40 Optimizing Optimization
Similar to various concepts in Economics such as price elasticity of supply
or income elasticity of demand, the concept of constraint elasticity is unit inde-
pendent. In other words, by working with percentage changes, and not “ unit ”
changes, in constraint bounds it allows us to assess the impact of changing a
heterogeneous set of constraints such as tracking error constraints, turnover
constraint, industry, asset, sector, style bound constraints, etc. Recall that these
constraints can have widely varying bounds that can differ by orders of mag-
nitude. Consequently, comparing models obtained by perturbing these con-
straints by a single unit hardly gives any insights into their marginal impact on
the overall model.
Next , we discuss an example to illustrate the concept of constraint elasticity
and the manner in which it can be used to answer the three questions raised
above. Consider the following strategy, referred to as “ ActiveLongOnly ” strat-
egy in the remainder of this chapter.
Maximize expected return
s .t
● Budget constraint
● Tracking error w.r.t a benchmark (maximum 5%)
● Turnover constraint (maximum 30%)
● Asset, industry, sector, style bound constraints
● Threshold holding and trading constraints
● Average daily volume holding and trading constraint
● Beta constraint
Table 2.2 displays the elasticities of various constraints in the ActiveLongOnly
strategy.
The first column of the table reports the constraint name, while the second
column reports the elasticity of the respective constraint. Thus, according to
the following table, 1% change in the tracking error bound from 5% to 5.05%
will produce a 0.759% change in the optimal expected return. Similarly, the
elasticities of the remaining constraints quantitatively assess the impact of
changing the constraint bound on the optimal objective value.
Table 2.2 Constraint elasticities (ActiveLongOnly strategy)Constraint ElasticityTracking error 0.75902
Maximum turnover 0.04825
Asset bounds 0.01277
Sector bounds 0.00291
Threshold trade 0.00003
Beta 0.00000
Style bounds 0.00000
Industry bounds 0.00000
Threshold holding 0.00000