60 Optimizing Optimization
BITA Robust applies an interior point method to solve a “ second-order
cone problem ”. This has enabled the new functionality of applying quadratic
constraints. Documentation on the method is in Appendix A. Using BITA
Robust Optimizer to apply quadratic inequality constraints, the above can be
reformulated as:
Uw w wT
(,)λα( B) (3.2)subject to
Awdand the risk constraint (s):
jk, ()( )()www wQ TEjBjkBkijn
∑ 1
^2
(3.3)or
σψψσε^2222
11 1
^2
j k jk i i Bi
inkmjm
∑∑∑ ()()ww TE
(3.4)in the structured factor model case, where:
σ 2 variance of portfolio p ;
ψ (^) j sensitivity of portfolio p to factor j βij i Bi
i
n
()ww
1
∑ ;
σjk^2 covariance of factor j and factor k ;
w i weight of security i in the portfolio;
w Bi weight of security i in the benchmark;
i error term (residual risk);
TE benchmark relative portfolio tracking error.
Equations (3.3) and (3.4) are identical, but Equation (3.3) expresses TE in
terms of the full covariance matrix, while Equation (3.4) expresses it in terms
of factor and residual risk. This eliminates the need for the problematic scal-
ing coefficient λ. Being in the units of “ marginal expected return with respect
to variance, ” λ is difficult to evaluate with confidence. This ambiguity in λ is
exacerbated by instability in alpha magnitudes when the binding mandate con-
straint is TE. In practice, users of optimization methods that require an objec-
tive function of the form in Equation (3.1) tend to make an initial guess at a
value for λ , and subsequently tweak the value to achieve the tracking error
required. Many products, including BITA Optimizer, mechanize this by putting
an optimization script in place that automates this process.
Using a risk constraint rather than a risk term in the objective function ena-
bles the user to put a straightforward constraint on tracking error and the user
will know, prior to execution, whether or not the problem is feasible.