Optimal solutions for optimization in practice 61
3.4.4 BITA Robust applications to controlling FE
Regardless of whether risk is a term in the objective function, or applied as a
constraint, both Equations (3.1) and (3.2) suffer from robustness issues. Both
posit that the precise weights, w , can be extracted as outputs from this process
given the uncertain vector of alphas, α. As α is not known with certainty and is
subject to frequent and dramatic change, the output portfolio composition, w , is
disproportionately unstable. In practical situations, a portfolio manager would
still need to implement a turnover constraint to both methods to attenuate
the effect of changing inputs on the output portfolio.
3.4.5 FE constraints
As one potential solution to this issue, we propose to introduce the concept of
constraining FE. FE is the difference between the ex ante expectation of return
and the realized return in the subsequent corresponding period. The FE on a
security is easily measured and is the most straightforward statistic for how
well an alpha predicts actual return of a security. It is not dependent on alpha
building methodology — whether an alpha is built via multifactor model or
expert opinion, a FE can be calculated. Even in cases where the alpha is a rank-
ing or a sign, an analogous term can be constructed.
Observing FE for a historic period of at least 30 observations is recom-
mended. This can be days, weeks, months, or any other period, but should cor-
respond to the investment horizon of the strategy.
In parametric statistics, mean and standard deviation go a long way toward
describing the behavior of a distribution. In a “ perfect world ” where alphas
always correspond to subsequent returns, the mean and standard deviation of
the FE sample would both be zero.
In the “ real world, ” Robust Optimization is about constraining these two
parameters in the realized portfolio to be below the respective constants, which
we denote by M and S :
k ()[]wwIk Ok FE Mkn
∑ 1 ()
μ(^22)
(3.5)
and
jk, ()( )[,]wIj wOj wIk wOk FE FEj k S
n
∑ 1 σ^2
(3.6)
where
w I initial weight;
w O optimal weight;
μ [ FE k ] mean FE;
σ [ FE j , FE k ] covariance between FE j and k.