# Documentation

## Working with CVaR Portfolio Constraints

The final element for a complete specification of a portfolio optimization problem is the set of feasible portfolios, which is called a portfolio set. A portfolio set $X\subset {R}^{n}$ is specified by construction as the intersection of sets formed by a collection of constraints on portfolio weights. A portfolio set necessarily and sufficiently must be a nonempty, closed, and bounded set.

When setting up your portfolio set, ensure that the portfolio set satisfies these conditions. The most basic or "default" portfolio set requires portfolio weights to be nonnegative (using the lower-bound constraint) and to sum to `1` (using the budget constraint). For information on the workflow when using PortfolioCVaR objects, see PortfolioCVaR Object Workflow.

### Setting Default Constraints for Portfolio Weights Using PortfolioCVaR Object

The "default" CVaR portfolio problem has two constraints on portfolio weights:

• Portfolio weights must be nonnegative.

• Portfolio weights must sum to `1`.

Implicitly, these constraints imply that portfolio weights are no greater than `1`, although this is a superfluous constraint to impose on the problem.

#### Setting Default Constraints Using the PortfolioCVaR Function

Given a portfolio optimization problem with `NumAssets` = `20` assets, use the `PortfolioCVaR` function to set up a default problem and explicitly set bounds and budget constraints:

```p = PortfolioCVaR('NumAssets', 20, 'LowerBound', 0, 'Budget', 1); disp(p);```
``` PortfolioCVaR with properties: BuyCost: [] SellCost: [] RiskFreeRate: [] ProbabilityLevel: [] Turnover: [] BuyTurnover: [] SellTurnover: [] NumScenarios: [] Name: [] NumAssets: 20 AssetList: [] InitPort: [] AInequality: [] bInequality: [] AEquality: [] bEquality: [] LowerBound: [20x1 double] UpperBound: [] LowerBudget: 1 UpperBudget: 1 GroupMatrix: [] LowerGroup: [] UpperGroup: [] GroupA: [] GroupB: [] LowerRatio: [] UpperRatio: []```

#### Setting Default Constraints Using the setDefaultConstraints Function

An alternative approach is to use the `setDefaultConstraints` function. If the number of assets is already known in a PortfolioCVaR object, use `setDefaultConstraints` with no arguments to set up the necessary bound and budget constraints. Suppose that you have 20 assets to set up the portfolio set for a default problem:

```p = PortfolioCVaR('NumAssets', 20); p = setDefaultConstraints(p); disp(p);```
``` PortfolioCVaR with properties: BuyCost: [] SellCost: [] RiskFreeRate: [] ProbabilityLevel: [] Turnover: [] BuyTurnover: [] SellTurnover: [] NumScenarios: [] Name: [] NumAssets: 20 AssetList: [] InitPort: [] AInequality: [] bInequality: [] AEquality: [] bEquality: [] LowerBound: [20x1 double] UpperBound: [] LowerBudget: 1 UpperBudget: 1 GroupMatrix: [] LowerGroup: [] UpperGroup: [] GroupA: [] GroupB: [] LowerRatio: [] UpperRatio: []```

If the number of assets is unknown, `setDefaultConstraints` accepts `NumAssets` as an optional argument to form a portfolio set for a default problem. Suppose that you have 20 assets:

```p = PortfolioCVaR; p = setDefaultConstraints(p, 20); disp(p);```
``` PortfolioCVaR with properties: BuyCost: [] SellCost: [] RiskFreeRate: [] ProbabilityLevel: [] Turnover: [] BuyTurnover: [] SellTurnover: [] NumScenarios: [] Name: [] NumAssets: 20 AssetList: [] InitPort: [] AInequality: [] bInequality: [] AEquality: [] bEquality: [] LowerBound: [20x1 double] UpperBound: [] LowerBudget: 1 UpperBudget: 1 GroupMatrix: [] LowerGroup: [] UpperGroup: [] GroupA: [] GroupB: [] LowerRatio: [] UpperRatio: []```

### Working with Bound Constraints Using PortfolioCVaR Object

Bound constraints are optional linear constraints that maintain upper and lower bounds on portfolio weights (see Bound Constraints). Although every CVaR portfolio set must be bounded, it is not necessary to specify a CVaR portfolio set with explicit bound constraints. For example, you can create a CVaR portfolio set with an implicit upper bound constraint or a CVaR portfolio set with average turnover constraints. The bound constraints have properties `LowerBound` for the lower-bound constraint and `UpperBound` for the upper-bound constraint. Set default values for these constraints using the `setDefaultConstraints` function (see Setting Default Constraints for Portfolio Weights Using PortfolioCVaR Object).

