The copula method can be employed with every given marginal
distribution. In order to make a comparison with the classical VaR
procedure, we will concentrate on Gaussian marginal densities.
Numerous copulas exist in the two-dimensional case,
Nelsen (1999). A selection is given in Table
16.1.
After choosing the copula, the parameter
needs to be
determined. This occurs on the basis of a given financial time
series
,
and log-returns
The time
series
represent the
risk factors and under the
assumption that
is normal, they are themselves log-normally
distributed. The variance of the Gaussian distribution is
estimated using the method shown in Section 3.2,
.
A very simple method for estimating the parameter
is the
least-squares method. In the case of determining the density
functions this means that the distance between the empirical
density function of the log-returns and the chosen parameterized
function from Table 16.1 is minimized. This can, for
example, be done using the Newton method, with however the
disadvantage that the estimation is concentrated on the region
containing the most data points. In risk management this is the
least interesting region.
The maximum likelihood method corrects this characteristic by
maximizing the likelihood function
Here
represents one of the densities resulting from the
combination of the marginal Gaussian distribution with a copula
from Table 16.1.
Assume that a copula
has been selected. Analytical methods to
calculate the VaR only exist in a few cases, e.g., for the
Gaussian copula. Therefore, one has to rely on Monte Carlo
simulations, for generating the random variables according to the
density schema
(after estimating
with
). Such simulations create scenarios for the VaR
analysis.
From Theorem 16.3 we know that the
partial derivative
exists and
Table 16.1:
A Selection of Copulas.
# |
 |
 |
1 |
![$ \max\Big([u^{-\theta}+v^{-\theta}-1]^{-1/\theta},0\Big)$](sfehtmlimg3340.gif) |
 |
2 |
![$ \max\Big(1-[(1-u)^{\theta}+(1-v)^{\theta}-1]^{1/\theta},0\Big)$](sfehtmlimg3342.gif) |
 |
3 |
 |
 |
4 |
![$ \Big.\exp\left(-[(-\ln u)^\theta+(-\ln v)^\theta]^{1/\theta}\right)$](sfehtmlimg3346.gif) |
 |
5 |
 |
 |
6 |
![$ 1-\Big[(1-u)^\theta+(1-v)^\theta-(1-u)^\theta(1-v)^\theta)\Big]^{1/\theta}$](sfehtmlimg3349.gif) |
 |
7 |
![$ \max\Big[\theta uv+(1-\theta)(u+v-1),0\Big]$](sfehtmlimg3350.gif) |
![$ (0,1]$](sfehtmlimg3351.gif) |
8 |
![$ \max\Big[{\theta^2uv-(1-u)(1-v)\over \theta^2-(\theta-1)^2(1-u)(1-v)},0\Big]$](sfehtmlimg3352.gif) |
![$ (0,1]$](sfehtmlimg3351.gif) |
9 |
.
 |
![$ (0,1]$](sfehtmlimg3351.gif) |
10 |
![$ uv/\Big[1+(1-u^\theta)(1-v^\theta)\Big]^{1/\theta}$](sfehtmlimg3354.gif) |
![$ (0,1]$](sfehtmlimg3351.gif) |
11 |
![$ \max\Big(\Big[u^\theta v^\theta-2(1-u^\theta)(1-v^\theta)\Big]^{1/\theta},0\Big)$](sfehtmlimg3355.gif) |
![$ (0,1/2]$](sfehtmlimg3356.gif) |
12 |
![$ \Big(1+\Big[(u^{-1}-1)^\theta+(v^{-1}-1)^\theta\Big]^{1/\theta}\Big)^{-1}$](sfehtmlimg3357.gif) |
 |
13 |
![$ \exp \Big(1- \Big[ (1-\ln u)^\theta + (1-\ln v)^\theta -1 \Big]^{1/\theta} \Big)$](sfehtmlimg3358.gif) |
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14 |
![$ \Big( 1+ \Big[ (u^{-1/\theta} -1)^\theta + (v^{-1/\theta} -1)^\theta \Big]^{1/\theta} \Big)^{-\theta}$](sfehtmlimg3360.gif) |
 |
15 |
![$ \max \Big( \Big\{ 1- \Big[ (1-u^{1/\theta})^\theta +
(1-v^{1/\theta})^\theta
\Big]^{1/\theta} \Big\}^\theta ,0 \Big)$](sfehtmlimg3361.gif) |
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16 |
 |
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|
21 |
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|
![$ \hookrightarrow S(u)=\Big[1-(1-u)^\theta\Big]^{1/\theta}$](sfehtmlimg3366.gif) |
|
|
is strictly monotone increasing. Now we can generate
the desired dependence structure using the following steps:
- Generate 2 independent uniformly distributed (pseudo) random numbers
. Fix
.
- Calculate the inverse of
,
which in general is dependent on the copula and the parameter
. Set
. The pair
has the desired
joint density.
Here we show possible applications of the Gumbel-Hougaard copula.
The form of this copula for
is sketched in Figure
16.1.
Selecting from Table 16.1 copulas 4, 5, 6 and 12 and
setting the parameters
,
,
, the
varying dependency structures in Figure 16.2 are
created.
The parameter
controls the form of the copula and thus
the dependency. Figure 16.3 displays the
connection between
(for
) and the correlation for
normally distributed, two-dimensional variables. One can clearly
see the extent of the variation relative to
. The parameter
parameterizes the non-linear dependence structure.
