Hedge fund return predictability in the presence of model risk*

Figures & data

Table 1. Summary statistics of monthly hedge fund returns.

Category	N	AUM	Mean	Std.Dev	Skewness	Kurtosis
Panel A: All Funds
All	1969	731.02	0.72	3.50	−0.55	5.54
CTA	55	95.98	0.77	4.22	0.56	4.39
Emerging Market	164	54.04	0.91	5.89	−0.48	6.18
Event Driven	152	86.71	0.80	3.05	−0.59	5.25
Fund of Funds	581	162.50	0.47	2.02	−1.07	5.80
Global Macro	92	29.01	0.82	3.78	0.34	4.40
Equity Long Only	65	26.16	0.81	4.83	−0.14	2.43
Equity Long/Short	444	97.89	0.86	4.37	−0.03	3.11
Market Neutral	36	4.76	0.53	2.03	0.10	3.26
Multi-Strategy	93	95.29	0.75	2.76	−0.70	6.41
Relative Value	172	59.97	0.74	3.08	−1.45	13.90
Sector	115	18.71	0.86	5.09	−0.14	3.76
Panel B: Alive Funds
All	926	429.54	0.73	3.55	−0.51	5.38
CTA	27	91.79	0.81	4.16	0.54	5.16
Emerging Market	89	27.71	0.86	5.82	−0.35	4.78
Event Driven	66	23.93	0.75	2.96	−0.47	5.48
Fund of Funds	265	93.70	0.48	2.02	−1.05	6.14
Global Macro	42	18.81	0.92	3.99	0.35	4.29
Equity Long Only	34	10.12	0.83	5.03	−0.23	2.61
Equity Long/Short	211	54.12	0.87	4.39	−0.10	2.89
Market Neutral	18	3.32	0.62	1.82	0.04	3.21
Multi-Strategy	41	54.79	0.82	2.97	−0.49	5.63
Relative Value	71	40.04	0.81	3.01	−1.36	14.87
Sector	62	11.22	0.77	4.77	−0.26	3.30
Panel C: Graveyard Funds
All	1043	301.48	0.71	3.45	−0.58	5.68
CTA	28	4.20	0.72	4.28	0.59	3.66
Emerging Market	75	26.33	0.98	5.98	−0.63	7.84
Event Driven	86	62.78	0.83	3.12	−0.68	5.07
Fund of Funds	316	68.80	0.47	2.02	−1.08	5.52
Global Macro	50	10.20	0.73	3.61	0.33	4.49
Equity Long Only	31	16.04	0.79	4.61	−0.04	2.22
Equity Long/Short	233	43.77	0.86	4.36	0.04	3.31
Market Neutral	18	1.44	0.45	2.25	0.17	3.32
Multi-Strategy	52	40.50	0.70	2.60	−0.87	7.02
Relative Value	101	19.93	0.70	3.14	−1.51	13.21
Sector	53	7.49	0.96	5.46	0.01	4.29

Notes: The table reports the total number and the total asset under management (AUM in $bn.) of funds under each category. The summary statistics are the average monthly return, standard deviation, skewness and excess kurtosis. Sample period: Jan 1994–Dec 2014.

Table 2. Statistical evaluation – Theil's U.

