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

Novel approaches to portfolio construction 29

factor model risk constraint or by raising the max asset bound ( X ) constraint.
The most diversified portfolios are obtained by doing both simultaneously.
Figures 2.2 and 2.3 show contours of the cumulative (in this case, annual),
active return and the single, worst, monthly active return over the course of the
12 rebalancings. Neither of these results includes transaction costs or market
impact.
Adding the Japanese statistical factor model risk constraint generally
increases the portfolio return and improves the worst monthly return. The
strategy parameters of X  1.0% and Y  2.3% produce a large cumulative
return and the smallest worst monthly return. This solution performs better
than any of the solutions without the secondary statistical factor model risk
constraint (the white region above the shaded regions). These parameter values
are the suggested strategy calibration for TO  30%.
Figures 2.4 – 2.6 below show the same calibration results for TO  15%,
30%, and 60%. The figures show that TO has a profound effect on the port-
folios obtained: tighter TO constraints narrow the region over which both risk
models are simultaneously binding. As a practical matter, for strategies similar
to TO  15%, it may be difficult to keep a strategy in the dual-binding region
unless it is calibrated explicitly. (The shaded regions shown here are averages
over all 12 rebalancings.)

4.0

2.2

1

X = Active asset bound (%)

Fundamental risk constraint not binding

Statistical risk constraint not binding

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2.4

2.6

2.8

Y

= Active statistical risk (% Ann.)

3.0

3.2

3.4

3.6

3.8

17.3–17.5

16.8–17.0

16.3–16.5

15.8–16.0

15.3–15.5

14.8–15.0

14.3–14.5

13.3–13.5

13.8–14.0

12.8–13.0

12.3–12.5

11.8–12.0

11.3–11.5

Figure 2.2 Cumulative, active return (%) when both risk models are binding. TO  30%.