Analysing assets’ performance inside a portfolio: From crossed beta to the net risk premium ratio

Figures & data

Table 1. Net risk premium ratio

Asset	Mean-variance								Mean-Gini
	Treynor (%)	Rank	η	Rank	η^*	Rank	$(\mathit{\varpi })$	Rank	Treynor (%)	Rank	η	Rank	η^*	Rank	$(\mathit{\varpi })$	Rank
AXP	0.07	21	0.45	21	1.043	24	−0.592	21	0.07	22	0.429	22	1.027	25	−0.597	22
BA	−0.05	26	−0.31	26	1.064	21	−1.376	26	−0.05	26	−0.306	26	1.033	23	−1.339	26
CAT	0.07	22	0.44	22	1.045	23	−0.601	22	0.08	20	0.464	20	1.023	28	−0.559	20
CSCO	0.56	5	3.48	6	1.069	20	2.408	6	0.60	4	3.654	4	1.047	17	2.607	4
CVX	−0.09	29	−0.58	29	1.150	5	−1.727	28	−0.09	29	−0.546	29	1.069	10	−1.615	29
DD	0.27	15	1.61	15	1.031	29	0.577	15	0.28	13	1.620	15	1.017	29	0.603	15
DIS	0.51	8	3.13	8	1.071	18	2.058	8	0.52	7	3.186	7	1.052	16	2.134	7
GE	−0.08	28	−0.52	28	1.118	9	−1.639	27	−0.08	28	−0.495	28	1.079	8	−1.573	27
GS	0.16	17	0.95	17	1.040	27	−0.088	17	0.16	17	0.950	17	1.024	26	−0.074	17
HD	0.56	6	3.45	7	1.070	19	2.384	7	0.49	8	2.948	8	1.037	21	1.910	8
IBM	−0.43	30	−3.07	30	1.239	2	−4.313	30	−0.45	30	−2.943	30	1.127	2	−4.071	30
INTC	0.65	4	4.08	4	1.084	14	2.991	4	0.64	3	3.838	3	1.045	19	2.793	3
JNJ	0.30	12	1.91	12	1.100	10	0.815	12	0.31	11	1.952	11	1.078	9	0.874	11
JPM	0.12	18	0.79	18	1.121	7	−0.329	18	0.12	18	0.767	18	1.066	11	−0.299	18
KO	0.18	16	1.19	16	1.134	6	0.059	16	0.17	16	1.082	16	1.079	7	0.003	16
MCD	0.02	25	0.09	25	1.041	26	−0.950	25	0.02	25	0.105	25	1.045	18	−0.940	25
MMM	0.29	13	1.71	14	1.036	28	0.675	14	0.29	12	1.695	13	1.023	27	0.672	12
MRK	0.34	10	2.16	10	1.099	11	1.057	10	0.35	10	2.156	10	1.058	13	1.098	10
MSFT	0.55	7	3.77	5	1.196	3	2.579	5	0.52	6	3.380	6	1.116	3	2.264	6
NKE	0.81	1	5.05	1	1.075	16	3.970	1	0.97	1	5.883	1	1.055	15	4.828	1
PFE	0.09	19	0.58	19	1.099	12	−0.515	19	0.09	19	0.551	19	1.059	12	−0.508	19
PG	0.65	3	4.14	3	1.096	13	3.045	3	0.56	5	3.549	5	1.091	5	2.458	5
T	0.04	23	0.28	23	1.121	8	−0.844	23	0.04	23	0.270	23	1.082	6	−0.813	23
TRV	0.42	9	2.44	9	1.011	30	1.433	9	0.41	9	2.393	9	1.009	30	1.384	9
UNH	0.71	2	4.25	2	1.043	25	3.210	2	0.66	2	3.909	2	1.027	24	2.883	2
UTX	0.08	20	0.50	20	1.050	22	−0.549	20	0.08	21	0.454	21	1.033	22	−0.578	21
V	0.28	14	1.76	13	1.082	15	0.678	13	0.27	14	1.645	14	1.038	20	0.608	14
VZ	0.02	24	0.14	24	1.071	17	−0.933	24	0.02	24	0.147	24	1.057	14	−0.910	24
WMT	0.31	11	2.09	11	1.171	4	0.923	11	0.27	15	1.750	12	1.105	4	0.645	13
XOM	−0.07	27	−0.47	27	1.257	1	−1.730	29	−0.07	27	−0.452	27	1.137	1	−1.589	28