Panel A: Equal weighted	CTA	EM	ED	FoF	GM	LO	LS	MN	MS	RV	Sector
OLS-AR(1)	1.676	11.156	3.140	1.923	1.829	12.358	6.526	0.390	1.889	1.847	7.030
BMA ($\alpha =\lambda =1$, $\kappa =0.92$)	0.960${}^{\ast }$	1.030	0.990	0.987${}^{\ast }$	1.003${}^{\ast }$	1.011	1.003	1.015${}^{\ast }$	0.976${}^{\ast }$	0.959${}^{\ast }$	1.021
BMA ($\alpha =\lambda =1$,$\kappa =0.97$)	0.974${}^{\ast }$	1.056	1.011	0.990${}^{\ast }$	1.002${}^{\ast }$	1.036	1.021	1.052${}^{\ast }$	0.976${}^{\ast }$	1.045	1.044
BMS ($\alpha =\lambda =1$, $\kappa =0.92$)	0.960${}^{\ast }$	1.021	1.001	0.989${}^{\ast }$	1.044	1.012	1.004	1.051	0.997${}^{\ast }$	0.972${}^{\ast }$	1.042
BMS ($\alpha =\lambda =1$, $\kappa =0.97$)	0.975${}^{\ast }$	1.050	1.013	1.003${}^{\ast }$	1.003${}^{\ast }$	1.045	1.037	1.134	1.013	1.044	1.062
DMA ($\alpha =\lambda =0.996$, $\kappa =0.92$)	0.942${}^{\ast }$	1.048	0.997	0.986${}^{\ast }$	1.008	1.030	1.013	1.011${}^{\ast }$	0.971${}^{\ast }$	0.990	1.027
DMA ($\alpha =\lambda =0.996$, $\kappa =0.97$)	0.948${}^{\ast }$	1.077	1.017	0.996${}^{\ast }$	1.007	1.056	1.032	1.033${}^{\ast }$	0.975${}^{\ast }$	1.062	1.043
DMA ($\alpha =\lambda =0.985$, $\kappa =0.92$)	0.950${}^{\ast }$	1.136	1.036	1.016${}^{\ast }$	1.047	1.085	1.046${}^{\ast }$	1.017${}^{\ast }$	0.990${}^{\ast }$	1.132	1.059${}^{\ast }$
DMA ($\alpha =\lambda =0.985$, $\kappa =0.97$)	0.933${}^{\ast }$	1.158	1.056	1.059${}^{\ast }$	1.036	1.107	1.057	1.011${}^{\ast }$	1.007${}^{\ast }$	1.240	1.057${}^{\ast }$
DMS ($\alpha =\lambda =0.996$, $\kappa =0.92$)	0.947${}^{\ast }$	1.024	1.038	1.019	1.024	1.065	1.034	1.074	1.018	1.064	1.051
DMS ($\alpha =\lambda =0.996$, $\kappa =0.97$)	0.948${}^{\ast }$	1.085	1.018	1.090	1.006${}^{\ast }$	1.041	1.051	1.138	1.013	1.086	1.117
DMS ($\alpha =\lambda =0.985$,$\kappa =0.92$)	0.945${}^{\ast }$	1.157	1.094	1.041${}^{\ast }$	1.088	1.100	1.044	1.110${}^{\ast }$	1.084	1.130	1.158
DMS ($\alpha =\lambda =0.985$, $\kappa =0.97$)	0.944${}^{\ast }$	1.241	1.085	1.172${}^{\ast }$	1.076	1.135	1.084	1.049${}^{\ast }$	1.061${}^{\ast }$	1.237	1.112${}^{\ast }$
TVP-ALL ($\lambda =0.996$, $\kappa =0.92$)	1.143${}^{\ast }$	1.349	1.206	1.158	1.133	1.166	1.153	1.293	1.061${}^{\ast }$	1.324	1.185
TVP-ALL ($\lambda =0.996$, $\kappa =0.97$)	1.128${}^{\ast }$	1.381	1.190	1.178	1.110	1.168	1.149	1.347	1.059${}^{\ast }$	1.332	1.200
TVP-ALL ($\lambda =0.985$, $\kappa =0.92$)	1.119${}^{\ast }$	1.556	1.293	1.328${}^{\ast }$	1.225	1.278	1.270	1.376${}^{\ast }$	1.146${}^{\ast }$	1.532	1.285${}^{\ast }$
TVP-ALL ($\lambda =0.985$, $\kappa =0.97$)	1.073${}^{\ast }$	1.581	1.285	1.387${}^{\ast }$	1.198	1.269	1.253	1.381	1.156${}^{\ast }$	1.575	1.297
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.92$)	0.942${}^{\ast }$	1.014	0.987${}^{\ast }$	0.994${}^{\ast }$	0.994	0.986${}^{\ast }$	0.988${}^{\ast }$	0.957${}^{\ast }$	0.990${}^{\ast }$	0.956${}^{\ast }$	0.977${}^{\ast }$
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.97$)	0.948${}^{\ast }$	1.005	0.991${}^{\ast }$	0.996	0.993${}^{\ast }$	0.991${}^{\ast }$	0.995	0.966${}^{\ast }$	0.993${}^{\ast }$	0.970	0.986
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.92$)	0.916${}^{\ast }$	1.030	0.982	1.007	1.002	0.998	0.997	0.926${}^{\ast }$	0.997	0.961	0.984
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.97$)	0.912${}^{\ast }$	1.024	0.988	1.011	0.995	1.003	1.002	0.915${}^{\ast }$	0.999	1.000	0.991
Panel B: Value weighted	CTA	EM	ED	FoF	GM	LO	LS	MN	MS	RV	Sector
