Optimal solutions for optimization in practice 59
As mentioned above, it is generally accepted that the most unstable input
to an optimization is the vector of expected returns. BITA Robust addresses
instability in alpha using its enhanced quadratic constraint functionality to set
nonlinear boundaries on forecast error (FE) mean and variance. In essence, the
process discounts the impact of past FE in alphas, thereby producing a more
stable output-reducing turnover.
As mentioned, BITA Robust employs quadratic constraints (not to be con-
fused with the basics of quadratic optimization that uses linear constraints),
which allow a number of key investment issues to be addressed:
● Uncertainty in forecast:
● Expected alpha entered as a range with a degree of confidence.
● Ranked expected returns.
● Minimizing expected FE based upon historic FE.
● Risk budgeting
● Allowing factor, or asset, or sector risk contributions to be set as constraints, ena-
bling detailed risk budgeting within the portfolio construction.
This is considered, and has indeed been proved to offer a better approach
than other variations on standard mean – variance optimization such as resam-
pling and bootstrapping.
For mathematical details on BITA Robust, please see Appendix A.
3.4.3 Reformulation of mean–variance optimization
Traditional mean – variance optimization can be couched in terms of maximiz-
ing the following expression:
Uw(,)λαT(wwBB) (wwQww) (T B)λ
(^2)
(3.1)
subject to
Awd
where
U utility;
w vector of asset weights in the portfolio;
λ degree of risk aversion.
α vector of expected returns;
w B vector of asset weights in the benchmark;
C covariance matrix of asset returns;
A matrix of linear constraints;
d vector of limits.
The mean – variance optimization problem is to find the values of w that maxi-
mize U given a degree of risk aversion λ , return expectations α , covariance
matrix C , and benchmark b , subject to the set of linear constraints.