Validate the MAD Portfolio Problem

Sometimes, you may want to validate either your inputs to, or outputs from, a portfolio optimization problem. Although most error checking that occurs during the problem setup phase catches most difficulties with a portfolio optimization problem, the processes to validate MAD portfolio sets and portfolios are time consuming and are best done offline. So, the portfolio optimization tools have specialized functions to validate MAD portfolio sets and portfolios. For information on the workflow when using PortfolioMAD objects, see PortfolioMAD Object Workflow.

Validating a MAD Portfolio Set

Since it is necessary and sufficient that your MAD portfolio set must be a nonempty, closed, and bounded set to have a valid portfolio optimization problem, the estimateBounds function lets you examine your portfolio set to determine if it is nonempty and, if nonempty, whether it is bounded. Suppose that you have the following MAD portfolio set which is an empty set because the initial portfolio at 0 is too far from a portfolio that satisfies the budget and turnover constraint:

p = PortfolioMAD('NumAssets', 3, 'Budget', 1);
p = setTurnover(p, 0.3, 0);

If a MAD portfolio set is empty, estimateBounds returns NaN bounds and sets the isbounded flag to []:

[lb, ub, isbounded] = estimateBounds(p)

lb =

   NaN
   NaN
   NaN

ub =

   NaN
   NaN
   NaN

isbounded =

     []

Suppose that you create an unbounded MAD portfolio set as follows:

p = PortfolioMAD('AInequality', [1 -1; 1 1 ], 'bInequality', 0);
[lb, ub, isbounded] = estimateBounds(p)

lb =

  -Inf
  -Inf

ub =

  1.0e-008 *

   -0.3712
       Inf

isbounded =

     0

In this case, estimateBounds returns (possibly infinite) bounds and sets the isbounded flag to false. The result shows which assets are unbounded so that you can apply bound constraints as necessary.

Finally, suppose that you created a PortfolioMAD object that is both nonempty and bounded. estimateBounds not only validates the set, but also obtains tighter bounds which are useful if you are concerned with the actual range of portfolio choices for individual assets in your portfolio:

p = PortfolioMAD;
p = setBudget(p, 1,1);
p = setBounds(p, [ -0.1; 0.2; 0.3; 0.2 ], [ 0.5; 0.3; 0.9; 0.8 ]);
        
[lb, ub, isbounded] = estimateBounds(p)

lb =

   -0.1000
    0.2000
    0.3000
    0.2000

ub =

    0.3000
    0.3000
    0.7000
    0.6000

isbounded =

     1

In this example, all but the second asset has tighter upper bounds than the input upper bound implies.

Validating MAD Portfolios

Given a MAD portfolio set specified in a PortfolioMAD object, you often want to check if specific portfolios are feasible with respect to the portfolio set. This can occur with, for example, initial portfolios and with portfolios obtained from other procedures. The checkFeasibility function determines whether a collection of portfolios is feasible. Suppose that you perform the following portfolio optimization and want to determine if the resultant efficient portfolios are feasible relative to a modified problem.

First, set up a problem in the PortfolioMAD object p, estimate efficient portfolios in pwgt, and then confirm that these portfolios are feasible relative to the initial problem:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0; 
    0.00408 0.0289 0.0204 0.0119;
    0.00192 0.0204 0.0576 0.0336;
    0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;

AssetScenarios = mvnrnd(m, C, 20000);

p = PortfolioMAD;
p = setScenarios(p, AssetScenarios);
p = setDefaultConstraints(p);

pwgt = estimateFrontier(p);

checkFeasibility(p, pwgt)

ans =

     1     1     1     1     1     1     1     1     1     1

Next, set up a different portfolio problem that starts with the initial problem with an additional a turnover constraint and an equally weighted initial portfolio:

q = setTurnover(p, 0.3, 0.25);
checkFeasibility(q, pwgt)

ans =

     0     0     1     1     1     0     0     0     0     0

In this case, only two of the 10 efficient portfolios from the initial problem are feasible relative to the new problem in PortfolioMAD object q. Solving the second problem using checkFeasibility demonstrates that the efficient portfolio for PortfolioMAD object q is feasible relative to the initial problem:

qwgt = estimateFrontier(q);
checkFeasibility(p, qwgt)

ans =

     1     1     1     1     1     1     1     1     1     1