#### Setting Bounds Using the PortfolioCVaR Function

The properties for bound constraints are set through the `PortfolioCVaR` function. Suppose that you have a balanced fund with stocks that can range from 50 to 75% of your portfolio and bonds that can range from 25 to 50% of your portfolio. The bound constraints for a balanced fund are set with:

```lb = [ 0.5; 0.25 ]; ub = [ 0.75; 0.5 ]; p = PortfolioCVaR('LowerBound', lb, 'UpperBound', ub); disp(p.NumAssets); disp(p.LowerBound); disp(p.UpperBound);```
```2 0.5000 0.2500 0.7500 0.5000```

To continue with this example, you must set up a budget constraint. For details, see Working with Budget Constraints Using PortfolioCVaR Object.

#### Setting Bounds Using the setBounds Function

You can also set the properties for bound constraints using `setBounds`. Suppose that you have a balanced fund with stocks that can range from 50% to 75% of your portfolio and bonds that can range from 25% to 50% of your portfolio. Given a PortfolioCVaR object `p`, use `setBounds` to set the bound constraints:

```lb = [ 0.5; 0.25 ]; ub = [ 0.75; 0.5 ]; p = PortfolioCVaR; p = setBounds(p, lb, ub); disp(p.NumAssets); disp(p.LowerBound); disp(p.UpperBound);```
``` 2 0.5000 0.2500 0.7500 0.5000```

#### Setting Bounds Using the PortfolioCVaR Function or setBounds Function

Both the `PortfolioCVaR` function and `setBounds` function implement scalar expansion on either the `LowerBound` or `UpperBound` properties. If the `NumAssets` property is already set in the PortfolioCVaR object, scalar arguments for either property expand to have the same value across all dimensions. In addition, `setBounds` lets you specify `NumAssets` as an optional argument. Suppose that you have a universe of 500 assets and you want to set common bound constraints on all assets in your universe. Specifically, you are a long-only investor and want to hold no more than 5% of your portfolio in any single asset. You can set these bound constraints in any of these equivalent ways:

`p = PortfolioCVaR('NumAssets', 500, 'LowerBound', 0, 'UpperBound', 0.05);`

or

```p = PortfolioCVaR('NumAssets', 500); p = setBounds(p, 0, 0.05);```

or

```p = PortfolioCVaR; p = setBounds(p, 0, 0.05, 500);```

To clear bound constraints from your PortfolioCVaR object, use either the `PortfolioCVaR` function or `setBounds` with empty inputs for the properties to be cleared. For example, to clear the upper-bound constraint from the PortfolioCVaR object `p` in the previous example:

`p = PortfolioCVaR(p, 'UpperBound', []);`

### Working with Budget Constraints Using PortfolioCVaR Object

The budget constraint is an optional linear constraint that maintains upper and lower bounds on the sum of portfolio weights (see Budget Constraints). Budget constraints have properties `LowerBudget` for the lower budget constraint and `UpperBudget` for the upper budget constraint. If you set up a CVaR portfolio optimization problem that requires portfolios to be fully invested in your universe of assets, you can set `LowerBudget` to be equal to `UpperBudget`. These budget constraints can be set with default values equal to `1` using `setDefaultConstraints` (see Setting Default Constraints Using the PortfolioCVaR Function).

