Novel approaches to portfolio construction 41
Next , let us try to understand the impact of jointly varying the upper bounds
on tracking error and turnover. Figure 2.10 gives a heatmap representation of the
optimal expected return subject to these perturbations. The information displayed
in Figure 2.10 has two interesting ramifications. First, it gives a graphical display
of the optimal expected return terrain for various combinations of tracking error
and turnover. Such a terrain can be very helpful in understanding trade-offs and
making an informed decision. Furthermore, if the specific values of tracking error
and turnover used in the original strategy were tentative, then the heatmap can be
used as a guiding tool in choosing more desirable values for these characteristics.
Second , consider the level curves of the heatmap displayed in Figure 2.10.
These level curves plot combinations of tracking error and turnover values that
give rise to the same optimal expected return. Note that the level curves of Figure
2.10 are arcs of concentric ellipses centered on the top right corner of the graph.
Alternatively, this means that as we decrease the tracking error, we lose a certain
portion of optimal expected return that we can recover by slightly increasing the
turnover. Since the elasticities of the tracking error constraint is roughly 15 times
higher than that of the turnover constraint, the arcs of ellipses representing the
level curves in Figure 2.10 are elongated along the turnover axis and compressed
along the tracking error axis. Note that as the ratio of these elasticities increases
so does the distortion of ellipses representing the level curves. In the extreme case
when the ratio of elasticities gets significantly high, the elliptic level curves trans-
form into flat lines representing complete dominance of one constraint in deter-
mining the optimal expected return. For instance, consider the heatmap shown in
Figure 2.11 obtained by jointly perturbing the sector bounds constraint and the
tracking error constraint. Recall that the elasticity of the tracking error constraint
(0.759) is almost 260 times higher than that of the sector bounds constraint
(0.0029). Consequently, the level curves in Figure 2.11 are flat lines that are insen-
sitive to variations in the sector bounds constraint. Clearly, heatmaps such as the
one presented in Figure 2.11 offer very little additional insight to a PM.
To summarize, constraint elasticities can be used as yardsticks to assess the
relative importance of various constraints. Furthermore, jointly perturbing con-
straints with comparable elasticities elucidates interesting aspects of the opti-
mal portfolio terrain. Perturbing constraints with vastly different elasticities,
on the other hand, offers little insights. Thus, constraint elasticities can also be
used to identify pairs of constraints that when perturbed jointly will possibly
lead to insightful outcomes.
For certain kinds of strategies that give rise to convex optimization models,
constraint elasticities can be computed using the constraint attribution utility
in our product suite. For strategies that do not satisfy this criterion, we use
more sophisticated techniques that go beyond the scope of this discussion.
2.3.2 Intractable metrics
A PM typically evaluates a portfolio using a wide variety of metrics. While
some of these metrics, such as expected return, tax characteristics, market