## Working with 'Conditional'BoundType, MinNumAssets, and MaxNumAssets Constraints Using PortfolioCVaR Objects

When any one, or any combination of 'Conditional' BoundType, MinNumAssets, or MaxNumAssets constraints are active, the portfolio problem is formulated by adding NumAssets binary variables, where 0 indicates not invested, and 1 is invested. For example, to explain the 'Conditional' BoundType and MinNumAssets and MaxNumAssets constraints, assume that your portfolio has a universe of 100 assets that you want to invest:

• 'Conditional' BoundType (also known as semicontinuous constraints), set by setBounds, is often used in situations where you do not want to invest small values. A standard example is a portfolio optimization problem where many small allocations are not attractive because of transaction costs. Instead, you prefer fewer instruments in the portfolio with larger allocations. This situation can be handled using'Conditional' BoundType constraints for a PortfolioCVaR object.

For example, the weight you invest in each asset is either 0 or between [0.01, 0.5]. Generally, a semicontinuous variable x is a continuous variable between bounds [lb, ub] that also can assume the value 0, where lb > 0, lbub. Applying this to portfolio optimization requires that very small or large positions should be avoided, that is values that fall in (0, lb) or are more than ub.

• MinNumAssets and MaxNumAssets (also known as cardinality constraints), set by setMinMaxNumAssets, limit the number of assets in a PortfolioCVaR object. For example, if you have 100 assets in your portfolio and you want the number of assets allocated in the portfolio to be from 40 through 60. Using MinNumAssets and MaxNumAssets you can limit the number of assets in the optimized portfolio, which allows you to limit transaction and operational costs or to create an index tracking portfolio.

### Setting 'Conditional'BoundType Constraints Using the setBounds Function

Use setBounds with a 'conditional' BoundType to set xi = 0 or 0.02 <= xi <= 0.5 for all i=1,...NumAssets:

p = PortfolioCVaR;
p = setBounds(p, 0.02, 0.5,'BoundType', 'Conditional', 'NumAssets', 3)
p =

PortfolioCVaR with properties:

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

### Setting the Limits on the Number of Assets Invested Using the setMinMaxNumAssets Function

You can also set the MinNumAssets and MaxNumAssets properties to define a limit on the number of assets invested using setMinMaxNumAssets. For example, by setting MinNumAssets=MaxNumAssets=2, only two of the three assets are invested in the portfolio.

p = PortfolioCVaR;
p = setBounds(p, 0.02, 0.5,'BoundType', 'Conditional', 'NumAssets', 3)
p = setMinMaxNumAssets(p, 2, 2)
p =

PortfolioCVaR with properties:

SellCost: []
RiskFreeRate: []
ProbabilityLevel: []
Turnover: []
SellTurnover: []
NumScenarios: []
Name: []
NumAssets: 3
AssetList: []
InitPort: []
AInequality: []
bInequality: []
AEquality: []
bEquality: []
LowerBound: [3×1 double]
UpperBound: [3×1 double]
LowerBudget: []
UpperBudget: []
GroupMatrix: []
LowerGroup: []
UpperGroup: []
GroupA: []
GroupB: []
LowerRatio: []
UpperRatio: []
MinNumAssets: 2
MaxNumAssets: 2
BoundType: [3×1 categorical]