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Extremal Dependence between Return Risk
and Liquidity Risk: An Analysis for the Swiss
Market
Christian Buhl
University of Basel
Christian Reich
University of Basel
Patrick Wegmann
University of Basel
April 2002
Corresponding Author:
Christian Buhl
University of Basel
WWZ/Department of Finance
Holbeinstrasse 12
4051 Basel
christian.buhl@unibas.ch
Phone: +41 (0)61 267 3198
WWZ/Department of Finance, Working Paper No. 6/02
Extremal Dependence between Return Risk and
Liquidity Risk: an Analysis for the Swiss Market
Christian Buhl, Christian Reich and Patrick Wegmann
Wirtschaftswissenschaftliches Zentrum der
Universität Basel,
Abteilung Finanzmarkttheorie,
Holbeinstrasse 12,
CH-4051 Basel
April 4, 2002
Abstract
We study the extremal dependence of market and liquidity risk, the former being measured through the market return and the latter being measured through
the relative bid-ask spread. We apply a non-parametrical approach to measure bivariate exceedance probabilities and the respective dependence function.
Our analysis for the Swiss Market indicates moderate tail dependence in that
roughly 10% of the exceedances of the 99% quantile are co-exceedances. As our
hypothesis tests for independence are rejected for confidence levels of 5% and
1% in almost all cases, we conclude that extreme dependence between negative
market returns and liquidity is existing in the empirical data and may be relevant for firm-wide risk management. In addition, we test for causality and find
decreasing extremal dependence when adding both positive and negative lags,
respectively.
This is a preliminary draft. All comments are welcome.
1
Introduction
Dependencies between different kinds of risks are an important issue to be considered in firm-wide risk management. It is possible, for example, that strong
adverse market movements are accompanied by insufficient market liquidity and
an increase in credit risk premia. The accurate modelling of dependencies is a
challenge on the way to an integrated risk measurement framework including
market, liquidity, credit, and possibly other risks.
Risk management does not usually take into account the whole distribution
of the value of a portfolio but only makes predictions about the loss potential
in stress situations. As a consequence, it does make sense to concentrate on
the tails of the respective distributions which mainly determine the stress loss
potential both for single risks and for a combination of risks. The classical
technique to model the tails of distributions is the Extreme Value Theory (EVT),
which has often been applied in risk management. In addition to the modelling
of extreme events of single time series, EVT also permits to model and determine
multivariate extremal dependencies between different kinds of risk.
In this paper, we focus on the extremal dependencies between market and
liquidity risk. We measure the market risk by the market return and the liquidity
risk by the relative spread, respectively. We are interested in adverse market
movement situations and want to infer something about the liquidity in such
(extreme) situations. In other words, we want to calculate a linkage measure
which indicates whether markets move together in turbulent periods or not. The
results can provide useful insights for risk and economic capital calculations, e.g.
value-at-risk.
The pitfalls of using correlations in the analysis of tail dependence between
risks are described in Embrechts / McNeil / Straumann (2000). Linear correlation is a natural dependence measure for multivariate normally and, more
generally, elliptically distributed risks. For other distributions, however, linear correlations can be misleading. Several empirical studies show that the
multivariate normal distribution underestimates the frequency of joint extreme
market exceedances. In addition, the linear correlation can be zero even if there
is a high spill-over probability. Mainly for this reason we do not want to rely
on the statistical concept of correlation as a measure of the interdependency
between markets during times of stress.
In order to avoid the potential problems of the linear correlation measure,
multivariate EVT have been employed. EVT proposes possible types of limiting
distributions for sample maxima or for sample observations exceeding a certain
high threshold. These limiting distributions have been applied extensively to
various time series of financial returns. There are parametric and nonparametric
approaches to estimate the parameters of the respective distributions. We,
know, however, of only a few applications of multivariate EVT to stock returns.
On the one hand, Longin / Solnik (2000) and Starica (1999) apply parametric
models, on the other hand, Straetmans (1998) and Hartmann / Straetmans /
De Vries (2000) use non-parametric approaches. Bac / Karolyi / Stulz (2001) go
another line of research and apply a (nonlinear) multinomial logistic regression
1
model. Their approach is, however, similar to the multivariate extreme value
approach.
We apply the model proposed by Straetmans (1998) and Hartmann / Straetmans / De Vries (2000), using a non-parametric approach. However, as indicated, we calculate extremal dependence between market returns and relative
spreads. We directly measure and report the expected number of exceedances
of a certain threshold, conditional on the event there is at least one exceedance.
Because of the use of non-parametric estimators for the limiting dependence
function, the probability law of the joint process can be left unspecified, this in
contrast to correlation-based approaches which measure market linkages.