OLS-AR(1)	6.448	7.181	2.671	1.557	1.872	18.679	5.429	0.557	0.982	1.293	6.938
BMA ($\alpha =\lambda =1$, $\kappa =0.92$)	1.071	1.028	0.970${}^{\ast }$	1.007	0.972${}^{\ast }$	1.036	0.996${}^{\ast }$	0.941${}^{\ast }$	0.966${}^{\ast }$	0.930${}^{\ast }$	1.017
BMA ($\alpha =\lambda =1$,$\kappa =0.97$)	1.139	1.074	0.971${}^{\ast }$	0.998${}^{\ast }$	0.971${}^{\ast }$	1.043	1.008	1.006${}^{\ast }$	0.991${}^{\ast }$	1.014	1.029
BMS ($\alpha =\lambda =1$, $\kappa =0.92$)	1.114	1.010	0.961${}^{\ast }$	1.010	0.951${}^{\ast }$	1.063	1.012	0.981${}^{\ast }$	0.963${}^{\ast }$	0.927${}^{\ast }$	1.025${}^{\ast }$
BMS ($\alpha =\lambda =1$, $\kappa =0.97$)	1.176	1.050	0.963${}^{\ast }$	0.988${}^{\ast }$	0.981${}^{\ast }$	1.025	1.018	1.052${}^{\ast }$	1.022	1.077	1.043
DMA ($\alpha =\lambda =0.996$, $\kappa =0.92$)	1.116	1.045	0.978${}^{\ast }$	0.996${}^{\ast }$	0.983${}^{\ast }$	1.052	1.007	0.927${}^{\ast }$	0.964${}^{\ast }$	0.945	1.023
DMA ($\alpha =\lambda =0.996$, $\kappa =0.97$)	1.151	1.086	0.979${}^{\ast }$	1.005${}^{\ast }$	0.977${}^{\ast }$	1.062	1.020	0.975${}^{\ast }$	1.002${}^{\ast }$	1.012	1.027
DMA ($\alpha =\lambda =0.985$, $\kappa =0.92$)	1.260	1.131	1.042${}^{\ast }$	1.017${}^{\ast }$	1.043${}^{\ast }$	1.112	1.046	0.932${}^{\ast }$	1.017${}^{\ast }$	1.072	1.052${}^{\ast }$
DMA ($\alpha =\lambda =0.985$, $\kappa =0.97$)	1.253	1.183	1.030${}^{\ast }$	1.066${}^{\ast }$	1.007${}^{\ast }$	1.117	1.059	0.946${}^{\ast }$	1.084${}^{\ast }$	1.114	1.036${}^{\ast }$
DMS ($\alpha =\lambda =0.996$, $\kappa =0.92$)	1.168	1.009	1.002${}^{\ast }$	1.021	0.953${}^{\ast }$	1.099	1.014	0.976${}^{\ast }$	1.080	0.945${}^{\ast }$	1.086
DMS ($\alpha =\lambda =0.996$, $\kappa =0.97$)	1.208	1.191	1.008${}^{\ast }$	1.014	0.973${}^{\ast }$	1.091	1.061	1.024${}^{\ast }$	1.105	0.999	1.093
DMS ($\alpha =\lambda =0.985$,$\kappa =0.92$)	1.403	1.153	1.080${}^{\ast }$	1.083${}^{\ast }$	1.120${}^{\ast }$	1.137	1.069	1.007${}^{\ast }$	1.129	1.106	1.134${}^{\ast }$
DMS ($\alpha =\lambda =0.985$, $\kappa =0.97$)	1.344	1.279	1.080${}^{\ast }$	1.143${}^{\ast }$	1.015${}^{\ast }$	1.215	1.101	0.990${}^{\ast }$	1.242	1.262	1.136
TVP-ALL ($\lambda =0.996$, $\kappa =0.92$)	1.219	1.360	1.218	1.273	1.494${}^{\ast }$	1.226	1.134	1.226${}^{\ast }$	1.130${}^{\ast }$	1.378	1.258
TVP-ALL ($\lambda =0.996$, $\kappa =0.97$)	1.209	1.477	1.151${}^{\ast }$	1.303	1.543	1.215	1.135	1.315${}^{\ast }$	1.156${}^{\ast }$	1.396	1.253
TVP-ALL ($\lambda =0.985$, $\kappa =0.92$)	1.354	1.571	1.393	1.420${}^{\ast }$	1.514${}^{\ast }$	1.324	1.260	1.312${}^{\ast }$	1.294${}^{\ast }$	1.499	1.376
TVP-ALL ($\lambda =0.985$, $\kappa =0.97$)	1.340	1.720	1.293${}^{\ast }$	1.483${}^{\ast }$	1.496${}^{\ast }$	1.294	1.250	1.324${}^{\ast }$	1.367${}^{\ast }$	1.521	1.364
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.92$)	0.998	1.015	0.996	0.998	0.953${}^{\ast }$	0.995	0.994	0.889${}^{\ast }$	0.962${}^{\ast }$	0.902${}^{\ast }$	0.988
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.97$)	1.005	1.012	0.995	0.996	0.937${}^{\ast }$	0.997	0.995	0.935${}^{\ast }$	0.970${}^{\ast }$	0.950${}^{\ast }$	0.996
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.92$)	1.028	1.038	1.006	1.012	0.965${}^{\ast }$	1.003	1.006	0.859${}^{\ast }$	0.964${}^{\ast }$	0.902${}^{\ast }$	0.996
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.97$)	1.059	1.036	1.004	1.016	0.946${}^{\ast }$	1.008	1.006	0.883${}^{\ast }$	0.971${}^{\ast }$	0.951${}^{\ast }$	1.004

