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

34 Optimizing Optimization

For the three different, second risk model constraints examples considered,
the largest out-of-sample improvement over the use of a single risk model
occurred in the second case, when the second risk constraint, using the statisti-
cal factor model, was used to limit the total risk of the portfolio.

2.2.2 Discussion and conclusions

In our examples, we have added the second risk model as a risk constraint. It
could also be incorporated into the objective function of the portfolio construc-
tion strategy, as could be done with the primary risk model constraint as well.
When a risk model is added to the objective function, it always affects the solu-
tion. There are no solutions corresponding to the nonbinding risk constraint
case. This can be advantageous in that it is impossible to choose calibration
parameters that render a risk model superfluous. In fact, many PMs mistak-
enly believe that the only way to obtain solutions on the efficient frontier is to
include the risk term(s) in the objective function.^2
There are, however, disadvantages to including risk in the objective function.
First and foremost, the actual risk predicted is unknown. It may or may not be
close to the targeted tracking error. In addition, although the risk terms always
affect the solution, it can be difficult to determine by how much they affect the
solution or when the other portfolio construction constraints are dominating

Table 2.1 Out-of-sample results from October 31, 2006 to October 31, 2007

Second risk model

Statistical act. risk Statistical tot. risk Specific risk

Asset bound 1.0% 1.0% 1.2% 1.2% 1.1% 1.1%

Second risk

bound

None 2.3% None 14.5% None 1.8%

Average assets

held

128.4 149.33 128.4 126.17 127.3 169.33

Annual active

return

3.71% 4.23% 3.58% 8.34% 3.528% 5.218%

Worst monthly

return

 1.68%  2.00%  1.79%  1.98%  1.866%  1.888%

Monthly

volume

1.43% 1.29% 1.55% 1.59% 1.464% 1.500%

Information

ratio

0.748 0.949 0.668 1.518 0.696 1.004

Transfer

coefficient

41.6% 42.9% 42.0% 41.5% 42.1% 42.6%

2 This is likely a legacy of the successful marketing of quadratic optimization programs that could
only handle risk in the objective function. SOCP solvers do not suffer this handicap.