Copernican Journal of Finance & Accounting

 e-ISSN 2300-3065
p-ISSN 2300-12402017, volume 6, issue 4

 Date of submission: March 2, 2018; date of acceptance: April 10, 2018.
* Contact information: p.jaworski@mimuw.edu.pl, Institute of Mathematics, Uni-

versity of Warsaw, Banacha 2, 02-097 Warszawa, Poland, phone: +48 22 554 45 23.
** Contact information: corresponding author, kliber@sgh.waw.pl, Institute of Fi-

nance, Warsaw School of Economics, Madalińskiego 6/8, 02-513 Warszawa, Poland, 
phone: +48 22 564 93 20.

*** Contact information: mliber@sgh.waw.pl, Institute of Finance, Warsaw School of 
Economics, Madalińskiego 6/8, 02-513 Warszawa, Poland, phone: +48 22 564 93 20; 
ORCID ID: https://orcid.org/0000-0001-8236-004X.

  The authors are very grateful for the financial support from Polish National Sci-
ence Center programme No 2015/17/B/HS4/00911.

Jaworski, P., Liberadzki, K., & Liberadzki, M. (2017). Contagion and divergence on sovereign bond 
markets. Copernican Journal of Finance & Accounting, 6(4), 39–68. http://dx.doi.org/10.12775/
CJFA.2017.022

piotr jaworski*

University of Warsaw

kaMIl lIberadzkI**

Warsaw School of Economics

MarcIn lIberadzkI***

Warsaw School of Economics

contagion and divergence  
on sovereign bond markets

Keywords: sovereign credit risk, global debt market, contagion effect, divergence ef-
fect, stochastic modeling, copulas.

J E L Classification: C, C5, G, G1, G15.

Abstract: The paper gives evidence for significant interdependency between sovereign 
bond yields during pre-crisis period as well as after the recent global debt crisis broke 
out. These interdependencies can be classified either as contagion or divergence. Both 
effects should be understood as an increase in interdependency among different assets 



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki40

as a result of a significant shock. Contagion refers to positive interdependency, while 
the negative correlation illustrates divergence effect. Basing on selected bonds, we fo-
und out that divergence effect prevails over contagion. What is more, there has been 
more divergence among the European Monetary Union (EMU) countries than betwe-
en the EMU sovereigns and Japanese or the US bonds. Despite the increase of contagion 
level after the crisis, there was no worldwide retreat from bond markets as a result of 
global turbulences. While there is a numerous literature on sovereign bond contagion, 
we propose our methodology for bond divergence effect measuring. The paper organi-
zes and presents main concepts for contagion effect modeling within both spatial and 
time approach. As the same index is utilized for contagion and divergence modeling, it 
is possible to analyze and compare these two opposite effects together basing on selec-
ted sovereign bond market data.

 Introduction

Overview

Portfolio selection has developed the methods of limiting the total portfolio 
risk by means of diversification. The occurrence of contagion effect devastated 
that approach since financial crises have underlined that the markets tend to 
become more dependent during crises than they are during calmer periods. Not 
taking this effect, called ‘contagion’ into consideration, may lead to significant 
underestimation of portfolio risk in case of extreme value changes of differ-
ent instruments included. Understanding the idea of contagion provides useful 
benefits when dealing with financial risk. For instance, diversification strate-
gies for building portfolios may fail under contagion, i.e., when there is a shift 
in the dependence among assets in crisis period (Durante, Foscolo, Jaworski 
& Wang, 2014, p. 4023). Bradley and Taqqu (2004, p. 8) suggested to focus on 
probability distribution functions of target financial time series and, in par-
ticular, at the discrepancies between the probability mass concentrated in tail 
and central sets of these distributions. What is also new in our article is that we 
formulate the notion of ‘divergence’ effect that states in opposition to ‘comove-
ment’ effect (which we explain in detail further in the text). 

The new approach to portfolio risk management (Durante & Jaworski, 
2010, p. 551) applied in this article is innovative in the way that instead of 
using Person correlation coefficient we rely on information based on copu-
la-class functions that fully describe the interdependence between returns 
on different assets considered in a portfolio. This of interdependency eval-
uation will be used for analyzing the selected sovereign bond yields data 



 ConTagion and divergence on sovereign bond markeTs 41

in the pre-crises and post-crises periods. The practical importance of the 
research and our findings is that it may lead to improving of calculation of 
sovereign bond portfolio risk. As we refer strictly to portfolio theory, we 
do not carry out fundamental analysis of bond issuers. Selecting the sam-
ple based on heuristic analysis, we conduct statistical evaluation of actual 
interdependencies between the instruments. Initially we tried to heuristi-
cally pre-select the bond sample including the issuers that might be catego-
rized into: “safe haven countries” (Germany, US, Australia), “emerging mar-
kets” (Poland) and “troubled economies” (Portugal, Italy) and Japan (not 
categorized). We restrict ourselves from investigating the fundamental 
reasons for observed price behavior, we rather follow the stress-testing ap-
proach and answer the question: what happens with the yields of different 
sovereign bonds when there is a turmoil on the global market, not inquiring 
for the fundamental reasons. 

The importance and practical value for this research may be confirmed 
by what happened during the recent financial crisis, which broke out in the 
US and spread globally, resulted in series of shocks across the markets. The 
last couple of years have seen the transformation of the global financial cri-
sis into a sovereign debt crisis. Sovereign bond markets have become in-
creasingly integrated both domestically and internationally long before 
year 2008. The nature of these interdependencies and the shock transmis-
sion channels are still not well understood. Interdependency may be trans-
lated either into contagion or divergence effect. The thesis of our paper is, 
that there is a significant interdependency between sovereign bond pric-
es both during tranquil and contraction periods. This interdependency in-
cludes both contagion and divergence whereas at turbulent times conta-
gion tends to increase while divergence remains still at significant level. 

The further goal of our research is to quantify contagion effect and di-
vergence effect referring to selected sovereign bond data before and after 
the outbreak of the recent global crisis. Therefore we propose the construc-
tion of proper contagion and divergence index with respect to bond yields 
or prices. 

The European government bond market integration was one of the posi-
tive results of the European Monetary Union from its advent till late 2008. 
However, it has been found out at that time that the global shocks affect 
Germany and the rest of euro bond markets in various levels (Balli, 2009, 
p. 331). The literature has responded to the EMU crisis through a series of 



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki42

empirical studies. It has been observed during global contraction, that the 
global risk perception with investment in sovereign bonds relative to safe 
havens of US and Germany increases. When it comes to the EMU countries, 
there is fundamental explanation for this: eliminating the exchange rate 
parity, no adjustment mechanism was retained to correct fundamental di-
vergence (Mazier & Petit, 2013, p. 513). To stir things up, the EMU countries 
were slow even to track rising divergences (Muellbauer, 2013, p. 610). As 
a result, German government bonds operated as a flight-to- quality asset, 
a feature apparent in figure 2. Ehrmann, Fratzscher and Rigobon (2005) an-
alyze the degree of financial transmission between money, bond and eq-
uity markets and exchange rates within and between the United States 
and the euro area. They find that asset prices react strongest to other do-
mestic asset price shocks, and that there are also substantial internation-
al spillovers, both within and across asset classes. The results under-
line the dominance of US markets as the main driver of global financial 
markets: US financial markets explain, on average, more than 25% of move-
ments in euro area financial markets, whereas euro area markets account 
only for about 8% of US asset price changes.

Arghyrou and Kontonikas (2012, p. 658) present the concept of crisis 
transmission through the two channels (periphery-to-core and periph-
ery-to-periphery transmission): increasing core-periphery divergence sig-
nals an increased probability of aggregating fiscal risks at the EU-level and 
may increase future borrowing requirements from the core group to cov-
er the potential support efforts. Hence, increasing divergence may finally 
cause contagion from the periphery group to the core group.

Mink and de Haan (2013, p. 102) discover contagion between Greece and 
indebted peripheral EMU countries (Portugal, Ireland, Spain) as a result of 
the fiscal crisis. Antonakakis and Vergos (2013, p. 258) give examples of 
empirical studies showing convergence of spreads in EMU countries before 
the 2007 and subsequent divergence of bond yields in countries with weak-
er fundamentals. 

Our contribution is that we focus equally on sovereign bond contagion 
as well as on divergence as we find the latter effect (divergence) being 
somehow overlooked in the literature. Therefore we propose a nonlinear 
contagion and divergence index enabling direct comparison of these two 
effects. What is more, working on bond yields rather than prices, we ana-
lyze the yields on each yield curve short end, its middle part and long seg-



 ConTagion and divergence on sovereign bond markeTs 43

ment. Therefore it is possible to observe the yield curve behavior as a re-
sult of shocks and contagion/divergence effects between bonds of different 
maturities.

There is a broad array of techniques applied for quantifying contagion ef-
fect, thus the paper somehow organizes these models inspired first and fore-
most by approach of Durante and Jaworski (2010, p. 551) and Jaworski and 
Pitera (2014, p. 303). Furthermore, the paper examines how these techniques 
may also be used for modeling bond yield divergence effect.

