Improving the volatility of the optimal weights of the Markowitz model

Figures & data

Figure 1. Capital market line.

Source: Authors.

Figure 2. Density of the eigenvalues of the covariance matrix Σ and of a random matrix.

Source: Authors.

Figure 3. Out-of-sample Sharpe ratio for different shrinkage levels for a sample size of 200.

Source: Authors.

Figure 4. Out-of-sample Sharpe ratio for different shrinkage levels for a sample size of 300.

Source: Authors.

Figure 5. Out-of-sample Sharpe ratio for different shrinkage levels for a sample size of 400.

Source: Authors.

Figure 6. Out-of-sample Sharpe ratio for different shrinkage levels for a sample size of 500.

Source: Authors.

Table 1. Means and standard deviations of the portfolio weights.

Values of weights
Markowitz rule without adjustment					Markowitz rule with optimal Adjustment
	Sample size 200		Sample size 500		Sample size 200		Sample size 500
	Mean	Standard deviation	Mean	Standard deviation	Mean	Standard deviation	Mean	Standard deviation
1	0.61	4.11	0.29	0.19	0.08	0.06	0.13	0.05
2	−0.58	5.88	−0.15	0.21	0.04	0.16	0.00	0.11
3	0.38	3.72	0.25	0.09	0.10	0.10	0.20	0.04
4	−0.04	1.14	−0.21	0.28	−0.01	0.05	−0.07	0.06
5	0.27	2.87	0.21	0.40	0.03	0.09	0.07	0.11
6	−0.63	1.92	−0.54	0.27	0.00	0.03	−0.07	0.03
7	−0.18	1.89	−0.16	0.26	0.00	0.05	−0.08	0.05
8	−0.24	2.77	0.29	0.17	0.06	0.08	0.15	0.06
9	0.31	1.56	0.24	0.34	0.04	0.06	0.08	0.06
10	−0.21	3.90	−0.16	0.16	−0.01	0.07	−0.10	0.05
11	0.41	5.59	−0.21	0.16	0.01	0.02	−0.04	0.02
12	−1.11	5.06	−0.38	0.21	0.00	0.04	−0.07	0.03
13	0.12	1.30	0.16	0.24	0.02	0.03	0.01	0.03
14	0.44	8.09	−0.09	0.24	0.02	0.05	−0.02	0.04
15	−0.31	3.27	−0.02	0.23	0.00	0.04	−0.05	0.04
16	0.27	1.59	0.26	0.15	0.04	0.06	0.07	0.05
17	0.36	0.88	0.16	0.13	0.03	0.09	0.06	0.05
18	0.15	0.81	0.04	0.12	0.03	0.11	0.03	0.06
19	0.53	3.18	0.12	0.23	0.06	0.12	0.11	0.11
20	−0.19	6.34	0.20	0.17	0.15	0.21	0.28	0.17
21	0.63	7.28	0.31	0.21	0.05	0.09	0.14	0.08
22	0.35	1.03	0.23	0.16	0.04	0.05	0.09	0.06
23	−0.13	3.61	−0.06	0.26	0.03	0.05	0.01	0.04
24	0.37	2.08	0.29	0.26	0.03	0.07	0.07	0.05
25	−0.32	2.59	−0.23	0.10	0.01	0.04	−0.05	0.02
26	−0.29	2.65	−0.16	0.24	0.02	0.06	−0.03	0.05
27	0.20	1.15	0.07	0.18	0.03	0.04	0.02	0.03
28	−0.12	0.74	−0.11	0.15	0.00	0.06	−0.06	0.07
29	0.20	2.03	0.39	0.45	0.07	0.14	0.12	0.11
30	−0.24	4.43	−0.01	0.17	0.03	0.04	0.00	0.03

Source: Authors.

Table 2. The effect of the eigenvalues of the covariance matrix.

Percentage of times
	Sample size 200		Sample size 500
	Without adjust	With optimal adjust	Without Adjust	With optimal Adjust
Eigenvalues less than inferior limit	77.47%	0.00%	88.00%	0.00%
Eigenvalues higher than superior limit	3.56%	3.33%	3.33%	3.33%

Source: Authors.