Notes: The values of the actual Treynor ratio, the actual net risk premium ratio (η), the required net risk premium ratio (η^*) and the excess net risk premium/market ratio $(\tilde{\mathit{\omega }})$ for the Mean-Variance and the Mean-Gini models. It shows as well the rankings that stem from these calculations, which are commented in the text of the article.

Table 2. Breaking down the portfolio weights

Asset	Mean-Variance					Mean-Gini
	TRR	πM (%)	SD (%)	CRR	RSKC (%)	TRR	πM (%)	SD (%)	CRR	RSKC (%)	%
AXP	2.43	4.94	4.2680	0.4754	−2.2389	1.59	3.22	2.6539	0.7646	−0.6248	2.0291
BA	3.61	7.78	6.4123	0.3362	−4.2564	1.86	4.02	3.2562	0.6621	−1.1004	2.1558
CAT	4.13	5.03	4.4520	0.2735	−3.2345	2.01	2.45	2.2680	0.5368	−1.0506	1.2174
CSCO	2.26	5.69	6.8726	0.3668	−4.3520	1.47	3.71	4.7103	0.5351	−2.1896	2.5207
CVX	3.89	19.63	15.0009	0.3366	−9.9517	1.90	9.58	6.9285	0.7288	−1.8793	5.0492
DD	2.98	3.77	3.1145	0.4065	−1.8486	1.62	2.05	1.7046	0.7426	−0.4388	1.2659
DIS	2.09	5.71	7.1242	0.3832	−4.3942	1.49	4.07	5.1766	0.5274	−2.4467	2.7299
GE	2.07	12.31	11.7627	0.5067	−5.8020	1.46	8.68	7.8613	0.7582	−1.9006	5.9608
GS	2.66	4.49	3.9548	0.4270	−2.2662	1.65	2.78	2.4411	0.6917	−0.7525	1.6887
HD	2.48	6.05	6.9789	0.3492	−4.5417	1.56	3.81	3.7498	0.6500	−1.3125	2.4372
IBM	2.65	11.00	23.9282	0.1737	−19.7712	1.47	6.11	12.7189	0.3268	−8.5619	4.1570
INTC	3.15	8.52	8.3824	0.3233	−5.6722	1.79	4.86	4.4969	0.6027	−1.7866	2.7103
JNJ	1.67	9.08	9.9607	0.5451	−4.5307	1.29	7.00	7.8373	0.6929	−2.4072	5.4301
JPM	3.11	14.36	12.0897	0.3822	−7.4689	1.75	8.10	6.6030	0.6998	−1.9823	4.6207
KO	2.35	9.01	13.4266	0.2856	−9.5923	1.53	5.86	7.9151	0.4844	−4.0808	3.8343
MCD	1.00	2.02	4.0689	0.4987	−2.0396	0.95	1.93	4.5046	0.4505	−2.4753	2.0293
MMM	2.34	4.58	3.6117	0.5414	−1.6563	1.50	2.93	2.2889	0.8543	−0.3335	1.9554
MRK	2.70	8.31	9.9031	0.3110	−6.8233	1.57	4.83	5.7558	0.5351	−2.6761	3.0798
MSFT	2.27	14.83	19.6184	0.3327	−13.0919	1.50	9.78	11.5843	0.5634	−5.0579	6.5264
NKE	2.48	3.64	7.4834	0.1961	−6.0156	1.56	2.29	5.4802	0.2678	−4.0124	1.4678
PFE	2.48	10.37	9.9020	0.4214	−5.7294	1.57	6.55	5.9090	0.7061	−1.7365	4.1726
PG	0.86	4.02	9.6441	0.4823	−4.9928	0.95	4.43	9.1075	0.5107	−4.4562	4.6513
T	2.10	8.17	12.0760	0.3224	−8.1828	1.47	5.72	8.2418	0.4724	−4.3486	3.8932
TRV	1.34	0.93	1.0742	0.6450	−0.3814	1.13	0.78	0.8820	0.7856	−0.1891	0.6928
UNH	2.23	3.49	4.2633	0.3668	−2.6996	1.52	2.38	2.6700	0.5857	−1.1062	1.5637
UTX	2.00	4.40	4.9885	0.4395	−2.7958	1.45	3.17	3.2650	0.6716	−1.0724	2.1927
V	3.90	9.23	8.2362	0.2875	−5.8686	1.90	4.50	3.7513	0.6311	−1.3837	2.3676
VZ	1.80	5.31	7.1252	0.4149	−4.1691	1.34	3.97	5.6684	0.5215	−2.7123	2.9561
WMT	1.90	10.19	17.0563	0.3138	−11.7044	1.40	7.52	10.5147	0.5090	−5.1628	5.3519
XOM	2.97	27.61	25.7375	0.3610	−16.4451	1.65	15.37	13.7090	0.6778	−4.4167	9.2923