Market contagion definitions

Contagion is defined commonly in the literature as the change in the way coun-
tries own fundamentals or other factors are priced during a crisis period, i.e. 
a change in the reaction of financial markets either in response to observable 
factors, such as changes in sovereign risk among neighboring countries, or due 
to unobservable ones such as herding behavior of market participants (Beirne 
& Fratzscher, 2013, p. 60). The World Bank (2016) makes use of following defi-
nitions:
 ■ Broad definition: Financial contagion is usually referred to as a cross-

-market transmission of shocks or the general cross-market spillover ef-
fects. It can take place both during good times and bad times. Thus, con-
tagion does not need to be related to crises. However, it is emphasized 
during crisis times. If present, it may mitigate the benefits of diversifi-
cation precisely when those benefits are needed most and have serious 
consequences for investors. Therefore, understanding this highly nonli-
near effect is of great interest not only to financial theorists but to prac-
titioners as well (Jaworski & Pitera, 2014, p. 303; Durante & Jaworski, 
2010, p. 555).

 ■ Restrictive Definition: Contagion is the transmission of shocks to other 
countries or the cross-country correlation, beyond any fundamental link 
among the countries and beyond common shocks. This definition is usu-
ally referred as excess co-movement, commonly explained by herding 
behavior. The World Bank recognizes following categories of fundamen-
tal links: financial links, real links and political links.

 ■ Very Restrictive Definition: Contagion occurs when cross-country cor-
relations increase during “crisis times” relative to correlations during 
”tranquil times”.



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki44

According to its probabilistic definition contagion is a significant growth 
of probability of shock in one country (default of a company respectively) on 
condition that shock occurs in other country (other company defaults respec-
tively).

Contagion process can be translated into three effects: 1) spillover, 2) trans-
mission, 3) co-movement. To avoid confusions, we propose the following de-
scription of these above-mentioned effects:

Spillover: the contagion spreads as a jump process. Beyond a threshold the 
changes become leap like (often deferred) usually strengthened by common 
level of investors stop loss orders.

Transmission: the spread occurs continuously. The market participants lev-
els of stop loss orders are usually dispersed. Positions are closed at different 
time.

Co-movement: the changes occur simultaneously (with no time delay). The 
same effect however would be classified as spillover if data is measured fre-
quently or co-movement if data is gathered at more time intervals.

There is however much disagreement among economists about what con-
tagion is and how it should be tested for empirically. Mink and de Haan (2013, 
p. 102) refer to following definition: contagion is an episode in which there 
are significant immediate effects in a number of countries following an event 
that is, when the consequences are fast and furious and evolve over a matter of 
hours or days. When the effect of the event is gradual, they refer to this as spill-
overs rather than contagion. Also in other cases one may question whether the 
label of contagion is adequate. For instance, in a widely used approach, conta-
gion is inferred by a significant rise in the correlation of asset returns in crisis 
periods compared to tranquil periods (Mink & de Haan, 2013, p. 102).

Basically, these studies examine whether cross correlations between mar-
kets increase significantly during a major crisis (crisis period) or after a crisis 
(post-crisis period) in comparison to periods before the crisis (pre-crisis peri-
od). If the correlations increased significantly during and after crisis periods, 
in comparison to pre-crisis periods, the authors conclude evidence in favor of 
contagion (Yunus, 2013, p. 327).



 ConTagion and divergence on sovereign bond markeTs 45

The research methodology and the course of the research process

The framework for market contagion modeling

Figure 1. Central (on the right, around v1) and extreme values (on the left, around v2)

 

ly during a major crisis (crisis period) or after a crisis (post-crisis period) in comparison to periods 
before the crisis (pre-crisis period). If the correlations increased significantly during and after crisis 
periods, in comparison to pre-crisis periods, the authors conclude evidence in favor of contagion 
(Yunus, 2013, p. 327). 

 
The research methodology and the course of the research process 
 
The framework for market contagion modeling 
 
Figure 1. Central (on the right, around v1) and extreme values (on the left, around v2) 
 

 
 
Source: own calculations. 
 
Co-movements, the main objective of our study, may be approximated by static models like copula 

class models. Modeling contagion may refer either to spatial analysis approach or time analysis approach. 
Within the spatial approach, the random variable changes grouped into moderate and sig-

nificant ones. In the spatial framework, according to Bradley and Taqqu (2004, p. 9), “ there is 
contagion from market X1 to market X2 if there is more dependence between X1 and X2 when X1 is 
doing badly than when X1 exhibits typical performance”. 

Let us consider Xt to be a two-dimensional stochastic process. Assume the Xt is stationary and ergodic. 
If m(t) is a type I regression function: 

,2 ,1( ) ( | ).t tm v E X X v           (1) 
There is a linear contagion effect referring to intervals [a, b] and [c, d], a b c d      , 

when: 
,1 ,1

( ) ( ) ( ) ( )( [ , ]) 0 ( [ , ]) and .t t
m b m a m d m cP X a b P X c d

b a d c
     
 

    (2) 
In case of normal joint probability distribution of Xt, both the type I and type II regressions are linear: 

2
,1 ,2 1 ,2 ,1( ) , Cov( , ) / ( ), ( ( )).t t t tm v v X X D X E X E X              (3) 

Since the ( )m v    is constant, there is no contagion effect unless we go beyond the normal distribu-
tion assumption. Bradley and Taqqu (2004, p. 9) point out that if the first derivative is equal , the conta-
gion exists only if  increases for extreme value interval (figure 1). 

Coefficient  should be calculated separately for each ‘stripe’ around the value vi (as presented on the 
figure 1). Maintaining the assumption on m(v) linearity within each value interval, it is recommended to 
use the difference quotient instead of the first derivative. 

Another approach within the spatial framework utilizes copula theory. The term ‘copula’ was coined by 
Sklar (1959, p. 229), who proved that a collection of marginal distributions may be merged together by 
using copula transformation to form a model of multivariate distribution. Any joint distribution F of a set 
of random variables 1 2, , , nX X XK  can be separated into two parts: the first is the combination of the 
marginals that (if expressed in cumulative distribution form) are ()iF   where (( ) )i iF P X xx    and the 

X2 

X1 

v2 v1 

interval of extreme X1 value

S o u r c e : own calculations.

Co-movements, the main objective of our study, may be approximated by 
static models like copula class models. Modeling contagion may refer either to 
spatial analysis approach or time analysis approach.

Within the spatial approach, the random variable changes grouped into 
moderate and significant ones. In the spatial framework, according to Bradley 
and Taqqu (2004, p. 9), “there is contagion from market X1 to market X2 if there 
is more dependence between X1 and X2 when X1 is doing badly than when X1 ex-
hibits typical performance”.

Let us consider Xt to be a two-dimensional stochastic process. Assume the Xt 
is stationary and ergodic. If m(t) is a type I regression function:

 

ly during a major crisis (crisis period) or after a crisis (post-crisis period) in comparison to periods 
before the crisis (pre-crisis period). If the correlations increased significantly during and after crisis 
periods, in comparison to pre-crisis periods, the authors conclude evidence in favor of contagion 
(Yunus, 2013, p. 327). 

 
The research methodology and the course of the research process 
 
The framework for market contagion modeling 
 
Figure 1. Central (on the right, around v1) and extreme values (on the left, around v2) 
 

 
 
Source: own calculations. 
 
Co-movements, the main objective of our study, may be approximated by static models like copula 

class models. Modeling contagion may refer either to spatial analysis approach or time analysis approach. 
Within the spatial approach, the random variable changes grouped into moderate and sig-

nificant ones. In the spatial framework, according to Bradley and Taqqu (2004, p. 9), “ there is 
contagion from market X1 to market X2 if there is more dependence between X1 and X2 when X1 is 
doing badly than when X1 exhibits typical performance”. 

Let us consider Xt to be a two-dimensional stochastic process. Assume the Xt is stationary and ergodic. 
If m(t) is a type I regression function: 

,2 ,1( ) ( | ).t tm v E X X v           (1) 
There is a linear contagion effect referring to intervals [a, b] and [c, d], a b c d      , 

when: 
,1 ,1

( ) ( ) ( ) ( )( [ , ]) 0 ( [ , ]) and .t t
m b m a m d m cP X a b P X c d

b a d c
     
 

    (2) 
In case of normal joint probability distribution of Xt, both the type I and type II regressions are linear: 

2
,1 ,2 1 ,2 ,1( ) , Cov( , ) / ( ), ( ( )).t t t tm v v X X D X E X E X              (3) 

Since the ( )m v    is constant, there is no contagion effect unless we go beyond the normal distribu-
tion assumption. Bradley and Taqqu (2004, p. 9) point out that if the first derivative is equal , the conta-
gion exists only if  increases for extreme value interval (figure 1). 

Coefficient  should be calculated separately for each ‘stripe’ around the value vi (as presented on the 
figure 1). Maintaining the assumption on m(v) linearity within each value interval, it is recommended to 
use the difference quotient instead of the first derivative. 

Another approach within the spatial framework utilizes copula theory. The term ‘copula’ was coined by 
Sklar (1959, p. 229), who proved that a collection of marginal distributions may be merged together by 
using copula transformation to form a model of multivariate distribution. Any joint distribution F of a set 
of random variables 1 2, , , nX X XK  can be separated into two parts: the first is the combination of the 
marginals that (if expressed in cumulative distribution form) are ()iF   where (( ) )i iF P X xx    and the 

X2 

X1 

v2 v1 

interval of extreme X1 value

 (1)

There is a linear contagion effect referring to intervals [a, b] and [c, d], 

 

ly during a major crisis (crisis period) or after a crisis (post-crisis period) in comparison to periods 
before the crisis (pre-crisis period). If the correlations increased significantly during and after crisis 
periods, in comparison to pre-crisis periods, the authors conclude evidence in favor of contagion 
(Yunus, 2013, p. 327). 

