Value-at-risk in the presence of asset price bubbles

Figures & data

Table 1. Contingency table of fisher’s exact test for VaR measures in different periods

	VaR success	VaR failure
Period A	$v_0^A$	$v_1^A$
Period B	$v_0^B$	$v_1^B$

Table 2. Stock exchanges and the respective indices used

Country/Region	Exchange Name	Index Used	Market Capitalisation	Sample Period^c
Australia (AU)	Australian Securities Exchange	S&P/ASX 200	1,454,171.20^a	2000M4 to 2019M10
Canada (CA)	Toronto Stock Exchange	S&P/TSX Composite	2,286,818.24^a	1993M5 to 2019M10
China (CN)	Shanghai Stock Exchange	SSE Composite	4,776,752.70^a	1997M5 to 2019M10
Germany (DE)	Deutsche Borse	DAX 30 Performance	1,949,134.03^a	1997M5 to 2019M10
Hong Kong (HK)	Hong Kong Stock Exchange	Hang Seng	4,189,279.03^a	1993M9 to 2019M10
Spain (ES)	BME Spanish Exchange	IBEX 35	774,949.68^a	1991M1 to 2019M10
Japan (JP)	Japan Exchange Group	NIKKEI 225	5,614,217.12^a	1993M5 to 2019M10
United Kingdom (UK)	London Stock Exchange	FTSE 100	4,903,809.99^b	2002M2 to 2019M10
United States (US)	New York Stock Exchange	S&P500	24,230,840.32^a	1978M2 to 2019M10

^aData on market capitalisation obtained from World Federation of Exchanges, June 2019; values are in US$ million.

^bData obtained from the London Stock Exchange Main Market Factsheet, June 2019; values are in US$ million, converted at 1.26993 USD/$\mathit{\pounds }$ 1.

^cM1, M2, …, M12 represent the months January to December, respectively.

Table 3. GSADF tests for the log price–dividend ratio and log dividend difference in the nine markets

				Critical Values
Country/Region	Sample Size ($\mathit{T}$)	log $\mathit{PD}$ Ratio	log $\mathrm{\Delta D}$	10%	5%	1%
Australia (AU)	235	2.47	–2.74	1.79	2.07	2.61
Canada (CA)	318	2.02	0.73	1.92	2.18	2.77
China (CN)	264	3.39	–1.27	1.81	2.10	2.62
Germany (DE)	270	2.60	–1.70	1.82	2.14	2.70
Hong Kong (HK)	314	2.07	–2.53	1.91	2.16	2.77
Spain (ES)	346	3.63	–1.87	1.92	2.21	2.76
Japan (JP)	318	2.52	–3.64	1.92	2.18	2.77
United Kingdom (UK)	213	2.85	–2.70	1.68	1.94	2.60
United States (US)	318	4.11	–1.63	1.92	2.18	2.77

We used Monte Carol Simulations to obtain the critical values with 2,000 replications following Phillips et al. (2015). The data generation process is $y_t=\mathit{d}{T}^{-\mathit{\eta }}+y_{\mathit{t}-1}+{\in }_t,{\in }_t\sim \mathit{N}(0,1),\mathit{d}=\mathit{\eta }=1$

Table 4. Bubble periods obtained by the PSY test and the modified date-stamping method

Date-stamping			Country/Region
method			AU	CA	CN	DE	ES	HK	JP	UK	US
	Period 1	From	2008M11	2008M4	2007 M2	2008M11	1997M6	2007M11	2008M4	2008M11	1996 M2
		to	2009M7	2009M7	2007M10	2009M8	1997M11	2008 M3	2008M8	2009M7	1996M7
	Period 2	From					1998M4		2008M11		1996M12
PSY		to					1998M8		2009M7		2000M9
method	Period 3	From					2000M10				2008M11
		to					2001 M2				2009M9
	Period 4	From					2008M10
		to					2009M10
	Period 1	From	2003M5	2002M11	2004M8	2005M5	1994 M2	2004M9	2003M6	2004M4	1989M8
Modified		to	2009 M2	2009 M2	2007M5	2009 M3	2000M9	2007M10	2009 M2	2009 M2	2000 M3
method	Period 2	From					2004M8				2003 M3
		to					2009M5				2009 M3

Figure 1. Bubble identification results for the hong kong stock market.

