Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Staying ahead on downside risk 155

that, in the limit as ω approaches 0.5, the dynamic model simplifies to a lin-
ear state space model where returns are assumed to follow a random walk
plus noise. As a result, the estimated expectile in the limit (i.e., the estimated
time-varying mean) is obtained from the KFS. While a random walk is argu-
ably a good approximation to the dynamics of the expectiles in the tails, it is
clearly an unrealistic model of the time-varying mean. Without constraints on
the alphas, the optimizer would simply put 100% of the weight on the FTSE
100, because it turns out to have the highest estimated mean at the end of the
sample if a random walk plus noise model is fitted to the data. In my exam-
ple, some exposure to the DAX is needed in order to satisfy the constraint on
the alphas. In other words, there is a conflict between the two estimates of
the expected returns (i.e., the KFS and the simulated alphas) and the optimizer
finds a compromise in maximizing expected returns at the portfolio level.

1.2

1

Weight

0.8

0.6

0.4

0.2

0

0.05 0.06 0.07 0.08 0.09 0.1 0.125

Omega (risk tolerance)

0.15 0.175 0.2 0.25 0.3 0.4 0.5

spx dax nky ukx

Figure 6.4 Country allocation frontier.

Table 6.3 Descriptive statistics for returns over the last

6 months in the sample period

(^) spx ukx
Kurtosis 3.85 2.31
Worst return  15.20%  12.00%
Rank 2  10.55%  8.46%
Rank 3  7.83%  7.70%