Notes: The table reports Theil's U statistics of each model calculated based on the out-of-sample MSFEs of equal- and AUM-weighted hedge fund strategy indices. The first row of Panel A and B reports the MSFE of the benchmark model. For the remaining rows, Theil's U less than 1 (in boldface) indicates better performance of the underlying model compared to the benchmark model OLS-AR(1). ${}^{\ast }$indicates statistical significance at the 5% significance level, based on the Clark and West (2007) test. Out-of-sample forecasting period: Jan 2002–Dec 2014.

Table 3. Economic evaluation – CER.

Panel A: Equal weighted	CTA	EM	ED	FoF	GM	LO	LS	MN	MS	RV	Sector
OLS-AR(1)	142.06	113.59	139.53	91.77	171.01	115.08	112.60	141.87	129.83	−0.3	88.76
BMA ($\alpha =\lambda =1$, $\kappa =0.92$)	161.42	130.68	147.99	150.41	149.69	121.13	123.71	169.49	163.58	154.48	119.67
BMA ($\alpha =\lambda =1$,$\kappa =0.97$)	175.94	135.04	150.92	145.60	168.31	125.18	134.43	200.36	157.82	143.14	130.26
BMS ($\alpha =\lambda =1$, $\kappa =0.92$)	159.97	129.42	146.16	148.41	147.60	124.48	125.52	156.36	158.80	142.69	118.18
BMS ($\alpha =\lambda =1$, $\kappa =0.97$)	175.66	132.30	144.92	145.73	162.45	127.87	132.65	185.19	136.96	115.39	131.18
DMA ($\alpha =\lambda =0.996$, $\kappa =0.92$)	162.88	131.98	146.27	151.37	151.68	120.79	124.07	175.79	165.33	155.01	125.12
DMA ($\alpha =\lambda =0.996$, $\kappa =0.97$)	176.70	134.72	152.39	144.60	167.16	125.32	134.43	200.50	158.82	149.14	133.77
DMA ($\alpha =\lambda =0.985$, $\kappa =0.92$)	163.76	138.05	151.43	150.09	155.43	124.91	132.27	191.23	162.84	163.34	133.97
DMA ($\alpha =\lambda =0.985$, $\kappa =0.97$)	173.17	134.89	157.87	145.85	163.50	129.93	140.41	198.07	156.16	156.69	139.60
DMS ($\alpha =\lambda =0.996$, $\kappa =0.92$)	161.09	130.04	138.20	146.34	115.55	121.21	118.05	151.27	148.18	146.33	122.37
DMS ($\alpha =\lambda =0.996$, $\kappa =0.97$)	176.37	134.03	142.66	139.01	162.77	129.74	130.20	182.05	131.14	123.04	130.95
DMS ($\alpha =\lambda =0.985$,$\kappa =0.92$)	162.84	125.83	121.68	137.98	100.24	121.39	123.50	159.38	122.92	63.78	94.77
DMS ($\alpha =\lambda =0.985$, $\kappa =0.97$)	173.39	119.55	144.97	127.05	134.88	126.93	131.20	192.93	126.81	89.42	123.17
TVP-ALL ($\lambda =0.996$, $\kappa =0.92$)	154.69	128.51	118.17	136.44	143.04	112.42	122.82	176.42	173.95	161.28	129.39
TVP-ALL ($\lambda =0.996$, $\kappa =0.97$)	155.05	123.85	136.19	130.46	152.82	118.16	133.84	168.62	177.17	179.56	132.78
TVP-ALL ($\lambda =0.985$, $\kappa =0.92$)	144.00	136.73	155.30	144.98	150.11	122.85	133.55	187.10	167.29	164.40	136.33
TVP-ALL ($\lambda =0.985$, $\kappa =0.97$)	148.53	132.76	161.85	142.03	154.43	127.38	139.30	177.33	169.89	175.28	137.99
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.92$)	160.07	130.81	148.54	146.34	162.39	124.68	126.12	173.20	162.12	198.83	124.04
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.97$)	176.62	139.77	159.55	144.06	176.45	130.21	134.95	210.38	152.99	171.22	133.60
