Working with Group Ratio Constraints Using PortfolioMAD 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 PortfolioMAD Function

The properties for group ratio constraints are set using PortfolioMAD object. 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 nonfinancial 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 = PortfolioMAD('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 = PortfolioMAD('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 nonfinancial companies (assets 4–6). Given a PortfolioMAD 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 = PortfolioMAD;
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 that 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 = PortfolioMAD;
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 PortfolioMAD object, 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.