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
for
various
(for notational convenience we drop the index
).
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The first data set () contains negative daily stock
-returns of BMW
and Deutsche Bank for the time period 1992-2001. The second data set (
) consists of negative daily
exchange rate
-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
against the thresholds
shows
a characteristic plateau for small
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
can be also observed in Figures 3.6 and 3.7. In particular, a small
comes along
with a large variance of the TDC estimator, whereas increasing
results in a strong bias. In the presence of tail dependence,
is chosen such that the TDC estimate
lies on the plateau between the decreasing variance and the
increasing bias. Thus for the data set
we take
between
and
which provides a TDC estimate of
whereas for
we choose
between
and
which yields
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.
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