32 Optimizing Optimization
used as the primary risk model and constrains the active risk (tracking error) of
the portfolio to 4%, as in the above case.
The motivation for this strategy is the fact that statistical factor risk mod-
els normally have fewer factors than fundamental factor risk models. Since
the minimum half-life of the historical time series is affected by the number of
factors, statistical models can use a shorter half-life and can therefore respond
more quickly to market movements, which, in turn, may be captured more by
total risk predictions than by active risk predictions.
We test the same universe and rebalancing frequency as in the previous exam-
ple. For sake of brevity, we will only report results for TO 30%. Figure 2.7
shows contour plots of the cumulative and worst monthly active returns as func-
tions of the asset bound X and the specific risk model total risk constraint level Y.
The results are similar to those reported in the previous example. The best
performance generally occurs as the second risk model constraint is tightened
until the primary risk model constraint is almost nonbinding. The best solution
for this case is approximately X 1.2% and Y 14.5%.
Example 3. Constraining specific risk as the second risk model constraint
For our third example, the second risk model is the specific risk predicted by
Axioma’s Japanese, fundamental factor model. This is a diagonal (uncorre-
lated) risk model that is meant to give results that are similar to those obtained
using classical robust portfolio construction with a diagonal, estimation error
matrix. Specific risk is a convenient “ second ” risk model to use since it is
already specified by any factor model (both fundamental and statistical mod-
els). Nevertheless, such strategies may benefit from a well-chosen, diagonal,
second risk model.
(^13123456710123456710)
14
15
16
17
18
(^19) 16.42–16.70
15.85–16.13
15.28–15.56
14.71–15.00
14.14–14.43
13.58–13.86
13.01–13.29
11.87–12.16
12.44–12.72
11.30–11.59
10.74–11.02
9.60–9.88
10.17–10.45
–7.64 to –7.55
–7.81 to –7.72
–7.98 to –7.90
–8.16 to –8.07
–8.33 to –8.24
–8.50 to –8.42
–8.68 to –8.59
–9.03 to –8.94
–8.85 to –8.77
–9.20 to –9.11
–9.37 to –9.29
–9.72 to –9.63
–9.55 to –9.46
18
Y^ =
Total statistical risk (%)
13
14
15
16
17
19
X = Active asset bound (%) X = Active asset bound (%)
Cumulative active return (%) Worst monthly active return (%)
Figure 2.7 The cumulative and worst-case monthly active returns versus asset bound
and total, statistical risk limit for TO 30%.