Notes: TRR = Total risk ratio; ${\mathit{\pi }}_M$ = Required net risk premium/market ratio; SD = Short sales distance; CRR = Crossed risk ratio; RSKC = Risk correction. This table explains the portfolio weights as the addition of the net risk premium ratio effect and the risk correction, following Equations (20), (26) and (27) for the Mean-Variance model, and Equations (40), (44) and (45) for the Mean-Gini model.

Table 3. Ex-post optimal weights

Mean-Variance
Asset	Weight	β	βC	TRR	πM	η^*	CRR	SD	Risk correction
CSCO	16.34%	1.0440	0.5322	3.1327	0.5118	1.9617	0.1699	0.9617	−0.7983
HD	3.89%	1.0855	0.9516	3.4440	0.1339	1.1408	0.2763	0.1408	−0.1019
INTC	10.34%	1.5021	1.0510	4.3620	0.4511	1.4292	0.2409	0.4292	−0.3258
MSFT	2.99%	1.0342	0.9401	3.1509	0.0941	1.1001	0.2984	0.1001	−0.0702
NKE	10.82%	0.8893	0.5173	3.4385	0.3720	1.7190	0.1505	0.7190	−0.6108
PG	30.97%	0.6252	0.2544	1.1975	0.3709	2.4581	0.2124	1.4581	−1.1484
UNH	24.66%	1.2619	0.4986	3.0958	0.7634	2.5311	0.1611	1.5311	−1.2845
	100.00%	1		RM	24.84%		σM	9.29%
Mean-Gini
CSCO	19.04%	1.0487	0.8490	1.0487	0.1997	1.2352	0.8096	0.2352	−0.0448
HD	2.04%	1.1188	1.0960	1.1188	0.0228	1.0208	0.9796	0.0208	−0.0004
INTC	10.10%	1.5060	1.3540	1.5060	0.1521	1.1123	0.8990	0.1123	−0.0113
MSFT	7.57%	1.0327	0.9545	1.0327	0.0782	1.0820	0.9243	0.0820	−0.0062
NKE	12.40%	0.8789	0.7699	0.8789	0.1089	1.1415	0.8760	0.1415	−0.0175
PG	28.08%	0.6283	0.4519	0.6283	0.1764	1.3904	0.7192	0.3904	−0.1096
UNH	20.77%	1.2605	0.9986	1.2605	0.2618	1.2622	0.7923	0.2622	−0.0545
	100%	1		RM			Gini
Analysing differences
	cd	g	g-cd		Distance differences		Risk correction		Difference
CSCO	−0.8301	−0.1904	0.6397		−0.7265		0.7536		2.70%
HD	−0.7237	−0.0204	0.7033		−0.1199		0.1014		−1.85%
INTC	−0.7591	−0.1010	0.6581		−0.3169		0.3145		−0.24%
MSFT	−0.7016	−0.0757	0.6259		−0.0181		0.0640		4.59%
NKE	−0.8495	−0.1240	0.7256		−0.5775		0.5933		1.58%
PG	−0.7876	−0.2808	0.5068		−1.0677		1.0388		−2.89%
UNH	−0.8389	−0.2077	0.6312		−1.2689		1.2300		−3.88%

Notes: The ex-post optimal portfolio weights for Mean-Variance and Mean-Gini models, with short sales not allowed. The first two sections also show the values of the parameters that explain the optimal weights, following the same path than Table 2, but with the new figures. The last section presents the analysis of the differences between the weights that stem from both models (Mean-Gini minus Mean-Variance) according to Equation (48).

