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Table 1. Descriptive Statistics Statistics
ASII BMRI
PTBA INTP
Mean 0.0009
0.0006 0.0006
0.0009 Standard
deviation 0.0276
0.0271 0.0317 0.0307 Skewness
0.1306 0.3690 ‐0.2404 0.370
Kurtosis 10.0123
8.2807 13.0502 45.0897 Jarque
‐Bera Test 5 1
1 1
1 1
= reject H Jarque
‐Bera Statistics 35700 20612 73398 12848 Jarque
‐Bera p‐Values 0.001
0.001 0.001 0.001
Jarque ‐Bera Crit‐Value 5.9586 5.9586 5.9586 5.9586
In this case the feasibility set is X defined on region satisfying 23 and 24. Set X is convex it is polyhedral, due to linearity in constraint 23 and 24. The
optimization problem 22-24 are a convex programming. This problem is solved using PortfolioCVaR class of MATLAB R2013a.
4. Empirical studies
In this section, the models of stock portfolio risk measurement and management described in the previous sections are implemented to a hypothetical
portfolio composed by 4 Indonesian Blue Chip stocksμ ASSI=Astra International Tbk, BMRI=Bank Mandiri Persero Tbk., PTBA=Tambang Batubara Bukit Asam
Tbk., IσTP=Indocement Tunggal Perkasa Tbk. The historical data are recorded during the period of 6 σovember 2006 to 2 August 2013. Precisely, the statistics of
the 4 stocks are described in Table 1. Table 1 reflects the mean, standard deviation, skewness, and kurtosis of the
daily returns of four stocks over the period from 6 σovember 2006 to 2 August 2013. The kurtosis of all stock returns exceeds the kurtosis of normal distribution
3.0 substantially. The Jarque-Bera tests of the null hypothesis that the return distribution follow a normal distribution against the alternative that the return do
not come from a normal distribution. As seen from Table 1. that the test statistics exceed the critical value at the 5 level of significant, this means that all stock
returns are not normally distributed, it shows a fat tailed distribution. This means that the probability of extreme events is higher than the probability of extreme
events under the normal distribution. Therefore, capturing the stock returns using normal distribution could be underestimated. When the skewness of the return data
are considered, it shows that all returns are right skewed except for PTBA. This suggests that the returns are not symmetry. Again, capturing by normal distribution
may result in misleading conclusions. Using results in Table 2 and Table 3, the value of c
13
,c
14
, and c
24
can be calculated using Equation 8-10, giving the correlation matrix ρ. Using ρ, the
273
Table 2. Estimated parameters and the standard errors from D-vine models D
‐vine copula Clayton
SJC c
12
0.3057 0.012
0.3374 0.036
0.2270 0.038
c
23
0.2636 0.014
0.1480 0.018
0.0541 0.038
c
34
0.1810 0.015
0.0992 0.032
0.0749 0.033
c
13|2
0.1059 0.015
0.4167 0.017
0.3650 0.027
c
24|3
0.1793 0.015
0.2269 0.027
0.0863 0.028
c
14|23
0.0982 0.014
0.2212 0.024
0.0538 0.029
AIC BIC
LL ‐1483.7095
‐1871.350 ‐1450.9396
‐1805.8104 747.855
948.753 Table 3. Estimated parameters and the standard errors from C-vine
models D
‐vine copula Clayton
SJC c
12
0.3057 0.012
0.3374 0.036
0.2270 0.038
C
13
0.2337 0.014
0.2274 0.031
0.0704 0.031
C
14
0.2401 0.014
0.0415 0.027
0.0266 0.023
C
23|1
0.1542 0.015
0.4127 0.024
0.3112 0.030
c
24|1
0.1391 0.016
0.3133 0.029
0.1864 0.033
c
14|2
0.0464 0.014
0.1604 0.034
0.0005 0.003
AIC BIC
LL ‐1487.152
‐1873.5069 ‐1454.382
‐1807.9672 749.576
947.675 return of the portfolio r
p
consisting of 4 assets are simulated by Gaussian copularnd of MatLab function. The vector weight
w = [0.25 0.25 0.25 0.25] is
used to compose the portfolio. Then apply PortfolioCVaR class [16] to r
p
= w
T
r.
The results are summarized in Table 4. σote that, for t C-Vine model, we use stepwise semiparametric estimator SSP proposed by Haff [9] Eqn.11-14, to
construct correlation matrix ρ. Table 4 reflects the value of VaR and CVaR simulated by Clayton D-Vine, t
C-Vine, Symmetrized Joe-Clayton SJC D-Vine, Gaussian pairwise, t student pairwise, and multivariate normal at 1 and 5 significant levels. The multivariate
normal calculated using the standard Markowitz mean-variance MV framework. As it was shown in [22] that when the loss functions come from normal distribution
then VaR and CVaR are equivalent in the sense that they generate the same efficient frontier. However, in the case of nonnormal distribution, and especially
skewed distributions, CVaR and MV portfolio optimization approaches result in significant