#### Setting Budget Constraints Using the PortfolioCVaR Function

The properties for the budget constraint can also be set using the `PortfolioCVaR` function. Suppose that you have an asset universe with many risky assets and a riskless asset and you want to ensure that your portfolio never holds more than 1% cash, that is, you want to ensure that you are 99–100% invested in risky assets. The budget constraint for this portfolio can be set with:

```p = PortfolioCVaR('LowerBudget', 0.99, 'UpperBudget', 1); disp(p.LowerBudget); disp(p.UpperBudget);```
``` 0.9900 1```

#### Setting Budget Constraints Using the setBudget Function

You can also set the properties for a budget constraint using `setBudget`. Suppose that you have a fund that permits up to 10% leverage which means that your portfolio can be from 100 to 110% invested in risky assets. Given a PortfolioCVaR object `p`, use `setBudget` to set the budget constraints:

```p = PortfolioCVaR; p = setBudget(p, 1, 1.1); disp(p.LowerBudget); disp(p.UpperBudget);```
``` 1 1.1000```
If you were to continue with this example, then set the `RiskFreeRate` property to the borrowing rate to finance possible leveraged positions. For details on the `RiskFreeRate` property, see Working with a Riskless Asset. To clear either bound for the budget constraint from your PortfolioCVaR object, use either the `PortfolioCVaR` function or `setBudget` with empty inputs for the properties to be cleared. For example, clear the upper-budget constraint from the PortfolioCVaR object `p` in the previous example with:
`p = PortfolioCVaR(p, 'UpperBudget', []);`

### Working with Group Constraints Using PortfolioCVaR Object

Group constraints are optional linear constraints that group assets together and enforce bounds on the group weights (see Group Constraints). Although the constraints are implemented as general constraints, the usual convention is to form a group matrix that identifies membership of each asset within a specific group with Boolean indicators (either `true` or `false` or with `1` or `0`) for each element in the group matrix. Group constraints have properties `GroupMatrix` for the group membership matrix,` LowerGroup` for the lower-bound constraint on groups, and `UpperGroup` for the upper-bound constraint on groups.

#### Setting Group Constraints Using the PortfolioCVaR Function

The properties for group constraints are set through the `PortfolioCVaR` function. Suppose that you have a portfolio of five assets and want to ensure that the first three assets constitute no more than 30% of your portfolio, then you can set group constraints:

```G = [ 1 1 1 0 0 ]; p = PortfolioCVaR('GroupMatrix', G, 'UpperGroup', 0.3); disp(p.NumAssets); disp(p.GroupMatrix); disp(p.UpperGroup);```
```5 1 1 1 0 0 0.3000```

The group matrix `G` can also be a logical matrix so that the following code achieves the same result.

```G = [ true true true false false ]; p = PortfolioCVaR('GroupMatrix', G, 'UpperGroup', 0.3); disp(p.NumAssets); disp(p.GroupMatrix); disp(p.UpperGroup);```
```5 1 1 1 0 0 0.3000```

#### Setting Group Constraints Using the setGroups and addGroups Functions

You can also set the properties for group constraints using `setGroups`. Suppose that you have a portfolio of five assets and want to ensure that the first three assets constitute no more than 30% of your portfolio. Given a PortfolioCVaR object `p`, use `setGroups` to set the group constraints:

```G = [ true true true false false ]; p = PortfolioCVaR; p = setGroups(p, G, [], 0.3); disp(p.NumAssets); disp(p.GroupMatrix); disp(p.UpperGroup);```
```5 1 1 1 0 0 0.3000```

In this example, you would set the `LowerGroup` property to be empty (`[]`).

Suppose you want to add another group constraint to make odd-numbered assets constitute at least 20% of your portfolio. Set up an augmented group matrix and introduce infinite bounds for unconstrained group bounds or use the `addGroups` function to build up group constraints. For this example, create another group matrix for the second group constraint:

```p = PortfolioCVaR; G = [ true true true false false ]; % group matrix for first group constraint p = setGroups(p, G, [], 0.3); G = [ true false true false true ]; % group matrix for second group constraint p = addGroups(p, G, 0.2); disp(p.NumAssets); disp(p.GroupMatrix); disp(p.LowerGroup); disp(p.UpperGroup);```
``` 5 1 1 1 0 0 1 0 1 0 1 -Inf 0.2000 0.3000 Inf```
`addGroups` determines which bounds are unbounded so you only need to focus on the constraints that you want to set.

