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

Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 277

Table 11.4

Performance of the method of quantiles in the case of density misspecification

True: NIG

True: Pearson Type IV

N



100

N



250

N



500

N



100

N



250

N



500

Mean

0.64(21.83)

0.15(13.74)

0.07(9.85)

0.66(21.59)

0.69(13.82)

0.34(9.74)

Std

1.86(13.49)

0.69(7.33)

0.46(5.01)

0.62(11.29)



0.17(6.67)



0.39(4.60)

Skewness



9.03(143.60)

1.71(81.11)

3.91(54.70)



23.86(121.88)



17.04(70.19)



15.71(48.94)

Kurtosis

45.32(159.83)

18.38(51.10)

11.64(30.42)

16.88(94.76)



2.57(33.82)



8.34(21.25)

EU

0.22(4.59)

0.04(1.53)

0.02(1.08)

0.06(3.08)



0.07(1.50)



0.06(1.05)

L

1

0.1493

0.0943

0.0669

0.1477

0.0931

0.0650

L

2

0.0386

0.0149

0.0074

0.0372

0.0143

0.0068

KL

0.0353

0.0149

0.0076

0.0335

0.0143

0.0071

True: Skew-T

True: Johnson Unbounded

N



100

N



250

N



500

N



100

N



250

N



500

Mean



1.28(21.87)



1.24(13.93)



0.53(10.12)

1.10(21.62)

0.32(13.72)

0.21(9.76)

Std

1.13(11.35)

0.05(6.85)

0.16(4.79)

0.95(11.37)

0.14(6.52)



0.05(4.58)

Skewness



5.33(119.62)

0.42(72.11)

3.62(49.95)



16.06(135.36)



8.84(76.93)



5.36(52.64)

Kurtosis

15.17(77.20)



0.76(34.22)



4.26(22.57)

26.67(99.72)

6.13(36.46)

0.66(21.52)

EU

0.26(2.51)

0.14(1.53)

0.06(1.11)

0.02(2.47)



0.01(1.49)



0.02(1.06)

L

1

0.1472

0.0963

0.0691

0.1472

0.0926

0.0660

L

2

0.0367

0.0154

0.0078

0.0368

0.0140

0.0070

KL

0.0333

0.0147

0.0077

0.0331

0.0142

0.0073

This table describes the relative bias (RMSE) estimates of the first four moments and expected utility

, as well as the three distance

criteria using 10,000 replications for each time period,

N



{ 100, 250, 500 }. We set

z^



0.65 for convenience, and assume a coefficient

of relative risk aversion equal to two, i.e.,

α^



2, with no disappointment aversion, i.e.,

A



1. The parameter values for the four

distributions, using the notation given in each of the references, are:

α^



7.06,

β^





1.58,

δ^



0.26, and

μ^



0.16 (NIG);

a^



0.47,

λ^



0.22,

ν^



1.80, and

m



4.46 (Pearson Type IV);

μ^



0.1,

σ^



0.2,

η^



7.49, and

λ^





0.18 (Skew Student’s-

t ); and

γ^



0.54,

η^



2.12,

ε^



0.20, and

λ^



0.36 (Unbounded Johnson,

S U

).