Figure 7. Density of the eigenvalues for different degrees of compression.

Source: Authors.

Figure 8. Shrinkage for a sample size of 200 months.

Source: Authors.

Figure 9. Shrinkage for a sample size of 200 months.

Source: Authors.

Figure 10. Out of-sample Sharpe ratio versus sample size.

Source: Authors.

Table 3. Means and standard deviations of the portfolio weights.

Weight values
	Markowitz		Ledoit and Wolf (2004a,b)		OSME
	Mean	Standard deviation	Mean	Standard deviation	Mean	Standard deviation
1	0.61	4.11	0.13	0.19	0.08	0.04
2	−0.58	5.88	−0.14	0.43	0.04	0.09
3	0.38	3.72	0.22	0.30	0.10	0.06
4	−0.04	1.14	−0.12	0.16	−0.01	0.03
5	0.27	2.87	0.10	0.25	0.03	0.06
6	−0.64	1.92	−0.14	0.14	0.00	0.02
7	−0.18	1.89	−0.07	0.14	0.00	0.03
8	−0.25	2.77	0.18	0.19	0.06	0.04
9	0.31	1.56	0.08	0.20	0.04	0.03
10	−0.21	3.90	−0.14	0.20	−0.01	0.04
11	0.41	5.59	−0.05	0.11	0.01	0.01
12	−1.11	5.06	−0.16	0.17	0.00	0.02
13	0.12	1.30	0.01	0.10	0.02	0.02
14	0.44	8.09	−0.04	0.20	0.02	0.03
15	−0.32	3.27	−0.05	0.14	0.00	0.03
16	0.28	1.59	0.11	0.17	0.04	0.04
17	0.36	0.88	0.10	0.22	0.03	0.06
18	0.15	0.81	0.08	0.22	0.03	0.07
19	0.53	3.18	0.17	0.29	0.06	0.07
20	−0.19	6.34	0.21	0.46	0.15	0.14
21	0.63	7.28	0.22	0.26	0.05	0.05
22	0.35	1.03	0.10	0.11	0.04	0.03
23	−0.13	3.61	0.04	0.16	0.03	0.03
24	0.37	2.08	0.03	0.21	0.03	0.04
25	−0.32	2.59	−0.08	0.17	0.01	0.02
26	−0.29	2.65	−0.06	0.19	0.02	0.03
27	0.20	1.15	0.04	0.10	0.03	0.02
28	−0.12	0.74	−0.08	0.15	0.00	0.04
29	0.20	2.03	0.23	0.51	0.07	0.09
30	−0.24	443	0.0^4	0.10	0.03	0.03

Source: Authors.

Table 4. Out-of-sample performance.

Out-of-sample performance (Sharpe Ratio)
Sample Size	Markowitz	Rule 1/n	LW 2020	LW 2003	LW 2004 A	LW 2004 B	011ila 1	011ila 2	OSME
100	−0.0182	0.1692	−0.0249	−0.0278	0.0333	0.0395	0.0433	0.0152	0.1070
150	0.0356	0.1691	−0.0198	−0.0078	−0.0125	−0.0274	0.0320	0.0038	0.1125
200	0.0011	0.1725	0.0847	0.0835	0.0944	0.0468	0.1086	0.0982	0.1928
250	0.1138	0.1622	0.1250	0.1240	0.1325	0.1171	0.1389	0.1341	0.1890
300	0.1641	0.1581	0.1730	0.1710	0.1797	0.1696	0.1845	0.1800	0.2110
350	0.1525	0.1484	0.1589	0.1558	0.1624	0.1568	0.1675	0.1630	0.2013
400	0.1480	0.1424	0.1553	0.1536	0.1601	0.1574	0.1660	0.1608	0.1942

Source: Authors.

Table 5. Performance measures using the Sharpe and Sortino ratios.

