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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.

`PortfolioMAD`

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

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 = PortfolioMAD('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 = PortfolioMAD('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 `PortfolioMAD`

object
`p`

, use `setGroups`

to set the group
constraints:

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

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 = PortfolioMAD; 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 13 1 1 1 1 1 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 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 Columns 14 through 26 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 1 1 1 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 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 27 through 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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

`PortfolioMAD`

| `setBounds`

| `setBudget`

| `setDefaultConstraints`

| `setEquality`

| `setGroupRatio`

| `setGroups`

| `setInequality`

| `setOneWayTurnover`

| `setTurnover`

- Setting Default Constraints for Portfolio Weights Using PortfolioMAD Object
- Creating the PortfolioMAD Object
- Validate the MAD Portfolio Problem
- Estimate Efficient Portfolios Along the Entire Frontier for PortfolioMAD Object
- Estimate Efficient Frontiers for PortfolioMAD Object
- Asset Returns and Scenarios Using PortfolioMAD Object