Table 5. Full, pre-bubble, bubble, post-burst, and normal periods for the bubbles examined in the nine stock markets

		Country/Region
	Periods	AU	CA	CN	DE	ES	HK	JP	UK	US
Full		2000M4	1993M5	1997M11	1997M5	1991M1	1993M9	1993M5	2002 M2	1978 M2
		to	to	to	to	to	to	to	to	to
		2019M10	2019M10	2019M10	2019M10	2019M10	2019M10	2019M10	2019M10	2019M10
Pre-bubble	Period 1	2001M5	2000M11	2002M08	2003M5	1992 M2	2002M9	2001M6	2002M4	1987M8
		to	to	to	to	to	to	to	to	to
		2003M4	2002M10	2004M7	2005M4	1994M1	2004M8	2003M5	2004 M3	1989M7
	Period 2					2002M10				2002M4
						to				to
						2004M7*				2003 M2*
Bubble	Period 1	2003M5	2002M11	2004M8	2005M5	1994 M2	2004M9	2003M6	2004M4	1989M8
		to	to	to	to	to	to	to	to	to
		2009 M2	2009 M2	2007M5	2009 M3	2000M9	2007M10	2009 M2	2009 M2	2000 M3
	Period 2					2004M8				2003 M3
						to				to
						2009M5				2009 M3
Post-burst	Period 1	2009 M3	2009 M3	2007M6	2009M4	2000M10	2007M11	2009 M3	2009 M3	2000M4
		to	to	to	to	to	to	to	to	to
		2011 M2	2011 M2	2009M5	2011 M3	2002M9	2009M10	2011 M2	2011 M2	2002 M3
	Period 2					2009M6				2009M4
						to				to
						2011M5				2011 M3
Normal	Period 1	2000M4	1993M5	1997M11	1997M5	1991M1	1993M9	1993M5	2002 M2	1978 M2
		to	to	to	to	to	to	to	to	to
		2001M4	2000M10	2002M7	2003M4	1992M1	2002M8	2001M5	2002 M3	1987M7
	Period 2	2011 M3	2011 M3	2009M6	2011M4	2011M6	2009M11	2011 M3	2011 M3	2011M4
		to	to	to	to	to	to	to	to	to
		2019M10	2019M10	2019M10	2019M10	2019M10	2019M10	2019M10	2019M10	2019M10

^*As the duration between the bubble period is less than four years (two years for the first bubble’s post-burst period and two years for the second bubble’s pre-bubble period), the duration of the pre-bubble period of the second bubble is less than two years.

Table 6. Backtest results of riskmetrics VaR in the pre-bubble, bubble, and post-burst periods

