0.3067    0.3001
    0.3000    0.3000    0.3005    0.3032    0.3088    0.3163    0.3240    0.3423

  Columns 9 through 10

    0.3289    0.3000
    0.3000    0.3000
    0.3711    0.4000

Length of 'BoundType' Must Be Conformable with NumAssets

The setBounds optional name-value argument for 'BoundType' must be defined for all assets in a Portfolio, PortfolioCVaR, or PortfolioMAD object. By default, the 'BoundType' is 'Simple' and applies to all assets. Using setBounds, you can choose to define a 'BoundType' for each asset. In this case, the number of 'BoundType' specifications must match the number of assets (NumAssets) in the Portfolio, PortfolioCVaR, or PortfolioMAD object. The following example demonstrates the error when the number of 'BoundType' specifications do not match the number of assets in the Portfolio object.

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, 'Budget', 1);
p = setBounds(p, 0.1, 0.5, 'BoundType',["simple"; "conditional"])

Cannot create bound constraints.

Caused by:
    Error using internal.finance.PortfolioMixedInteger/checkBoundType (line 28)
    Length of 'BoundType' must be conformable with 'NumAssets'=3.

To correct this, modify the BoundType to include three specifications because the Portfolio object has 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, 'Budget', 1);
p = setBounds(p, 0.1, 0.5, 'BoundType',["simple"; "conditional";"conditional"])
p.BoundType

p = 

  Portfolio with properties:

          BuyCost: []
         SellCost: []
     RiskFreeRate: []
        AssetMean: [3×1 double]
       AssetCovar: [3×3 double]
    TrackingError: []
     TrackingPort: []
         Turnover: []
      BuyTurnover: []
     SellTurnover: []
             Name: []
        NumAssets: 3
        AssetList: []
         InitPort: []
      AInequality: []
      bInequality: []
        AEquality: []
        bEquality: []
       LowerBound: [3×1 double]
       UpperBound: [3×1 double]
      LowerBudget: 1
      UpperBudget: 1
      GroupMatrix: []
       LowerGroup: []
       UpperGroup: []
           GroupA: []
           GroupB: []
       LowerRatio: []
       UpperRatio: []
        BoundType: [3×1 categorical]
     MinNumAssets: []
     MaxNumAssets: []

ans = 

  3×1 categorical array

     simple 
     conditional 
     conditional 

Redundant Constraints from 'BoundType', 'MinNumAssets', 'MaxNumAssets' Constraints

When none of the constraints from 'BoundType', 'MinNumAssets', or 'MaxNumAssets' are active, the redundant constraints from 'BoundType', 'MinNumAssets', 'MaxNumAssets' warning occurs. This happens when you explicitly use setBounds and setMinMaxNumAssets but with values that are inactive. That is, the 'Conditional' BoundType has a LowerBound = [ ] or 0, 'MinNumAssets' is 0, or 'MaxNumAssets' is the same value as NumAssets. In other words, if any of these three are active, the warning will not show up when using the estimate functions or plotFrontier. The following two examples show the rationale.

The first example is when the BoundType is explicitly set as 'Conditional' but the LowerBound is 0, and no 'MinNumAssets' and 'MaxNumAssets' constraints are defined using setMinMaxNumAssets.

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, 'Budget', 1);
p = setBounds(p, 0, 0.5, 'BoundType', 'Conditional');   
estimateFrontier(p, 10)

Warning: Redundant constraints from 'BoundType', 'MinNumAssets', 'MaxNumAssets'. 
> In internal.finance.PortfolioMixedInteger/hasIntegerConstraints (line 24)
  In Portfolio/estimateFrontier (line 51) 

ans =

  Columns 1 through 8

    0.5000    0.3555    0.3011    0.3299    0.3586    0.3873    0.4160    0.4448
    0.5000    0.5000    0.4653    0.3987    0.3321    0.2655    0.1989    0.1323
         0    0.1445    0.2335    0.2714    0.3093    0.3471    0.3850    0.4229

  Columns 9 through 10

    0.4735    0.5000
    0.0657         0
    0.4608    0.5000

The second example is when you explicitly set the three constraints, but all with inactive values. In this example, the BoundType is 'Conditional' and the LowerBound is 0, thus specifying ineffective 'Conditional' BoundType constraints, and the 'MinNumAssets' and 'MaxNumAssets' values are 0 and 3, respectively. The setMinMaxNumAssets function specifies ineffective 'MinNumAssets' and 'MaxNumAssets' constraints.

