17.2 The Calculation of VaR and Copulas

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 $ \theta$ needs to be determined. This occurs on the basis of a given financial time series $ \{S_t\}_{t=1}^T$, $ S_t \in \mathbb{R}^p$ and log-returns $ X_{t j}=\log (S_{t j} / S_{t-1, j})\; j=1,\cdots,p.$ The time series $ S_{t j}$ represent the $ p$ risk factors and under the assumption that $ X_j$ is normal, they are themselves log-normally distributed. The variance of the Gaussian distribution is estimated using the method shown in Section 3.2, $ \hat{\sigma}^2_j={1\over T-1}\sum_{t=2}^{T} X_{t j} ^2$.

A very simple method for estimating the parameter $ \theta$ 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

$\displaystyle L(\theta)=\prod_{t=2}^T f_{\theta}({ x}_{t}).$

Here $ f_\theta$ 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 $ C$ 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 $ F_{\hat{\theta}}$ (after estimating $ \theta$ with $ \hat{\theta}$). Such simulations create scenarios for the VaR analysis.

From Theorem 16.3 we know that the partial derivative $ C_u(v)$ exists and

Table 16.1: A Selection of Copulas.
# $ C_\theta(u,v)=$ $ \theta\in$
1 $ \max\Big([u^{-\theta}+v^{-\theta}-1]^{-1/\theta},0\Big)$ $ [-1,\infty)\backslash\{0\}$
2 $ \max\Big(1-[(1-u)^{\theta}+(1-v)^{\theta}-1]^{1/\theta},0\Big)$ $ [1,\infty)$
3 $ {uv\over 1-\theta(1-u)(1-v)}\Big.$ $ [-1,1)$
4 $ \Big.\exp\left(-[(-\ln u)^\theta+(-\ln v)^\theta]^{1/\theta}\right)$ $ [1,\infty)$
5 $ - {1\over\theta}\ln\left(1+{(e^{-\theta u}-1)(e^{-\theta v}-1)\over e^{-\theta}-1}\right)$ $ (-\infty,\infty)\backslash\{0\}$
6 $ 1-\Big[(1-u)^\theta+(1-v)^\theta-(1-u)^\theta(1-v)^\theta)\Big]^{1/\theta}$ $ [1,\infty)$
7 $ \max\Big[\theta uv+(1-\theta)(u+v-1),0\Big]$ $ (0,1]$
8 $ \max\Big[{\theta^2uv-(1-u)(1-v)\over \theta^2-(\theta-1)^2(1-u)(1-v)},0\Big]$ $ (0,1]$
9 . $ uv\exp(-\theta \ln u \ln v)$ $ (0,1]$
10 $ uv/\Big[1+(1-u^\theta)(1-v^\theta)\Big]^{1/\theta}$ $ (0,1]$
11 $ \max\Big(\Big[u^\theta v^\theta-2(1-u^\theta)(1-v^\theta)\Big]^{1/\theta},0\Big)$ $ (0,1/2]$
12 $ \Big(1+\Big[(u^{-1}-1)^\theta+(v^{-1}-1)^\theta\Big]^{1/\theta}\Big)^{-1}$ $ [1,\infty)$
13 $ \exp \Big(1- \Big[ (1-\ln u)^\theta + (1-\ln v)^\theta -1 \Big]^{1/\theta} \Big)$ $ (0,\infty)$
14 $ \Big( 1+ \Big[ (u^{-1/\theta} -1)^\theta + (v^{-1/\theta} -1)^\theta \Big]^{1/\theta} \Big)^{-\theta}$ $ [1,\infty)$
15 $ \max \Big( \Big\{ 1- \Big[ (1-u^{1/\theta})^\theta +
(1-v^{1/\theta})^\theta
\Big]^{1/\theta} \Big\}^\theta ,0 \Big)$ $ [1,\infty)$
16 $ {1 \over 2} \Big( S + \sqrt{S^2 + 4 \, \theta} \Big) $ $ [0,\infty)$
  $ \hookrightarrow S=u+v-1- \theta \Big( {1 \over u}+{1 \over v}-1 \Big)$  
21 $ 1- \Big( 1- \big\{ \max(S(u) + S(v) -1, 0) \big\}^\theta\Big)^{1\over\theta}$ $ [1,\infty)$
  $ \hookrightarrow S(u)=\Big[1-(1-u)^\theta\Big]^{1/\theta}$  


is strictly monotone increasing. Now we can generate the desired dependence structure using the following steps:
  1. Generate 2 independent uniformly distributed (pseudo) random numbers $ u,w\in[0,1]$. Fix $ u$.
  2. Calculate the inverse of $ C_u$, which in general is dependent on the copula and the parameter $ \theta$. Set $ v = C_u^{-1}(w)$. The pair $ (u,v)$ has the desired joint density.

