Constraints Analysis II
The Constraint Analysis II Worksheet is a useful tool for visualizing the effect of customized constraints on the optimal portfolios and thereby determining the appropriate bounds for customized constraints and asset bounds. (The other Constraint Analysis Worksheet provides a numerical approach to some of the same concerns.)
Select a customized constraint or a particular asset's bounds in one of the two drop down menus at the top of the worksheet.
A set of M constraints on N-asset portfolios with weights P (N by 1) are specified mathematically in the form L ≤ AP ≤ U, where L and U are vectors (M by 1) of lower and upper bounds, and A is a matrix (M by N) whose rows form the linear combinations for each constraint.
Practical application of constraints requires setting the upper and lower bounds appropriately. Because of the unique resampling framework of the NFA Optimizer, some of the simulations may trigger certain constraint bounds while others may not. Since the final optimal portfolio averages the individual simulations, adjusting a customized constraint bound will have a “soft” effect on the corresponding portfolio weight. Rather than pushed up against the hard boundary, the optimal portfolio weight will be gently moved but remain inside the constraint bounds as long as less than 100% of the resampled simulations are binding.
In order to set the constraint bounds properly, it is of interest to know how often simulated portfolios are hitting the constraint values. The Constraint Analysis II Worksheet shows this information graphically, showing a contour map of the frequencies of the simulated constraint portfolios AP on the vertical axis, charted for the portfolios spanning the efficient frontier on the horizontal axis. The contour map shows one constraint (one row of the matrix A and one scalar value each from L and U) at a time, selected by drop-down menu. The vertical axis ranges from the lower bound to the upper bound and shows the frequencies for simulated portfolios P whose constraint values AP fall between the bounds. Above and below this main contour map graphs display the percentage of time the corresponding bounds are attained. Particularly binding constraints will show a high percentage of simulations attaining that bound. This information can be extremely useful in adjusting the constraint bounds.
The figure shows the constraint chart for the Growth to Value constraint in the Vanguard Sample Case. As portfolio risk increases from left to right, the values of the constraint fan out rapidly, with the number of cases attaining the upper and lower bounds steadily increasing across the frontier. This constraint is somewhat binding both above and below, and is likely to have some impact on the optimal portfolio weights.