Testing expected shortfall: an application to emerging market stock indices

Abstract

In a recent paper, Acerbi and Székely (Risk Magazine, 76–81, 2014) presented three methods to test expected shortfall, and this is the first empirical application of that paper on emerging markets. We employ daily stock index returns from the Morgan Stanley Capital International Inc. Emerging Markets Index covering the 2000–2015 period, extending Acerbi and Székely (Risk Magazine, 76–81, 2014) results to derive the significance thresholds for the Student’s skewed-t distribution using two testing methods. We find that the thresholds for the Z₁ Test and Z₂ Test for skewed-t distribution are similar to the values obtained by Acerbi and Székely for Student’s t distribution. Therefore, the Z₁ and Z₂ thresholds are invariant to the skewed-t-shaped parameter values found in the emerging market stock indices. Empirical results show outperformance of Student’s skewed-t and Student’s t distributions over Gaussian distribution.

Appendices

Appendix A: Dataset description

Country index	Bloomberg ticker
Americas
Brazil	IBOV
Chile	IPSA
Colombia	COLCAP
Mexico	MEXBOL
Peru	IGBVL
Europe, Middle East & Africa
Czech Republic	PX
Egypt	HERMES
Greece	ASE
Hungary	BUX
Poland	WIG
Qatar	DSM
Russia	INDEXCF
South Africa	TOP40
Turkey	XU100
ASIA
China	SHSZ300
India	SENSEX
Indonesia	JCI
Korea	KRX100
Malaysia	FBMKLCI
Philippines	PCOMP
Taiwan	TWSE
Thailand	SET

1. Source Bloomberg and msci.com

Appendix B: Model selection when innovations are normally, t and skewed-t distributed

