262 Optimizing Optimization
Even though one may have access to vast quantities of historic data, most
practitioners would therefore caution against the use of the entire history for
modeling purposes or the na ï ve assumption that the information content of all
subsets of the data is identical. The popularity of “ moving window ” proce-
dures that ignore observations beyond a certain arbitrary distance in the past,
while equally weighting recent ones, is a testament to this.
Although we agree with the sentiment behind by the removal of old data, we
believe that information decay is a more gradual process than implied by the
binary nature of the moving window approach. To capture this idea, we intro-
duce a vector of nondecreasing weights π (^) t 0 ∀ t , such that Σ π (^) t 1 (cf.
Mittnik & Paolella, 2000 ). By giving more weight to recent observations, it is
hoped that parameter estimates will more closely reflect the “ current ” value of
the “ true ” parameter. It is interesting to note that the idea of weighting recent
events more heavily is embedded in the decision weight function from prospect
theory. Such “ recency effects ” have been discussed by Kahneman and Tversky
(1979) , Hogarth and Einhorn (1992) , and Kahneman (1995) who all argued that
recently sampled outcomes receive greater weight than earlier sampled ones.
To determine the (weighted) parameter estimates, we combine the stand-
ardized weights, { π (^) t } , with the quantile estimator described in Section 11.3.2.
Specifically, we sort the data into ascending order and then continue to sum
the weights of the sorted data until the appropriate quantile value has been
reached ( Boudoukh, Richardson, & Whitelaw, 1998 ). For instance, to calcu-
late the q th quantile, we start from the lowest return and keep accumulating
the weights until q is reached. Linear interpolation is used between adjacent
points to achieve a more accurate estimate of the desired quantile.
Following Mittnik and Paolella (2000) , we consider two weight schemes, a
geometric scheme, for which π (^) t ρ T t , and a hyperbolic scheme, for which
π (^) t ( T t 1) ρ^ ^1. In both cases, a value of ρ 1 ( ρ 1) means that the
most recent observations are given relatively more (less) weight than those
values far in the past, while ρ 1 corresponds to standard moving window
estimation. The two weight schemes are illustrated in Figure 11.3.
11.6 Alpha information
To the extent that they provide additional insights into the future evolution of
the price process, asset return forecasts ( “ alphas ” ) have long been an integral
part of portfolio optimization. 19 The objective of this section is therefore to
illustrate how such forecasts can be incorporated into the existing framework
via the method of Bayesian updating. Formally, if we let X denote the portfolio
return whose realization x is being forecasted, and if we let Y denote the point
19 In spite of mixed evidence surrounding the issue of return predictability ( Goyal & Welch, 2008 ;
Campbell & Thompson, 2008 ; Boudoukh, Richardson, & Whitelaw, 2008 ), financial forecasts
are still a crucial input into the asset allocation process.