			Tests
			Coverage Test		Independence Test		Joint Test
Period	Country/Region	Violations/nobs	POF	TUFF	Christoffersen	Mixed-Kupiec	Christoffersen	Mixed-Kupiec
	AU	6/507	0.687	0.968	0.704	0.220	0.858	0.297
	CA	7/503	0.405	0.783	0.002*	0.003*	0.006*	0.004*
	CN	2/482	0.144	0.500	0.897	0.337	0.341	0.229
	DE	2/510	0.116	0.028*	0.003*	0.001*	0.003*	0.001*
Pre-bubble	ES (Period 1)	9/504	0.110	0.871	0.144	0.000*	0.096	0.000*
	ES (Period 2)	2/458	0.173	0.102	0.895	0.185	0.391	0.155
	HK	6/496	0.650	0.205	0.701	0.125	0.838	0.178
	JP	3/492	0.348	0.764	0.848	0.039*	0.633	0.055
	UK	13/507 [12/507]	0.003* [0.009*]	0.542	0.038* [0.028*]	0.000*	0.002* [0.020*]	0.000*
	US (Period 1)	9/505	0.112	0.489	0.000*	0.000*	0.000*	0.000*
	US (Period 2)	5/232	0.125	0.745	0.083	0.004*	0.068	0.003*
	AU	43/1,478	0.000*	0.268	0.041*	0.000*	0.000*	0.000*
	CA	55/1,592 [54/1,592]	0.000*	0.992	0.001* [0.002*]	0.000*	0.000*	0.000*
	CN	17/684	0.001*	0.521	0.352	0.001*	0.003*	0.000*
	DE	33/998	0.000*	0.521	0.926	0.000*	0.000*	0.000*
Bubble	ES (Period 1)	39/1,675	0.000*	0.002*	0.002*	0.000*	0.000*	0.000*
	ES (Period 2)	33/1,223	0.000*	0.038*	0.176	0.000*	0.000*	0.000*
	HK	24/783	0.000*	0.307	0.763	0.000*	0.000*	0.000*
	JP	46/1,412	0.000*	0.820	0.261	0.000*	0.000*	0.000*
	UK	43/1,242 [42/1,242]	0.000*	0.255	0.074 [0.228]	0.000*	0.000*	0.000*
	US (Period 1)	53/2,697	0.000*	0.563	0.108	0.000*	0.000*	0.000*
	US (Period 2)	45/1,532	0.000*	0.102	0.193	0.000*	0.000*	0.000*
	AU	3/506	0.319	0.168	0.850	0.031*	0.598	0.042*
	CA	1/501	0.028*	0.130	0.950	0.130	0.089	0.028*
	CN	14/488	0.001*	0.011*	0.410	0.000*	0.002*	0.000*
	DE	4/511	0.608	0.215	0.801	0.125	0.849	0.187
Post-burst	ES (Period 1)	13/499	0.003*	0.068	0.404	0.001*	0.008*	0.000*
	ES (Period 2)	6/512	0.704	0.536	0.706	0.016*	0.866	0.028*
	HK	17/492	0.000*	0.011*	0.017*	0.000*	0.000*	0.000*
	JP	5/488	0.957	0.200	0.035*	0.005*	0.109	0.010*
	UK	6/505	0.680	0.191	0.704	0.001*	0.854	0.002*
	US (Period 1)	4/497	0.651	0.089	0.799	0.212	0.874	0.302
	US (Period 2)	6/505	0.680	0.219	0.704	0.035*	0.854	0.056

The backtest results of the RiskMetrics models that based on t-distribution do not differ from those with normal distribution. The only different results from t-distribution are shown in the square bracket[].

The decimal numbers in the cells are the p-values for assessing the null hypothesis that the model adequately measures the downside risk. Asterisk (*) indicates a p-value of less than 0.05.