Models	GARCH(1,1)			AR(1) GARCH(1,1)			ARMA(1,1) GARCH(1,1)
Stock index	Log likelihood	AIC	BIC	Log likelihood	AIC	BIC	Log likelihood	AIC	BIC
Panel A: Normal
Brazil	− 9703.64	5.11	5.10	− 9704.75	5.11	5.10	− 9705.45	5.10	5.10
Chile	− 12,112.69	6.37	6.37	− 12,132.88	6.38	6.37	− 12,138.10	6.39	6.38
Colombia	− 9727.28	6.27	6.27	− 9760.99	6.29	6.28	− 9761.02	6.29	6.28
Mexico	− 9679.34	5.09	5.09	− 9680.24	5.09	5.08	− 9680.70	5.09	5.08
Peru	− 9679.34	5.09	5.09	− 9680.24	5.09	5.08	− 9680.70	5.09	5.08
Czech Rep	− 10,954.30	5.76	5.76	− 10,954.63	5.76	5.75	− 10,955.25	5.76	5.75
Egypt	− 10,128.27	5.62	5.62	− 10,188.31	5.66	5.65	− 10,189.32	5.66	5.65
Greece	− 9968.73	5.24	5.24	− 9992.49	5.26	5.25	− 10,109.36	5.32	5.31
Hungary	− 10,529.39	5.54	5.53	− 10,531.51	5.54	5.53	− 10,532.80	5.54	5.53
Poland	− 9679.34	5.09	5.09	− 9680.24	5.09	5.08	− 9680.70	5.09	5.08
Qatar	− 11,755.57	6.19	6.18	− 11,823.58	6.22	6.21	− 11,823.75	6.22	6.21
Russia	− 8954.01	4.71	4.70	− 8954.06	4.71	4.70	− 8954.07	4.71	4.70
South Africa	− 9679.34	5.09	5.09	− 9680.24	5.09	5.08	− 9680.70	5.09	5.08
Turkey	− 9679.34	5.09	5.09	− 9680.24	5.09	5.08	− 9680.70	5.09	5.08
China	− 8809.01	5.50	5.50	− 8809.86	5.50	5.49	− 8810.71	5.50	5.49
India	− 10,364.87	5.45	5.45	− 10,367.15	5.45	5.45	− 10,396.30	5.47	5.46
Indonesia	− 10,876.98	5.72	5.72	− 10,893.35	5.73	5.72	− 10,894.53	5.73	5.72
Korea	− 10,145.89	5.80	5.79	− 10,146.14	5.79	5.79	− 10,147.57	5.80	5.78
Malaysia	− 9679.34	5.09	5.09	− 9680.24	5.09	5.08	− 9680.70	5.09	5.08
Phillipines	− 10,619.01	5.59	5.58	− 10,631.41	5.59	5.58	− 10,637.45	5.60	5.59
Taiwan	− 10,328.95	5.43	5.43	− 10,356.28	5.45	5.44	− 10,558.47	5.55	5.54
Thailand	− 10,931.66	5.75	5.74	− 10,941.49	5.76	5.75	− 10,943.82	5.76	5.75
Panel B: Student’s t
Brazil	− 10,073.49	5.30	5.29	− 10,074.55	5.30	5.29	− 10,076.03	5.30	5.29
Chile	− 12,539.57	6.60	6.59	− 12,591.60	6.62	6.61	− 12,592.91	6.62	6.61
Colombia	− 9791.16	6.31	6.30	− 9822.43	6.33	6.32	− 9822.48	6.33	6.32
Mexico	− 11,369.76	5.98	5.97	− 11,379.70	5.99	5.98	− 11,381.80	5.99	5.98
Peru	− 11,516.77	6.06	6.05	− 11,578.67	6.09	6.08	− 11,578.72	6.09	6.08
Czech Rep	− 11,384.93	5.99	5.98	− 11,387.01	5.99	5.98	− 11,387.14	5.99	5.98
Egypt	− 10,246.76	5.69	5.68	− 10,299.70	5.72	5.71	− 10,300.79	5.72	5.71
Greece	− 10,676.43	5.62	5.61	− 10,690.51	5.62	5.61	− 10,691.40	5.62	5.61
Hungary	− 10,762.09	5.66	5.65	− 10,763.42	5.66	5.65	− 10,764.55	5.66	5.65
Poland	− 11,348.47	5.97	5.96	− 11,354.27	5.97	5.96	− 11,355.22	5.97	5.96
