Research Article

Modeling and Risk Analysis Using Parametric Distributions with an Application in Equity-Linked Securities

Table 10

VaR violation backtesting for four indices. and are the unconditional coverage test likelihood ratio statistics and the conditional coverage test likelihood ratio statistics, respectively. The Cauchy and Laplace distributions are excluded from this test because they show terrible graphical tail test for the stock indices given by Figure 11. Exp.XVaR (0.01) = number of expected violations and Act.XVaR (0.01) = number of actual violations given 99% VaR estimated.

HSCEIKOSPI 200S&P 500EURO STOXX 50
Trading days (N)489492504512
Exp.XVaR (0.01)4455

Normal
Act.XVaR (0.01)961711
2.79550.223817.70645.1327
p value0.09450.6361≤0.00000.0234
3.1330.372218.8955.6168
p value0.20860.8301≤0.00000.0603

Student’s t
Act.XVaR (0.01)85128
1.6760.00136.99751.3969
p value0.19540.97110.0080.2372
1.94260.10417.58411.6514
p value0.37850.94920.02250.4379

Skew normal
Act.XVaR (0.01)1271311
7.430.78538.84375.1327
p value0.0060.37550.0020.0234
8.03510.98789.53365.6168
p value0.01790.61020.00850.0603

Skew Laplace
Act.XVaR (0.01)2168
2.2214.68480.17411.3969
p value0.13610.03040.67650.2372
2.23754.68890.31891.6514
p value0.32660.09580.85250.4379

Skew Student’s t
Act.XVaR (0.01)9488
2.79550.18561.49011.3969
p value0.09450.66650.2220.2372
3.13370.25131.74871.6514
p value0.20860.88190.41710.4379
Hyperbolic
Act.XVaR (0.01)8378
1.67600.87930.68671.3969
p value0.19540.34830.40720.2372
1.94260.91620.88431.6514
p value0.37850.63240.64260.4379

NIG
Act.XVaR (0.01)8379
1.67600.87930.68672.4230
p value0.19540.34830.40720.1195
1.94260.91620.88432.7457
p value0.37850.63240.64260.2533

Variance gamma
Act.XVaR (0.01)8399
1.67600.87932.54822.423
p value0.19540.34830.11040.1195
1.94260.91622.87612.7457
p value0.37850.63240.23730.2533

G. hyperbolic
Act.XVaR (0.01)8399
1.67600.87932.54822.423
p value0.19540.34830.11040.1195
1.94260.91622.87612.7457
p value0.37850.63240.23730.2533