The `PortfolioCVaR` function, `setGroups`, and `addGroups` implement scalar expansion on either the `LowerGroup` or `UpperGroup` properties based on the dimension of the group matrix in the property `GroupMatrix`. Suppose that you have a universe of 30 assets with 6 asset classes such that assets 1–5, assets 6–12, assets 13–18, assets 19–22, assets 23–27, and assets 28–30 constitute each of your asset classes and you want each asset class to fall from 0 to 25% of your portfolio. Let the following group matrix define your groups and scalar expansion define the common bounds on each group:

```p = PortfolioCVaR; G = blkdiag(true(1,5), true(1,7), true(1,6), true(1,4), true(1,5), true(1,3)); p = setGroups(p, G, 0, 0.25); disp(p.NumAssets); disp(p.GroupMatrix); disp(p.LowerGroup); disp(p.UpperGroup);```
```30 Columns 1 through 16 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 17 through 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500```

### Working with Group Ratio Constraints Using PortfolioCVaR Object

Group ratio constraints are optional linear constraints that maintain bounds on proportional relationships among groups of assets (see Group Ratio Constraints). Although the constraints are implemented as general constraints, the usual convention is to specify a pair of group matrices that identify membership of each asset within specific groups with Boolean indicators (either `true` or `false` or with `1` or `0`) for each element in each of the group matrices. The goal is to ensure that the ratio of a base group compared to a comparison group fall within specified bounds. Group ratio constraints have properties:

• `GroupA` for the base membership matrix

• `GroupB` for the comparison membership matrix

• `LowerRatio` for the lower-bound constraint on the ratio of groups

• `UpperRatio` for the upper-bound constraint on the ratio of groups

#### Setting Group Ratio Constraints Using the PortfolioCVaR Function

The properties for group ratio constraints are set using `PortfolioCVaR` function. For example, assume you want the ratio of financial to nonfinancial companies in your portfolios to never go above 50%. Suppose that you have six assets with three financial companies (assets 1–3) and three nonfinanical companies (assets 4–6). To set group ratio constraints:

```GA = [ 1 1 1 0 0 0 ]; % financial companies GB = [ 0 0 0 1 1 1 ]; % nonfinancial companies p = PortfolioCVaR('GroupA', GA, 'GroupB', GB, 'UpperRatio', 0.5); disp(p.NumAssets); disp(p.GroupA); disp(p.GroupB); disp(p.UpperRatio);```
```6 1 1 1 0 0 0 0 0 0 1 1 1 0.5000```

Group matrices `GA` and `GB` in this example can be logical matrices with `true` and `false` elements that yield the same result:

```GA = [ true true true false false false ]; % financial companies GB = [ false false false true true true ]; % nonfinancial companies p = PortfolioCVaR('GroupA', GA, 'GroupB', GB, 'UpperRatio', 0.5); disp(p.NumAssets); disp(p.GroupA); disp(p.GroupB); disp(p.UpperRatio);```
```6 1 1 1 0 0 0 0 0 0 1 1 1 0.5000```

#### Setting Group Ratio Constraints Using the setGroupRatio and addGroupRatio Functions

You can also set the properties for group ratio constraints using `setGroupRatio`. For example, assume that you want the ratio of financial to nonfinancial companies in your portfolios to never go above 50%. Suppose that you have six assets with three financial companies (assets 1–3) and three nonfinanical companies (assets 4–6). Given a PortfolioCVaR object `p`, use `setGroupRatio` to set the group constraints:

```GA = [ true true true false false false ]; % financial companies GB = [ false false false true true true ]; % nonfinancial companies p = PortfolioCVaR; p = setGroupRatio(p, GA, GB, [], 0.5); disp(p.NumAssets); disp(p.GroupA); disp(p.GroupB); disp(p.UpperRatio);```
```6 1 1 1 0 0 0 0 0 0 1 1 1 0.5000```
In this example, you would set the `LowerRatio` property to be empty (`[]`).

