Tables and figures

Table 1: Stock market parameters (1 year history). In the portfolio with 4 assets the Xetra DAX is treated like a single stock.

Share	$V_i(0)$	$\hat{\mu}_i$	$\hat{\sigma}_i$
Xetra DAX	2942.04	-0.54	0.45
Allianz	56.17	-1.46	0.78
BASF	38.16	-0.29	0.31
BMW	29.06	-0.50	0.32
Bayer	16.75	-0.80	0.66
Commerzbank	8.32	-0.92	0.79
DaimlerChrysler	28.90	-0.65	0.36
Deutsche Bank	44.86	-0.59	0.47
Lufthansa	8.79	-0.56	0.48
E.ON	41.80	-0.31	0.29
Hypovereinsbank	10.20	-1.41	0.92

Table 2: Estimates for the CIR-2-model (1 year history)

$\hat{a}_1$	$\hat{b}_1$	$\hat{\sigma}_1$	$\hat{\lambda}_1$
0.2648	0.0120	0.1236	-0.0647
$\hat{a}_2$	$\hat{b}_2$	$\hat{\sigma}_2$	$\hat{\lambda}_2$
1.7563	0.0145	0.1704	0.4968

Table: Adjusted covariance matrix $\hat{\Sigma}$ (4 assets)

Figure 3: Histogram of returns; normed portfolio with 4 assets, $10^4$ loops

Figure 4: Histogram of returns; ES-optimized portfolio with 4 assets (local GS), $10^4$ loops. Compared to the original portfolio in Figure 3, all values have tremendously been improved (please note the different scales).

Table 6: 4 assets; $\alpha = 0.05$; constraint $b=0$; SI-GS-optimized portfolios

Portfolio	Mean	VaR	ES	ES-RORC	ES-RORAC
normed	-29.21	148.56	169.90	-0.1719	-0.0250
Units	(256.03, 382.88, 0.0850, 4.4508)
Capital	(250.0, 250.0, 250.0, 250.0)
ES-opt.	2.20	8.02	11.67	0.1888	0.0022
Units	(575.22, 595.47, 0.0284, -0.6059)
Capital	(561.67, 388.81, 83.54, -34.03)
RORC-opt.	8.73	18.61	28.39	0.3071	0.0085
Units	(506.74, 650.08, 0.0856, -3.0470)
Capital	(494.81, 424.47, 251.87, -171.15)
RORAC-opt.	14535.32	34795.68	56428.72	0.2576	0.2531
Units	(89585.98, -170200.78, 102.93, -4952.51)
Capital	(87476.85, -111132.65, 302838.23, -278182.42)

Table 4: 4 assets; $\alpha = 0.05$; no constraints; locally GS-optimized portfolios; GS started at normed portfolio

Portfolio	Mean	VaR	ES	ES-RORC	ES-RORAC
normed	-29.21	148.56	169.90	-0.1719	-0.0250
Units	(256.03, 382.88, 0.0850, 4.4508
Capital	(250.0, 250.0, 250.0, 250.0)
ES-opt.	2.75	1.11	2.19	1.2555	0.0027
Units	(1758.10, -1049.55, -0.0115, 0.0413)
Capital	(1716.71, -685.31, -33.73, 2.32)
RORC-opt.	3.07	0.84	1.96	1.5643	0.0031
Units	(1533.04, -730.19, -0.0036, -0.1729)
Capital	(1496.95, -476.78, -10.46, -9.71)
RORAC-opt.	29454.12	70484.98	114339.65	0.2576	0.2554
Units	(186074.79, -353548.05, 208.71, -10038.90)
Capital	(181693.00, -230849.31, 614040.41, -563885.10)

Table 5: 4 assets; $\alpha = 0.05$; no constraints; SI-GS-optimized portfolios

Portfolio	Mean	VaR	ES	ES-RORC	ES-RORAC
normed	-29.21	148.56	169.90	-0.1719	-0.0250
Units	(256.03, 382.88, 0.0850, 4.4508)
Capital	(250.0, 250.0, 250.0, 250.0)
ES-opt.	1.81	4.19	6.26	0.2894	0.0018
Units	(1020.03, 0.22, 0.0013, 0.0001)
Capital	(996.01, 0.14, 3.84, 0.01)
RORC-opt.	1.88	4.29	6.37	0.2957	0.0019
Units	(1020.60, 4.22, 0.0000, 0.0119)
Capital	(996.57, 2.76, 0.01, 0.67)
RORAC-opt.	2.13	19.43	25.32	0.0843	0.0021
Units	(84.47, 1404.48, 0.0001, 0.0027)
Capital	(82.48, 917.06, 0.31, 0.15)

Table 9: 20 assets; $\alpha = 0.05$; constraint $b=0$; SI-GS-optimized portfolios

Portfolio	Mean	VaR	ES	ES-RORC	ES-RORAC
normed	-19.29	115.36	134.59	-0.1433	-0.0170
ES-opt.	0.32	6.93	8.87	0.0361	0.0003
RORC-opt.	7.13	8.28	13.79	0.5172	0.0070
RORAC-opt.	653.55	884.70	1369.21	0.4773	0.2759

Table 7: 20 assets; $\alpha = 0.05$; no constraints; locally GS-optimized portfolios; GS started at normed portfolio

Portfolio	Mean	VaR	ES	ES-RORC	ES-RORAC
normed	-19.29	115.36	134.59	-0.1433	-0.0170
ES-opt.	2.52	2.23	3.56	0.7064	0.0025
RORC-opt.	4.40	2.92	4.82	0.9129	0.0044
RORAC-opt.	651.75	842.94	1357.35	0.4802	0.2765

Table 8: 20 assets; $\alpha = 0.05$; no constraints; SI-GS-optimized portfolios

Portfolio	Mean	VaR	ES	ES-RORC	ES-RORAC
normed	-19.29	115.36	134.59	-0.1433	-0.0170
ES-opt.	-2.10	18.59	22.75	-0.0922	-0.0021
RORC-opt.	-2.22	23.76	27.84	-0.0799	-0.0021
RORAC-opt.	-2.1723	19.01	23.17	-0.0937	-0.0021