The paper is structured as follows. In the following section, we show how to
measure extremal dependence and we discuss asymptotic and in-sample properties of the Stable Tail Dependence Function (STDF). Estimation and test
procedures are discussed in the third section. In Section 4, we apply the model
to our data set, which consists of daily returns and daily relative spreads of 15
titles of the Swiss Market Index. We draw the conclusions in the last section.
2
2.1
Theory of Extremal Linkages
The Measurement of Extremal Linkages
In statistics, the concept of asymptotic dependence can be formally defined in
terms of the conditional exceedance probability. Let (X, Y ) represent the time
series of the asset returns and the bid-ask spread, respectively, and let y and x
be the excess levels above which we speak of an extreme event, e.g. of a market
crash (boom) and of an insufficient amount of liquidity1 . Furthermore, to study
market crashes, we adopt the convention to take the negative of a return, in
order to study all extreme events in the first quadrant. It follows that (X, Y )
are asymptotically or tail independent if the conditional exceedance probability
(or conditional crash probability) defined on the joint probability distribution
converges to zero, e.g.
lim
y,x→+∞
Pr {Y > y | X > x} = 0.
(1)
However, the ranking of asset return/spread pairs according to the criterion in
(1) about asymptotic dependency may be reversed by changing the conditioning time series. In order to have a numeraire invariant version for asymptotic
independence, we estimate the following:
Pr {X > x, Y > y}
= 0.
(2)
Pr {X > x or Y > y}
In words, asymptotic independence means that the probability of a simultaneous
”crash” (or extreme event) in the market value and the liquidity of a certain
lim
y,x→+∞
1 Throughout the rest of the paper, whenever we speak of a crash in the context of liquidity,
we refer to an (extremely) insufficient amount of liquidity.
2
asset, given that there is at least one crash, is equal to zero. This definition does
not allow, however, to make any inference about the causality of the crashes.
In practice, we may be interested in the likelihood of extremal but bounded
spillovers. Generally, let κ be the number of crashes that simultaneously occur
in a bivariate system. In a finite sample framework, we have the following
expression for the likelihood of simultaneous declines in κ different financial
markets
Pr {κ = 2 | κ ≥ 1} =
Pr {X > x, Y > y}
.
Pr {X > x or Y > y}
(3)
An alternative expression for the asymptotic dependence is E {κ | κ ≥ 1} , i.e.
the expected number of crashes or attacks that may simultaneously occur, given
that there is a crash somewhere (e.g. the conditional crash expectation), results
from:
Pr {κ = 1} + 2 Pr {κ = 2}
(4)
E {κ | κ ≥ 1} =
Pr {κ ≥ 1}
=
Pr {κ}
Pr {κ ≥ 1}
(5)
=
Pr {κ ≥ 1} + Pr {κ = 2}
Pr {κ ≥ 1}
(6)
= Pr {κ = 2 | κ ≥ 1} + 1.
(7)
In order to estimate the exceedance probabilities p1 , p2 and p12 , it is crucial
to choose the right distribution function, which should correctly capture the
empirical regularities of the respective market and spread returns, respectively.
We can proceed either parametrically or non-parametrically. The disadvantage
of the former is the nonnestedness of alternative parametric models in the parameter space. As a consequence, the estimates are dependent on the maintained
hypothesis. In order to circumvent this problem, Straetmans (1998) suggests
a non-parametric approach which is robust. The tail is estimated without any
knowledge about the underlying distribution function.
2.2
Multivariate Extreme Value Theory Results
The conditional crash probability (equation 3) and the conditional crash expectation (equation 7) are estimated by applying the (multivariate) EVT. According to Straetmans (1998), this results in a two-step approach. In the first
univariate step, the stylized fact of tail fatness is exploited in order to estimate
the univariate exceedance probabilities. For this purpose, the univariate EVT
provides the respective limit law (e.g. three extremal types of distributions) for
the maxima of a return series in which all fat tailed distributions are nested
with respect to their tail index. Conditional upon the estimates of the tail index, inverse quantile estimators are calculated in order to estimate the univariate
3
probabilities. In the second bivariate step, we identify the so-called Stable Tail
Dependence Function (STDF), which links the multivariate exceedance probability to the univariate exceedance probabilities. Consistently, the tail index as
well as the tail dependence function are estimated non-parametrically.
The probability that the pair (Xn,n , Yn,n ) of the first n random variables is
below a certain level x and y, respectively, is given by the following distribution
function:
Pr {Xn,n ≤ x, Yn,n ≤ y} = F n (x, y).