Method	Mean	Standard deviation	Downside deviation	Sharpe ratio	Sortino ratio
Markowitz	0.001	0.740	0.050	0.001	0.016
Rule 1/n	0.009	0.055	0.029	0.172	3.296
LW 2020	0.018	0.214	0.009	0.085	2.047
LW 2003	0.017	0.203	0.085	0.084	1.987
LW 2004 A	0.015	0.158	0.076	0.095	1.957
LW 2004 8	0.030	0.634	0.016	0.047	1.869
011ila 1	0.014	0.129	0.062	0.109	2.259
011ila 2	0.015	0.151	0.073	0.098	2.043
OSME	0.010	0.049	0.027	0.193	3.492

Source: Authors.

Table 6. Standard deviations of the optimal weights.

Standard deviations of Markowitz model optimal weights with different methods
Asset	Markowitz	LW 2003	LW 2004 A	LW 200413	LW 2020	011ila 1	011ila 2	OSME
1	4.12	1.08	0.61	2.11	0.94	0.44	0.57	0.06
2	5.89	1.59	1.23	4.36	1.75	0.99	1.18	0.16
3	3.72	0.88	0.66	2.87	1.01	0.52	0.63	0.10
4	1.15	0.47	0.50	0.58	0.54	0.40	0.48	0.05
5	2.87	0.94	0.71	1.88	0.93	0.58	0.69	0.09
6	1.92	1.08	0.62	1.15	0.81	0.46	0.59	0.03
7	1.90	0.50	0.43	0.78	0.53	0.34	0.41	0.05
8	2.78	0.73	0.56	1.07	0.84	0.46	0.54	0.08
9	1.57	0.84	0.60	1.09	0.78	0.47	0.58	0.06
10	3.90	0.97	0.69	2.57	1.06	0.54	0.66	0.07
11	5.59	1.49	0.88	2.35	1.31	0.60	0.83	0.02
12	5.07	1.30	0.91	2.41	1.29	0.66	0.85	0.04
13	1.30	0.67	0.41	0.54	0.59	0.31	0.40	0.03
14	8.09	1.92	1.20	4.09	1.92	0.87	1.12	0.05
15	3.28	0.85	0.54	1.56	0.81	0.41	0.51	0.04
16	1.59	0.42	0.38	1.06	0.50	0.32	0.37	0.06
17	0.89	0.45	0.48	0.73	0.54	0.41	0.47	0.09
18	0.81	0.32	0.33	0.69	0.35	0.30	0.32	0.11
19	3.18	0.82	0.68	2.37	0.97	0.57	0.66	0.12
20	6.34	1.38	1.01	4.70	1.53	0.77	0.96	0.21
21	7.28	1.68	1.20	4.78	1.84	0.90	1.13	0.09
22	1.04	0.41	0.31	0.75	0.42	0.26	0.30	0.05
23	3.61	1.01	0.92	1.56	1.22	0.70	0.88	0.05
24	2.08	0.83	0.63	0.92	0.73	0.49	0.60	0.07
25	2.59	0.76	0.56	2.03	0.84	0.45	0.54	0.04
26	2.65	0.94	0.68	1.64	0.92	0.54	0.65	0.06
27	1.15	0.47	0.33	0.47	0.43	0.26	0.32	0.04
28	0.74	0.35	0.33	0.41	0.39	0.30	0.33	0.06
29	2.03	1.11	0.90	1.39	1.09	0.72	0.87	0.14
30	4.43	0.97	0.56	1.90	0.91	0.38	0.52	0.04
Mean	3.12	0.91	0.66	1.83	0.93	0.51	0.63	0.07

Source: Authors.

Ledoit, O., & Wolf, M. (2004a). Honey, i shrunk the sample covariance matrix. The Journal of Portfolio Management, 30(4), 110–119. https://doi.org/10.3905/jpm.2004.110

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Ledoit, O., & Wolf, M. (2004b). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2), 365–411. https://doi.org/10.1016/S0047-259X(03)00096-4

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