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

Novel approaches to portfolio construction 33

Figure 2.8 shows contour plots of the cumulative and worst monthly active
returns as functions of the asset bound X and the specific risk model risk con-
straint level Y for TO  30%.
The results are similar to those reported in the previous examples. The best
solution occurs when X  1.1% and Y  1.8%.

2.2.1 Out-of-sample results

We performed an out-of-sample backtest from October 31, 2006 to October
31, 2007. We considered six different strategies, three without the second
risk model and three with the second risk model. The cases correspond to the
optimal parameters determined by the results above. These different cases are
summarized in the top of Table 2.1. TO  30% and the primary tracking error
from the fundamental factor model is 4%. When running the backtest, we
start from October 31, 2005 and run 24 monthly rebalancings so that the turn-
over for the first, out-of-sample month is meaningful. The results reported are
only for the 12 out-of-sample rebalancings.
Table 2.1 shows the performance statistics for these six cases. In all cases, the
addition of a properly calibrated second risk model constraint leads to supe-
rior portfolio performance as measured by either the strategy’s annual active
return or its information ratio. In general, the risk metrics — volatility or worst
monthly return — changed only slightly (in either direction) while the return
increases substantially. Of course, there are only 12 out-of-sample returns, so
the performance statistics have relatively large standard errors. Nevertheless,
the performance statistics have moved in a favorable direction when the second
risk model was added to the strategy and all the parameters were calibrated
adequately.

–8.84 to –8.80

–8.91 to –8.88

–8.99 to –8.95

–9.07 to –9.03

–9.14 to –9.10

–9.22 to –9.18

–9.29 to –9.26

–9.45 to –9.41

–9.37 to –9.33

–9.52 to –9.48

–9.60 to –9.56

–9.75 to –9.71

–9.67 to –9.64

15.91–16.10

15.52–15.72

15.14–15.33

14.76–14.95

14.37–14.56

13.99–14.18

13.60–13.80

12.84–13.03

13.22–13.41

12.45–12.64

12.07–12.26

11.30–11.49

11.68–11.88

1.6 123456710

1.8

2.0

2.2

2.4

2.6

Y^ =

Active specific risk (%)

X = Active asset bound (%)

1.6 12345

67 10

1.8

2.0

2.2

2.4

2.6

X = Active asset bound (%)

Cumulative active return (%) Worst monthly active return (%)

Figure 2.8 The cumulative and worst-case monthly active returns versus asset bound
and active specific risk limit for TO  30%.