 
The research methodology and the course of the research process 
 
The framework for market contagion modeling 
 
Figure 1. Central (on the right, around v1) and extreme values (on the left, around v2) 
 

 
 
Source: own calculations. 
 
Co-movements, the main objective of our study, may be approximated by static models like copula 

class models. Modeling contagion may refer either to spatial analysis approach or time analysis approach. 
Within the spatial approach, the random variable changes grouped into moderate and sig-

nificant ones. In the spatial framework, according to Bradley and Taqqu (2004, p. 9), “ there is 
contagion from market X1 to market X2 if there is more dependence between X1 and X2 when X1 is 
doing badly than when X1 exhibits typical performance”. 

Let us consider Xt to be a two-dimensional stochastic process. Assume the Xt is stationary and ergodic. 
If m(t) is a type I regression function: 

,2 ,1( ) ( | ).t tm v E X X v           (1) 
There is a linear contagion effect referring to intervals [a, b] and [c, d], a b c d      , 

when: 
,1 ,1

( ) ( ) ( ) ( )( [ , ]) 0 ( [ , ]) and .t t
m b m a m d m cP X a b P X c d

b a d c
     
 

    (2) 
In case of normal joint probability distribution of Xt, both the type I and type II regressions are linear: 

2
,1 ,2 1 ,2 ,1( ) , Cov( , ) / ( ), ( ( )).t t t tm v v X X D X E X E X              (3) 

Since the ( )m v    is constant, there is no contagion effect unless we go beyond the normal distribu-
tion assumption. Bradley and Taqqu (2004, p. 9) point out that if the first derivative is equal , the conta-
gion exists only if  increases for extreme value interval (figure 1). 

Coefficient  should be calculated separately for each ‘stripe’ around the value vi (as presented on the 
figure 1). Maintaining the assumption on m(v) linearity within each value interval, it is recommended to 
use the difference quotient instead of the first derivative. 

Another approach within the spatial framework utilizes copula theory. The term ‘copula’ was coined by 
Sklar (1959, p. 229), who proved that a collection of marginal distributions may be merged together by 
using copula transformation to form a model of multivariate distribution. Any joint distribution F of a set 
of random variables 1 2, , , nX X XK  can be separated into two parts: the first is the combination of the 
marginals that (if expressed in cumulative distribution form) are ()iF   where (( ) )i iF P X xx    and the 

X2 

X1 

v2 v1 

interval of extreme X1 value

 when:



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki46

 

ly during a major crisis (crisis period) or after a crisis (post-crisis period) in comparison to periods 
before the crisis (pre-crisis period). If the correlations increased significantly during and after crisis 
periods, in comparison to pre-crisis periods, the authors conclude evidence in favor of contagion 
(Yunus, 2013, p. 327). 

 
The research methodology and the course of the research process 
 
The framework for market contagion modeling 
 
Figure 1. Central (on the right, around v1) and extreme values (on the left, around v2) 
 

 
 
Source: own calculations. 
 
Co-movements, the main objective of our study, may be approximated by static models like copula 

class models. Modeling contagion may refer either to spatial analysis approach or time analysis approach. 
Within the spatial approach, the random variable changes grouped into moderate and sig-

nificant ones. In the spatial framework, according to Bradley and Taqqu (2004, p. 9), “ there is 
contagion from market X1 to market X2 if there is more dependence between X1 and X2 when X1 is 
doing badly than when X1 exhibits typical performance”. 

Let us consider Xt to be a two-dimensional stochastic process. Assume the Xt is stationary and ergodic. 
If m(t) is a type I regression function: 

,2 ,1( ) ( | ).t tm v E X X v           (1) 
There is a linear contagion effect referring to intervals [a, b] and [c, d], a b c d      , 

when: 
,1 ,1

( ) ( ) ( ) ( )( [ , ]) 0 ( [ , ]) and .t t
m b m a m d m cP X a b P X c d

b a d c
     
 

    (2) 
In case of normal joint probability distribution of Xt, both the type I and type II regressions are linear: 

2
,1 ,2 1 ,2 ,1( ) , Cov( , ) / ( ), ( ( )).t t t tm v v X X D X E X E X              (3) 

Since the ( )m v    is constant, there is no contagion effect unless we go beyond the normal distribu-
tion assumption. Bradley and Taqqu (2004, p. 9) point out that if the first derivative is equal , the conta-
gion exists only if  increases for extreme value interval (figure 1). 

Coefficient  should be calculated separately for each ‘stripe’ around the value vi (as presented on the 
figure 1). Maintaining the assumption on m(v) linearity within each value interval, it is recommended to 
use the difference quotient instead of the first derivative. 

Another approach within the spatial framework utilizes copula theory. The term ‘copula’ was coined by 
Sklar (1959, p. 229), who proved that a collection of marginal distributions may be merged together by 
using copula transformation to form a model of multivariate distribution. Any joint distribution F of a set 
of random variables 1 2, , , nX X XK  can be separated into two parts: the first is the combination of the 
marginals that (if expressed in cumulative distribution form) are ()iF   where (( ) )i iF P X xx    and the 

X2 

X1 

v2 v1 

interval of extreme X1 value

 (2)

In case of normal joint probability distribution of Xt, both the type I and type 
II regressions are linear:

 

ly during a major crisis (crisis period) or after a crisis (post-crisis period) in comparison to periods 
before the crisis (pre-crisis period). If the correlations increased significantly during and after crisis 
periods, in comparison to pre-crisis periods, the authors conclude evidence in favor of contagion 
(Yunus, 2013, p. 327). 

 
The research methodology and the course of the research process 
 
The framework for market contagion modeling 
 
Figure 1. Central (on the right, around v1) and extreme values (on the left, around v2) 
 

 
 
Source: own calculations. 
 
Co-movements, the main objective of our study, may be approximated by static models like copula 

class models. Modeling contagion may refer either to spatial analysis approach or time analysis approach. 
Within the spatial approach, the random variable changes grouped into moderate and sig-

nificant ones. In the spatial framework, according to Bradley and Taqqu (2004, p. 9), “ there is 
contagion from market X1 to market X2 if there is more dependence between X1 and X2 when X1 is 
doing badly than when X1 exhibits typical performance”. 

Let us consider Xt to be a two-dimensional stochastic process. Assume the Xt is stationary and ergodic. 
If m(t) is a type I regression function: 

,2 ,1( ) ( | ).t tm v E X X v           (1) 
There is a linear contagion effect referring to intervals [a, b] and [c, d], a b c d      , 

when: 
,1 ,1

( ) ( ) ( ) ( )( [ , ]) 0 ( [ , ]) and .t t
m b m a m d m cP X a b P X c d

b a d c
     
 

    (2) 
In case of normal joint probability distribution of Xt, both the type I and type II regressions are linear: 

2
,1 ,2 1 ,2 ,1( ) , Cov( , ) / ( ), ( ( )).t t t tm v v X X D X E X E X              (3) 

Since the ( )m v    is constant, there is no contagion effect unless we go beyond the normal distribu-
tion assumption. Bradley and Taqqu (2004, p. 9) point out that if the first derivative is equal , the conta-
gion exists only if  increases for extreme value interval (figure 1). 

Coefficient  should be calculated separately for each ‘stripe’ around the value vi (as presented on the 
figure 1). Maintaining the assumption on m(v) linearity within each value interval, it is recommended to 
use the difference quotient instead of the first derivative. 

Another approach within the spatial framework utilizes copula theory. The term ‘copula’ was coined by 
Sklar (1959, p. 229), who proved that a collection of marginal distributions may be merged together by 
using copula transformation to form a model of multivariate distribution. Any joint distribution F of a set 
of random variables 1 2, , , nX X XK  can be separated into two parts: the first is the combination of the 
marginals that (if expressed in cumulative distribution form) are ()iF   where (( ) )i iF P X xx    and the 

X2 

X1 

v2 v1 

interval of extreme X1 value

 (3)

Since the 

 

ly during a major crisis (crisis period) or after a crisis (post-crisis period) in comparison to periods 
before the crisis (pre-crisis period). If the correlations increased significantly during and after crisis 
periods, in comparison to pre-crisis periods, the authors conclude evidence in favor of contagion 
(Yunus, 2013, p. 327). 

 
The research methodology and the course of the research process 
 
The framework for market contagion modeling 
 
Figure 1. Central (on the right, around v1) and extreme values (on the left, around v2) 
 

 
 
Source: own calculations. 
 
Co-movements, the main objective of our study, may be approximated by static models like copula 

class models. Modeling contagion may refer either to spatial analysis approach or time analysis approach. 
Within the spatial approach, the random variable changes grouped into moderate and sig-

nificant ones. In the spatial framework, according to Bradley and Taqqu (2004, p. 9), “ there is 
contagion from market X1 to market X2 if there is more dependence between X1 and X2 when X1 is 
doing badly than when X1 exhibits typical performance”. 