Suppose you want to add another group ratio constraint to ensure that the weights in odd-numbered assets constitute at least 20% of the weights in nonfinancial assets your portfolio. You can set up augmented group ratio matrices and introduce infinite bounds for unconstrained group ratio bounds, or you can use the `addGroupRatio` function to build up group ratio constraints. For this example, create another group matrix for the second group constraint:

```p = PortfolioCVaR; GA = [ true true true false false false ]; % financial companies GB = [ false false false true true true ]; % nonfinancial companies p = setGroupRatio(p, GA, GB, [], 0.5); GA = [ true false true false true false ]; % odd-numbered companies GB = [ false false false true true true ]; % nonfinancial companies p = addGroupRatio(p, GA, GB, 0.2); disp(p.NumAssets); disp(p.GroupA); disp(p.GroupB); disp(p.LowerRatio); disp(p.UpperRatio);```
``` 6 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 1 -Inf 0.2000 0.5000 Inf```
Notice that `addGroupRatio` determines which bounds are unbounded so you only need to focus on the constraints you want to set.

The `PortfolioCVaR` function, `setGroupRatio`, and `addGroupRatio` implement scalar expansion on either the `LowerRatio` or `UpperRatio` properties based on the dimension of the group matrices in `GroupA` and `GroupB` properties.

### Working with Linear Equality Constraints Using PortfolioCVaR Object

Linear equality constraints are optional linear constraints that impose systems of equalities on portfolio weights (see Linear Equality Constraints). Linear equality constraints have properties `AEquality`, for the equality constraint matrix, and `bEquality`, for the equality constraint vector.

#### Setting Linear Equality Constraints Using the PortfolioCVaR Function

The properties for linear equality constraints are set using the `PortfolioCVaR` function. Suppose that you have a portfolio of five assets and want to ensure that the first three assets are 50% of your portfolio. To set this constraint:

```A = [ 1 1 1 0 0 ]; b = 0.5; p = PortfolioCVaR('AEquality', A, 'bEquality', b); disp(p.NumAssets); disp(p.AEquality); disp(p.bEquality);```
```5 1 1 1 0 0 0.5000```

#### Setting Linear Equality Constraints Using the setEquality and addEquality Functions

You can also set the properties for linear equality constraints using `setEquality`. Suppose that you have a portfolio of five assets and want to ensure that the first three assets are 50% of your portfolio. Given a PortfolioCVaR object `p`, use `setEquality` to set the linear equality constraints:

```A = [ 1 1 1 0 0 ]; b = 0.5; p = PortfolioCVaR; p = setEquality(p, A, b); disp(p.NumAssets); disp(p.AEquality); disp(p.bEquality);```
```5 1 1 1 0 0 0.5000```

Suppose you want to add another linear equality constraint to ensure that the last three assets also constitute 50% of your portfolio. You can set up an augmented system of linear equalities or use `addEquality` to build up linear equality constraints. For this example, create another system of equalities:

```p = PortfolioCVaR; A = [ 1 1 1 0 0 ]; % first equality constraint b = 0.5; p = setEquality(p, A, b); A = [ 0 0 1 1 1 ]; % second equality constraint b = 0.5; p = addEquality(p, A, b); disp(p.NumAssets); disp(p.AEquality); disp(p.bEquality);```
```5 1 1 1 0 0 0 0 1 1 1 0.5000 0.5000```

The `PortfolioCVaR` function, `setEquality`, and `addEquality` implement scalar expansion on the `bEquality` property based on the dimension of the matrix in the `AEquality` property.

### Working with Linear Inequality Constraints Using PortfolioCVaR Object

Linear inequality constraints are optional linear constraints that impose systems of inequalities on portfolio weights (see Linear Inequality Constraints). Linear inequality constraints have properties `AInequality` for the inequality constraint matrix, and` bInequality` for the inequality constraint vector.

#### Setting Linear Inequality Constraints Using the PortfolioCVaR Function

The properties for linear inequality constraints are set using the `PortfolioCVaR` function. Suppose that you have a portfolio of five assets and you want to ensure that the first three assets are no more than 50% of your portfolio. To set up these constraints:

```A = [ 1 1 1 0 0 ]; b = 0.5; p = PortfolioCVaR('AInequality', A, 'bInequality', b); disp(p.NumAssets); disp(p.AInequality); disp(p.bInequality);```
```5 1 1 1 0 0 0.5000```

