Optimal solutions for optimization in practice 65
3.4.7 Mean forecast intervals
Following a suggestion by one of their clients, the BITA team incorporated
mean forecast intervals into optimization processes. An example of such a fore-
cast would be, “ We assume that the mean return for equities lies between 6%
and 8% p.a. ” The forecast uncertainty is captured by the interval around the
expected mean (7% in the example forecast). Consistently, this allows assum-
ing zero FE for the expected mean.
The formulation of this problem is comparable to the application of FE
constraints. Equation (3.5), however, becomes redundant with regard to the
assumption that uncertainty is completely captured by adding a forecast inter-
val. Further assuming that FE across asset classes are independent, we can
reduce Equation (3.6) to:
k ()[]wwIk Ok FE Skn
∑ 1
^22 σ^2
(3.7)where
w I initial weight;
w O optimal weight;
σ 2 [ FE k ] variance of FE k.
To compute the FE standard deviation, we map the mean forecast interval to
a 95% confidence interval. 1 Assuming that asset returns follow a normal distri-
bution, we can induce the standard deviation from the relationship:
dFEkk196.[]σwhere
d k error from the estimation (half the interval).
For the example given above, we would have d k 0.01 and
σ [ FE k ] 0.51%.
Hence , adding Equation (3.7) as a constraint to the general mean – variance
optimization problem in Equation (3.2), where we use the interval mean as
the alphas, provides us with an additional application for BITA Robust. The
importance of this approach relative to FE constraints is that it needs much
lower data requirements. It still, however, allows the additional benefit of
robust optimization to add value.
3.4.8 Explicit risk budgeting
BITA Robust enables a generic functionality to apply quadratic constraints to an
objective function. The FE framework proposed above is not the only potential
1 Since the forecast interval can be as wide or narrow as the investor wishes (depending on his or
her confidence), we keep the confidence interval fixed at 95%. However, it would, in general, be
possible to vary the level to fit the user’s degree of confidence.