Let us consider Xt to be a two-dimensional stochastic process. Assume the Xt is stationary and ergodic. 
If m(t) is a type I regression function: 

,2 ,1( ) ( | ).t tm v E X X v           (1) 
There is a linear contagion effect referring to intervals [a, b] and [c, d], a b c d      , 

when: 
,1 ,1

( ) ( ) ( ) ( )( [ , ]) 0 ( [ , ]) and .t t
m b m a m d m cP X a b P X c d

b a d c
     
 

    (2) 
In case of normal joint probability distribution of Xt, both the type I and type II regressions are linear: 

2
,1 ,2 1 ,2 ,1( ) , Cov( , ) / ( ), ( ( )).t t t tm v v X X D X E X E X              (3) 

Since the ( )m v    is constant, there is no contagion effect unless we go beyond the normal distribu-
tion assumption. Bradley and Taqqu (2004, p. 9) point out that if the first derivative is equal , the conta-
gion exists only if  increases for extreme value interval (figure 1). 

Coefficient  should be calculated separately for each ‘stripe’ around the value vi (as presented on the 
figure 1). Maintaining the assumption on m(v) linearity within each value interval, it is recommended to 
use the difference quotient instead of the first derivative. 

Another approach within the spatial framework utilizes copula theory. The term ‘copula’ was coined by 
Sklar (1959, p. 229), who proved that a collection of marginal distributions may be merged together by 
using copula transformation to form a model of multivariate distribution. Any joint distribution F of a set 
of random variables 1 2, , , nX X XK  can be separated into two parts: the first is the combination of the 
marginals that (if expressed in cumulative distribution form) are ()iF   where (( ) )i iF P X xx    and the 

X2 

X1 

v2 v1 

interval of extreme X1 value

  is constant, there is no contagion effect unless we go 
beyond the normal distribution assumption. Bradley and Taqqu (2004, p. 9) 
point out that if the first derivative is equal β, the contagion exists only if β in-
creases for extreme value interval (figure 1).

Coefficient β should be calculated separately for each ‘stripe’ around the 
value vi (as presented on the figure 1). Maintaining the assumption on m(v) lin-
earity within each value interval, it is recommended to use the difference quo-
tient instead of the first derivative.

Another approach within the spatial framework utilizes copula theory. The 
term ‘copula’ was coined by Sklar (1959, p. 229), who proved that a collection 
of marginal distributions may be merged together by using copula transfor-
mation to form a model of multivariate distribution. Any joint distribution F of 
a set of random variables X1, X2, K, Xn can be separated into two parts: the first is 
the combination of the marginals that (if expressed in cumulative distribution 
form) are F1(·) where Fi(x) = P(XI ≤ x) and the second is the copula that describes 
the dependence structure between the random variables (Nelsen, 2006, p. 47), 
(Embrechts, 2009, p. 639), (Durante & Sempi, 2010, p. 42), (Kemp, 2011, p. 64), 
(Joe, 2014, p. 7). The decomposition relies on Sklar’s theorem, which states that 
there exists n-dimensional copula C such that:

 

second is the copula that describes the dependence structure between the random variables (Nelsen,  
2006, p. 47), ( Embrechts,  2009, p. 639), ( Durante &  Sempi, 2010, p. 42), ( Kemp,  2011, p. 64), 
( Joe, 2014, p.7). The decomposition relies on Sklar’s theorem, which states that there exists n-
dimensional copula C such that: 

 
1 1 1( , , ) ( ( ), , ( )) .nn n nF x x C F x F x x R  K K      (4) 

 
If marginal cumulative distribution functions 1, , nF FK  are continuous then C is unique. 
According to Durante and Jaworski (2010, p. 555), ‘There is contagion from market X to market Y if the 

conditional copula of the market returns X and Y, when X is smaller than certain quantile, dominates the 
conditional copula when X is around its median.’ 

Let us assume the two-dimensional case. If C is a copula for the joint distribution for X1 and X2, then the 
copula ,[ ]C   is a copula of a conditional distribution of the pair X1 and X2, given that: 

 
F1(X1) ϵ [α , β].         (5) 
 
Following Durante and Jaworski (2010, p. 555) there is a non-linear contagion – co-movement effect 

relative to intervals [, ] and [, ], 0 ≤ α < β ≤ γ < δ ≤ 1, when copula ,[ ]C   dominates over ,[ ]C    with 
respect to concordance order (the higher the copula values, the more intense the dependence), i.e.: 
 

2 2
[ , ] [ , ] 0 [ , ] 0 [ , ] 0[0,1] ( ) ( ) [0,1] ( ) ( ).p C p C p and p C p C p              (6) 

 
Referring once again to Figure 1 we simply compare conditional copula values for each value ‘stripe’ 

and therefore conclude that a higher copula value marks stronger interdependence. The above-defined 
non-linear contagion effect does not depend on marginal distribution. In addition, the copulas act inde-
pendently whenever we work on indexes, prices or logarithmic returns. The practical problem in this ap-
proach is that copula values must be verified at each point. 

Despite the very strong relation between comonotonic random variables, there will be no contagion ef-
fect, as their interdependence, although strong, remains stable. 

However, it is relatively difficult to estimate copula values, so it is more convenient to compare a copu-
la-dependent measure like Spearman rho or Kendall tau constructing thus a contagion index d: 
 

 d = ρ({xt : t ϵ A}) – ρ({xt : t ϵ B}) ,     (7) 
 
where  refers to the correlation coefficient (Spearman or  Kendall). 
 
Contagion versus divergence, the toy model 
Let Xt Yt and Zt be the daily changes of the yields of sovereign debts of three countries, respective-

ly X, Y and Z. X represents a country with a strong economy (a leading player), Y a country with 
economy shaken by a crisis and Z a ”safe haven”. Note that when the rating of a given country is 
decreasing then the prices of government securities are also decreasing but the yields are increasing. 
We present here a simplified ”toy” model which illustrates the interdependence between daily chang-
es of yields. We would like to capture two panic-driven effects. The first one is a standard one, due to 
some internal problems the global investors lose the confidence in bonds issued by “the leading play-
er” X and start selling them, which results in a decrease of the price and increase of yields. This causes 
an anxiety between the “the dependent player” Y bondholders and they start to close their positions 
although there are no direct “risks” as regards Y. 

The second one has a more complicated nature. The global investors lose confidence in “the 
local player” Y, close their positions in bonds issued by Y and move their wealth to the safe heav-
en Z. Thus, although there are no internal reasons, the prices of bonds issued by Z are going up 
and yields down. 

We consider a three factor model. The exogenous variables (risk factors) are εi, i = 1, 2, 3. For 
simplicity we assume that they are independent random variables with standard normal distri-

 (4)

If marginal cumulative distribution functions F1, K, Fn are continuous then 
C is unique.

According to Durante and Jaworski (2010, p. 555), ‘There is contagion from 
market X to market Y if the conditional copula of the market returns X and Y, 
when X is smaller than certain quantile, dominates the conditional copula when 
X is around its median.’



 ConTagion and divergence on sovereign bond markeTs 47

Let us assume the two-dimensional case. If C is a copula for the joint distri-
bution for X1 and X2, then the copula 

 

second is the copula that describes the dependence structure between the random variables (Nelsen,  
2006, p. 47), ( Embrechts,  2009, p. 639), ( Durante &  Sempi, 2010, p. 42), ( Kemp,  2011, p. 64), 
( Joe, 2014, p.7). The decomposition relies on Sklar’s theorem, which states that there exists n-
dimensional copula C such that: 

 
1 1 1( , , ) ( ( ), , ( )) .nn n nF x x C F x F x x R  K K      (4) 

 
If marginal cumulative distribution functions 1, , nF FK  are continuous then C is unique. 
According to Durante and Jaworski (2010, p. 555), ‘There is contagion from market X to market Y if the 

conditional copula of the market returns X and Y, when X is smaller than certain quantile, dominates the 
conditional copula when X is around its median.’ 

Let us assume the two-dimensional case. If C is a copula for the joint distribution for X1 and X2, then the 
copula ,[ ]C   is a copula of a conditional distribution of the pair X1 and X2, given that: 

 
F1(X1) ϵ [α , β].         (5) 
 
Following Durante and Jaworski (2010, p. 555) there is a non-linear contagion – co-movement effect 

relative to intervals [, ] and [, ], 0 ≤ α < β ≤ γ < δ ≤ 1, when copula ,[ ]C   dominates over ,[ ]C    with 
respect to concordance order (the higher the copula values, the more intense the dependence), i.e.: 
 

2 2
[ , ] [ , ] 0 [ , ] 0 [ , ] 0[0,1] ( ) ( ) [0,1] ( ) ( ).p C p C p and p C p C p              (6) 

 
Referring once again to Figure 1 we simply compare conditional copula values for each value ‘stripe’ 

and therefore conclude that a higher copula value marks stronger interdependence. The above-defined 
non-linear contagion effect does not depend on marginal distribution. In addition, the copulas act inde-
pendently whenever we work on indexes, prices or logarithmic returns. The practical problem in this ap-
proach is that copula values must be verified at each point. 

Despite the very strong relation between comonotonic random variables, there will be no contagion ef-
fect, as their interdependence, although strong, remains stable. 

However, it is relatively difficult to estimate copula values, so it is more convenient to compare a copu-
la-dependent measure like Spearman rho or Kendall tau constructing thus a contagion index d: 
 

 d = ρ({xt : t ϵ A}) – ρ({xt : t ϵ B}) ,     (7) 
 
where  refers to the correlation coefficient (Spearman or  Kendall). 
 