Table 4. Basic calculations

General data					Mean-Variance						Mean-Gini
Ticker	Company name	Cap 31/12/2013	Weights (%)	R (%)	σ	β	βC	π* (%)	π (%)	α (%)	${\mathrm{\Delta }}_i^{*}$ (%)	β^G	${\mathit{\beta }}_C^G$	π^G* (%)	${\mathit{\pi }}^G$ (%)	α^G (%)
	Portfolio	4,757,616.10	100.00	0.1732	1.5163	1.00	0.00	0.1732	0.1732	0.0000	0.8369	1.00	0.00	0.0000	0.0000	0.0000
AXP	American Express Co	96,536.7	2.03	0.0903	2.3654	1.21	1.16	0.0085	0.2003	−0.1186	1.3290	1.25	1.21	0.0056	0.2102	−0.1255
BA	Boeing Co.	102,565.8	2.16	−0.0655	2.8804	1.29	1.21	0.0135	0.2101	−0.2890	1.5599	1.27	1.23	0.0070	0.2137	−0.2861
CAT	Caterpillar Inc.	57,920.6	1.22	0.0868	3.0828	1.18	1.13	0.0087	0.1957	−0.1176	1.6842	1.10	1.08	0.0042	0.1870	−0.1045
CSCO	Cisco Systems	119,924.6	2.52	0.4987	2.2790	0.89	0.83	0.0099	0.1435	0.3454	1.2327	0.83	0.79	0.0064	0.1365	0.3558
CVX	Chevron Corp.	240,223.5	5.05	−0.1307	2.9901	1.51	1.31	0.0340	0.2266	−0.3913	1.5884	1.48	1.38	0.0166	0.2395	−0.3868
DD	DuPont	602,26.3	1.27	0.3372	2.6175	1.25	1.21	0.0065	0.2097	0.1209	1.3548	1.22	1.20	0.0035	0.2082	0.1254
DIS	The Walt Disney Co.	129,880	2.73	0.4342	2.1926	0.86	0.80	0.0099	0.1387	0.2856	1.2490	0.83	0.79	0.0071	0.1363	0.2908
GE	General Electric	283,589.9	5.96	−0.0945	2.1791	1.17	1.05	0.0213	0.1812	−0.2970	1.2181	1.19	1.10	0.0150	0.1911	−0.3006
GS	Goldman Sachs Group	80,339.7	1.69	0.1873	2.4739	1.18	1.14	0.0078	0.1968	−0.0172	1.3776	1.17	1.14	0.0048	0.1972	−0.0146
HD	Home Depot	115,953.4	2.44	0.5186	2.3896	0.93	0.87	0.0105	0.1502	0.3579	1.3082	1.05	1.02	0.0066	0.1759	0.3361
IBM	IBM Corp.	197,772.1	4.16	−0.2448	2.4671	0.57	0.46	0.0191	0.0796	−0.3434	1.2297	0.54	0.48	0.0106	0.0832	−0.3385
INTC	Intel Corp.	128,943.3	2.71	0.7176	2.6892	1.10	1.02	0.0148	0.1761	0.5268	1.4995	1.13	1.08	0.0084	0.1870	0.5222
JNJ	Johnson & Johnson	258,341.3	5.43	0.3020	1.9603	1.00	0.91	0.0157	0.1578	0.1285	1.0791	0.96	0.89	0.0121	0.1547	0.1352
JPM	JPMorgan Chase & Co.	219,837.4	4.62	0.1628	2.6731	1.33	1.19	0.0249	0.2057	−0.0677	1.4664	1.31	1.23	0.0140	0.2123	−0.0635
KO	The Coca Cola Co.	182,421.8	3.83	0.1387	2.3244	0.76	0.67	0.0156	0.1162	0.0069	1.2784	0.80	0.74	0.0101	0.1281	0.0004
MCD	McDonald’s Corp.	96,547.8	2.03	0.0078	1.5136	0.52	0.50	0.0035	0.0860	−0.0818	0.7967	0.45	0.43	0.0033	0.0743	−0.0698
MMM	3 M Company	93,028.7	1.96	0.3755	2.3200	1.31	1.27	0.0079	0.2194	0.1481	1.2536	1.31	1.28	0.0051	0.2216	0.1489
MRK	Merck & Co.	146,522.7	3.08	0.3131	2.4903	0.92	0.84	0.0144	0.1452	0.1535	1.3122	0.89	0.84	0.0084	0.1453	0.1595
