Working with Group Constraints Using PortfolioMAD 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 PortfolioMAD Function

The 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

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 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