(8)
Bivariate Extreme value theory studies the limiting distribution of the pair
of order statistics (Xn,n , Yn,n ), appropriately scaled. One is interested in the
suitable normalizing constants an > 0, cn > 0 and dn such that:
½
¾
Xn,n − bn
Yn,n − dn
lim Pr
≤ x,
≤ y = G(x, y),
(9)
n→+∞
an
cn
where G(x, y) is the bivariate extreme value distribution and F is said to be
in the domain of attraction of G. The class of limit distribution functions in
the above equation is the class of max-stable distributions. If the definition for
max-stability for G(x, y) holds, then it should also hold for the univariate counterparts G1 and G2 , for the same set of scaling constants (Straetmans (1998)).
Max-stable marginals are one of the three classical extreme value distributions
studied by Gnedenko (1943):
G(x) = exp(−e−x )
G(x) =
0
exp(−x−α )
Type III: G(x) = exp(−x−α )
1
Type I:
Type II:
-∞ < x < +∞
x≤0
x > 0, α > 0
x > 0, α > 0
x≥0
,
where α is called the tail index. Straetmans (1998) shows that the univariate
tail probabilities can be expressed in closed form when n becomes large. There
is, however, no closed form solution for the bivariate exceedance probability, e.g.
the dependence function. As indicated above, in order to nonparametrically
estimate the bivariate tail, we identify the extremal dependence structure by
means of the STDF, which allows us to specify G(x, y) non-parametrically.
2.3
The Tail Dependence Structure: Asymptotic and InSample
Suppose that F (x, y) is a bivariate probability distribution function with continuous and strictly increasing marginal distributions F1 and F2 . The dependence
function of F is defined by
4
DF (u, v) = F (F1−1 (u), F2−1 (v)), 0 ≤ u ≤ 1, 0 ≤ v ≤ 1,
(10)
Fi−1 (x) = inf {y | Fi (y) ≥ x} , i = 1, 2,
(11)
where
are the general inverse functions of F1 and F2 .Note that through the transformation in (10), the marginal distributions of F are uniformly distributed. In
order to make inferences about extremal dependencies, we need a tail version of
the dependence function DF , the Stable Tail Dependence Function lF ,as defined
in Xin (1992):
lF (u, v) := lim t−1 [1 − DF (1 − tu, 1 − tv)]
(12)
= lim t−1 [1 − F (Q1 (tu), Q2 (tv))]
(13)
t→+0
t→+0
= − ln G(u−1/α1 , v−1/α2 ),
(14)
−1
reprewhere F is the distribution function of (X, Y ), and Qi := (1 − Fi )
sents the corresponding marginal quantile functions for some positive u, v and
t. Estimators and properties for this function can be seen in Xin (1992) and
De Haan / De Ronde (1998). Applied to our conditional crash expectation
Pr(κ = 2 | κ ≥ 1),we have the following asymptotic conditional crash probability (Straetmans (1998))
u+v
Pr {X > Q1 (tu), Y > Q2 (tv)}
=
− 1,
Pr {X > Q1 (tu) or Y > Q2 (tv)}
lF (u, v)
(15)
and the respective asymptotic analogue for E(κ | κ ≥ 1),
Pr {κ = 2 | κ ≥ 1} → lim
t→+0
u+v
t−1 Pr {X > Q1 (tu)} + t−1 Pr {Y > Q2 (tv)}
=
.
−1
t→+0
t (1 − Pr {X ≤ Q1 (tu) , Y ≤ Q2 (tv)})
lF (u, v)
(16)
Exploiting the homogeneity property of the STDF, one can show that the bivariate excess probability and the marginal probabilities are related via the STDF
(see Hartmann / Straetmans / De Vries (2000)), e.g. the following approximation identity (for large n) results in a finite sample framework:
E {κ | κ ≥ 1} → lim
p12,n ≈ lF (p1,n ; p2,n ),
with
p1,n := 1 − F1 (an , y)
p2,n := 1 − F2 (cn , y)
p12,n := 1 − F (an x, cn y),
5
(17)
where p1,n and p2,n represent the sample exceedance probablilities. Consequently, the joint probability p12,n only depends on the marginal probabilities
p1,n and p2,n , once lF is known.
3
3.1
Estimating and Hypothesis Testing
Estimating the Tail Dependence and the Spillover Probabilities
In order to estimate the the (possible) out-of-sample probability Pr {X > x or Y > y},
we concentrate on the following estimator (see Straetmans (1998)):
pb12,n = b
lF (b
p1,n ; pb2,n ).
