Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Novel approaches to portfolio construction 39

the variation of the optimal expected return is completely determined by the
tracking error and is insensitive to the variation in the asset bound constraint.
Ideally, a PM would like to have at his or her disposal an automatic tool that
can discard such uninteresting pairs of constraints and help him or her focus
on pairs that directly impact the key performance determinants.
Next , we introduce the concept of “ Constraint Elasticity ” and discuss how it
can be used to address the questions raised in this section.

2.3.1 Constraint elasticity

Constraint elasticity measures the responsiveness of the optimal objective value
to perturbations in the constraint bounds. More precisely, the elasticity of the
model with respect to a constraint in the strategy is defined to be the percent-
age change in the optimal objective value that arises due to 1% change in the
constraint bounds.^3

Sector bounds

9

9.19.29.39.49.59.69.79.89.9^10 10.110.210.310.410.510.610.710.810.9^11

5

5.5

5.45

5.4

5.35

5.3

5.25

5.2

5.15

5.1

5.05

4.95

4.9

4.85

4.8

4.75

4.7

4.65

4.6

4.55

4.5

Color key

Value

0.82 0.9 Expected return

Tracking error

Figure 2.11 Sector bounds — tracking error — expected return heatmap.

3 Naturally, the concept of constraint elasticity gives useful results only for constraints with
nonzero bounds. Fortunately, most constraints of interest such as tracking error constraint, turn-
over constraint, etc. satisfy this condition.