Contagion versus divergence, the toy model 
Let Xt Yt and Zt be the daily changes of the yields of sovereign debts of three countries, respective-

ly X, Y and Z. X represents a country with a strong economy (a leading player), Y a country with 
economy shaken by a crisis and Z a ”safe haven”. Note that when the rating of a given country is 
decreasing then the prices of government securities are also decreasing but the yields are increasing. 
We present here a simplified ”toy” model which illustrates the interdependence between daily chang-
es of yields. We would like to capture two panic-driven effects. The first one is a standard one, due to 
some internal problems the global investors lose the confidence in bonds issued by “the leading play-
er” X and start selling them, which results in a decrease of the price and increase of yields. This causes 
an anxiety between the “the dependent player” Y bondholders and they start to close their positions 
although there are no direct “risks” as regards Y. 

The second one has a more complicated nature. The global investors lose confidence in “the 
local player” Y, close their positions in bonds issued by Y and move their wealth to the safe heav-
en Z. Thus, although there are no internal reasons, the prices of bonds issued by Z are going up 
and yields down. 

We consider a three factor model. The exogenous variables (risk factors) are εi, i = 1, 2, 3. For 
simplicity we assume that they are independent random variables with standard normal distri-

 is a copula of a conditional distribu-
tion of the pair X1 and X2, given that:

F1(X1) ϵ [α , β]. (5)

Following Durante and Jaworski (2010, p. 555) there is a non-linear conta-
gion – co-movement effect relative to intervals [α, β] and [γ, δ], 0 ≤ α < β ≤ γ < δ ≤ 1, 
when copula 

 

second is the copula that describes the dependence structure between the random variables (Nelsen,  
2006, p. 47), ( Embrechts,  2009, p. 639), ( Durante &  Sempi, 2010, p. 42), ( Kemp,  2011, p. 64), 
( Joe, 2014, p.7). The decomposition relies on Sklar’s theorem, which states that there exists n-
dimensional copula C such that: 

 
1 1 1( , , ) ( ( ), , ( )) .nn n nF x x C F x F x x R  K K      (4) 

 
If marginal cumulative distribution functions 1, , nF FK  are continuous then C is unique. 
According to Durante and Jaworski (2010, p. 555), ‘There is contagion from market X to market Y if the 

conditional copula of the market returns X and Y, when X is smaller than certain quantile, dominates the 
conditional copula when X is around its median.’ 

Let us assume the two-dimensional case. If C is a copula for the joint distribution for X1 and X2, then the 
copula ,[ ]C   is a copula of a conditional distribution of the pair X1 and X2, given that: 

 
F1(X1) ϵ [α , β].         (5) 
 
Following Durante and Jaworski (2010, p. 555) there is a non-linear contagion – co-movement effect 

relative to intervals [, ] and [, ], 0 ≤ α < β ≤ γ < δ ≤ 1, when copula ,[ ]C   dominates over ,[ ]C    with 
respect to concordance order (the higher the copula values, the more intense the dependence), i.e.: 
 

2 2
[ , ] [ , ] 0 [ , ] 0 [ , ] 0[0,1] ( ) ( ) [0,1] ( ) ( ).p C p C p and p C p C p              (6) 

 
Referring once again to Figure 1 we simply compare conditional copula values for each value ‘stripe’ 

and therefore conclude that a higher copula value marks stronger interdependence. The above-defined 
non-linear contagion effect does not depend on marginal distribution. In addition, the copulas act inde-
pendently whenever we work on indexes, prices or logarithmic returns. The practical problem in this ap-
proach is that copula values must be verified at each point. 

Despite the very strong relation between comonotonic random variables, there will be no contagion ef-
fect, as their interdependence, although strong, remains stable. 

However, it is relatively difficult to estimate copula values, so it is more convenient to compare a copu-
la-dependent measure like Spearman rho or Kendall tau constructing thus a contagion index d: 
 

 d = ρ({xt : t ϵ A}) – ρ({xt : t ϵ B}) ,     (7) 
 
where  refers to the correlation coefficient (Spearman or  Kendall). 
 
Contagion versus divergence, the toy model 
Let Xt Yt and Zt be the daily changes of the yields of sovereign debts of three countries, respective-

ly X, Y and Z. X represents a country with a strong economy (a leading player), Y a country with 
economy shaken by a crisis and Z a ”safe haven”. Note that when the rating of a given country is 
decreasing then the prices of government securities are also decreasing but the yields are increasing. 
We present here a simplified ”toy” model which illustrates the interdependence between daily chang-
es of yields. We would like to capture two panic-driven effects. The first one is a standard one, due to 
some internal problems the global investors lose the confidence in bonds issued by “the leading play-
er” X and start selling them, which results in a decrease of the price and increase of yields. This causes 
an anxiety between the “the dependent player” Y bondholders and they start to close their positions 
although there are no direct “risks” as regards Y. 

The second one has a more complicated nature. The global investors lose confidence in “the 
local player” Y, close their positions in bonds issued by Y and move their wealth to the safe heav-
en Z. Thus, although there are no internal reasons, the prices of bonds issued by Z are going up 
and yields down. 

We consider a three factor model. The exogenous variables (risk factors) are εi, i = 1, 2, 3. For 
simplicity we assume that they are independent random variables with standard normal distri-

 dominates over 

 

second is the copula that describes the dependence structure between the random variables (Nelsen,  
2006, p. 47), ( Embrechts,  2009, p. 639), ( Durante &  Sempi, 2010, p. 42), ( Kemp,  2011, p. 64), 
( Joe, 2014, p.7). The decomposition relies on Sklar’s theorem, which states that there exists n-
dimensional copula C such that: 

 
1 1 1( , , ) ( ( ), , ( )) .nn n nF x x C F x F x x R  K K      (4) 

 
If marginal cumulative distribution functions 1, , nF FK  are continuous then C is unique. 
According to Durante and Jaworski (2010, p. 555), ‘There is contagion from market X to market Y if the 

conditional copula of the market returns X and Y, when X is smaller than certain quantile, dominates the 
conditional copula when X is around its median.’ 

Let us assume the two-dimensional case. If C is a copula for the joint distribution for X1 and X2, then the 
copula ,[ ]C   is a copula of a conditional distribution of the pair X1 and X2, given that: 

 
F1(X1) ϵ [α , β].         (5) 
 
Following Durante and Jaworski (2010, p. 555) there is a non-linear contagion – co-movement effect 

relative to intervals [, ] and [, ], 0 ≤ α < β ≤ γ < δ ≤ 1, when copula ,[ ]C   dominates over ,[ ]C    with 
respect to concordance order (the higher the copula values, the more intense the dependence), i.e.: 
 

2 2
[ , ] [ , ] 0 [ , ] 0 [ , ] 0[0,1] ( ) ( ) [0,1] ( ) ( ).p C p C p and p C p C p              (6) 

 
Referring once again to Figure 1 we simply compare conditional copula values for each value ‘stripe’ 

and therefore conclude that a higher copula value marks stronger interdependence. The above-defined 
non-linear contagion effect does not depend on marginal distribution. In addition, the copulas act inde-
pendently whenever we work on indexes, prices or logarithmic returns. The practical problem in this ap-
proach is that copula values must be verified at each point. 

Despite the very strong relation between comonotonic random variables, there will be no contagion ef-
fect, as their interdependence, although strong, remains stable. 

However, it is relatively difficult to estimate copula values, so it is more convenient to compare a copu-
la-dependent measure like Spearman rho or Kendall tau constructing thus a contagion index d: 
 

 d = ρ({xt : t ϵ A}) – ρ({xt : t ϵ B}) ,     (7) 
 
where  refers to the correlation coefficient (Spearman or  Kendall). 
 
Contagion versus divergence, the toy model 
Let Xt Yt and Zt be the daily changes of the yields of sovereign debts of three countries, respective-

ly X, Y and Z. X represents a country with a strong economy (a leading player), Y a country with 
economy shaken by a crisis and Z a ”safe haven”. Note that when the rating of a given country is 
decreasing then the prices of government securities are also decreasing but the yields are increasing. 
We present here a simplified ”toy” model which illustrates the interdependence between daily chang-
es of yields. We would like to capture two panic-driven effects. The first one is a standard one, due to 
some internal problems the global investors lose the confidence in bonds issued by “the leading play-
er” X and start selling them, which results in a decrease of the price and increase of yields. This causes 
an anxiety between the “the dependent player” Y bondholders and they start to close their positions 
although there are no direct “risks” as regards Y. 

The second one has a more complicated nature. The global investors lose confidence in “the 
local player” Y, close their positions in bonds issued by Y and move their wealth to the safe heav-
en Z. Thus, although there are no internal reasons, the prices of bonds issued by Z are going up 
and yields down. 

