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

Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 267

Although instructive, the optimal ρ implied by these figures was calibrated using
6 months of historical data, i.e., T  168, and so there is no guarantee that it will
still be optimal for other data lengths. In response, we derive a data-dependent
formula for selecting the optimal decay factor, ρ  ρ ( T ), using the generalized
logistic transform. Specifically, we repeat the above simulation procedure, but this
time we randomly choose between 3 and 18 months of data, i.e., T ~ Uniform

[56, 504], for each of the N  250 repetitions. The resulting Tii
i

N

{ ,ρ*} 1 are then

used to estimate the parameters C 1 and C 2 in the logistic transform:

ρ*

exp( ( ))









UL

CCTT

L

1 12 (11.13)

where T 280 and L and U are the upper and lower asymptotes of ρ *,
respectively. After linearizing the transform:

ln

*

*

ln

UL

L

CCTT







ρ

ρ

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟ ()^12 ( )

(11.14)

we set L  0.5 and U  1 and then apply standard least-squares estimation to
yield Cˆ 1 0.2612 and Cˆ 2 0.0094. Figure 11.6 provides a graphical illus-
tration of the resulting rule for choosing the decay factor.

0 100 200 300 400 500 600 700 800 900 1000

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

No. of observations,T

Optimal decay coefficient, Rho

Figure 11.6 Optimal hyperbolic decay coefficient, ρ.