3.6 Tail Dependence of Asset and FX Returns

Tail dependence is indeed often found in financial data series. Consider two scatter plots of daily negative log-returns of a tuple of financial securities and the corresponding upper TDC estimate $ \hat{\lambda}^{(1)}_U$ for various $ k$ (for notational convenience we drop the index $ m$).

Figure: Scatter plot of BMW versus Deutsche Bank negative daily stock log-returns (2347 data points) and the corresponding TDC estimate $ \hat{\lambda}^{(1)}_U$ for various thresholds $ k.$
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\includegraphics[width=0.72\defpicwidth]{DataFig17BmwDtB.ps} \includegraphics[width=0.72\defpicwidth]{BmwDtBankTDCest.ps}

The first data set ($ D_1$) contains negative daily stock $ \log$-returns of BMW and Deutsche Bank for the time period 1992-2001. The second data set ($ D_2$) consists of negative daily exchange rate $ \log$-returns of DEM/USD and JPY/USD (so-called FX returns) for the time period 1989-2001. For modelling reasons we assume that the daily log-returns are i.i.d. observations. Figures 3.6 and 3.7 show the presence of tail dependence and the order of magnitude of the tail-dependence coefficient. Tail dependence is present if the plot of TDC estimates $ \hat{\lambda}^{(1)}_U$ against the thresholds $ k$ shows a characteristic plateau for small $ k.$ The existence of this plateau for tail-dependent distributions is justified by a regular variation property of the tail distribution; we refer the reader to Peng (1998) or Schmidt and Stadtmüller (2003) for more details. By contrast, the characteristic plateau is not observable if the distribution is tail independent.

The typical variance-bias problem for various thresholds $ k$ can be also observed in Figures 3.6 and 3.7. In particular, a small $ k$ comes along with a large variance of the TDC estimator, whereas increasing $ k$ results in a strong bias. In the presence of tail dependence, $ k$ is chosen such that the TDC estimate $ \hat{\lambda}^{(1)}_U$ lies on the plateau between the decreasing variance and the increasing bias. Thus for the data set $ D_1$ we take $ k$ between $ 80$ and $ 110$ which provides a TDC estimate of $ \hat{\lambda}^{(1)}_{U,D_1}=0.31,$ whereas for $ D_2$ we choose $ k$ between $ 40$ and $ 90$ which yields $ \hat{\lambda}^{(1)}_{U,D_2}=0.17.$

The importance of the detection and the estimation of tail dependence becomes clear in the next section. In particular, we show that the Value at Risk estimation of a portfolio is closely related to the concept of tail dependence. A proper analysis of tail dependence results in an adequate choice of the portfolio's loss distribution and leads to a more precise assessment of the Value at Risk.

Figure: Scatter plot of DEM/USD versus JPY/USD negative daily exchange rate log-returns (3126 data points) and the corresponding TDC estimate $ \hat{\lambda}^{(1)}_U$ for various thresholds $ k.$
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\includegraphics[width=0.72\defpicwidth]{DataFig18DMYEN.ps} \includegraphics[width=0.72\defpicwidth]{DMYENUSDTDCest.ps}