Here we show possible applications of the Gumbel-Hougaard copula. The form of this copula for $ \theta=3$ is sketched in Figure 16.1.

Fig.: $ C_4(u,v)$ für $ \theta=3$ 28808 SFEaccvar1.xpl
\includegraphics[width=1\defpicwidth]{copplot1.ps}

Selecting from Table 16.1 copulas 4, 5, 6 and 12 and setting the parameters $ \sigma_1=1$, $ \sigma_2=1$, $ \theta=3$, the varying dependency structures in Figure 16.2 are created.

Fig.: 10000 Simulations with $ \sigma_1=1$, $ \sigma_2=1$, $ \theta=3$ 28812 SFEaccvar2.xpl
\includegraphics[width=1\defpicwidth]{copt2.ps}

The parameter $ \theta$ controls the form of the copula and thus the dependency. Figure 16.3 displays the connection between $ \theta$ (for $ C_4$) and the correlation for normally distributed, two-dimensional variables. One can clearly see the extent of the variation relative to $ \rho$. The parameter $ \theta$ parameterizes the non-linear dependence structure.

Fig.: Plot of $ \theta$ against $ \rho$ for $ C_4$ 28816 SFEaccvar3.xpl
\includegraphics[width=0.85\defpicwidth]{copt3.ps}

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:

Value$\displaystyle (a_i,t)[EUR] = a_{i,1} \times$   USD$\displaystyle _t - a_{i,2} \times$   GBP$\displaystyle _t \; .$ (17.13)

with $ a_1=(-3,-2)$, $ a_2=(3,-2)$, $ a_3=(-3,2)$, $ a_4=(3,2)$. The VaR is calculated with a confidence level of $ 1-\alpha_i$, $ \alpha_1=0.1$, $ \alpha_2=0.05$, $ \alpha_3=0.01$ and $ T=250$ 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 $ 1-\alpha_i,\;
i=1,\cdots,3$ 28822 SFMaccvar4.xpl
Copula in Table 16.1
$ \alpha\!\!=$ $ a\!\!=$ his vcv 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 21
.10 $ a_1$ .103 .084 .111 .074 .100 .086 .080 .086 .129 .101 .128 .129 .249 .090 .087 .084 .073 .104 .080
.05 $ a_1$ .053 .045 .066 .037 .059 .041 .044 .040 .079 .062 .076 .079 .171 .052 .051 .046 .038 .061 .041
.01 $ a_1$ .015 .019 .027 .013 .027 .017 .020 .016 .032 .027 .033 .034 .075 .020 .022 .018 .015 .027 .018
.10 $ a_2$ .092 .078 .066 .064 .057 .076 .086 .062 .031 .049 .031 .031 .011 .086 .080 .092 .085 .065 .070
.05 $ a_2$ .052 .044 .045 .023 .033 .041 .049 .031 .012 .024 .012 .013 .003 .051 .046 .054 .049 .039 .032
.01 $ a_2$ .010 .011 .016 .002 .007 .008 .009 .006 .002 .002 .002 .002 .001 .015 .010 .018 .025 .011 .005
.10 $ a_3$ .099 .086 .126 .086 .064 .088 .096 .073 .032 .054 .033 .031 .016 .094 .086 .105 .133 .070 .086
.05 $ a_3$ .045 .048 .093 .047 .032 .052 .050 .040 .017 .026 .017 .016 .009 .049 .047 .058 .101 .034 .050
.01 $ a_3$ .009 .018 .069 .018 .012 .018 .016 .012 .007 .009 .006 .006 .002 .018 .015 .018 .073 .013 .020
.10 $ a_4$ .103 .090 .174 .147 .094 .095 .086 .103 .127 .094 .129 .127 .257 .085 .085 .085 .136 .088 .111
.05 $ a_4$ .052 .058 .139 .131 .056 .060 .058 .071 .084 .068 .084 .085 .228 .053 .054 .051 .114 .053 .098
.01 $ a_4$ .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: $ \vert\hat\alpha_1-\alpha_1\vert+5
\vert\hat\alpha_2-\alpha_2\vert+10 \vert\hat\alpha_3-\alpha_3\vert$. 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.