MSFT	Microsoft Corp.	310,503	6.53	0.4941	2.2856	0.90	0.76	0.0257	0.1309	0.3375	1.2540	0.94	0.84	0.0169	0.1462	0.3310
NKE	Nike, Inc.	69,832.3	1.47	0.4248	2.3877	0.52	0.49	0.0063	0.0842	0.3343	1.3033	0.44	0.42	0.0040	0.0722	0.3487
PFE	Pfizer Inc.	198,515.2	4.17	0.1058	2.3900	1.15	1.05	0.0179	0.1813	−0.0934	1.3147	1.17	1.11	0.0113	0.1921	−0.0976
PG	Procter & Gamble	221,291.2	4.65	0.2987	1.4090	0.46	0.42	0.0070	0.0721	0.2196	0.7964	0.53	0.49	0.0077	0.0841	0.2068
T	AT&T Inc.	185,222.9	3.89	0.0324	2.1971	0.76	0.68	0.0142	0.1172	−0.0989	1.2305	0.75	0.69	0.0099	0.1203	−0.0977
TRV	The Travelers Companies	32,962.7	0.69	0.3665	1.7571	0.88	0.87	0.0016	0.1500	0.2149	0.9423	0.89	0.88	0.0014	0.1532	0.2120
UNH	United Health Group Inc.	74,396.4	1.56	0.6029	2.2656	0.85	0.82	0.0060	0.1418	0.4551	1.2726	0.91	0.89	0.0041	0.1542	0.4445
UTX	United Technologies	104,318.9	2.19	0.0764	2.1469	0.93	0.88	0.0076	0.1526	−0.0837	1.2111	1.00	0.97	0.0055	0.1683	−0.0973
V	Visa Inc.	112,642.7	2.37	0.3418	2.9944	1.21	1.12	0.0160	0.1941	0.1317	1.5910	1.24	1.20	0.0078	0.2078	0.1263
VZ	Verizon Communications	140,638.7	2.96	0.0178	2.0323	0.80	0.75	0.0092	0.1290	−0.1204	1.1233	0.74	0.70	0.0069	0.1212	−0.1102
WMT	Wal-Mart Stores	254,622.9	5.35	0.2167	2.0928	0.70	0.60	0.0177	0.1035	0.0956	1.1756	0.79	0.72	0.0130	0.1238	0.0799
XOM	Exxon Mobil Corp.	442,093.6	9.29	−0.0878	2.6137	1.35	1.07	0.0478	0.1857	−0.3213	1.3842	1.27	1.12	0.0266	0.1941	−0.3085

Notes: The results of the basic calculations that display Tables 1–3. First, it shows the corporations listed in DJIA at the end of 2013. For each one of them we have calculated its capitalisation multiplying the number of shares by the share price. Next, it shows the weight of each corporation that stems from this capitalisation. The rates of return that follow are the average of the weekly rates of return of each corporation. The same data has been used to estimate the standard deviation, beta and crossed beta. Next, the table displays the values of the net risk premium $(\mathit{\pi })$, required net risk premium (π^*) and alpha. The Gini-Mean differences $({\mathrm{\Delta }}_i^{*})$ have been calculated as twice the assets’ rates of return and their cumulative distribution function, approached through the rankings of their rates of return. Next, the Table shows the values of beta, crossed beta, actual and required net risk premia, and Gini’s alpha.