Qatar	− 12,427.34	6.54	6.53	− 12,554.03	6.60	6.59	− 12,556.00	6.60	6.59
Russia	− 9784.28	5.15	5.14	− 9784.40	5.15	5.14	− 9784.41	5.15	5.13
South Africa	− 10,073.49	5.30	5.29	− 10,074.55	5.30	5.29	− 10,076.03	5.30	5.29
Turkey	− 12,539.57	6.60	6.59	− 12,591.60	6.62	6.61	− 12,592.91	6.62	6.61
China	− 11,369.76	5.98	5.97	− 11,379.70	5.99	5.98	− 11,381.80	5.99	5.98
India	− 11,516.77	6.06	6.05	− 11,578.67	6.09	6.08	− 11,578.72	6.09	6.08
Indonesia	− 11,384.93	5.99	5.98	− 11,387.01	5.99	5.98	− 11,387.14	5.99	5.98
Korea	− 10,198.86	5.83	5.82	− 10,198.88	5.82	5.81	− 10,200.89	5.83	5.81
Malaysia	− 10,676.43	5.62	5.61	− 10,690.51	5.62	5.61	− 10,691.40	5.62	5.61
Philippines	− 10,762.09	5.66	5.65	− 10,763.42	5.66	5.65	− 10,764.55	5.66	5.65
Taiwan	− 11,348.47	5.97	5.96	− 11,354.27	5.97	5.96	− 11,355.22	5.97	5.96
Thailand	− 12,427.34	6.54	6.53	− 12,554.03	6.60	6.59	− 12,556.00	6.60	6.59
Panel C: Skewed-t
Brazil	− 10,084.16	5.30	5.29	− 10,085.46	5.30	5.29	− 10,089.21	5.31	5.29
Chile	− 12,552.49	6.60	6.59	− 12,597.93	6.63	6.62	− 12,599.91	6.63	6.61
Colombia	− 9797.18	6.32	6.31	− 9826.46	6.34	6.32	− 9826.58	6.33	6.32
Mexico	− 11,386.78	5.99	5.98	− 11,393.55	5.99	5.98	− 11,397.00	5.99	5.98
Peru	− 11,519.84	6.06	6.05	− 11,580.20	6.09	6.08	− 11,580.27	6.09	6.08
Czech Rep	− 11,398.14	6.00	5.99	− 11,399.26	6.00	5.98	− 11,399.41	6.00	5.98
Egypt	− 10,264.72	5.70	5.69	− 10,309.90	5.72	5.71	− 10,312.17	5.72	5.71
Greece	− 10,682.95	5.62	5.61	− 10,694.87	5.63	5.61	− 10,696.05	5.63	5.61
Hungary	− 10,762.41	5.66	5.65	− 10,763.64	5.66	5.65	− 10,764.77	5.66	5.65
Poland	− 11,351.24	5.97	5.96	− 11,356.31	5.97	5.96	− 11,357.36	5.97	5.96
Qatar	− 12,427.63	6.54	6.53	− 12,554.05	6.60	6.59	− 12,556.04	6.60	6.59
Russia	− 9798.18	5.15	5.14	− 9798.40	5.15	5.14	− 9799.31	5.15	5.14
South Africa	− 11,264.57	5.93	5.92	− 11,264.80	5.93	5.91	− 11,264.83	5.92	5.91
Turkey	− 9767.90	5.14	5.13	− 9768.24	5.14	5.13	− 9770.02	5.14	5.12
China	− 8892.87	5.55	5.54	− 8893.46	5.55	5.54	− 8894.54	5.55	5.54
India	− 10,925.33	5.75	5.74	− 10,933.50	5.75	5.74	− 10,935.47	5.75	5.74
Indonesia	− 11,151.90	5.87	5.86	− 11,157.38	5.87	5.86	− 11,159.18	5.87	5.86
Korea	− 10,209.91	5.83	5.82	− 10,210.22	5.83	5.82	− 10,211.99	5.83	5.82
Malaysia	− 13,305.95	7.00	6.99	− 13,338.38	7.02	7.01	− 13,338.68	7.02	7.00
Philippines	− 11,347.10	5.97	5.96	− 11,365.51	5.98	5.97	− 11,365.70	5.98	5.96
Taiwan	− 11,149.59	5.87	5.86	− 11,152.82	5.87	5.85	− 11,153.16	5.87	5.85
Thailand	− 11,288.38	5.94	5.93	− 11,293.07	5.94	5.93	− 11,294.90	5.94	5.93