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.92$)	164.15	127.96	147.68	143.20	164.44	124.56	129.18	180.95	162.84	204.10	127.49
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.97$)	177.47	137.37	159.69	139.45	175.22	130.76	137.15	208.34	153.82	179.57	133.91
Panel B: Value weighted	CTA	EM	ED	FoF	GM	LO	LS	MN	MS	RV	Sector
OLS-AR(1)	35.63	92.90	175.59	94.15	161.44	121.74	144.55	204.98	77.47	−2.6	90.70
BMA ($\alpha =\lambda =1$, $\kappa =0.92$)	130.08	164.01	169.47	169.24	171.22	114.95	156.06	224.44	193.74	209.68	149.30
BMA ($\alpha =\lambda =1$,$\kappa =0.97$)	135.14	148.76	178.26	163.02	193.42	116.68	156.88	254.00	235.75	200.93	159.84
BMS ($\alpha =\lambda =1$, $\kappa =0.92$)	124.87	168.21	165.67	169.23	172.96	115.77	157.79	232.76	191.10	223.52	150.50
BMS ($\alpha =\lambda =1$, $\kappa =0.97$)	136.19	140.23	198.30	165.09	197.29	116.36	154.69	251.74	240.99	199.15	163.40
DMA ($\alpha =\lambda =0.996$, $\kappa =0.92$)	139.38	166.94	165.33	172.19	176.20	118.78	153.15	230.31	212.59	216.25	155.97
DMA ($\alpha =\lambda =0.996$, $\kappa =0.97$)	141.94	149.26	171.40	164.30	191.23	119.85	154.85	251.60	244.27	204.88	162.25
DMA ($\alpha =\lambda =0.985$, $\kappa =0.92$)	136.87	172.15	170.02	171.25	186.52	124.91	153.09	241.21	238.53	204.91	167.81
DMA ($\alpha =\lambda =0.985$, $\kappa =0.97$)	138.66	150.21	178.34	165.61	184.47	128.49	156.28	240.78	244.55	204.26	169.53
DMS ($\alpha =\lambda =0.996$, $\kappa =0.92$)	132.18	166.19	158.07	161.91	172.78	113.56	155.27	225.95	162.45	224.74	130.78
DMS ($\alpha =\lambda =0.996$, $\kappa =0.97$)	144.75	137.05	165.03	161.51	197.10	113.16	150.98	267.78	209.99	191.75	165.04
DMS ($\alpha =\lambda =0.985$,$\kappa =0.92$)	136.63	164.16	167.83	152.97	153.70	122.08	133.99	208.17	141.62	178.64	140.96
DMS ($\alpha =\lambda =0.985$, $\kappa =0.97$)	139.34	123.86	117.02	145.11	196.53	120.89	137.68	251.04	165.83	172.65	163.19
TVP-ALL ($\lambda =0.996$, $\kappa =0.92$)	137.17	152.34	149.86	138.85	164.74	121.98	147.60	225.35	234.63	196.83	156.54
TVP-ALL ($\lambda =0.996$, $\kappa =0.97$)	139.48	133.00	169.49	130.89	154.55	120.93	150.87	196.46	246.79	209.31	155.73
TVP-ALL ($\lambda =0.985$, $\kappa =0.92$)	136.62	155.74	182.06	155.37	160.87	128.40	152.74	221.83	232.49	172.37	157.70
TVP-ALL ($\lambda =0.985$, $\kappa =0.97$)	137.79	139.42	192.09	150.12	151.30	131.28	154.77	197.60	226.12	189.57	155.97
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.92$)	101.57	163.61	177.91	165.21	172.78	125.17	158.58	229.15	197.35	226.64	151.24
TVP-AR(1) ($\lambda =0.996$, $\kappa =0.97$)	86.67	158.50	193.27	162.63	196.66	129.78	157.22	255.34	233.66	203.11	163.17
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.92$)	104.29	160.33	176.14	160.02	174.65	124.28	155.69	228.90	215.76	236.38	154.48
TVP-AR(1) ($\lambda =0.985$, $\kappa =0.97$)	102.22	155.32	193.11	157.19	194.45	129.46	153.73	245.28	238.07	208.15	160.94