(18)
The estimation is done in two steps. In the first step, the univariate exceedance
probabilities for fat tailed marginals are estimated as follows (see Dekkers (1991)
and De Haan et al. (1994)):
µ
¶α
mi Xn−mi ,n i
pbi,n =
, i = 1, 2,
(19)
n
s
where Xn−mi ,n is the (n − m)-th ascending order statistic from a sample of size
n, such that lim(1/m(n)) = 0, but m = o(n), and where the extreme probabilityquantile combination (b
pi,n , s) has to be such that x > Xn−mi ,n . The idea behind
the univariate probability estimate is that it extends the empirical distribution
function outside the domain of the sample by means of its asymptotic Pareto
tail. However, in risk management we are interested not only in the exceedance
probability, but also in the value at risk of our position. Therefore, we apply
the following estimator for the value of the quantile
µ
¶−1/αb
i
n
x
bi,p =
(1 − p)
Xi,n , i = 1, 2,
(20)
mi
where p is the respecitve quantile (e.g. the 0.99 quantile). The quantile estimators are still conditional upon knowledge of the tail indexes α1 and α2 , which
can be estimated by use of the popular Hill estimator γ̂ i (see Hill (1975), Jansen
/ De Vries (1991) or Embrechts / Klüppelberg / Mikosch (2001)):
γ̂ i =
mi −1
1
1 X
Xn−j,n
=
ln
, i = 1, 2,
α
bi
mi j=0
Xn−mi
(21)
where m equals the number of highest order statistics used in the estimation
of the univariate exceedance probability, and where α
b i is the corresponding
tail index estimate. Note that the lower the alpha, the fatter the tails of the
distribution. Furthermore, for a t-distribution, the tail index αi equals the
number of degrees of freedom.
6
To determine the choice of m, the number of highest order statistics, Xin
(1992) shows that one can pick m such that it is in the range which minimizes
the respective asymptotic mean squared error. Consequently, minimizing the
sample mean squared error is an appropriate selection criterion. Alternatively,
one can simply compute α
b for different m and select the threshold in the region
over which α
b is more or less constant.
In the second step, we estimate the bivariate tail dependence function. This
can be done by either adopting a specific functional form for the STDF (see Longin / Solnik (2000)) or by proceeding non-parametrically (Straetmans (1998)).
Recall definition (13) from the last section, here in terms of (b
p1,n ; pb2,n ) :
½
µ
µ
¶
¶¾
n
n 1X
kb
p1,n
kb
p2,n
b
p1,n ; pb2,n ) = ·
I X > Q1
or Y > Q2
, (22)
lF (b
k n i=1
n
n
where I stands for the indicator function and the summation represents the
number of points in the area
µ
µ
½
¶
¶¾
kb
p1,n
kb
p2,n
(Xi , Yi )i=1,...,n | Xi > Q1
or Yi > Q2
.
(23)
n
n
Q1 and Q2 are so large that there may be no observations left in the area defined
by equation Y. However, the homogeneity property allows us to scale up the
arguments of the quantile function in order to decrease the quantile values and
increase the number of excesses (Straetmans (1998)). Naturally, the estimator
should then be premultiplied by the inverse of this ’scaling up’ factor in order
to leave b
lF invariant. A popular candidate for this operation is the following
polar transformation (see Xin (1992)):
pb1,n = b
ρn cos b
θn and pb2,n = b
ρn sin b
θn .
(24)
b
p1,n ; pb2,n ) ≈ b
ρnb
θn , sin b
θn ).
lF (cos b
lF (b
(25)
The approximate homogeneity of b
lF results in:
Furthermore, conditional upon knowledge of the inverse quantile estimates, the
angle θ and the corresponding radius ρ can now be consistently estimated by
(see Straetmans (1998)):
q
b
θn = arctan(b
p2,n /b
p1,n ) and b
ρn = pb21,n + pb22,n .
(26)
Again, b
lF (cos b
θn , sin b
θn ) can be estimated non-parametrically by the number of
points in the area
½
µ
¶
µ
¶¾
k
k
b
b
(Xi , Yi )i=1,...,n | Xi > Q1
or Yi > Q2
,
cos θn
sin θn
n
n
7
(27)
divided by n. Next, we replace Q1 and Q2 by their empirical counterparts, it
follows the estimator of the dependence function which was first introduced by
Xin (1992):
b
p1,n ; pb2,n ) = k−1b
ρn
lF (b
n
X
i=1
n
o
I Xi > X[n−k cos bθn ],n or Yi > Y[n−k sin bθn ],n .