We consider a three factor model. The exogenous variables (risk factors) are εi, i = 1, 2, 3. For 
simplicity we assume that they are independent random variables with standard normal distri-

 with respect to concordance order (the 
higher the copula values, the more intense the dependence), i.e.:

 

second is the copula that describes the dependence structure between the random variables (Nelsen,  
2006, p. 47), ( Embrechts,  2009, p. 639), ( Durante &  Sempi, 2010, p. 42), ( Kemp,  2011, p. 64), 
( Joe, 2014, p.7). The decomposition relies on Sklar’s theorem, which states that there exists n-
dimensional copula C such that: 

 
1 1 1( , , ) ( ( ), , ( )) .nn n nF x x C F x F x x R  K K      (4) 

 
If marginal cumulative distribution functions 1, , nF FK  are continuous then C is unique. 
According to Durante and Jaworski (2010, p. 555), ‘There is contagion from market X to market Y if the 

conditional copula of the market returns X and Y, when X is smaller than certain quantile, dominates the 
conditional copula when X is around its median.’ 

Let us assume the two-dimensional case. If C is a copula for the joint distribution for X1 and X2, then the 
copula ,[ ]C   is a copula of a conditional distribution of the pair X1 and X2, given that: 

 
F1(X1) ϵ [α , β].         (5) 
 
Following Durante and Jaworski (2010, p. 555) there is a non-linear contagion – co-movement effect 

relative to intervals [, ] and [, ], 0 ≤ α < β ≤ γ < δ ≤ 1, when copula ,[ ]C   dominates over ,[ ]C    with 
respect to concordance order (the higher the copula values, the more intense the dependence), i.e.: 
 

2 2
[ , ] [ , ] 0 [ , ] 0 [ , ] 0[0,1] ( ) ( ) [0,1] ( ) ( ).p C p C p and p C p C p              (6) 

 
Referring once again to Figure 1 we simply compare conditional copula values for each value ‘stripe’ 

and therefore conclude that a higher copula value marks stronger interdependence. The above-defined 
non-linear contagion effect does not depend on marginal distribution. In addition, the copulas act inde-
pendently whenever we work on indexes, prices or logarithmic returns. The practical problem in this ap-
proach is that copula values must be verified at each point. 

Despite the very strong relation between comonotonic random variables, there will be no contagion ef-
fect, as their interdependence, although strong, remains stable. 

However, it is relatively difficult to estimate copula values, so it is more convenient to compare a copu-
la-dependent measure like Spearman rho or Kendall tau constructing thus a contagion index d: 
 

 d = ρ({xt : t ϵ A}) – ρ({xt : t ϵ B}) ,     (7) 
 
where  refers to the correlation coefficient (Spearman or  Kendall). 
 
Contagion versus divergence, the toy model 
Let Xt Yt and Zt be the daily changes of the yields of sovereign debts of three countries, respective-

ly X, Y and Z. X represents a country with a strong economy (a leading player), Y a country with 
economy shaken by a crisis and Z a ”safe haven”. Note that when the rating of a given country is 
decreasing then the prices of government securities are also decreasing but the yields are increasing. 
We present here a simplified ”toy” model which illustrates the interdependence between daily chang-
es of yields. We would like to capture two panic-driven effects. The first one is a standard one, due to 
some internal problems the global investors lose the confidence in bonds issued by “the leading play-
er” X and start selling them, which results in a decrease of the price and increase of yields. This causes 
an anxiety between the “the dependent player” Y bondholders and they start to close their positions 
although there are no direct “risks” as regards Y. 

The second one has a more complicated nature. The global investors lose confidence in “the 
local player” Y, close their positions in bonds issued by Y and move their wealth to the safe heav-
en Z. Thus, although there are no internal reasons, the prices of bonds issued by Z are going up 
and yields down. 

We consider a three factor model. The exogenous variables (risk factors) are εi, i = 1, 2, 3. For 
simplicity we assume that they are independent random variables with standard normal distri-

 (6)

Referring once again to Figure 1 we simply compare conditional copula val-
ues for each value ‘stripe’ and therefore conclude that a higher copula value 
marks stronger interdependence. The above-defined non-linear contagion ef-
fect does not depend on marginal distribution. In addition, the copulas act in-
dependently whenever we work on indexes, prices or logarithmic returns. The 
practical problem in this approach is that copula values must be verified at each 
point.

Despite the very strong relation between comonotonic random variables, 
there will be no contagion effect, as their interdependence, although strong, 
remains stable.

However, it is relatively difficult to estimate copula values, so it is more con-
venient to compare a copula-dependent measure like Spearman rho or Kendall 
tau constructing thus a contagion index d:

 d = ρ({xt : t ϵ A}) – ρ({xt : t ϵ B}),  (7)

where ρ refers to the correlation coefficient (Spearman or τ Kendall).

Contagion versus divergence, the toy model

Let Xt Yt and Zt be the daily changes of the yields of sovereign debts of three 
countries, respectively X, Y and Z. X represents a country with a strong 
economy (a leading player), Y a country with economy shaken by a cri-
sis and Z a ”safe haven”. Note that when the rating of a given country is de-



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki48

creasing then the prices of government securities are also decreasing but 
the yields are increasing. We present here a simplified ”toy” model which 
illustrates the interdependence between daily changes of yields. We would 
like to capture two panic-driven effects. The first one is a standard one, due 
to some internal problems the global investors lose the confidence in bonds 
issued by “the leading player” X and start selling them, which results in a de-
crease of the price and increase of yields. This causes an anxiety between 
the “the dependent player” Y bondholders and they start to close their posi-
tions although there are no direct “risks” as regards Y.

The second one has a more complicated nature. The global investors 
lose confidence in “the local player” Y, close their positions in bonds is-
sued by Y and move their wealth to the safe heaven Z. Thus, although 
there are no internal reasons, the prices of bonds issued by Z are going 
up and yields down.

We consider a three factor model. The exogenous variables (risk fac-
tors) are εi, i = 1, 2, 3. For simplicity we assume that they are independ-
ent random variables with standard normal distribution N (0, 1). We put:

 

bution N (0, 1). We put: 
�� � �����, 

�� � ����� � ������ − 0.842��, 
�� � ����� − ������ − 0.842�� 

            (8) 
where σX, σY , σZ, δ1 and δ2 are positive constants. The strike 0.842 is selected as a 80% quantile 

of the standard normal distribution N (0, 1). 
Note, that under the condition Xt < 0.841 · σX the daily changes of yields Xt and Yt are inde-

pendent, while under the condition Xt > 0,841 · σX the daily changes Xt and Yt are positively de-
pendent. 

Similarly, under the condition ��� < 0.841 · σX the daily changes Yt and Zt are independent, 
while under the condition ��� > 0.841 · σX the daily changes Yt and Zt are negatively dependent. 

Hence our toy model is capturing the contagion between X and Y, and divergence between Y 
and Z. 

Basing on the toy model, the figure 2 and figure 3 present simulated scatter plots from the bivariate 
margins of the trivariate copula of (X,Y,Z). For simulation we assumed the values for σX= σY = σZ = 0,1, 
whereas δ1=0,3 and δ2=0,4. 

Please note, that the concentration of the points at the upper right corner of the first scatter-
plot (see figure 2) indicates that the dependence between extreme value is bigger than the one 
between moderate values. Similar effect can be seen on the scatterplot in Figure 3. The differ-
ence is that in the former case the extra dependence is between the big values of X and Y (conta-
gion effect) while in the latter the extra dependence is for the big values of Y and small values of 
Z. 

 
Contagion and divergence at the bond market 
The formal definition of contagion and divergence depends whether we deal with bond prices 

or yields. In the first case contagious is the drop of the prices hence we compare the lower tail 
and median part. In the second description contagious is the increase of yields hence we compare 
the upper tail and median. We will follow the approach presented in Durante and Jaworski 
(2010, p. 551) and Jaworski and Pitera (2014, p. 303) adjusted to yield description. Instead of low-
er tail we consider the upper tail of the distribution. We restate the definition. 

Let Xt and Yt be the daily changes of yield-based indices of markets X and Y. We assume that in a given 
period of time they are stationary. 

Let [ ],a bC  denote the copula of the conditional distribution of ( ),t tX Y  under the condition 
( ) [ , ]X tF X a b . 
Definition 1. 
We say that there is contagion from market X to market Y with respect to intervals [1,1] and [1, 2], if 

the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the first one dominates in concord-
ance ordering the second one, i.e.:  

 

       (9) 
 
We say that there is divergence between market X and market Y with respect to intervals [1,1] and [1, 

2], if the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the second one dominates the 
first one, i.e.: 

 

      (10) 
 
The comparison of copulas in concordance ordering, when we have only their estimates based on em-

pirical data, is rather challenging. Therefore we switch to concordance measures. We choose Spearman . 
Furthermore we reduce the number of parameters. Since the distributions of daily changes are (nearly) 
symmetric in the median part, we introduce a new parameter   (0.5, 1) and put 2= and 1=1- . Next, to 
avoid the loss of empirical data or the counting of some of them twice, we put =1-1= . This leads to the 
following definition of a contagion index: 

 
1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ),u v C u v C u v    

1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ).u v C u v C u v    

 (8)

where σX, σY , σZ, δ1 and δ2 are positive constants. The strike 0.842 is se-
lected as a 80% quantile of the standard normal distribution N (0, 1).

Note, that under the condition Xt < 0.841 · σX the daily changes of 
yields Xt and Yt are independent, while under the condition Xt > 0.841 · σX 

the daily changes Xt and Yt are positively dependent.
Similarly, under the condition 

 

bution N (0, 1). We put: 
�� � �����, 

�� � ����� � ������ − 0.842��, 
�� � ����� − ������ − 0.842�� 

            (8) 
where σX, σY , σZ, δ1 and δ2 are positive constants. The strike 0.842 is selected as a 80% quantile 

of the standard normal distribution N (0, 1). 
Note, that under the condition Xt < 0.841 · σX the daily changes of yields Xt and Yt are inde-

pendent, while under the condition Xt > 0,841 · σX the daily changes Xt and Yt are positively de-
pendent. 