Notes: The table reports the CER of a portfolio consisting of a risk-free asset and one risky asset (i.e. the underlying hedge fund strategy index). CERs of competing models are computed based on the model forecast expected returns and variance. Boldface indicates superiority of the proposed model over the benchmark OLS-AR(1) model. Out-of-sample forecasting period: Jan 2002–Dec 2014.

Table 4. Out-of-sample performance of top expected return portfolios.

Panel A: Equal weighted	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.999	3.998	0.743	1.997	0.640	1.662	0.281	0.706	13.918
BMA	1.031	4.311	0.793	2.451	0.771	1.827	0.350	0.773	11.183
BMS	0.991	4.062	0.743	2.278	0.728	1.761	0.324	0.751	12.046
DMA	1.011	4.204	0.791	2.481	0.769	1.805	0.344	0.771	11.353
DMS	1.009	4.206	0.774	2.397	0.755	1.785	0.340	0.774	12.108
TVP-ALL	0.940	3.875	0.722	2.417	0.762	1.805	0.348	0.781	10.908
TVP-AR(1)	1.088	4.621	0.841	2.477	0.779	1.858	0.365	0.791	12.438
Panel B: Mean-Variance	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.881	3.517	0.659	2.488	0.785	1.885	0.337	0.718	8.335
BMA	0.895	3.801	0.744	4.400	1.275	2.576	0.630	1.029	5.755
BMS	0.821	3.385	0.661	3.599	1.044	2.221	0.467	0.852	6.140
DMA	0.920	3.862	0.783	4.171	1.161	2.426	0.533	0.905	6.783
DMS	0.829	3.397	0.674	3.390	0.956	2.105	0.417	0.799	7.948
TVP-ALL	0.826	3.405	0.684	4.385	1.263	2.647	0.643	1.032	5.096
TVP-AR(1)	0.983	4.220	0.802	3.673	1.073	2.306	0.504	0.891	7.607
Panel C: Mean-CVaR	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.927	3.751	0.689	2.539	0.812	1.915	0.350	0.732	8.335
BMA	0.920	3.955	0.752	4.258	1.250	2.568	0.627	1.028	5.799
BMS	0.886	3.725	0.705	3.622	1.080	2.299	0.512	0.907	6.709
DMA	1.008	4.431	0.851	4.385	1.249	2.591	0.623	1.015	6.373
DMS	0.963	4.118	0.774	3.541	1.046	2.270	0.517	0.928	6.886
TVP-ALL	0.823	3.342	0.652	3.430	1.006	2.175	0.470	0.869	6.736
TVP-AR(1)	1.080	4.905	0.890	4.112	1.219	2.553	0.635	1.043	6.453

Notes: The table reports the out-of-sample performance of portfolios constructed based on the expected future return. Asset weights are allocated by 1/N method for the equal weighted portfolios, by the mean-variance approach and the mean-CVaR approach, with a target of 2% monthly return. The performance measures reported are the average of each portfolio's individual measures within a specific family of methods: average monthly return (AR), end of period value (EPV), annualized Sharpe ratio, Omega ratio, Sortino ratio, Upside Potential ratio, and 1% VaR. Out-of-sample forecasting period: Jan 2009–Dec 2014.

Table 5. Out-of-sample performance of top t-statistics portfolios.