(28)
Finally, we estimate our two linkage measures, e.g. the following estimator for
(3) (conditional probability)
as well as for (7)
c {κ = 2 | κ ≥ 1} := pb1 + pb2 − 1,
Pr
b
p1 , pb2 )
lF (b
c {κ | κ ≥ 1} := pb1 + pb2 .
Pr
b
p1 , pb2 )
lF (b
(29)
(30)
Note that if both returns are completely dependent in the tails, E {κ | κ ≥ 1} ≈
2 and the markets co-crash with certainty. On the other side, without extreme
co-movements in the two markets E {κ | κ ≥ 1} ≈ 1 (Hartmann / Straetmans /
De Vries (2000)).
3.2
Hypothesis Testing
Peng (1999), proposes a hypothesis test for asymptotic independence under the
null hypothesis of asymptotic dependence. As the power of this test is very low,
we simply examine the number of simultaneous extreme observations in order
to test for asymptotic dependence. The null hypothesis H0 states that the
(relative) spread and the return distribution are independent. Under the null
hypothesis, the number of observations of spreads and returns simultaneously
exceeding the p-quantile is binomially distributed. Furthermore, the expected
number of values for an independent sample consisting of n datapoints in the
bivariate exceedance region is n×(1−p)2 . Hence, for our given sample consisting
of 2210 datapoints and a 99% quantile, we expect 0.22 observations to be in the
bivariate exceedance region, and we reject the null hypothesis of independence
whenever we observe more than one datapoint at the 5% significance level or
two datapoints at the 1% significance level, respectively.
4
Analysis for the Swiss Market
The data set consists of 2210 daily stock returns of 15 titles included (or included
in the past) in the Swiss Market Index, the time period being from March 1993
to December 2001 provided by Datastream.
8
Title
ABB
ADECCO
BALOISE
CREDIT SUISSE
EMS
JULIUS BAER
NESTLE
NOVARTIS
RICHEMONT
ROCHE
SULZER
SWISS RE
UBS
UNAXIS
ZURICH
Min
-0.2037
-0.2234
-0.0922
-0.1406
-0.0867
-0.1001
-0.0604
-0.0748
-0.1358
-0.0725
-0.1839
-0.1902
-0.1542
-0.1384
-0.2257
Max
0.1262
0.1630
0.1219
0.1281
0.0870
0.0948
0.0593
0.1823
0.1136
0.0725
0.1586
0.1243
0.1054
0.1062
0.1408
m left
35
18
29
33
20
30
50
40
40
23
41
40
22
30
28
m right
27
25
60
34
39
25
47
20
35
34
40
39
52
53
20
Tail left
2.5940
3.4037
2.6702
2.8902
3.2154
3.5932
3.1328
3.1308
3.2289
4.0469
2.6745
2.6490
2.9386
3.4388
2.5880
Tail right
2.7367
4.2517
3.3613
3.2362
3.3636
5.2576
3.7258
3.3987
3.4176
4.5825
2.7996
2.9700
3.0309
3.4676
2.7760
Figure 1: Univariate analysis for market values
4.1
Univariate Analysis
Figure 1 shows the minimum and maximum values of the continuously compounded returns of the considered SMI-titles. For example, the daily maximum
(negative) return in the given period was 16.30% (22.57%). Furthermore, we
show the tail-index estimators α̂i for the right hand side (e.g. gains) and for
the left hand side (e.g. losses) of the respective empirical distributions. As it is
common practice, the number of highest order statistics used in the estimation,
m, is selected in the region over which the Hill-plot is more or less constant.
We separate between the left and right tails in order to examine the different
tail characteristics of positive and negative returns. Note that the tail-index
estimates for the left tail, i.e. the loss side estimates, are in all cases smaller
than their right tail counterparts, indicating fatter tails on the left hand side.
This is consistent (in almost every case) with the observed asymmetry between
minimal and maximal stock returns, as shown in the left part of the table.
Figure 2, on the other hand, shows the estimates for different quantiles,
applying formula (20). We calculate the 95%, 99% and the 99.9% quantile
values. These estimates are particularly interesting for risk management, as they
can be regarded as the (daily) value-at-risk measures for the respective SMItitles. The values are well in line with the empirical quantiles. Furthermore,
we calculate the value-at-risk based on the normal distribution assumption.
The respective results are shown in Figure 3. As can be seen, the risk, as
measured through the VaR based on the normal distribution assumption, is
underestimated for high quantiles, such as the 99% and the 99.9% quantiles,
while it is adequately measured for the 95% quantile.