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, 'Budget', 1);
p = setBounds(p, 0, 0.5, 'BoundType', 'Conditional');   
p = setMinMaxNumAssets(p, 0, 3);   
estimateFrontier(p, 10)

Warning: Redundant constraints from 'BoundType', 'MinNumAssets', 'MaxNumAssets'. 
> In internal.finance.PortfolioMixedInteger/hasIntegerConstraints (line 24)
  In Portfolio/estimateFrontier (line 51) 

ans =

  Columns 1 through 8

    0.5000    0.3555    0.3011    0.3299    0.3586    0.3873    0.4160    0.4448
    0.5000    0.5000    0.4653    0.3987    0.3321    0.2655    0.1989    0.1323
         0    0.1445    0.2335    0.2714    0.3093    0.3471    0.3850    0.4229

  Columns 9 through 10

    0.4735    0.5000
    0.0657         0
    0.4608    0.5000

Infeasible Problem with Conditional (Semi-Continuous) Bounds, Cardinality Constraints, or Conditional Budget Constraints

The Portfolio, PortfolioCVaR, or PortfolioMAD object performs validations of all the constraints before solving any specific optimization problems. The Portfolio, PortfolioCVaR, or PortfolioMAD object first considers all constraints other than conditional bounds, cardinality constraints, and conditional budget constraints and issues an error message if they are not compatible. Then the Portfolio, PortfolioCVaR, or PortfolioMAD object adds the remaining constraints to check if they are compatible with the already checked constraints. This separation is natural because conditional bounds, cardinality constraints, and conditional budget constraints require additional binary variables in the mathematical formulation that leads to an MINLP, while other constraints only need continuous variables. You can follow the error messages to check when the infeasible problem occurs and take actions to fix the constraints.

One possible scenario is when the BoundType is 'Conditional' and Groups are defined for the Portfolio object. In this case, the Group definitions are themselves in conflict. Consequently, the 'Conditional' bound constraint cannot be applied when running estimateFrontierLimits.

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, 'Budget', 1);
p = setBounds(p, 0.1, 0.5, 'BoundType','Conditional');
p = setGroups(p, [1,1,0], 0.3, 0.5);
p = addGroups(p, [0,1,0], 0.6, 0.7);
pwgt = estimateFrontierLimits(p)

Error using Portfolio/buildMixedIntegerProblem (line 31)
Infeasible problem prior to considering constraints involving integer variables. 
Verify compatibility of continuous constraints.

Error in Portfolio/estimateFrontierLimits>int_frontierLimits (line 93)
ProbStruct = buildMixedIntegerProblem(obj);

Error in Portfolio/estimateFrontierLimits (line 73)
    pwgt = int_frontierLimits(obj, minsolution, maxsolution);

To correct this error, change the LowerGroup in the addGroups function to also be 0.3 to match the GroupMatrix input from setGroups.

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, 'Budget', 1);
p = setBounds(p, 0.1, 0.5, 'BoundType','Conditional');
p = setGroups(p, [1,1,0], 0.3, 0.5);
p = addGroups(p, [0,1,0], 0.3, 0.7);
pwgt = estimateFrontierLimits(p)

pwgt =

         0    0.2000
    0.5000    0.3000
    0.5000    0.5000

A second possible scenario is when the BoundType is 'Conditional' and the setEquality function is used with the bEquality parameter set to 0.04. This sets an equality constraint to have x1 + x3 = 0.04. At the same time, setBounds also set the semicontinuous constraints to have xi = 0 or 0.1 <= xi <= 2.5, which lead to x1 + x3 = 0 or 0.1 <= x1 + x3 <= 5. The semicontinuous constraints are not compatible with the equality constraint because there is no way to get x1 + x3 to equal 0.04. Therefore, the error message is displayed.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ];
AssetCovar = [ 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('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1);
A = [ 1 0 1 0 ];
b = 0.04;
p = setEquality(p, A, b);
p = setBounds(p, 0.1, 2.5, 'BoundType','Conditional');
p = setMinMaxNumAssets(p, 2, 2);  
pwgt = estimateFrontierLimits(p)