Panel A: Equal weighted	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.570	2.392	0.402	4.249	1.258	2.763	0.583	0.913	4.277
BMA	0.625	2.573	0.457	3.947	1.151	2.712	0.546	0.864	3.771
BMS	0.630	2.596	0.456	3.978	1.166	2.714	0.559	0.886	4.197
DMA	0.664	2.731	0.491	4.031	1.191	2.871	0.618	0.948	4.069
DMS	0.647	2.655	0.472	3.759	1.123	2.587	0.529	0.862	5.055
TVP-ALL	0.718	2.947	0.537	3.814	1.142	2.646	0.608	0.978	4.671
TVP-AR(1)	0.648	2.671	0.467	4.171	1.271	2.848	0.633	0.975	3.356
Panel B: Mean-Variance	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.628	2.580	0.421	2.986	0.946	2.161	0.395	0.735	5.887
BMA	0.742	3.069	0.609	5.832	1.659	3.753	0.846	1.153	3.222
BMS	0.718	2.966	0.573	5.865	1.688	3.751	0.932	1.271	3.117
DMA	0.783	3.254	0.641	5.523	1.584	3.477	0.813	1.140	4.002
DMS	0.680	2.784	0.523	4.410	1.283	2.805	0.601	0.931	5.098
TVP-ALL	0.710	2.917	0.564	5.703	1.665	3.467	1.008	1.418	2.341
TVP-AR(1)	0.732	3.001	0.558	4.645	1.424	3.116	0.776	1.147	3.926
Panel C: Mean-CVaR	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.627	2.577	0.419	2.967	0.942	2.160	0.395	0.735	5.887
BMA	0.791	3.293	0.650	5.425	1.539	3.451	0.751	1.057	3.335
BMS	0.732	3.015	0.582	5.044	1.433	3.140	0.698	1.024	3.444
DMA	0.866	3.678	0.715	5.523	1.574	3.499	0.844	1.182	4.017
DMS	0.773	3.189	0.600	4.349	1.271	2.856	0.636	0.978	5.141
TVP-ALL	0.684	2.761	0.512	3.532	1.026	2.288	0.491	0.870	5.395
TVP-AR(1)	0.766	3.163	0.581	4.542	1.397	3.044	0.799	1.190	3.825

Notes: The table reports the out-of-sample performance of portfolios constructed based on the forecasting accuracy (t-statistics) of the expected return. Asset weights are allocated by 1/N method for the equal weighted portfolios, by the mean-variance approach and the mean-CVaR approach, with a target of 2% monthly return. The performance measures reported are the average of each portfolio's individual measures within a specific family of methods: average monthly return (AR), end of period value (EPV), annualized Sharpe ratio, Omega ratio, Sortino ratio, Upside Potential ratio, and 1% VaR. Out-of-sample forecasting period: Jan 2009–Dec 2014.

Table 6. Crisis period – out-of-sample performance of top expected return portfolios.

Panel A: Equal weighted	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.667	1.100	−0.327	−0.262	0.133	1.112	0.054	0.536	20.658
BMA	0.821	1.192	−0.142	−0.137	0.246	1.213	0.104	0.593	15.886
BMS	0.740	1.157	−0.190	−0.176	0.194	1.165	0.080	0.565	17.155
DMA	0.796	1.196	−0.131	−0.132	0.243	1.207	0.100	0.582	15.695
DMS	0.788	1.192	−0.138	−0.142	0.232	1.196	0.096	0.586	15.268
TVP-ALL	0.733	1.191	−0.159	−0.183	0.230	1.194	0.099	0.613	13.358
TVP-AR(1)	1.024	1.268	0.093	0.081	0.331	1.296	0.148	0.646	15.690
Panel B: Mean-Variance	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.482	1.128	−0.268	−0.300	0.064	1.056	0.024	0.449	18.679
BMA	0.616	1.215	0.036	0.093	0.355	1.325	0.134	0.547	7.522
BMS	0.246	1.061	−0.340	−0.651	−0.155	0.885	−0.051	0.393	10.321
DMA	0.609	1.204	−0.071	−0.124	0.258	1.237	0.096	0.505	10.677
DMS	0.257	1.058	−0.365	−0.598	−0.124	0.906	−0.041	0.389	12.172
TVP-ALL	0.570	1.202	−0.039	−0.106	0.285	1.246	0.128	0.634	5.489
TVP-AR(1)	0.745	1.257	0.156	0.229	0.320	1.294	0.125	0.555	12.645
Panel C: Mean-CVaR	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.493	1.131	−0.290	−0.324	0.071	1.064	0.026	0.439	19.207
BMA	0.738	1.261	0.122	0.291	0.515	1.522	0.204	0.602	8.631
BMS	0.555	1.182	−0.071	−0.142	0.201	1.185	0.074	0.488	10.156
DMA	0.861	1.305	0.163	0.322	0.551	1.551	0.226	0.641	9.468
DMS	0.684	1.221	−0.049	−0.093	0.262	1.247	0.115	0.569	10.223
TVP-ALL	0.376	1.103	−0.332	−0.624	−0.002	1.003	0.002	0.503	9.805
TVP-AR(1)	1.010	1.389	0.417	0.657	0.607	1.645	0.274	0.694	10.568

Notes: The table reports the out-of-sample performance of portfolios constructed based on the expected future return. Asset weights are allocated by 1/N method for the equal weighted portfolios, by the mean-variance approach and the mean-CVaR approach, with a target of 2% monthly return. The performance measures reported are the average of each portfolio's individual measures within a specific family of methods: average monthly return (AR), end of period value (EPV), annualized Sharpe ratio, Omega ratio, Sortino ratio, Upside Potential ratio, and 1% VaR. Out-of-sample forecasting period: Jan 2007–Dec 2009.