To measure liquidity risk, we calculate the relative spread,e.g.
s=
PA − PB
,
(PA + PB )/2
9
(31)
Title
ABB
ADECCO
BALOISE
CREDIT SUISSE
EMS
JULIUS BAER
NESTLE
NOVARTIS
RICHEMONT
ROCHE
SULZER
SWISS RE
UBS
UNAXIS
ZURICH
VaR 95%
left tail
0.0326
0.0405
0.0250
0.0320
0.0176
0.0317
0.0219
0.0219
0.0345
0.0232
0.0324
0.0210
0.0292
0.0389
0.0304
VaR 99%
left tail
0.0607
0.0649
0.0457
0.0559
0.0291
0.0495
0.0366
0.0365
0.0568
0.0345
0.0591
0.0386
0.0505
0.0622
0.0566
VaR 99.9%
left tail
0.1475
0.1277
0.1082
0.1239
0.0595
0.0940
0.0763
0.0762
0.1159
0.0609
0.1399
0.0920
0.1105
0.1214
0.1379
VaR 95%
right tail
0.0279
0.0502
0.0299
0.0337
0.0181
0.0402
0.0221
0.0235
0.0358
0.0243
0.0285
0.0253
0.0276
0.0390
0.0272
VaR 99%
right tail
0.0503
0.0733
0.0482
0.0554
0.0292
0.0545
0.0341
0.0378
0.0573
0.0345
0.0507
0.0435
0.0470
0.0620
0.0485
Figure 2: VaR analysis for market values
Title
ABB
ADECCO
BALOISE
CREDIT SUISSE
EMS
JULIUS BAER
NESTLE
NOVARTIS
RICHEMONT
ROCHE
SULZER
SWISS RE
UBS
UNAXIS
ZURICH
Normal VaR 95%
0.0335
0.0420
0.0273
0.0319
0.0172
0.0294
0.0212
0.0221
0.0342
0.0210
0.0323
0.0257
0.0289
0.0372
0.0305
Normal VaR 99%
0.0473
0.0594
0.0386
0.0451
0.0242
0.0415
0.0299
0.0313
0.0482
0.0296
0.0457
0.0363
0.0408
0.0526
0.0431
Normal VaR 99.9%
0.0630
0.0790
0.0513
0.0599
0.0323
0.0552
0.0398
0.0416
0.0642
0.0394
0.0608
0.0483
0.0543
0.0699
0.0574
Figure 3: VaR based on normal distribution assumption
10
VaR 99.9%
right tail
0.1166
0.1260
0.0957
0.1129
0.0579
0.0845
0.0632
0.0744
0.1123
0.0571
0.1153
0.0945
0.1004
0.1205
0.1112
Title
ABB
ADECCO
BALOISE
CREDIT SUISSE
EMS
JULIUS BAER
NESTLE
NOVARTIS
RICHEMONT
ROCHE
SULZER
SWISS RE
UBS
UNAXIS
ZURICH
Min
0.0000
0.0005
0.0003
0.0007
0.0005
0.0002
0.0003
0.0003
0.0002
0.0003
0.0008
0.0002
0.0006
0.0010
0.0006
Max
0.0535
0.0348
0.0533
0.0162
0.0389
0.0476
0.0355
0.0725
0.1238
0.0388
0.0785
0.0180
0.0169
0.0392
0.0164
m
94
85
75
95
47
44
80
55
37
50
83
53
92
65
85
Tail index
2.4582
4.3271
3.0921
2.5800
3.6496
3.1736
5.4855
3.5549
3.8329
3.3156
2.7042
3.4176
7.6336
5.2549
4.8497
95% quant
0.0052
0.0129
0.0092
0.0053
0.0107
0.0097
0.0042
0.0036
0.0208
0.0035
0.0090
0.0042
0.0076
0.0155
0.0056
99% quant
0.0100
0.0187
0.0155
0.0099
0.0166
0.0161
0.0056
0.0057
0.0317
0.0057
0.0163
0.0068
0.0093
0.0210
0.0078
99.9% quant
0.0256
0.0319
0.0326
0.0243
0.0312
0.0332
0.0085
0.0108
0.0578
0.0114
0.0381
0.0133
0.0126
0.0326
0.0126
Figure 4: Univariate and quantile analysis for spread values
where PB and PA are the bid and ask price, respectively. The minimum values
are around the value of zero, indicating a very high liquidity. On the other side,
high values for the relative spread indicate low market liquidity. The spread
results are shown in Figure 4.
If we compare the tail-index-estimates for the spread with the tail-indexestimates for the market value, we find mixed results, which do not justify any
conclusion. The quantile estimates show the (relative) spread values which are
exceeded in 5%, 1% and 0.1% of the cases, respectively. Note that calculating
the normal-based spread quantile values does not make sense, since the spread
is not normally distributed.
