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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.

`Portfolio`

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

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 = Portfolio('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 = Portfolio('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 `Portfolio`

object
`p`

, use `setGroups`

to set the group
constraints:

G = [ true true true false false ]; p = Portfolio; 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 = Portfolio; 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 `Portfolio`

object and `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 six 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
six 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 = Portfolio; 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

`Portfolio`

| `setBounds`

| `setBudget`

| `setDefaultConstraints`

| `setEquality`

| `setGroupRatio`

| `setGroups`

| `setInequality`

| `setOneWayTurnover`

| `setTrackingError`

| `setTrackingPort`

| `setTurnover`

- Creating the Portfolio Object
- Working with Portfolio Constraints Using Defaults
- Validate the Portfolio Problem for Portfolio Object
- Estimate Efficient Portfolios for Entire Efficient Frontier for Portfolio Object
- Estimate Efficient Frontiers for Portfolio Object
- Constraint Specification Using a Portfolio Object
- Asset Allocation Case Study
- Portfolio Optimization Examples
- Portfolio Optimization with Semicontinuous and Cardinality Constraints
- Black-Litterman Portfolio Optimization
- Portfolio Optimization Using Factor Models

- Portfolio Object
- Portfolio Optimization Theory
- Portfolio Object Workflow
- Setting Up a Tracking Portfolio