Table 7. Crisis Period -Out-of-sample performance of top t-statistics portfolios.

Panel A: Equal weighted	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.542	1.189	0.030	0.078	0.258	1.228	0.105	0.565	5.449
BMA	0.619	1.200	0.047	0.052	0.282	1.287	0.128	0.571	7.955
BMS	0.595	1.192	0.014	0.009	0.265	1.254	0.116	0.573	7.071
DMA	0.751	1.255	0.133	0.192	0.419	1.500	0.219	0.664	6.841
DMS	0.581	1.186	−0.022	−0.054	0.239	1.234	0.104	0.556	7.503
TVP-ALL	0.861	1.297	0.242	0.448	0.558	1.645	0.302	0.773	5.800
TVP-AR(1)	0.653	1.238	0.108	0.236	0.396	1.436	0.190	0.626	6.313
Panel B: Mean-Variance	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.462	1.144	−0.071	−0.131	0.086	1.071	0.033	0.494	8.210
BMA	0.632	1.234	0.130	0.426	0.512	1.596	0.214	0.573	6.226
BMS	0.671	1.253	0.194	0.713	0.671	1.730	0.287	0.687	4.726
DMA	0.562	1.206	0.047	0.175	0.358	1.468	0.160	0.555	5.517
DMS	0.337	1.114	−0.117	−0.250	−0.033	1.072	0.013	0.414	6.589
TVP-ALL	0.856	1.339	0.401	1.546	1.209	3.072	0.862	1.294	1.868
TVP-AR(1)	0.679	1.268	0.243	1.353	0.935	2.123	0.444	0.857	3.188
Panel C: Mean-CVaR	AR	EPV	α	$t_{\alpha }$	SR	Omega	Sortino	Upside	VaR 1%
OLS AR(1)	0.474	1.149	−0.066	−0.121	0.100	1.083	0.038	0.501	8.210
BMA	0.789	1.286	0.242	0.565	0.610	1.644	0.244	0.629	8.153
BMS	0.679	1.245	0.140	0.341	0.453	1.438	0.182	0.592	7.219
DMA	0.880	1.326	0.314	0.693	0.700	1.788	0.322	0.748	6.266
DMS	0.729	1.259	0.172	0.348	0.424	1.473	0.192	0.626	6.814
TVP-ALL	0.502	1.155	−0.080	−0.114	0.159	1.193	0.090	0.553	7.984
TVP-AR(1)	0.782	1.310	0.312	1.000	0.809	1.994	0.469	0.947	3.411

Notes: The table reports the out-of-sample performance of portfolios constructed based on the forecasting accuracy (t-statistics) of the expected return. Asset weights are allocated by 1/N method for the equal weighted portfolios, by the mean-variance approach and the mean-CVaR approach, with a target of 2% monthly return. The performance measures reported are the average of each portfolio's individual measures within a specific family of methods: average monthly return (AR), end of period value (EPV), annualized Sharpe ratio, Omega ratio, Sortino ratio, Upside Potential ratio, and 1% VaR. Out-of-sample forecasting period: Jan 2007–Dec 2009.

Figure 1. Optimal portfolio composition (expected returns).

Note: The figure shows the composition of portfolios selected based on the forecasted expected value of fund returns in Jan 2002, Jan 2006, Jan 2009 and Jan 2013. Selected models are OLS-AR(1) BMA, DMA and TVP-AR(1).

Figure 2. Optimal portfolio composition (t-statistics).

Note: The figure shows the composition of portfolios selected based on t-statistics of expected returns in Jan 2002, Jan 2006, Jan 2009 and Jan 2013. Selected models are OLS-AR(1), BMA, DMA and TVP-AR(1).

Clark, T. E., and K. D. West. 2007. “Approximately Normal Tests for Equal Predictive Accuracy in Nested Models.” Journal of Econometrics 138 (1): 291–311.

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