Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

3.6 Establishing the uncertainty of estimated upper and lower tail indexes
ˆν by creating bs = 1000 bootstrap samples of historical return data for
S&P 500 and included assets and calculation of bootstrap mean value
νmean with rel. error from the original value, deviation quantiles and
standard deviation from νmean, and most extreme deviation value. The
tail represents the most extreme 4 % of the return values during a time
interval ranging from January 1991 to December 2000 (k = 100 values)
and from July 1985 to April 2008 (k = 229 values). . . . . . . . . . . . 29
3.7 Establishing the uncertainty of estimated upper and lower tail indexes ˆ b γ n
by creating bootstrap samples (bs) of historical return data for S&P 500
and included assets and calculation of bootstrap mean value νmean with
rel. error from the original value, deviation quantiles and standard deviation
from bγ mean , and most extreme deviation value. The tail represents
the most extreme 4 % of the return values during a time interval ranging
from January 1991 to December 2000 (k = 100 values, bs = 1000) and
from July 1985 to April 2008 (k = 229 values, bs = 650). . . . . . . . . 30
3.8 Estimated values of upper and lower tail dependence for S&P 500 index
with a set of nine major assets traded on the New York stock Exchange
calculated by a non-parametric approach: ˆ λ +,− �
= 1/ max 1, l
�ν on β
the left and by a parametric approach: ˆ λ +,− �
= 1/ 1 + β−ν · Cε
�
on CY
the right. The tail represents the most extreme 4% of the return values
during a time interval from January 1991 to December 2000. ’tail’ shows
the calculation interval of l with c = 13 and k = 100 and ∗ denotes
negative ˆ β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.9 Estimated values of upper and lower tail dependence for S&P 500 index
with a set of nine major assets traded on the New York stock Exchange
calculated by a non-parametric approach: ˆ λ +,− �
= 1/ max 1, l
� γ
bn on β
the left and by a parametric approach: ˆ λ +,− �
= 1/ 1 + β−bγ �
n Cε · on CY
the right. The tail represents the most extreme 4% of the return values
during a time interval from January 1991 to December 2000. ’tail’ shows
the calculation interval of l with c = 13 and k = 100 and ∗ denotes
negative ˆ β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Estimated values of upper and lower tail dependence for S&P 500 index
with a set of nine major assets traded on the New York stock Exchange
calculated by a non-parametric approach:
35
ˆ λ +,− �
= 1/ max 1, l
�ν on β
the left and by a parametric approach: ˆ λ +,− �
= 1/ 1 + β−ν · Cε
�
on CY
the right. The tail represents the most extreme 4% of the return values
during a time interval from July 1985 to April 2008. ’tail’ shows the
calculation interval of l with c = 29 and k = 229. . . . . . . . . . . . . 35