4.2
Bivariate Analysis
In Figure 6, we show the results of the hypothesis testing (fifth column), as
proposed in the last section. We begin with choosing the 99%-quantile as the
crash level for the market returns and for the relative spreads. Figure 5 shows
the example of Swiss Re, where we observe four bivariate exceedances, which
can be seen in the upper left corner. Over the whole sample, thirteen out of
fifteen titles have more than one observation in the bivariate exceedance region,
hence we can reject the null hypothesis of independence for 87% of all the titles,
at a significance level of 5%. Similarly, we can reject the null hypothesis at a
significance level of 1% for ten of the titles in our sample (or 66%). In order to
justify the application of the extremal dependence measure as presented above,
we test for dependency in smaller quantiles, e.g. the 98% and the 97% quantiles,
respectively. When choosing the 98%-quantile (97% quantile) as the crash level,
we can reject the null hypothesis for a sample fraction of 73% (67%), at a
significance level of 5%. Hence, the higher the quantile, the more reason we
have to reject the null hypothesis of independence.
Column seven shows the extremal dependence, according to the proposed
11
2.00%
1.50%
1.00%
0.50%
-20.00%
-15.00%
-10.00%
0.00%
0.00%
-5.00%
5.00%
10.00%
15.00%
Figure 5: Scatterplot between market returns and relative spreads
linkage measure, e.g. equation (30), and thereby set the univariate exceedance
probabilities equal to 0.01. This means that we do not choose the exceedance
probabilities over a given level x and y as crash levels, but we rather take the
exceedance probability as given (one percent, meaning the 1% value at risk
level). In words, the linkage measure has the following interpretation: Given
that there is at least one crash (return or relative spread, respectively), meaning
that the one percent quantile is exceeded, how much is the probability that there
is a second crash. We find that roughly one out of ten crashes is a co-crash.
The correlations between the time series of the market values and the relative
spreads are shown in column two. In addition, we show the correlations of
the positive (negative) returns with their respective spreads, see column two
and three. Note that the correlation measure reported in the third and fourth
column, respectively, might easily read as suggesting higher linkage values than
the results in the last column of Figure 6. However, as the correlation calculation
tends to assume a normal distribution, we would find values of the linkage
measure that are much less than the calculated ones. Hence, the (left and right)
conditional correlations underestimate the extremal dependence between the
market returns and the relative spreads.
Figure 7 shows the values of the linkage measure for different values of the
threshold level k (in equation (28)), for the example of Sulzer. It can be inferred
that there is a trade off between the validity of the applied measure for extremal
dependence (e.g. small values for the number of highest order statistics k) and
the stability of the linkage measure.
12
Title
ABB
ADECCO
BALOISE
CREDIT SUISSE
EMS
JULIUS BAER
NESTLE
NOVARTIS
RICHEMONT
ROCHE
SULZER
SWISS RE
UBS
UNAXIS
ZURICH
Corr
-0.034
-0.014
-0.055
-0.105
-0.051
0.055
-0.019
-0.052
-0.061
-0.050
-0.153
-0.066
-0.042
-0.074
-0.050
Corr left
-0.295
-0.141
-0.267
-0.107
-0.151
-0.100
-0.160
-0.129
-0.208
-0.145
-0.377
-0.259
-0.085
-0.147
-0.201
Corr right
0.258
0.130
0.180
-0.050
0.151
0.213
0.181
0.076
0.175
0.110
0.058
0.185
-0.007
0.042
0.125
Test
3
0
3
1
1
2
2
0
1
2
3
4
2
2
3
k
46
38
31
21
29
23
24
50
28
26
21
25
42
26
20
Linkage M.
1.0842
1.0443
1.1241
1.0999
1.0516
1.0842
1.0949
1.0544
1.0702
1.0815
1.1422
1.1405
1.0999
1.0815
1.0879
Figure 6: Bivariate Analysis: Correlation and Extremal dependence
Figure 7: Linkage measure for different values of k
13
In order to look for causalilty, we want to calculate leads and lags of the
relative spreads and apply basically the same bivariate analysis as above. However, the iid-assumption (identically independent distributed random variables)
is often violated for the relative spread, due to autocorrelation in the series.
In order to reduce autocorrelation, we chose a new dataset, consisting of daily
stock returns of the100 most often traded titles included in the Swiss Performance Index, the time period being from January 1996 to December 1997. The
procedure is as follows. In a first step, we apply the simple test (as introduced
in section 3.2, adjusted to the new sample) for extremal dependence between
market returns and relative spread, for every single title. Hence, for every single
stock we decide whether there is or is not extremal dependence, on a significance level of 95%. In the second step, the whole sample (e.g. the 100 mostly
traded SPI-titles) is ”cross-sectionally” analysed for dependence. Under the null
hypothesis of independence, the relative number of stocks which showed dependence in the first step is binomially distributed with mean 5%. If we observe
extremal dependence for more than 9.4% or 11.2% of the whole sample, we can
reject the null hypothesis of independence on significance levels of 95% and 99%,
respectively.