#### Setting Linear Inequality Constraints Using the setInequality and addInequality Functions

You can also set the properties for linear inequality constraints using `setInequality`. Suppose that you have a portfolio of five assets and you want to ensure that the first three assets constitute no more than 50% of your portfolio. Given a PortfolioCVaR object `p`, use `setInequality` to set the linear inequality constraints:

```A = [ 1 1 1 0 0 ]; b = 0.5; p = PortfolioCVaR; p = setInequality(p, A, b); disp(p.NumAssets); disp(p.AInequality); disp(p.bInequality);```
```5 1 1 1 0 0 0.5000```

Suppose you want to add another linear inequality constraint to ensure that the last three assets constitute at least 50% of your portfolio. You can set up an augmented system of linear inequalities or use the `addInequality` function to build up linear inequality constraints. For this example, create another system of inequalities:

```p = PortfolioCVaR; A = [ 1 1 1 0 0 ]; % first inequality constraint b = 0.5; p = setInequality(p, A, b); A = [ 0 0 -1 -1 -1 ]; % second inequality constraint b = -0.5; p = addInequality(p, A, b); disp(p.NumAssets); disp(p.AInequality); disp(p.bInequality);```
```5 1 1 1 0 0 0 0 -1 -1 -1 0.5000 -0.5000```

The `PortfolioCVaR` function, `setInequality`, and `addInequality` implement scalar expansion on the `bInequality` property based on the dimension of the matrix in the `AInequality` property.

### Working with Average Turnover Constraints Using PortfolioCVaR Object

The turnover constraint is an optional linear absolute value constraint (see Average Turnover Constraints) that enforces an upper bound on the average of purchases and sales. The turnover constraint can be set using the `PortfolioCVaR` function or the `setTurnover` function. The turnover constraint depends on an initial or current portfolio, which is assumed to be zero if not set when the turnover constraint is set. The turnover constraint has properties `Turnover`, for the upper bound on average turnover, and `InitPort`, for the portfolio against which turnover is computed.

#### Setting Average Turnover Constraints Using the PortfolioCVaR Function

The properties for the turnover constraints are set using the `PortfolioCVaR` function. Suppose that you have an initial portfolio of 10 assets in a variable `x0` and you want to ensure that average turnover is no more than 30%. To set this turnover constraint:

```x0 = [ 0.12; 0.09; 0.08; 0.07; 0.1; 0.1; 0.15; 0.11; 0.08; 0.1 ]; p = PortfolioCVaR('Turnover', 0.3, 'InitPort', x0); disp(p.NumAssets); disp(p.Turnover); disp(p.InitPort);```
```10 0.3000 0.1200 0.0900 0.0800 0.0700 0.1000 0.1000 0.1500 0.1100 0.0800 0.1000```
Note if the `NumAssets` or `InitPort` properties are not set before or when the turnover constraint is set, various rules are applied to assign default values to these properties (see Setting Up an Initial or Current Portfolio).

#### Setting Average Turnover Constraints Using the setTurnover Function

You can also set properties for portfolio turnover using `setTurnover` to specify both the upper bound for average turnover and an initial portfolio. Suppose that you have an initial portfolio of 10 assets in a variable `x0` and want to ensure that average turnover is no more than 30%. Given a PortfolioCVaR object `p`, use `setTurnover` to set the turnover constraint with and without the initial portfolio being set previously:

```x0 = [ 0.12; 0.09; 0.08; 0.07; 0.1; 0.1; 0.15; 0.11; 0.08; 0.1 ]; p = PortfolioCVaR('InitPort', x0); p = setTurnover(p, 0.3); disp(p.NumAssets); disp(p.Turnover); disp(p.InitPort);```
```10 0.3000 0.1200 0.0900 0.0800 0.0700 0.1000 0.1000 0.1500 0.1100 0.0800 0.1000```

or

```x0 = [ 0.12; 0.09; 0.08; 0.07; 0.1; 0.1; 0.15; 0.11; 0.08; 0.1 ]; p = PortfolioCVaR; p = setTurnover(p, 0.3, x0); disp(p.NumAssets); disp(p.Turnover); disp(p.InitPort);```
```10 0.3000 0.1200 0.0900 0.0800 0.0700 0.1000 0.1000 0.1500 0.1100 0.0800 0.1000 ```

`setTurnover` implements scalar expansion on the argument for the initial portfolio. If the `NumAssets` property is already set in the PortfolioCVaR object, a scalar argument for `InitPort` expands to have the same value across all dimensions. In addition, `setTurnover` lets you specify `NumAssets` as an optional argument. To clear turnover from your PortfolioCVaR object, use the `PortfolioCVaR` function or `setTurnover` with empty inputs for the properties to be cleared.