Table 7. Backtest results of riskmetrics VaR in the normal and full periods

			Tests
			Coverage Test		Independence Test		Joint Test
Period	Country/Region	Violations/nobs	POF	TUFF	Christoffersen	Mixed-Kupiec	Christoffersen	Mixed-Kupiec
	AU (Period 1)	1/270	0.233	0.100	0.931	0.100	0.489	0.127
	AU (Period 2)	47/2,196	0.000*	0.686	0.000*	0.000*	0.000*	0.000*
	CA (Period 1)	51/1,888	0.000*	0.499	0.216	0.000*	0.000*	0.000*
	CA (Period 2)	58/2,177 [57/2,177]	0.000*	0.068	0.000*	0.000*	0.000*	0.000*
	CN (Period 1)	24/1,141	0.001*	0.459	0.100	0.000*	0.001*	0.000*
	CN (Period 2)	55/2,536	0.000*	0.595	0.007*	0.000*	0.000*	0.000*
	DE (Period 1)	37/1,513	0.000*	0.176	0.012*	0.000*	0.000*	0.000*
	DE (Period 2)	53/2,175	0.000*	0.735	0.000*	0.000*	0.000*	0.000*
	ES (Period 1)	2/269	0.658	0.624	0.862	0.884	0.893	0.931
Normal	ES (Period 2)	35/2,155	0.008*	0.467	0.282	0.000*	0.016*	0.000*
	HK (Period 1)	40/2,222	0.001*	0.909	0.000*	0.000*	0.000*	0.000*
	HK (Period 2)	53/2,465	0.000*	0.199	0.891	0.000*	0.000*	0.000*
	JP (Period 1)	37/1,992	0.001*	0.789	0.187	0.000*	0.001*	0.000*
	JP (Period 2)	45/2,125	0.000*	0.089	0.016*	0.000*	0.000*	0.000*
	UK (Period 1)	0/40	0.370	0.423	N/A	N/A	N/A	N/A
	UK (Period 2)	46/2,192	0.000*	0.945	0.000*	0.000*	0.000*	0.000*
	US (Period 1)	33/2,402	0.081	0.070	0.477	0.000*	0.170	0.000*
	US (Period 2)	57/2,161	0.000*	0.455	0.004*	0.000*	0.000*	0.000*
	AU	100/4,957	0.000*	0.100	0.000*	0.000*	0.000*	0.000*
	CA	172/6,661 [170/6,661]	0.000*	0.499	0.000*	0.000*	0.000*	0.000*
	CN	112/5,331	0.000*	0.459	0.003*	0.000*	0.000*	0.000*
	DE	129/5,707	0.000*	0.176	0.000*	0.000*	0.000*	0.000*
Full	ES	139/7,295	0.000*	0.624	0.070	0.000*	0.000*	0.000*
	HK	140/6,458	0.000*	0.909	0.000*	0.000*	0.000*	0.000*
	JP	136/6,509	0.000*	0.789	0.001*	0.000*	0.000*	0.000*
	UK	108/4,486 [106/4,486]	0.000*	0.926	0.000*	0.000*	0.000*	0.000*
	US	212/10,531	0.000*	0.070	0.000*	0.000*	0.000*	0.000*

The backtest results of the RiskMetrics models that based on t-distribution do not differ from those with normal distribution. The only different results from t-distribution are shown in the square bracket[].

The decimal numbers in the cells are the p-values for assessing the null hypothesis that the model adequately measures the downside risk. Asterisk (*) indicates a p-value of less than 0.05.

Table 8. Backtest Results of RiskMetrics 2006 VaR in the Pre-bubble, Bubble, and Post-burst Periods

			Tests
			Coverage Test		Independence Test		Joint Test
Period	Country/Region	Violations/nobs	POF	TUFF	Christoffersen	Mixed-Kupiec	Christoffersen	Mixed-Kupiec
	AU	4/507	0.620	0.968	0.801	0.179	0.857	0.258
	CA	2/503	0.122	0.783	0.899	0.898	0.300	0.457
	CN	0/482	0.002*	0.033*	N/A	N/A	N/A	N/A
	DE	1/510	0.026*	0.362	0.950	0.362	0.083	0.055
Pre-bubble	ES (Period 1)	1/504	0.027*	0.070	0.950	0.070	0.087	0.017*
	ES (Period 2)	1/458	0.042*	0.102	0.947	0.102	0.126	0.033*
	HK	3/496	0.340	0.205	0.848	0.327	0.623	0.359
	JP	1/492	0.030*	0.764	0.949	0.764	0.096	0.092
	UK	2/507	0.118	0.304	0.900	0.321	0.293	0.195
	US (Period 1)	5/505	0.982	0.489	0.034*	0.020*	0.105	0.037*
	US (Period 2)	0/232	0.031*	0.326	N/A	N/A	N/A	N/A
	AU	11/1,478	0.301	0.268	0.685	0.190	0.589	0.196
	CA	13/1,592	0.447	0.992	0.644	0.507	0.673	0.539
	CN	5/684	0.458	0.047*	0.786	0.187	0.732	0.236
	DE	5/998	0.080	0.235	0.820	0.719	0.210	0.429
Bubble	ES (Period 1)	10/1,675	0.073	0.338	0.729	0.674	0.189	0.465
	ES (Period 2)	13/1,223	0.827	0.536	0.597	0.197	0.849	0.251
	HK	5/783	0.276	0.307	0.800	0.878	0.535	0.813
	JP	15/1,412	0.816	0.820	0.570	0.060	0.828	0.082
	UK	11/1,242	0.680	0.077	0.657	0.010*	0.832	0.015*
	US (Period 1)	25/2,697	0.700	0.563	0.494	0.172	0.735	0.204
	US (Period 2)	6/1,532	0.006*	0.003*	0.828	0.029*	0.024*	0.003*
	AU	0/506	0.001*	0.027*	N/A	N/A	N/A	N/A
	CA	0/501	0.002*	0.028*	N/A	N/A	N/A	N/A
	CN	5/488	0.957	0.011*	0.035*	0.001*	0.109	0.002*
	DE	1/510	0.026*	0.029*	0.950	0.029*	0.082	0.008*
Post-burst	ES (Period 1)	2/499	0.126	0.531	0.899	0.749	0.308	0.404
	ES (Period 2)	3/512	0.307	0.536	0.851	0.175	0.584	0.200
	HK	4/492	0.667	0.553	0.798	0.096	0.882	0.153
	JP	0/488	0.002*	0.032*	N/A	N/A	N/A	N/A
	UK	0/505	0.001*	0.027*	N/A	N/A	N/A	N/A
	US (Period 1)	4/497	0.651	0.089	0.799	0.424	0.834	0.539
	US (Period 2)	0/505	0.001*	0.027*	N/A	N/A	N/A	N/A

