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

36 Optimizing Optimization

Portfolio optimization is as much about gaining insights into various
components — objectives and constraints — of the strategy as it is about con-
structing the final set of asset holdings. Consequently, our multisolution gen-
erator is geared toward generating a set of portfolios that not only provides
the PM a large universe of portfolios to select from, but also gives him or her
additional information to gain insights into the mechanics of portfolio con-
struction. In our attempt to design such a generator, we identified the following
three questions that every PM strives to answer.

1. What is the impact of modifying a constraint bound on the optimal portfolio?


2. What are the trade-offs in jointly varying pairs of constraint bounds?


3. Is there a good way of detecting pairs of constraints whose joint variation has non-
   trivial impact on the overall model?

What is the impact of modifying a constraint bound on the optimal portfolio?

For the sake of illustration, consider a strategy aimed at minimizing tax liability
subject to a set of constraints that includes among others a constraint that the
expected return cannot be less than a certain predetermined level, say ER 
2.14%. A PM might be interested in understanding the marginal impact of
modifying ER on the minimum attainable tax liability. Figure 2.9 shows a
typical tax liability expected return frontier.
As it is evident from the figure, a significant decrease in tax liability can be
obtained by reducing ER* by a small amount. It is exactly this kind of insight
that we endeavor to capture through our multisolution generator and that can-
not be obtained from a single optimal portfolio.

What are the trade-offs in jointly varying pairs of constraint bounds?

Consider a PM trying to rebalance his or her portfolio under a 5% tracking
error constraint and a 20% turnover constraint, and let us suppose that the

Tax liability (%)

Expected return (%)

4.50

1.95 2.00 2.05 2.10 2.15

4.00

3.50

3.00

2.50

2.00

Tax liability

Figure 2.9 Tax liability — expected return frontier.