1. AIC represents the Akaike Information Criterion, whereas BIC stands for Bayesian Information Criterion

Appendix C. 99%-VaR Backtesting for different periods

See Appendix Tables 4, 5, 6.

Table 4 99%− VaR backtesting analysis for total period

Table 5 99%-VaR backtesting analysis for pre-crisis period

Table 6 99%-VaR backtesting analysis for crisis period

Appendix D: 97.5%-VaR backtesting for different periods

See Appendix Tables 7, 8, 9.

Table 7 97.5%-VaR backtesting analysis for total period

Table 8 97.5%-VaR backtesting analysis for pre-crisis period

Table 9 97.5%-VaR backtesting analysis for crisis period

Appendix E: Range of the parameters estimated for Student’s t and skewed-t distributions

Distribution	Student’s t (degrees of freedom)			Skewed-t (degrees of freedom; shape parameters)
Period	Total period	Pre-crisis	Crisis	Total period	Pre-crisis	Crisis
Brazil	3.43–10	7.28–10	4.93–10	3.56–10; 0.8–1.04	7.93–10; 0.81–1.04	4.95–10; 0.79–1.03
Chile	4.65–10	8.66–10	4.92–10	5–10; 0.78–1.07	8.66–10; 0.81–1.08	4.99–10; 0.77–1
Colombia	3.59–10	4.24–10	3.59–8.73	3.72–10; 0.77–1.06	4.27–10; 0.77–1.06	3.72–8.78; 0.87–1.05
Mexico	3.22–10	4.54–10	4.04–9.71	3.15–10; 0.8–1.16	4.52–10; 0.8–1.15	4.08–10; 0.82–0.94
Peru	3.53–10	4.78–10	3.95–10	3.56–10; 0.75–1.14	4.77–10; 0.92–1.13	3.94–10; 0.75–1.14
Czech Republic	4.1–10	4.22–10	6.62–10	4.26–10;0.7–1.1	4.26–10; 0.7–1.1	6.6–10; 0.72–1.08
Egypt	4.06–10	4.35–10	4.04–9.43	4.07–10; 0.72–1.13	4.37–10; 0.9–1.13	4.17–10; 0.74–0.9
Greece	3.77–10	4–10	5.27–10	4.01–10; 0.71–1.21	4.01–10; 0.75–1.15	5.45–10; 0.71–1.21
Hungary	5.21–10	6.06–10	5.22–10	5.23–10; 0.92–1.09	6.96–10; 0.92–1.09	5.23–10; 0.95–10
Poland	3.62–10	6.05–10	6.41–10	3.76–10; 0.86–1.2	6.56–10; 0.86–1.2	6.47–10; 0.87–1.06
Qatar	2.32–10	2.33–10	2.49–8.65	2.29–10; 0.86–1.17	2.28–10; 0.91–1.16	2.49–8.74; 0.85–1.06
Russia	3.32–10	3.94–10	4.75–10	3.48–10; 0.77–1.03	4.01–10; 0.77–0.98	4.97–10; 0.8–1.03
South Africa	3.56–10	6.62–10	7.23–10	3.68–10; 0.71–1.14	6.5–10; 0.7–1.14	7.19–10; 0.73–1
Turkey	4.57–10	5.13–10	5.1–10	4.8–10; 0.8–1.14	5.14–10; 0.81–1.15	5.07–10; 0.8–1.1
China	3.45–10	3.9–8.33	3.45–10	3.63–10; 0.72–1.3	4.11–10; 0.8–1.3	3.97–10; 0.72–0.9
India	3.14–10	5.65–10	4.17–10	3.31–10; 0.7–1.12	6.28–10; 0.7–1.02	4.58–10; 0.84–1.12
Indonesia	3.16–10	4.71–10	4.21–10	3.6–10; 0.76–1.12	4.9–10; 0.85–1.12	4.54–10; 0.77–1
Korea	5.47–10	6.35–10	5.46–10	5.79–10; 0.79–1.04	6.21–10; 0.83–0.98	5.8–10; 0.8–1
Malaysia	4.37–10	5.12–8.8	4.38–10	4.4–10; 0.77–1.14	5–8.95; 0.94–1.13	4.4–10; 0.77–1.13
Philippines	3.13–10	3.17–10	4.68–10	3.16–10	3.17–10; 0.96–1.14	5.12–10; 0.79–1
Taiwan	3.06–10	4.5–10	4.46–10	3.02–10; 0.77–1.19	4.5–10; 0.85–1.19	4.48–10; 0.76–0.95
Thailand	4.47–10	4.58–10	4.52–10	3.5–10; 0.83–1.12	4.57–10; 0.92–1.12	4.47–10; 0.83–1.06