The decimal numbers in the cells are the p-values for assessing the null hypothesis that the model adequately measures the downside risk. Asterisk (*) indicates a p-value of less than 0.05.

Table 9. Backtest results of riskmetrics 2006 VaR in the full and normal periods

			Tests
			Coverage Test		Independence Test		Joint Test
Period	Country/Region	Violations/nobs	POF	TUFF	Christoffersen	Mixed-Kupiec	Christoffersen	Mixed-Kupiec
	AU (Period 1)	2/270	0.654	0.100	0.863	0.258	0.891	0.405
	AU (Period 2)	18/2,196	0.381	0.923	0.008*	0.004*	0.019*	0.005*
	CA (Period 1)	20/1,888	0.798	0.499	0.513	0.892	0.781	0.919
	CA (Period 2)	22/2,177	0.961	0.930	0.503	0.078	0.798	0.100
	CN (Period 1)	7/1,141	0.158	0.532	0.769	0.682	0.353	0.557
	CN (Period 2)	32/2,536	0.203	0.456	0.366	0.006*	0.295	0.006*
	DE (Period 1)	5/1,513	0.002*	0.810	0.856	0.006*	0.010*	0.000*
	DE (Period 2)	11/2,175	0.011*	0.917	0.738	0.112	0.036*	0.024*
	ES (Period 1)	2/269	0.658	0.624	0.862	0.884	0.893	0.931
Normal	ES (Period 2)	13/2,155	0.046*	0.542	0.691	0.468	0.126	0.271
	HK (Period 1)	16/2,222	0.163	0.909	0.103	0.009*	0.100	0.008*
	HK (Period 2)	15/2,465	0.035*	0.199	0.668	0.146	0.099	0.067
	JP (Period 1)	9/1,992	0.006*	0.517	0.775	0.029*	0.022*	0.004*
	JP (Period 2)	18/2,125	0.467	0.089	0.008*	0.008*	0.023*	0.010*
	UK (Period 1)	0/40	0.370	0.423	N/A	N/A	N/A	N/A
	UK (Period 2)	12/2,192	0.020*	0.938	0.054	0.027*	0.010*	0.008*
	US (Period 1)	12/2,402	0.006*	0.070	0.728	0.047*	0.023*	0.007*
	US (Period 2)	25/2,161	0.475	0.445	0.002*	0.000*	0.008*	0.000*
	AU	34/4,957	0.028*	0.100	0.025*	0.000*	0.007*	0.000*
	CA	57/6,661	0.225	0.499	0.321	0.141	0.293	0.135
	CN	49/5,331	0.548	0.532	0.476	0.000*	0.647	0.000*
	DE	23/5,707	0.000*	0.810	0.666	0.000*	0.000*	0.000*
Full	ES	45/7,295	0.000*	0.624	0.455	0.048*	0.001*	0.005*
	HK	43/6,458	0.004*	0.909	0.296	0.006*	0.009*	0.001*
	JP	43/6,509	0.003*	0.517	0.034*	0.000*	0.001*	0.000*
	UK	25/4,486	0.001*	0.213	0.131	0.000*	0.002*	0.000*
	US	77/10,531	0.004*	0.070	0.003*	0.000*	0.000*	0.000*

The decimal numbers in the cells are the p-values for assessing the null hypothesis that the model adequately measures the downside risk. Asterisk (*) indicates a p-value of less than 0.05.