Similarly, under the condition ��� < 0.841 · σX the daily changes Yt and Zt are independent, 
while under the condition ��� > 0.841 · σX the daily changes Yt and Zt are negatively dependent. 

Hence our toy model is capturing the contagion between X and Y, and divergence between Y 
and Z. 

Basing on the toy model, the figure 2 and figure 3 present simulated scatter plots from the bivariate 
margins of the trivariate copula of (X,Y,Z). For simulation we assumed the values for σX= σY = σZ = 0,1, 
whereas δ1=0,3 and δ2=0,4. 

Please note, that the concentration of the points at the upper right corner of the first scatter-
plot (see figure 2) indicates that the dependence between extreme value is bigger than the one 
between moderate values. Similar effect can be seen on the scatterplot in Figure 3. The differ-
ence is that in the former case the extra dependence is between the big values of X and Y (conta-
gion effect) while in the latter the extra dependence is for the big values of Y and small values of 
Z. 

 
Contagion and divergence at the bond market 
The formal definition of contagion and divergence depends whether we deal with bond prices 

or yields. In the first case contagious is the drop of the prices hence we compare the lower tail 
and median part. In the second description contagious is the increase of yields hence we compare 
the upper tail and median. We will follow the approach presented in Durante and Jaworski 
(2010, p. 551) and Jaworski and Pitera (2014, p. 303) adjusted to yield description. Instead of low-
er tail we consider the upper tail of the distribution. We restate the definition. 

Let Xt and Yt be the daily changes of yield-based indices of markets X and Y. We assume that in a given 
period of time they are stationary. 

Let [ ],a bC  denote the copula of the conditional distribution of ( ),t tX Y  under the condition 
( ) [ , ]X tF X a b . 
Definition 1. 
We say that there is contagion from market X to market Y with respect to intervals [1,1] and [1, 2], if 

the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the first one dominates in concord-
ance ordering the second one, i.e.:  

 

       (9) 
 
We say that there is divergence between market X and market Y with respect to intervals [1,1] and [1, 

2], if the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the second one dominates the 
first one, i.e.: 

 

      (10) 
 
The comparison of copulas in concordance ordering, when we have only their estimates based on em-

pirical data, is rather challenging. Therefore we switch to concordance measures. We choose Spearman . 
Furthermore we reduce the number of parameters. Since the distributions of daily changes are (nearly) 
symmetric in the median part, we introduce a new parameter   (0.5, 1) and put 2= and 1=1- . Next, to 
avoid the loss of empirical data or the counting of some of them twice, we put =1-1= . This leads to the 
following definition of a contagion index: 

 
1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ),u v C u v C u v    

1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ).u v C u v C u v    

 < 0.841 · σX the daily changes Yt and 
Zt are independent, while under the condition 

 

bution N (0, 1). We put: 
�� � �����, 

�� � ����� � ������ − 0.842��, 
�� � ����� − ������ − 0.842�� 

            (8) 
where σX, σY , σZ, δ1 and δ2 are positive constants. The strike 0.842 is selected as a 80% quantile 

of the standard normal distribution N (0, 1). 
Note, that under the condition Xt < 0.841 · σX the daily changes of yields Xt and Yt are inde-

pendent, while under the condition Xt > 0,841 · σX the daily changes Xt and Yt are positively de-
pendent. 

Similarly, under the condition ��� < 0.841 · σX the daily changes Yt and Zt are independent, 
while under the condition ��� > 0.841 · σX the daily changes Yt and Zt are negatively dependent. 

Hence our toy model is capturing the contagion between X and Y, and divergence between Y 
and Z. 

Basing on the toy model, the figure 2 and figure 3 present simulated scatter plots from the bivariate 
margins of the trivariate copula of (X,Y,Z). For simulation we assumed the values for σX= σY = σZ = 0,1, 
whereas δ1=0,3 and δ2=0,4. 

Please note, that the concentration of the points at the upper right corner of the first scatter-
plot (see figure 2) indicates that the dependence between extreme value is bigger than the one 
between moderate values. Similar effect can be seen on the scatterplot in Figure 3. The differ-
ence is that in the former case the extra dependence is between the big values of X and Y (conta-
gion effect) while in the latter the extra dependence is for the big values of Y and small values of 
Z. 

 
Contagion and divergence at the bond market 
The formal definition of contagion and divergence depends whether we deal with bond prices 

or yields. In the first case contagious is the drop of the prices hence we compare the lower tail 
and median part. In the second description contagious is the increase of yields hence we compare 
the upper tail and median. We will follow the approach presented in Durante and Jaworski 
(2010, p. 551) and Jaworski and Pitera (2014, p. 303) adjusted to yield description. Instead of low-
er tail we consider the upper tail of the distribution. We restate the definition. 

Let Xt and Yt be the daily changes of yield-based indices of markets X and Y. We assume that in a given 
period of time they are stationary. 

Let [ ],a bC  denote the copula of the conditional distribution of ( ),t tX Y  under the condition 
( ) [ , ]X tF X a b . 
Definition 1. 
We say that there is contagion from market X to market Y with respect to intervals [1,1] and [1, 2], if 

the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the first one dominates in concord-
ance ordering the second one, i.e.:  

 

       (9) 
 
We say that there is divergence between market X and market Y with respect to intervals [1,1] and [1, 

2], if the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the second one dominates the 
first one, i.e.: 

 

      (10) 
 
The comparison of copulas in concordance ordering, when we have only their estimates based on em-

pirical data, is rather challenging. Therefore we switch to concordance measures. We choose Spearman . 
Furthermore we reduce the number of parameters. Since the distributions of daily changes are (nearly) 
symmetric in the median part, we introduce a new parameter   (0.5, 1) and put 2= and 1=1- . Next, to 
avoid the loss of empirical data or the counting of some of them twice, we put =1-1= . This leads to the 
following definition of a contagion index: 

 
1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ),u v C u v C u v    

1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ).u v C u v C u v    

 > 0.841 · σX the daily 
changes Yt and Zt are negatively dependent.

Hence our toy model is capturing the contagion between X and Y, and 
divergence between Y and Z.

Basing on the toy model, the figure 2 and figure 3 present simulated scatter 
plots from the bivariate margins of the trivariate copula of (X, Y, Z). For simula-
tion we assumed the values for σX= σY = σZ = 0.1, whereas δ1=0.3 and δ2=0.4.



 ConTagion and divergence on sovereign bond markeTs 49

Please note, that the concentration of the points at the upper right 
corner of the first scatterplot (see figure 2) indicates that the depend-
ence between extreme value is bigger than the one between moderate 
values. Similar effect can be seen on the scatterplot in figure 3. The dif-
ference is that in the former case the extra dependence is between the 
big values of X and Y (contagion effect) while in the latter the extra de-
pendence is for the big values of Y and small values of Z.

Contagion and divergence at the bond market

The formal definition of contagion and divergence depends whether we deal 
with bond prices or yields. In the first case contagious is the drop of the pric-
es hence we compare the lower tail and median part. In the second descrip-
tion contagious is the increase of yields hence we compare the upper tail 
and median. We will follow the approach presented in Durante and Jaworski 
(2010, p. 551) and Jaworski and Pitera (2014, p. 303) adjusted to yield de-
scription. Instead of lower tail we consider the upper tail of the distribution. 
We restate the definition.

Let Xt and Yt be the daily changes of yield-based indices of markets X and Y. 
We assume that in a given period of time they are stationary.

Let C[a,b] denote the copula of the conditional distribution of (Xt, Yt) under the 
condition  

 

bution N (0, 1). We put: 
�� � �����, 

�� � ����� � ������ − 0.842��, 
�� � ����� − ������ − 0.842�� 

            (8) 
where σX, σY , σZ, δ1 and δ2 are positive constants. The strike 0.842 is selected as a 80% quantile 

of the standard normal distribution N (0, 1). 
Note, that under the condition Xt < 0.841 · σX the daily changes of yields Xt and Yt are inde-

pendent, while under the condition Xt > 0,841 · σX the daily changes Xt and Yt are positively de-
pendent. 

Similarly, under the condition ��� < 0.841 · σX the daily changes Yt and Zt are independent, 
while under the condition ��� > 0.841 · σX the daily changes Yt and Zt are negatively dependent. 

Hence our toy model is capturing the contagion between X and Y, and divergence between Y 
and Z. 

Basing on the toy model, the figure 2 and figure 3 present simulated scatter plots from the bivariate 
margins of the trivariate copula of (X,Y,Z). For simulation we assumed the values for σX= σY = σZ = 0,1, 
whereas δ1=0,3 and δ2=0,4. 

Please note, that the concentration of the points at the upper right corner of the first scatter-
plot (see figure 2) indicates that the dependence between extreme value is bigger than the one 
between moderate values. Similar effect can be seen on the scatterplot in Figure 3. The differ-
ence is that in the former case the extra dependence is between the big values of X and Y (conta-
gion effect) while in the latter the extra dependence is for the big values of Y and small values of 
Z. 