Appendix F: Results for Z1 and Z2 tests

Innovations	Normal		Student’s t		Skewed-t
Country index	Z1	Z2	Z1	Z2	Z1	Z2
Panel A: Total period (January, 2000–February, 2015)
Brazil	− 0.17	− 0.68	− 0.08	− 0.40	− 0.17	0.23
Chile	− 0.25	− 0.60	− 0.16	− 0.47	− 0.22	0.03
Colombia	− 0.13	− 0.90	0.00	− 0.61	0.04	− 0.22
Mexico	− 0.42	− 0.88	− 0.26	− 0.74	− 0.28	− 0.17
Peru	− 0.49	− 0.77	− 0.36	− 0.55	− 0.35	− 0.43
Czech Rep	− 0.34	− 0.55	− 0.19	− 0.43	− 0.17	0.00
Egypt	− 0.22	− 0.78	− 0.09	− 0.62	− 0.09	− 0.17
Greece	− 0.32	− 0.39	− 0.21	− 0.40	− 0.19	0.02
Hungary	− 0.23	− 0.21	− 0.16	− 0.06	− 0.15	− 0.09
Poland	− 0.40	− 0.49	− 0.25	− 0.47	− 0.27	− 0.29
Qatar	− 0.30	− 0.67	− 0.09	− 0.46	− 0.11	− 0.53
Russia	− 0.34	− 0.72	− 0.17	− 0.59	− 0.16	− 0.09
South Africa	− 0.35	− 0.43	− 0.23	− 0.40	− 0.21	0.15
Turkey	− 0.12	− 0.38	− 0.07	− 0.08	− 0.03	0.04
China	− 0.14	− 0.38	− 0.00	− 0.10	− 0.02	− 0.01
India	− 0.26	− 0.62	− 0.12	− 0.45	− 0.12	0.03
Indonesia	− 0.21	− 0.78	− 0.08	− 0.55	− 0.13	0.01
Korea	0.04	− 0.65	0.05	− 0.40	0.07	0.33
Malaysia	− 0.42	− 0.73	− 0.25	− 0.58	− 0.25	− 0.24
Philippines	− 0.29	− 0.53	− 0.15	− 0.39	− 0.16	− 0.22
Taiwan	− 0.32	− 0.78	− 0.19	− 0.66	− 0.19	− 0.20
Thailand	− 0.29	− 0.61	− 0.11	− 0.40	− 0.14	− 0.18
Panel B: Pre-crisis period (January, 2000–July, 2007)
Brazil	− 0.05	− 0.56	0.02	− 0.37	0.01	0.26
Chile	− 0.11	− 0.36	− 0.04	− 0.13	− 0.03	− 0.00
Colombia	− 0.15	− 1.18	− 0.04	− 0.93	− 0.00	− 0.17
Mexico	− 0.08	− 0.74	− 0.00	− 0.48	0.00	0.00
Peru	− 0.12	− 0.47	− 0.02	− 0.11	− 0.03	− 0.27
Czech Rep	− 0.29	− 0.44	− 0.14	− 0.24	− 0.12	0.00
Egypt	− 0.17	− 0.33	− 0.04	− 0.15	− 0.07	− 0.32
Greece	− 0.13	− 0.17	− 0.03	− 0.03	0.02	0.19
Hungary	− 0.13	0.07	− 0.06	0.18	− 0.06	0.15
Poland	− 0.19	− 0.19	− 0.08	− 0.11	− 0.05	− 0.23
Qatar	− 0.45	− 1.24	− 0.21	− 0.73	− 0.18	− 1.20
Russia	− 0.16	− 0.66	0.00	− 0.45	0.04	0.15
South Africa	− 0.11	− 0.33	− 0.03	− 0.17	0.00	0.09
Turkey	− 0.14	− 0.14	− 0.13	− 0.30	− 0.11	− 0.34
China	− 0.16	− 0.35	− 0.07	0.04	− 0.02	− 0.54
India	− 0.18	− 0.68	− 0.09	− 0.47	− 0.04	0.25
Indonesia	− 0.16	− 0.82	− 0.06	− 0.54	− 0.07	− 0.38
Korea	− 0.04	− 0.54	0.03	− 0.31	0.08	0.43
Malaysia	− 0.17	− 0.24	− 0.03	− 0.09	− 0.06	− 0.06
Philippines	− 0.10	− 0.32	− 0.03	− 0.58	− 0.04	− 0.31
Taiwan	− 0.18	− 0.52	− 0.09	− 0.22	− 0.07	− 0.40
Thailand	− 0.23	− 0.51	− 0.13	− 0.29	− 0.11	− 0.34
Panel C: Crisis period (July, 2007–February, 2015)
Brazil	− 0.08	− 0.54	− 0.00	− 0.26	− 0.01	0.33
Chile	− 0.11	− 0.74	− 0.01	− 0.53	− 0.02	0.09
Colombia	− 0.10	− 0.60	0.06	− 0.34	0.07	− 0.24
Mexico	− 0.17	− 0.71	− 0.02	− 0.49	0.05	0.03
Peru	− 0.19	− 0.70	− 0.09	− 0.43	− 0.06	− 0.28
Czech Rep	− 0.12	− 0.47	− 0.02	− 0.37	0.00	0.00
Egypt	− 0.19	− 0.75	− 0.07	− 0.50	− 0.05	0.23
Greece	− 0.05	− 0.48	0.02	− 0.26	− 0.02	0.00
Hungary	− 0.14	− 0.40	− 0.07	− 0.16	− 0.07	− 0.22
Poland	− 0.07	− 0.59	− 0.01	− 0.35	− 0.01	− 0.04
Qatar	− 0.15	− 0.61	0.07	− 0.25	0.07	0.07
Russia	− 0.16	− 0.72	− 0.06	− 0.42	− 0.02	− 0.11
South Africa	− 0.09	− 0.31	− 0.02	− 0.13	0.04	0.37
Turkey	− 0.07	− 0.47	− 0.04	− 0.13	− 0.00	− 0.03
China	− 0.12	− 0.74	0.02	− 0.48	− 0.03	0.44
India	− 0.06	− 0.51	0.05	− 0.16	0.05	− 0.12
Indonesia	− 0.18	− 0.79	− 0.04	− 0.51	− 0.07	0.27
Korea	− 0.06	− 0.82	0.03	− 0.57	0.05	0.30
Malaysia	− 0.15	− 1.01	− 0.07	− 0.66	− 0.03	− 0.21
Philippines	− 0.14	− 0.69	− 0.04	− 0.43	− 0.04	− 0.42
Taiwan	− 0.10	− 0.98	0.01	− 0.69	0.01	0.24
Thailand	− 0.13	− 0.58	0.02	− 0.46	0.00	− 0.02

1. Values of Z₁ and Z₂ statistics are shown from Column 2–7. These values are obtained using Eqs. (11) and (13), respectively. To this end, VaR and ES are calculated by employing Eq. (8), after filtering the returns by an AR(1)-GARCH(1,1) model for three different innovation distributions: normal (Columns 2 and 3), Student’s t (Columns 4 and 5), and skewed-t (Columns 6 and 7). The critical value of Test 1 for Gaussian distribution is − 0.07 (− 0.04) when T = 1400 (3300), i.e., pre-crisis and crisis periods (total period). For Test 2, the critical value for Gaussian, Student’s t, and skewed-t distributions is − 0.30 (− 0. 1 9), when T = 1400 (3300), i.e., pre-crisis and crisis periods (total period). Then, a model underpredicts risk if the statistic value is ≤ the corresponding critical value