Table A.3.

VaR and ES forecasts at the 99% level of confidence for equally and optimally weighted portfolios, whose marginal returns follow AR(1)-APARCH(1,1) models with a normal or Student’s $t$ innovation.

Panel A: Equally Weighted Portfolio			Estimative VaR			Improved VaR			Estimative ES			Improved ES
			VaR	CCP	${\mathrm{RR}}_{\mathrm{V\; aR}}$	VaR	CCP	${\mathrm{RR}}_{\mathrm{V\; aR}}$	ES	$Z_{\mathrm{MF}}$ ($p$-value)	${\mathrm{RR}}_{\mathrm{ES}}$	ES	$Z_{\mathrm{MF}}$ ($p$-value)	${\mathrm{RR}}_{\mathrm{ES}}$
Before COVID-19
OIL, BTC	$N$	(50.000%, 50.000%)	4.572	99.363%	1.193	4.304	99.079%	1.088	5.239	−0.201 (0.873)	1.192	4.998	−0.078 (0.583)	1.109
	$t$	(50.000%, 50.000%)	6.057	99.847%	1.493	5.221	99.652%	1.290	6.950	0.868 (0.179)	1.404	6.201	1.011 (0.182)	1.256
OIL, BTC, USDT	$N$	(33.333%, 33.333%, 33.333%)	3.090	99.379%	0.676	2.901	99.086%	0.674	3.540	0.000 (0.358)	0.676	3.370	−0.282 (0.884)	0.674
	$t$	(33.333%, 33.333%, 33.333%)	4.078	99.852%	0.673	3.513	99.660%	0.673	4.678	0.835 (0.184)	0.673	4.172	0.978 (0.197)	0.673
OIL, BTC, USD	$N$	(33.333%, 33.333%, 33.333%)	3.052	99.354%	0.665	2.877	99.074%	0.665	3.498	−0.033 (0.567)	0.665	3.341	0.087 (0.305)	0.665
	$t$	(33.333%, 33.333%, 33.333%)	4.038	99.858%	0.665	3.473	99.677%	0.665	4.633	1.492 (0.127)	0.665	4.127	1.636 (0.135)	0.665
During COVID-19
OIL, BTC	$N$	(50.000%, 50.000%)	5.944	99.246%	0.902	5.704	99.036%	0.663	6.827	−0.012 (0.585)	0.904	6.610	−0.124 (0.729)	0.703
	$t$	(50.000%, 50.000%)	5.276	99.595%	0.829	4.727	99.256%	0.737	6.109	1.169 (0.333)	0.689	5.616	1.301 (0.365)	0.630
OIL, BTC, USDT	$N$	(33.333%, 33.333%, 33.333%)	3.961	99.247%	0.666	3.800	99.036%	0.666	4.549	−0.004 (0.573)	0.666	4.404	−0.116 (0.723)	0.666
	$t$	(33.333%, 33.333%, 33.333%)	3.516	99.595%	0.667	3.150	99.256%	0.667	4.072	1.180 (0.333)	0.667	3.743	1.312 (0.364)	0.667
OIL, BTC, USD	$N$	(33.333%, 33.333%, 33.333%)	3.882	99.250%	0.659	3.722	99.037%	0.658	4.459	−0.078 (0.655)	0.659	4.315	−0.035 (0.530)	0.659
	$t$	(33.333%, 33.333%, 33.333%)	3.379	99.609%	0.659	3.018	99.270%	0.659	3.915	1.051 (0.301)	0.659	3.591	1.359 (0.280)	0.659
Panel B: Optimally Weighted Portfolio			Estimative VaR			Improved VaR			Estimative ES			Improved ES
			VaR	CCP	${\mathrm{RR}}_{\mathrm{V\; aR}}$	VaR	CCP	${\mathrm{RR}}_{\mathrm{V\; aR}}$	ES	$Z_{\mathrm{MF}}$ ($p$-value)	${\mathrm{RR}}_{\mathrm{ES}}$	ES	$Z_{\mathrm{MF}}$ ($p$-value)	${\mathrm{RR}}_{\mathrm{ES}}$
Before COVID-19
OIL, BTC	$N$	(82.052%, 17.948%)	3.468	99.043%	0.905	3.444	99.002%	0.871	3.978	−0.070 (0.612)	0.905	3.956	−0.063 (0.596)	0.878
	$t$	(89.751%, 10.249%)	3.610	99.416%	0.889	3.362	99.117%	0.831	4.146	0.044 (0.513)	0.838	3.924	0.255 (0.378)	0.795
OIL, BTC, USDT	$N$	(7.291%, 0.465%, 92.244%)	0.953	99.739%	0.275	0.841	99.437%	0.244	1.089	−0.053 (0.574)	0.274	0.989	−0.168 (0.696)	0.250
	$t$	(6.349%, 0.201%, 93.450%)	0.901	99.880%	0.250	0.776	99.681%	0.231	1.030	−0.147 (0.931)	0.248	0.918	−0.364 (0.980)	0.234
OIL, BTC, USD	$N$	(2.370%, 0.749%, 96.881%)	0.553	99.160%	0.159	0.539	99.014%	0.156	0.636	−0.138 (0.767)	0.159	0.622	−0.264 (0.902)	0.157
	$t$	(2.626%, 0.549%, 96.825%)	0.583	99.256%	0.161	0.559	99.040%	0.166	0.670	0.000 (0.395)	0.161	0.648	0.000 (0.418)	0.165
During COVID-19
OIL, BTC	$N$	(70.106%, 29.894%)	5.511	98.478%	0.837	5.981	99.067%	0.695	6.324	−0.118 (0.738)	0.838	6.750	0.000 (0.346)	0.718
	$t$	(72.485%, 27.515%)	4.765	99.379%	0.748	4.448	99.097%	0.694	5.521	1.385 (0.244)	0.623	5.236	1.460 (0.261)	0.587
OIL, BTC, USDT	$N$	(0.112%, 0.080%, 99.808%)	0.147	98.606%	0.027	0.157	98.961%	0.026	0.170	0.173 (0.156)	0.027	0.179	0.250 (0.073)	0.026
	$t$	(0.044%, 0.029%, 99.928%)	0.071	99.239%	0.015	0.068	99.062%	0.015	0.081	1.529 (0.303)	0.015	0.079	1.731 (0.288)	0.015
OIL, BTC, USD	$N$	(2.855%, 1.727%, 95.418%)	0.759	99.334%	0.139	0.717	99.066%	0.121	0.872	0.194 (0.167)	0.139	0.834	0.354 (0.107)	0.125
	$t$	(3.121%, 1.733%, 95.146%)	0.752	99.401%	0.162	0.702	99.099%	0.163	0.866	0.313 (0.162)	0.161	0.820	0.065 (0.408)	0.162

Note: CCP refers to the conditional coverage probability of the VaR forecast computed using the simulation method of Kabaila and Syuhada (2008). The test statistic $Z_{\mathrm{MF}}$ of McNeil and Frey (2000) was utilized for ES backtesting we carried out through bootstrapping. The resulting $p$-value in italics indicates that the corresponding null hypothesis fails to be rejected at the 5% level of significance. Meanwhile, a relative ratio (RR) in boldface is below one, indicating portfolio risk reduction.