### Working with One-way Turnover Constraints Using PortfolioCVaR Object

One-way turnover constraints are optional constraints (see One-way Turnover Constraints) that enforce upper bounds on net purchases or net sales. One-way turnover constraints can be set using the `PortfolioCVaR` function or the `setOneWayTurnover` function. One-way turnover constraints depend upon an initial or current portfolio, which is assumed to be zero if not set when the turnover constraints are set. One-way turnover constraints have properties `BuyTurnover`, for the upper bound on net purchases, `SellTurnover`, for the upper bound on net sales, and `InitPort`, for the portfolio against which turnover is computed.

#### Setting One-way Turnover Constraints Using the PortfolioCVaR Function

The Properties for the one-way turnover constraints are set using the `PortfolioCVaR` function. Suppose that you have an initial portfolio with 10 assets in a variable `x0` and you want to ensure that turnover on purchases is no more than 30% and turnover on sales is no more than 20% of the initial portfolio. To set these turnover constraints:

```x0 = [ 0.12; 0.09; 0.08; 0.07; 0.1; 0.1; 0.15; 0.11; 0.08; 0.1 ]; p = PortfolioCVaR('BuyTurnover', 0.3, 'SellTurnover', 0.2, 'InitPort', x0); disp(p.NumAssets); disp(p.BuyTurnover); disp(p.SellTurnover); disp(p.InitPort); ```
``` 10 0.3000 0.2000 0.1200 0.0900 0.0800 0.0700 0.1000 0.1000 0.1500 0.1100 0.0800 0.1000```

If the `NumAssets` or `InitPort` properties are not set before or when the turnover constraint is set, various rules are applied to assign default values to these properties (see Setting Up an Initial or Current Portfolio).

#### Setting Turnover Constraints Using the setOneWayTurnover Function

You can also set properties for portfolio turnover using `setOneWayTurnover` to specify to the upper bounds for turnover on purchases (`BuyTurnover`) and sales (`SellTurnover`) and an initial portfolio. Suppose that you have an initial portfolio of 10 assets in a variable `x0` and want to ensure that turnover on purchases is no more than 30% and that turnover on sales is no more than 20% of the initial portfolio. Given a PortfolioCVaR object `p`, use `setOneWayTurnover` to set the turnover constraints with and without the initial portfolio being set previously:

```x0 = [ 0.12; 0.09; 0.08; 0.07; 0.1; 0.1; 0.15; 0.11; 0.08; 0.1 ]; p = PortfolioCVaR('InitPort', x0); p = setOneWayTurnover(p, 0.3, 0.2); disp(p.NumAssets); disp(p.BuyTurnover); disp(p.SellTurnover); disp(p.InitPort);```
``` 10 0.3000 0.2000 0.1200 0.0900 0.0800 0.0700 0.1000 0.1000 0.1500 0.1100 0.0800 0.1000```
or
```x0 = [ 0.12; 0.09; 0.08; 0.07; 0.1; 0.1; 0.15; 0.11; 0.08; 0.1 ]; p = PortfolioCVaR; p = setOneWayTurnover(p, 0.3, 0.2, x0); disp(p.NumAssets); disp(p.BuyTurnover); disp(p.SellTurnover); disp(p.InitPort);```
``` 10 0.3000 0.2000 0.1200 0.0900 0.0800 0.0700 0.1000 0.1000 0.1500 0.1100 0.0800 0.1000```

`setOneWayTurnover` implements scalar expansion on the argument for the initial portfolio. If the `NumAssets` property is already set in the PortfolioCVaR object, a scalar argument for `InitPort` expands to have the same value across all dimensions. In addition, `setOneWayTurnover` lets you specify `NumAssets` as an optional argument. To remove one-way turnover from your PortfolioCVaR object, use the`PortfolioCVaR` function or `setOneWayTurnover` with empty inputs for the properties to be cleared.