Table 10. Fisher exact test for the significance level of the difference of VaR failure rate by time period

Period	Pre-bubble	Bubble	Post-burst	Normal	Full	Pre-bubble	Bubble	Post-burst	Normal	Full
Pre-bubble		1.000	0.124	1.000	0.776		0.544	0.500	0.226	0.438
Bubble	0.031*		0.076	1.000	0.859	0.015*		0.047*	0.547	1.000
Post-burst	0.506	0.002*		0.036*	0.070	0.069	0.000*		0.012*	0.032*
Normal	0.278	0.062	0.036*		0.563	0.097	0.134	0.000*		0.351
Full	0.237	0.044*	0.024*	0.861		0.104	0.060	0.000*	0.756
	(a) Australia					(b) Canada
Period	Pre-bubble	Bubble	Post-burst	Normal	Full	Pre-bubble	Bubble	Post-burst	Normal	Full
Pre-bubble		0.081	0.062	0.019*	0.032*		0.670	1.000	0.712	0.717
Bubble	0.005*		0.750	0.535	0.829	0.000*		0.670	0.790	0.597
Post-burst	0.004*	0.715		1.000	0.803	0.687	0.002*		0.712	0.717
Normal	0.005*	0.570	0.326		0.515	0.001*	0.146	0.015*		0.870
Full	0.005*	0.484	0.255	0.882		0.002*	0.057	0.024*	0.576
	(c) China					(d) Germany
Period	Pre-bubble	Bubble	Post-burst	Normal	Full	Pre-bubble	Bubble	Post-burst	Normal	Full
Pre-bubble		0.061	0.453	0.178	0.166		1.000	0.725	1.000	1.000
Bubble	0.014*		0.394	0.515	0.345	0.037*		0.740	1.000	1.000
Post-burst	0.201	0.332		0.808	0.828	0.021*	0.745		0.570	0.573
Normal	0.519	0.015*	0.462		1.000	0.300	0.061	0.046*		1.000
Full	0.122	0.076	1.000	0.253		0.192	0.125	0.081	0.546
	(e) Spain					(f) Hong Kong
Period	Pre-bubble	Bubble	Post-burst	Normal	Full	Pre-bubble	Bubble	Post-burst	Normal	Full
Pre-bubble		0.087	1.000	0.356	0.368		0.369	0.500	1.000	1.000
Bubble	0.001*		0.016*	0.154	0.120	0.372		0.040*	0.275	0.221
Post-burst	0.504	0.008*		0.107	0.072	0.163	0.010*		0.139	0.170
Normal	0.031	0.010*	0.160		1.000	0.497	0.014*	0.277		1.000
Full	0.018*	0.011*	0.131	0.779		0.762	0.045*	0.085	0.388
	(g) Japan					(h) United Kingdom
Period	Pre-bubble	Bubble	Post-burst	Normal	Full
Pre-bubble		1.000	0.507	0.796	1.000
Bubble	0.590		0.289	0.432	1.000
Post-burst	0.145	0.006*		0.639	0.320
Normal	1.000	0.269	0.035*		0.392
Full	1.000	0.253	0.022*	0.899
			(i) United States

The decimal numbers in the cells are the p-values for assessing the null hypothesis that the VaR failure rates show no significant differences between the periods. The results of RiskMetrics are shown in the lower triangular matrix (non-shaded) and the results of RiskMetrics 2006 are shown in the upper triangular matrix (shaded). Asterisk (*) indicates a p-value of less than 0.05. P-value of Fisher test larger than 0.05 is indicative no statistically significant relationship of the VaR failure between two chosen periods.

Phillips, P., Shi, S., & Yu, J. (2015). Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500. International Economic Review, 56(4), 1043–1078.

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