Error using Portfolio/buildMixedIntegerProblem (line 109)
Infeasible problem when considering constraints involving integer variables. 
Verify constraints compatibility or decrease 'AbsoluteGapTolerance' or 'RelativeGapTolerance'.

Error in Portfolio/estimateFrontierLimits>int_frontierLimits (line 93)
ProbStruct = buildMixedIntegerProblem(obj);

Error in Portfolio/estimateFrontierLimits (line 73)
    pwgt = int_frontierLimits(obj, minsolution, maxsolution);
 

To correct this error, change the bEquality parameter from 0.04 to .4.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ];
AssetCovar = [ 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('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1);
A = [ 1 0 1 0 ];
b = 0.4;
p = setEquality(p, A, b);
p = setBounds(p, 0.1, 2.5, 'BoundType','Conditional');
p = setMinMaxNumAssets(p, 2, 2);  
pwgt = estimateFrontierLimits(p)

pwgt =

    0.4000         0
    0.6000         0
         0    0.4000
         0    0.6000

Redundant Conditional Budget Constraint

This redundant conditional budget constraint warning occurs when you are using setConditionalBudget with a Portfolio, PortfolioCVaR, or PortfolioMAD object and the conditional threshold of all assets is greater than or equal to their respective upper bounds. This happens when the UpperBound ≤ ConditionalBudgetThreshold. In this case, assets are not allowed to invest more than the conditional threshold, which makes the conditional budget constraint unnecessary. The following example shows the case where this happens:

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 = setBounds(p,0,0.5);
p = setConditionalBudget(p,0.5,0.75);   
estimateFrontier(p)

Warning: Redundant conditional budget constraint. 
> In internal.finance.PortfolioMixedInteger/hasIntegerConstraints (line 59)
In Portfolio/estimateFrontier (line 54) 

ans =

         0    0.0454    0.0909    0.1825    0.3186    0.4547    0.5000    0.5000    0.5000    0.5000
         0    0.2008    0.4015    0.5000    0.5000    0.5000    0.5000    0.5000    0.5000    0.5000
         0    0.0169    0.0338    0.0491    0.0629    0.0766    0.1574    0.2716    0.3858    0.5000

To correct this warning, use setBounds to specify an UpperBound value that is > than the ConditionalBudgetThreshold that you specify using setConditionalBudget.

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 = setBounds(p,0,0.55);
p = setConditionalBudget(p,0.5,0.75);   
estimateFrontier(p)

ans =

    0.0000    0.0477    0.0953    0.1799    0.3226    0.4653    0.5000    0.5000    0.5000    0.5000
    0.0001    0.2107    0.4212    0.5500    0.5500    0.5500    0.5500    0.5500    0.5500    0.5000
    0.0000    0.0177    0.0355    0.0519    0.0663    0.0808    0.1749    0.2947    0.4144    0.5500

Unbounded Portfolio Problem

This error occurs when you are using a Portfolio, PortfolioCVaR, or PortfolioMAD object and there is no UpperBound defined in setBounds and you are using setMinMaxNumAssets. In this case, this is formulated as a mixed integer programming problem and an UpperBound is required to enforce MinNumAssets and MaxNumAssets constraints.

The optimizer first attempts to estimate the upper bound of each asset, based on all the specified constraints. If the UpperBound cannot be found, an error message occurs which instructs you to set an explicit UpperBound. In most cases, as long as you set some upper bounds to the problem using any set function, the optimizer can successfully find a good estimation.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ];
AssetCovar = [ 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('AssetMean', AssetMean, 'AssetCovar', AssetCovar);
p = setBounds(p, 0.1, 'BoundType','Conditional');
p = setGroups(p, [1,1,1,0], 0.3, 0.5);
p = setMinMaxNumAssets(p, 3, 3);
pwgt = estimateFrontierLimits(p)

Error using Portfolio/buildMixedIntegerProblem (line 42)
Unbounded portfolio problem. Upper bounds cannot be inferred from the existing
constraints. Set finite upper bounds using 'setBounds'.