An application of the method on FX rates USD/EUR and GBP/EUR (time
period: Jan. 2, 1991 through Mar. 9, 2000) and a simple linear
portfolio is described in the following. The portfolio has the
structure:
with
,
,
,
. The
VaR is calculated with a confidence level of
,
,
,
and
trading days. Within the framework of backtesting one is
interested in the number of outliers crossing the barriers. The
results are given in Table 16.2.
Table:
Number of outliers in backtesting
for the VaR with a confidence level of
SFMaccvar4.xpl
|
|
|
|
Copula in Table 16.1 |
|
 |
 |
his |
vcv |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
21 |
|
.10 |
|
.103 |
.084 |
.111 |
.074 |
.100 |
.086 |
.080 |
.086 |
.129 |
.101 |
.128 |
.129 |
.249 |
.090 |
.087 |
.084 |
.073 |
.104 |
.080 |
|
.05 |
|
.053 |
.045 |
.066 |
.037 |
.059 |
.041 |
.044 |
.040 |
.079 |
.062 |
.076 |
.079 |
.171 |
.052 |
.051 |
.046 |
.038 |
.061 |
.041 |
|
.01 |
|
.015 |
.019 |
.027 |
.013 |
.027 |
.017 |
.020 |
.016 |
.032 |
.027 |
.033 |
.034 |
.075 |
.020 |
.022 |
.018 |
.015 |
.027 |
.018 |
|
.10 |
|
.092 |
.078 |
.066 |
.064 |
.057 |
.076 |
.086 |
.062 |
.031 |
.049 |
.031 |
.031 |
.011 |
.086 |
.080 |
.092 |
.085 |
.065 |
.070 |
|
.05 |
|
.052 |
.044 |
.045 |
.023 |
.033 |
.041 |
.049 |
.031 |
.012 |
.024 |
.012 |
.013 |
.003 |
.051 |
.046 |
.054 |
.049 |
.039 |
.032 |
|
.01 |
|
.010 |
.011 |
.016 |
.002 |
.007 |
.008 |
.009 |
.006 |
.002 |
.002 |
.002 |
.002 |
.001 |
.015 |
.010 |
.018 |
.025 |
.011 |
.005 |
|
.10 |
|
.099 |
.086 |
.126 |
.086 |
.064 |
.088 |
.096 |
.073 |
.032 |
.054 |
.033 |
.031 |
.016 |
.094 |
.086 |
.105 |
.133 |
.070 |
.086 |
|
.05 |
|
.045 |
.048 |
.093 |
.047 |
.032 |
.052 |
.050 |
.040 |
.017 |
.026 |
.017 |
.016 |
.009 |
.049 |
.047 |
.058 |
.101 |
.034 |
.050 |
|
.01 |
|
.009 |
.018 |
.069 |
.018 |
.012 |
.018 |
.016 |
.012 |
.007 |
.009 |
.006 |
.006 |
.002 |
.018 |
.015 |
.018 |
.073 |
.013 |
.020 |
|
.10 |
|
.103 |
.090 |
.174 |
.147 |
.094 |
.095 |
.086 |
.103 |
.127 |
.094 |
.129 |
.127 |
.257 |
.085 |
.085 |
.085 |
.136 |
.088 |
.111 |
|
.05 |
|
.052 |
.058 |
.139 |
.131 |
.056 |
.060 |
.058 |
.071 |
.084 |
.068 |
.084 |
.085 |
.228 |
.053 |
.054 |
.051 |
.114 |
.053 |
.098 |
|
.01 |
|
.011 |
.020 |
.098 |
.108 |
.017 |
.025 |
.025 |
.035 |
.042 |
.056 |
.041 |
.042 |
.176 |
.016 |
.017 |
.016 |
.087 |
.015 |
.071 |
|
.10 |
Avg |
.014 |
.062 |
.145 |
.123 |
.085 |
.055 |
.052 |
.082 |
.193 |
.104 |
.194 |
.194 |
.478 |
.045 |
.061 |
.045 |
.110 |
.082 |
.075 |
|
.05 |
Avg |
.011 |
.021 |
.154 |
.124 |
.051 |
.030 |
.016 |
.060 |
.134 |
.080 |
.132 |
.136 |
.387 |
.006 |
.012 |
.017 |
.127 |
.041 |
.075 |
|
.01 |
Avg |
.007 |
.029 |
.169 |
.117 |
.028 |
.031 |
.032 |
.036 |
.065 |
.071 |
.065 |
.067 |
.249 |
.029 |
.025 |
.029 |
.160 |
.026 |
.083 |
|
Avg |
Avg |
.009 |
.028 |
.163 |
.120 |
.039 |
.032 |
.028 |
.047 |
.095 |
.076 |
.094 |
.096 |
.306 |
.022 |
.023 |
.026 |
.147 |
.034 |
.080 |
|
Rank |
1 |
6 |
18 |
16 |
9 |
7 |
5 |
10 |
14 |
11 |
13 |
15 |
19 |
2 |
3 |
4 |
17 |
8 |
12 |
|
|
As a benchmark the variance-covariance method from
Deutsch and Eller (1999) is used. This is based on the multivariate
normal distribution. In the last row the ranking of the weighted
error is given:
. According to
this ranking the historical simulation method (based on an
empirical density function) performs quite well. The copulas 5,
12, 13 and 14, however, lie altogether in front of the
variance-covariance method.