276 Optimizing Optimization
Table 11.3 Choice of z : bias and RMSEParameter N z 0.5 z 0.6 z 0.7 z 0.8 z 0.9 z z *
100 14.98 3.29 5.41 6.32 7.91 3.00
(^) 311.14 241.92 218.69 225.31 377.29 173.77
(^250) 5.13 3.13 2.22 0.31 2.03 0.88
(^) 136.55 116.00 113.52 128.81 156.60 88.66
(^500) 0.40 0.78 2.22 0.53 3.29 1.72
(^) 80.07 71.68 72.95 85.29 111.90 58.63
λ 100 31.31 14.64 5.16 0.15 35.62 3.10
(^) 96.59 61.43 36.18 31.15 74.24 33.70
(^250) 8.74 4.48 1.73 0.08 1.39 1.22
(^) 39.82 25.43 20.49 18.70 19.73 19.09
(^500) 4.42 1.47 0.85 0.29 0.00 0.32
(^) 23.69 16.14 14.03 13.26 14.07 12.99
γ 100 36.22 6.34 4.72 0.66 10.42 69.84
(^) 434.60 428.61 437.53 457.36 733.37 381.55
(^250) 8.07 3.68 8.97 5.91 1.82 25.11
(^) 271.76 255.29 258.10 277.38 345.05 203.42
(^500) 4.77 3.58 5.47 3.54 4.34 13.09
(^) 176.27 159.42 167.90 189.21 234.98 142.34
η 100 1.73 2.28 4.11 2.47 25.80 3.56
(^) 28.48 26.88 24.52 23.22 42.38 24.16
(^250) 3.14 4.04 2.81 1.73 1.21 1.96
(^) 23.77 19.60 16.36 14.80 15.67 14.65
(^500) 3.77 1.76 1.39 1.16 1.43 0.79
(^) 18.71 12.95 11.07 10.48 11.35 9.92
This table describes the relative bias (upper entry) and RMSE (lower entry) estimates (%) for
the parameters of the unbounded Johnson density, S U , based on 10,000 replications for each
time period and choice of z. The choice of z * is based on a 20-point grid-search procedure with
support [0.4, 1] and the Lilliefors (1967) test for normality.
where f is the true density, fˆ is the estimated Johnson density, and log denotes
the base 2 logarithm. The results are reported in Table 11.4.
In accordance with our discussion in Section 11.3.1, the results indicate that
the method of quantiles can capture the four-moment patterns of the three
alternative densities — NIG, Pearson Type IV, and Skew-T — with a high degree
of accuracy, especially for N 250. For our purposes, however, the more per-
tinent finding is that the relative bias and RMSE estimates of expected utility
are small. In each case, the estimated relative bias is less than 0.25% with a
relative RMSE of no more than 5%. To interpret these magnitudes, we calcu-
late the certainty equivalent return differential, CED, along the lines of Chopra
and Ziemba (1993) :
CED
)
(^100)
1
0
1
1
0
UU
U
() (
()
UU
U
ˆ
(11.21)