Error in Portfolio/estimateFrontierLimits>int_frontierLimits (line 93)
ProbStruct = buildMixedIntegerProblem(obj);

Error in Portfolio/estimateFrontierLimits (line 73)
    pwgt = int_frontierLimits(obj, minsolution, maxsolution);

To correct this error, specify an UpperBound value for setBounds.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ];
AssetCovar = [ 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('AssetMean', AssetMean, 'AssetCovar', AssetCovar);
p = setBounds(p, 0.1, .9, 'BoundType','Conditional');
p = setGroups(p, [1,1,1,0], 0.3, 0.5);
p = setMinMaxNumAssets(p, 3, 3);
pwgt = estimateFrontierLimits(p)

pwgt =

    0.1000         0
    0.1000    0.1000
    0.1000    0.4000
         0    0.9000

Total Number of Portfolio Weights with a Value > 0 Are Greater Than MaxNumAssets Specified

When using a Portfolio, PortfolioCVaR, or PortfolioMAD object, the optimal allocation w may contain some very small values that leads to sum(w>0) larger than MaxNumAssets, even though the MaxNumAssets constraint is specified using setMinMaxNumAssets. For example, in the following code when setMinMaxNumAssets is used to set MaxNumAssets to 15, the sum(w>0) indicates that there are 19 assets. A close examination of the weights shows that the weights are extremely small and are actually 0.

T = readtable('dowPortfolio.xlsx');
symbol = T.Properties.VariableNames(3:end);
assetReturn = tick2ret(T{:,3:end});
p = Portfolio('AssetList', symbol, 'budget', 1);
p = setMinMaxNumAssets(p, 10, 15);
p = estimateAssetMoments(p,assetReturn);
p = setBounds(p,0.01,0.5,'BoundType','Conditional','NumAssets',30);
p = setTrackingError(p,0.05,ones(1, p.NumAssets)/p.NumAssets);
w = estimateFrontierLimits(p,'min'); % minimum risk portfolio

sum(w>0)   % Number of assets that are allocated in the optimal portfolio
w(w<eps)   % Check the weights of the very small weighted assets

ans =

    19


ans =

   1.0e-20 *

   -0.0000
         0
         0
    0.0293
         0
    0.3626
    0.2494
         0
    0.0926
   -0.0000
         0
    0.0020
         0
         0
         0
         0

This situation only happens when the OuterApproximation algorithm is used with setSolverMINLP to solve a MINLP portfolio optimization problem. The OuterApproximation internally fixes the latest solved integer variables and runs an NLP with quadprog or fmincon, which introduces numerical issues and leads to weights that are very close to 0.

If you do not want to deal with very small values, you can use setSolverMINLP to select a different algorithm. In this example, the 'TrustRegionCP' algorithm is specified.

T = readtable('dowPortfolio.xlsx');
symbol = T.Properties.VariableNames(3:end);
assetReturn = tick2ret(T{:,3:end});
p = Portfolio('AssetList', symbol, 'budget', 1);
p = setMinMaxNumAssets(p, 10, 15);
p = estimateAssetMoments(p,assetReturn);
p = setBounds(p,0.01,0.5,'BoundType','Conditional','NumAssets',30);
p = setTrackingError(p,0.05,ones(1, p.NumAssets)/p.NumAssets);
p = setSolverMINLP(p,'TrustRegionCP');
w = estimateFrontierLimits(p,'min'); % minimum risk portfolio 

sum(w>0)   % Number of assets that are allocated in the optimal portfolio
w(w<eps)   % The weights of the very small weighted assets are strictly zeros

ans =

    14


ans =

     0
     0
     0
     0
     0
     0
     0
     0
     0
     0
     0
     0
     0
     0
     0
     0