Figure 8 shows the results for the 5% quantiles. There are positive and
negative lags shown, respectively. The former indicates an influence of the
market return today on the liqudity (e.g. relative bid-ask spread) in the future,
e.g. a lag of the relative spreads. The latter, negative lag, indicates an influence
of the relative bid-ask spread on the market return, e.g. a lead of the relative
spreads. The number of lags is from zero up to 50 lags. The dotted lines
correspond to the 9.4% (e.g. the 95% significance level) and the 11.2% (e.g. the
99% significance level), respectively. We can infer from Figure 8 that dependence
is highest at the lag of zero. Nearly 50% of the analysed titles show extremal
dependence between the 5% most extreme spreads and the 5% most extreme
market returns. If the conditional spreads and market returns were independent,
only (up to) 5% of the titles were to show dependence.
Furthermore, as indicated, we want to adress the question of causality. The
extremal dependence decreases with an increasing lag, however, the latter effect
is more significant on the left hand side than on the right hand side, indicating
that there is a stronger influence of extreme market returns on liquidity than
the reverse.
In order to further examine the latter result, we repeat the two steps for
different quantiles. Figure 9 depicts extremal dependence for values within
three additional quantiles (e.g. the 1%,10% and 15% quantiles), the number of
maximal lags being 10. For the positive lag, we note that up to 5 trading days,
there is a significant influence of the (extreme) market returns on the liquidity,
for all the chosen quantiles. One explanation for this is that the uncertainty of
the market participants after an extreme adverse movement results in a lasting
(one week) effect on the supply of liquidity.
The negative lags show the relation between the market return today and
the relative spreads in the past, e.g. the influence of the relative spread today
14
Dependence [%]
50%
25%
0%
-50
-30
-10
10
30
50
Lag [Trading Days]
Figure 8: 5% quantile dependence: test significance 95%
on the market return in the future. With the exception of the 1% quantile, only
the lags -1 and -2, respectively, show extremal dependence. If we assume that
prices are mainly driven by the respective bid and ask quotes, we expect the
observed (extremal) depencence, because an increase in the spread naturally
should result in a bigger movement in the market return.
5
Conclusion and Outlook
Dependence between asset returns and their liquidity influences the dynamic
liquidation of asset positions and is thus relevant for firm-wide risk management. In this paper we focus on extreme events and characterize the linkage
between return and relative bid-ask-spread by their asymptotic tail dependence.
We calculate non-parametric estimates for the expected number of exceedances
of a high threshold given that there is at least one exceedance, i.e. the asset
return is strongly negative or the bid-ask-spread is very high. We find moderate
tail dependence in that roughly 10% of the exceedances of the 99% quantile are
co-exceedances. Furthermore, we test non-parametrically whether the number
of bivariate threshold exceedances suggests tail dependence of return and liquidity. Significant dependence is found for 13 out of 15 stocks on the 5% confidence
level and for 10 out of 15 on the 1% level. We thus conclude that extreme dependence between negative market returns and liquidity is existing in the empirical
data and may be relevant for firm-wide risk management. Finally, we test for
causality and find decreasing extremal dependence when adding both positive
15
1% Quantil Dependence: Test Significance 95%
5% Quantil Dependence: Test Significance 95%
50%
Depe nde nce [%]
Depe nde nce [%]
50%
25%
25%
0%
-10
-5
0%
0
5
10
-10
-5
Lag [ Trading Days ]
10% Quantil Dependence: Test Significance 95%
Depe nde nce [%]
Depe ndence [%]
10
50%
25%
25%
0%
-5
5
15% Quantil Dependence: Test Significance 95%
50%
-10
0
Lag [ Trading Days ]
0
5
0%
10
-10
Lag [Trading Days ]
-5
0
5
10
Lag [ Trading Days ]
Figure 9: Quantile dependence between market returns and relative spreads, for
positive and negative lags
16
and negative lags, respectively.
There are a number of directions in which the analysis of this paper can
be extended in future research. One possibility is the parametric modelling of
return and spread distributions and the examination of parametric measures
of dependence, e.g. copula functions. Another possibility is the consideration
of a larger data set in order to reveal whether tail dependence is a prevalent
phenomenon in asset markets.
6
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19