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

Novel approaches to portfolio construction 31

For TO  15%, the best cumulative returns are given by either a max asset
holdings bound constraint ( X )  1.0% or X  10%. The best, worst-case
return is achieved for similar values of X , with X  1.0% being slightly better.
We therefore conclude that the best calibration is X  1.0% and Y  2.6%.
This solution has the largest number of assets held (about 150).
For TO  60%, the story is similar. The region over which both risk models
are binding is substantially broader. Cumulative return is maximized by taking
a relatively low asset holdings bound ( X  1.1%) and a tight statistical risk
model risk constraint ( Y  1.7%).

Example 2. Constraining the total risk predicted by the statistical risk model

For our next example, the second risk model is, once again, Axioma’s Japanese
statistical factor model. In this case, however, this second model is used to con-
strain the total risk of the portfolio instead of the active risk as was done in the
first example. The first risk model, Axioma’s fundamental factor model, is still

4.0

Y =

Active statistical risk (%)

3.5

3.0

2.5

2.0

1.5

123

15% monthly TO

4567

17.14–17.50

16.43–16.79

15.72–16.08

15.01–15.36

14.30–14.65

13.58–13.94

12.87–13.23

12.16–12.52

11.45–11.80

10.74–11.09

10.02–10.38

9.31–9.67

8.60–8.96

10 1 2 3 4 5 6 7 10 1 2 34567 10

30% monthly TO 60% monthly TO

X = Active asset bound (%) X = Active asset bound (%) X = Active asset bound (%)

Figure 2.5 The cumulative, active return (%) versus asset bound and the statistical risk
model risk constraint for TO  15%, 30%, and 60%.

4.0

Y

= Active statistical risk (%)

3.5

3.0

2.5

2.0

1.5

123

15% monthly TO

4567

–7.98 to –7.90

–8.15 to –8.07

–8.32 to –8.24

–8.49 to –8.40

–8.66 to –8.57

–8.82 to –8.74

–8.99 to –8.91

–9.16 to –9.08

–9.33 to –9.24

–9.50 to –9.41

–9.66 to –9.58

–9.83 to –9.75

–10.00 to –9.92

10 1 2 3 4 5 6 7 10 1 2 34567 10

30% monthly TO 60% monthly TO

X = Active asset bound (%) X = Active asset bound (%) X = Active asset bound (%)

Figure 2.6 The worst-case monthly active return (%) versus asset bound and the
statistical risk model risk constraint for TO  15%, 30%, and 60%.