Constraint Analysis

The optimization process and the post-optimization process both allow constraints.  In addition to bounding individual assets to values between lower and upper bounds, the user can limit arbitrarily weighted combinations of assets to values between lower and upper bounds using customized constraints.  When many assets and many constraints are involved, the full set of asset and linear bounds imply combined bounds for each asset, which may not be simple to determine from a cursory examination of the full set of bounds. Furthermore, it becomes progressively easier to specify an infeasible set of bounds.  

To facilitate construction, as well as understanding, of a large constraint set, we offer two tools.  This worksheet uses numbers to show the combined effect of constraints.  Another worksheet, the Constraint Analysis II Worksheet, graphically displays where the simulated frontier portfolios are in constraint space and how often each constraint is binding.

Select a Customized Constraint and Portfolio

Choose which customized constraint to examine with the two drop down menus at the top left of the Constraint Analysis Worksheet.  The first menu selects between optimization and post-optimization constraints, and the second one selects a specific constraint by the name assigned to it on the Constraints or Investability Constraints Worksheet.  The tables on the Investability Constraints Worksheet display the analysis of the selected constraint.  The third drop down menu, Portfolio Type, in concert with the Portfolio slider, permits you to choose a portfolio to compare to the implied bounds.  The portfolio selected does not impact the displayed implied bounds in any way.

A Note on the Budget Constraint

The first optimization and post-optimization constraint is always specified by default as a vector of ones with upper and lower bounds equal to 100%.  Labeled "[portfolio]" on the Constraints and Investability Constraints Worksheets, this constraint is also known as the budget constraint.  It essentially specifies that the sum of asset allocations must be 100%, as 100% of your budget will be allocated into the various assets available in the investable universe.  This constraint should not normally be modified, and is treated differently from other constraints in the constraint analysis.  The list of selectable constraints in the constraint analysis worksheet does not include the budget constraint, because it will be assumed present on all of the analyses except for the bounds implied by the selected constraint and portfolio.

Analysis

The middle columns show any asset bounds entered and the currently selected customized constraint (see above).

• Unlike on the Constraints Worksheet, the customized constraint is expressed as percentages, so expense ratios that may appear as zeros on the Constraints Worksheet show up as fractions of a percent here.  Similarly, coefficients of "1" appear as "100%".
• The Current % Field at the very bottom is the current calculation of the middle of the constraint inequality in relation to the portfolio selected by the Portfolio slider.  Keeping in mind that Min ≤ C1W1 + C2W2 + …. CnWn ≤ Max, where C is the Constraint Value, W is the weight of the asset, and there are n assets in the problem, the Current % Field is equal to C₁W₁ + C₂W₂ + …. C_nW_n  = . This information updates immediately, so you can always see the range of values in the inequality for the currently applied constraints or investability constraints.

The right side of the display shows the results of the analysis, summarized here sequentially:

• The bounds implied by all constraints.  The first two columns show the lower and upper bounds implied by the full constraint set.  This determines the smallest possible n-dimensional box which surrounds the asset weights when constrained by all of the constraints. All resampled portfolios which are averaged to make the Michaud Resampled Efficient Frontier must have weights inside these ranges. If the constraint set is feasible, these will appear as numbers for each asset.  If not, some of the numbers will display as N/A. The other parts of the display are useful in diagnosing problems with constraint sets.

• The bounds implied by all constraints except the selected one. This analysis is similar to the all-constraints analysis, but with the selected constraint omitted. If the results of this analysis contain no N/A values, but the all-constraint analysis does contain N/A values, then it is the current constraint which is causing the infeasibility.

Note that it is possible for a constraint to limit the available combinations of weights without changing the smallest bounding box, which is shown in these analyses. Thus, even if a constraint does not show a difference between the analysis without that constraint and the all-constraint analysis, it is still possible that some combinations of asset weights are disallowed by the constraint. For example, with three assets, the unconstrained bounding box will be in the shape of a cube. A constraint might limit values on one vertex of that cube, effectively slicing off a corner of that cube. The smallest bounding box with the constraint, however, will remain the same cube as without the constraint.

Infeasible Constraints

When problems arise with constraint sets, an error message will be shown when optimization or post-optimization is attempted. This is often the result of a recently added constraint.  In such cases, the recently added constraints can be examined on this worksheet. When a constraint is found which has no N/A values in the Bounds Implied by All Constraints Except the [Constraint Name] Columns, this is the constraint causing the problems. The bounds for that constraint can then be relaxed until a suitable bound is determined where the N/A values in the other rows disappear.

For a more visual, rather than numerical review of your constraints, see the Constraint Analysis II Worksheet.