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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.

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Notes

  1. Acerbi and Székely employed − 0.17 when T = 250, see Table 2, Panel A1.

  2. We are grateful to the Reviewer for pointing this out.

  3. P is the number of the random samples.

  4. For Colombian stock index 3101 observations, for Chinese stock index 3201 observations, for Korean stock index 3501 observations, and for Egyptian stock index 3601 observations.

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Acknowledgements

The authors are very grateful for the comments of the Editor and the Reviewer, which helped to improve the original version of the paper.

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Corresponding author

Correspondence to Daniel Velásquez-Gaviria.

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

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Cardona, E., Mora-Valencia, A. & Velásquez-Gaviria, D. Testing expected shortfall: an application to emerging market stock indices. Risk Manag 21, 153–182 (2019). https://doi.org/10.1057/s41283-018-0046-z

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Keywords

  • Value-at-risk
  • Expected shortfall
  • Backtesting
  • Skewed-t