3.7 Value at Risk - a Simulation Study

Value at Risk (VaR) estimations refer to the estimation of high target quantiles of single asset or portfolio loss distributions. Thus, VaR estimations are very sensitive towards the tail behavior of the underlying distribution model.

On the one hand, the VaR of a portfolio is affected by the tail distribution of each single asset. On the other hand, the general dependence structure and especially the tail-dependence structure among all assets have a strong impact on the portfolio's VaR, too. With the concept of tail dependence, we supply a methodology for measuring and modelling one particular type of dependence of extreme events.

What follows, provides empirical justification that the portfolio's VaR estimation depends heavily on a proper specification of the (tail-)dependence structure of the underlying asset-return vector. To illustrate our assertion we consider three financial data sets: The first two data sets $ D_1$ and $ D_2$ refer again to the daily stock $ \log$-returns of BMW and Deutsche Bank for the time period 1992-2001 and the daily exchange rate $ \log$-returns of DEM/USD and JPY/USD for the time period 1989-2001, respectively. The third data set ($ D_3$) contains exchange rate $ \log$-returns of FFR/USD and DEM/USD for the time period 1984-2002.

Typically, in practice, either a multivariate normal distribution or multivariate $ t$-distribution is fitted to the data in order to describe the random behavior (market riskiness) of asset returns. Especially multivariate $ t$-distributions have recently gained the attraction of practitioners due to their ability to model heavy tails while still having the advantage of being in the class of elliptically-contoured distributions. Recall that the multivariate normal distribution has thin tailed marginals which exhibit no tail-dependence, and the $ t$-distribution possesses heavy tailed marginals which are tail dependent (see Section 3.3.2). Due to the different tail behavior, one might pick one of the latter two distribution classes if the data are elliptically contoured. However, elliptically-contoured distributions require a very strong symmetry of the data and might not be appropriate in many circumstances.

Figure 3.8: Scatter plot of foreign exchange data (left panel) and simulated normal pseudo-random variables (right panel) of FFR/USD versus DEM/USD negative daily exchange rate log-returns (5189 data points). $ \textrm { }$
\includegraphics[width=0.72\defpicwidth]{DataScatterFFRDEMUSD.ps} \includegraphics[width=0.72\defpicwidth]{DataScatterFFRDEMUSDNormalFit.ps}

For example, the scatter plot of the data set $ D_3$ in Figure 3.8 reveals that its distributional structure does not seem to be elliptically contoured at all. To circumvent this problem, one could fit a distribution from a broader distribution class, such as a generalized hyperbolic distribution (Bingham and Kiesel; 2002; Eberlein and Keller; 1995). Alternatively, a split of the dependence structure and the marginal distribution functions via the theory of copulae (as described in Section 3.2) seems to be also attractive. This split exploits the fact that statistical (calibration) methods are well established for one-dimensional distribution functions.

For the data sets $ D_1, D_2,$ and $ D_3$, one-dimensional $ t$-distributions are utilized to model the marginal distributions. The choice of an appropriate copula function turns out to be delicate. Two structural features are important in the context of VaR estimations regarding the choice of the copula. First, the general structure (symmetry) of the chosen copula should coincide with the dependence structure of the real data. We visualize the dependence structure of the sample data via the respective empirical copula (Figure 3.9), i.e. the marginal distributions are standardized by the corresponding empirical distribution functions. Second, if the data show tail dependence than one must utilize a copula which comprises tail dependence. Especially VaR estimations at a small confidence level are very sensitive towards tail dependence. Figure 3.9 indicates that the FX data set $ D_3$ has significantly more dependence in the lower tail than the simulated data from a fitted bivariate normal copula. The data clustering in the lower left corner of the scatter plot of the empirical copula is a strong indication for tail dependence.

Based on the latter findings, we use a $ t$-copula (which allows for tail dependence, see Section 3.3.2) and $ t$-distributed marginals (which are heavy tailed). Note, the resulting common distribution is only elliptically contoured if the degrees of freedom of the $ t$-copula and the $ t$-margins coincide, since in this case the common distribution corresponds to a multivariate $ t$-distribution. The parameters of the marginals and the copula are separately estimated in two consecutive steps via maximum likelihood. For statistical properties of the latter procedure, which is called Inference Functions for Margins method (IFM), we refer to Joe and Xu (1996).

