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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.

`PortfolioCVaR`

FunctionThe properties for group constraints are set through the `PortfolioCVaR`

object. 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

`setGroups`

and `addGroups`

FunctionsYou 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 that 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`

object, `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

`PortfolioCVaR`

| `setDefaultConstraints`

| `setBounds`

| `setBudget`

| `setGroups`

| `setGroupRatio`

| `setEquality`

| `setInequality`

| `setTurnover`

| `setOneWayTurnover`

- Creating the PortfolioCVaR Object
- Working with CVaR Portfolio Constraints Using Defaults
- Validate the CVaR Portfolio Problem
- Estimate Efficient Portfolios for Entire Frontier for PortfolioCVaR Object
- Estimate Efficient Frontiers for PortfolioCVaR Object
- Asset Returns and Scenarios Using PortfolioCVaR Object
- Hedging Using CVaR Portfolio Optimization
- Compute Maximum Reward-to-Risk Ratio for CVaR Portfolio