The “family” of plots in Figure 2
shows how this probability changes for
the familiar bivariate Normal (Gaussian)
distribution. For example, if returns are
correlated 0.50, under the Normal model
the probability of observing a joint event
beyond the 95th percentile (given one
observation at or beyond the 95th percentile) is 25 percent. It’s worthwhile
noting that these probabilities decline
for the extreme percentiles.
How does this compare with real-world equity markets? As we have observed, it’s difficult to make strong
statements about the tails of distributions because so few observations are
available. They are “undersampled.”
Figure 2 plots the tail dependence measure averaged across all possible pairs of
a sample of large equity markets using
monthly price changes over the period
1958–2008. You can see that the observed
profile is quite different from what might
be expected under a joint Normal relationship. A CPJQE of 0.40 appears to be
a reasonable estimate across the entire
bottom decile of monthly returns, but is
consistent with a linear correlation of 60
percent at the 10th percentile and 80 percent at the 1st percentile.
What do Analysts say?
How have analysts responded to the
challenge of how to capture this strong
dependency in the tails of the distributions? Two schools of thought have
emerged—one statistical and an alternative that makes use of fundamental
knowledge. The pure statistical approach seeks to extend the description
of dependency using “copula” methods.
They offer the prospect of capturing
the entire dependency structure and
an ability to separate the description of
dependency from the “marginal” distributions of individual risk factors. Figure
3 provides a comparison in which sample
random variates are drawn from the same
figure. 1: Lower Tail of the Distribution of annual Excess Equity Returns
2008 Sou
2007 ire
2002 Swi
2008 Uk 2001 ita
2008 Swi 1992 Jap
2008 Can 1990 Swi
2002 net 1990 Fra
2002 ire 1990 aus
2002 Fra 1987 Swi
1992 Den 1984 Den
1990 Spa 1982 aus
1990 Bel 1981 Can
1987 ita 1974 ita
1987 Fra 1974 Fra
1977 ita 1974 Can
1976 Spa 1973 Jap
1974 US 1973 aus
1974 Bel 1970 ger
1974 aus 1970 aus
1973 Uk 1966 Bel
1948 Spa 1964 ita
2008 US 1947 Bel 1952 ire
2008 Spa 1937 US 1952 aus
2008 Swe 1932 Spa 1947 ita
2002 Swe 1932 net 1941 net
1990 Swe 1931 Swi 1931 Uk
1990 ire 1931 ger 1931 net
2008 Jap 1987 ger 1931 Fra 1931 ita
2008 ger 1977 Spa 1931 Can 1930 aus
2008 Fra 1974 Swi 1930 US 1921 Swe
2008 ita 2008 aus 1946 Jap 1930 Can 1920 Swi
1974 ire 2002 ger 1931 Swe 1920 Sou 1920 net
2008 Den 2008 Met 1945 ita 1990 Jap 1931 Bel 1914 Swe 1919 Swi
2008 ire 1974 Uk 2008 Bel 1920 Jap 1931 US 1926 ita 1907 US 1908 Jap
–70% or
worse
–70% to
–65%
–65% to
–60%
–60% to
–55%
–55% to
–50%
–50% to
–45%
–45% to
–40%
–40% to
–35%
–35% to
–30%
–30% to
–25%
Source: author
figure 2: a Comparison of Empirical Tail Dependency with the
normal Copula
frequency of joint extreme observation (normal & empirical)
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.975
0.95
0.9
0.8
0.7
0.6
0.5
0.4
Empirical
0.00
0.00
0.05
0.10
tail percentile
0.15
0.20