Tables 3.11, 3.12, and 3.13 compare the historical VaR estimates of the data sets $ D_1, D_2,$ and $ D_3$ with the average of $ 100$ VaR estimates which are simulated from different distributions. The fitted distribution is either a bivariate normal, a bivariate $ t$-distribution or a bivariate distribution with $ t$-copula and $ t$-marginals. The respective standard deviation of the VaR estimations are provided in parenthesis. For a better exposition, we have multiplied all numbers by $ 10^5.$

Figure 3.9: Lower left corner of the empirical copula density plots of real data (left panel) and simulated normal pseudo-random variables (right panel) of FFR/USD versus DEM/USD negative daily exchange rate log-returns (5189 data points).
\includegraphics[width=0.72\defpicwidth]{DataScatterFFRDEMUSDCopula.ps} \includegraphics[width=0.72\defpicwidth]{DataScatterFFRDEMUSDCopulaNormalFit.ps}


Table 3.11: Mean and standard deviation of $ 100$ VaR estimations (multiplied by $ 10^5$) from simulated data following different distributions which are fitted to the data set $ D_1.$
Quantile Historical Normal $ t$-distribution $ t$-copula &
  VaR distribution   $ t$-marginals
    Mean $ $ (Std) Mean $ $ (Std) Mean $ $ (Std)
         
$ \;\;0.01$ $ 489.93$ $ 397.66$ ($ 13.68$) $ 464.66$ ($ 39.91$) $ 515.98 $ ($ 36.54$)
$ \;\;0.025$ $ 347.42$ $ 335.28$ ($ 9.67$) $ 326.04$ ($ 18.27$) $ 357.40 $ ($ 18.67$)
$ \;\;0.05$ $ 270.41$ $ 280.69$ ($ 7.20$) $ 242.57$ ($ 10.35$) $ 260.27 $ ($ 11.47$)



Table 3.12: Mean and standard deviation of $ 100$ VaR estimations (multiplied by $ 10^5$) from simulated data following different distributions which are fitted to the data set $ D_2.$
Quantile Historical Normal $ t$-distribution $ t$-copula &
  VaR distribution   $ t$-marginals
    Mean $ $ (Std) Mean $ $ (Std) Mean $ $ (Std)
         
$ \;\;0.01$ $ 155.15$ $ 138.22$ ($ 4.47$) $ 155.01$ ($ 8.64$) $ 158.25$ ($ 8.24$)
$ \;\;0.025$ $ 126.63$ $ 116.30$ ($ 2.88$) $ 118.28$ ($ 4.83$) $ 120.08$ ($ 4.87$)
$ \;\;0.05$ $ \;\;98.27$ $ \;\;97.56$ ($ 2.26$) $ \;\;92.35$ ($ 2.83$) $ \;\;94.14$ ($ 3.12$)



Table 3.13: Mean and standard deviation of $ 100$ VaR estimations (multiplied by $ 10^5$) from simulated data following different distributions which are fitted to the data set $ D_3.$
Quantile Historical Normal $ t$-distribution $ t$-copula &
  VaR distribution   $ t$-marginals
    Mean $ $ (Std) Mean $ $ (Std) Mean $ $ (Std)
         
$ \;\;0.01$ $ 183.95$ $ 156.62$ ($ 3.65$) $ 179.18$ ($ 9.75$) $ 179.41$ ($ 6.17$)
$ \;\;0.025$ $ 141.22$ $ 131.54$ ($ 2.41$) $ 124.49$ ($ 4.43$) $ 135.21$ ($ 3.69$)
$ \;\;0.05$ $ 109.94$ $ 110.08$ ($ 2.05$) $ \;\;91.74$ ($ 2.55$) $ 105.67$ ($ 2.59$)


The results of the latter tables clearly show that the fitted bivariate normal-distribution does not yield an overall satisfying estimation of the VaR for all data sets $ D_1, D_2,$ and $ D_3$. The poor estimation results for the $ 0.01-$ and $ 0.025-$quantile VaR (i.e. the mean of the VaR estimates deviate strongly from the historical VaR estimate) are mainly caused by the thin tails of the normal distribution. By contrast, the bivariate $ t$-distribution provides good estimations of the historical VaR for the data sets $ D_1$ and $ D_2$ over all quantiles. However, both data sets are approximately elliptically-contoured distributed since the estimated parameters of the copula and the marginals are almost equal. For example for the data set $ D_1,$ the estimated degree of freedom of the $ t$-copula is $ 3.05$ whereas the estimated degrees of freedom of the $ t$-marginals are $ 2.99$ and $ 3.03,$ respectively. We have already discussed that the distribution of the data set $ D_3$ is not elliptically contoured. Indeed, the VaR estimation improves with a splitting of the copula and the marginals. The corresponding estimated degree of freedom of the $ t$-copula is $ 1.11$ whereas the estimated degrees of freedom of the $ t$-marginals are $ 4.63$ and $ 5.15.$ Finally, note that the empirical standard deviations do significantly differ between the VaR estimation based on the multivariate $ t$-distribution and the $ t$-copula, respectively.