 
Contagion and divergence at the bond market 
The formal definition of contagion and divergence depends whether we deal with bond prices 

or yields. In the first case contagious is the drop of the prices hence we compare the lower tail 
and median part. In the second description contagious is the increase of yields hence we compare 
the upper tail and median. We will follow the approach presented in Durante and Jaworski 
(2010, p. 551) and Jaworski and Pitera (2014, p. 303) adjusted to yield description. Instead of low-
er tail we consider the upper tail of the distribution. We restate the definition. 

Let Xt and Yt be the daily changes of yield-based indices of markets X and Y. We assume that in a given 
period of time they are stationary. 

Let [ ],a bC  denote the copula of the conditional distribution of ( ),t tX Y  under the condition 
( ) [ , ]X tF X a b . 
Definition 1. 
We say that there is contagion from market X to market Y with respect to intervals [1,1] and [1, 2], if 

the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the first one dominates in concord-
ance ordering the second one, i.e.:  

 

       (9) 
 
We say that there is divergence between market X and market Y with respect to intervals [1,1] and [1, 

2], if the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the second one dominates the 
first one, i.e.: 

 

      (10) 
 
The comparison of copulas in concordance ordering, when we have only their estimates based on em-

pirical data, is rather challenging. Therefore we switch to concordance measures. We choose Spearman . 
Furthermore we reduce the number of parameters. Since the distributions of daily changes are (nearly) 
symmetric in the median part, we introduce a new parameter   (0.5, 1) and put 2= and 1=1- . Next, to 
avoid the loss of empirical data or the counting of some of them twice, we put =1-1= . This leads to the 
following definition of a contagion index: 

 
1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ),u v C u v C u v    

1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ).u v C u v C u v    

Definition 1.
We say that there is contagion from market X to market Y with respect to inter-

vals [1−α, 1] and [β1, β2], if the conditional copulas C[1−α,1] and C[β1, β2] differ at least at 
one point, and the first one dominates in concordance ordering the second one, i.e.: 

 

bution N (0, 1). We put: 
�� � �����, 

�� � ����� � ������ − 0.842��, 
�� � ����� − ������ − 0.842�� 

            (8) 
where σX, σY , σZ, δ1 and δ2 are positive constants. The strike 0.842 is selected as a 80% quantile 

of the standard normal distribution N (0, 1). 
Note, that under the condition Xt < 0.841 · σX the daily changes of yields Xt and Yt are inde-

pendent, while under the condition Xt > 0,841 · σX the daily changes Xt and Yt are positively de-
pendent. 

Similarly, under the condition ��� < 0.841 · σX the daily changes Yt and Zt are independent, 
while under the condition ��� > 0.841 · σX the daily changes Yt and Zt are negatively dependent. 

Hence our toy model is capturing the contagion between X and Y, and divergence between Y 
and Z. 

Basing on the toy model, the figure 2 and figure 3 present simulated scatter plots from the bivariate 
margins of the trivariate copula of (X,Y,Z). For simulation we assumed the values for σX= σY = σZ = 0,1, 
whereas δ1=0,3 and δ2=0,4. 

Please note, that the concentration of the points at the upper right corner of the first scatter-
plot (see figure 2) indicates that the dependence between extreme value is bigger than the one 
between moderate values. Similar effect can be seen on the scatterplot in Figure 3. The differ-
ence is that in the former case the extra dependence is between the big values of X and Y (conta-
gion effect) while in the latter the extra dependence is for the big values of Y and small values of 
Z. 

 
Contagion and divergence at the bond market 
The formal definition of contagion and divergence depends whether we deal with bond prices 

or yields. In the first case contagious is the drop of the prices hence we compare the lower tail 
and median part. In the second description contagious is the increase of yields hence we compare 
the upper tail and median. We will follow the approach presented in Durante and Jaworski 
(2010, p. 551) and Jaworski and Pitera (2014, p. 303) adjusted to yield description. Instead of low-
er tail we consider the upper tail of the distribution. We restate the definition. 

Let Xt and Yt be the daily changes of yield-based indices of markets X and Y. We assume that in a given 
period of time they are stationary. 

Let [ ],a bC  denote the copula of the conditional distribution of ( ),t tX Y  under the condition 
( ) [ , ]X tF X a b . 
Definition 1. 
We say that there is contagion from market X to market Y with respect to intervals [1,1] and [1, 2], if 

the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the first one dominates in concord-
ance ordering the second one, i.e.:  

 

       (9) 
 
We say that there is divergence between market X and market Y with respect to intervals [1,1] and [1, 

2], if the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the second one dominates the 
first one, i.e.: 

 

      (10) 
 
The comparison of copulas in concordance ordering, when we have only their estimates based on em-

pirical data, is rather challenging. Therefore we switch to concordance measures. We choose Spearman . 
Furthermore we reduce the number of parameters. Since the distributions of daily changes are (nearly) 
symmetric in the median part, we introduce a new parameter   (0.5, 1) and put 2= and 1=1- . Next, to 
avoid the loss of empirical data or the counting of some of them twice, we put =1-1= . This leads to the 
following definition of a contagion index: 

 
1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ),u v C u v C u v    

1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ).u v C u v C u v    

. (9)

We say that there is divergence between market X and market Y with respect 
to intervals [1−α,1] and [β1, β2], if the conditional copulas C[1−α,1] and C[β1,β2] differ at 
least at one point, and the second one dominates the first one, i.e.:

 

bution N (0, 1). We put: 
�� � �����, 

�� � ����� � ������ − 0.842��, 
�� � ����� − ������ − 0.842�� 

            (8) 
where σX, σY , σZ, δ1 and δ2 are positive constants. The strike 0.842 is selected as a 80% quantile 

of the standard normal distribution N (0, 1). 
Note, that under the condition Xt < 0.841 · σX the daily changes of yields Xt and Yt are inde-

pendent, while under the condition Xt > 0,841 · σX the daily changes Xt and Yt are positively de-
pendent. 

Similarly, under the condition ��� < 0.841 · σX the daily changes Yt and Zt are independent, 
while under the condition ��� > 0.841 · σX the daily changes Yt and Zt are negatively dependent. 

Hence our toy model is capturing the contagion between X and Y, and divergence between Y 
and Z. 

Basing on the toy model, the figure 2 and figure 3 present simulated scatter plots from the bivariate 
margins of the trivariate copula of (X,Y,Z). For simulation we assumed the values for σX= σY = σZ = 0,1, 
whereas δ1=0,3 and δ2=0,4. 

Please note, that the concentration of the points at the upper right corner of the first scatter-
plot (see figure 2) indicates that the dependence between extreme value is bigger than the one 
between moderate values. Similar effect can be seen on the scatterplot in Figure 3. The differ-
ence is that in the former case the extra dependence is between the big values of X and Y (conta-
gion effect) while in the latter the extra dependence is for the big values of Y and small values of 
Z. 

 
Contagion and divergence at the bond market 
The formal definition of contagion and divergence depends whether we deal with bond prices 

or yields. In the first case contagious is the drop of the prices hence we compare the lower tail 
and median part. In the second description contagious is the increase of yields hence we compare 
the upper tail and median. We will follow the approach presented in Durante and Jaworski 
(2010, p. 551) and Jaworski and Pitera (2014, p. 303) adjusted to yield description. Instead of low-
er tail we consider the upper tail of the distribution. We restate the definition. 

Let Xt and Yt be the daily changes of yield-based indices of markets X and Y. We assume that in a given 
period of time they are stationary. 

Let [ ],a bC  denote the copula of the conditional distribution of ( ),t tX Y  under the condition 
( ) [ , ]X tF X a b . 
Definition 1. 
We say that there is contagion from market X to market Y with respect to intervals [1,1] and [1, 2], if 

the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the first one dominates in concord-
ance ordering the second one, i.e.:  

 

       (9) 
 
We say that there is divergence between market X and market Y with respect to intervals [1,1] and [1, 

2], if the conditional copulas C[1,1] and C[1,2] differ at least at one point, and the second one dominates the 
first one, i.e.: 

 

      (10) 
 
The comparison of copulas in concordance ordering, when we have only their estimates based on em-

pirical data, is rather challenging. Therefore we switch to concordance measures. We choose Spearman . 
Furthermore we reduce the number of parameters. Since the distributions of daily changes are (nearly) 
symmetric in the median part, we introduce a new parameter   (0.5, 1) and put 2= and 1=1- . Next, to 
avoid the loss of empirical data or the counting of some of them twice, we put =1-1= . This leads to the 
following definition of a contagion index: 

 
1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ),u v C u v C u v    

1 2

2
[1 ,1] [ , ]( , ) [0,1] ( , ) ( , ).u v C u v C u v      (10)



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki50

The comparison of copulas in concordance ordering, when we have only their 
estimates based on empirical data, is rather challenging. Therefore we switch 
to concordance measures. We choose Spearman ρ. Furthermore we reduce 
the number of parameters. Since the distributions of daily changes are (near-
ly) symmetric in the median part, we introduce a new parameter β ∈ (0.5, 1) 
and put β2 = β and β1= 1 – β. Next, to avoid the loss of empirical data or the count-
ing of some of them twice, we put α = 1-β1 = β . This leads to the following defini-
tion of a contagion index:

Definition 2.
Let C be a copula and β ∈ (0.5, 1).
Δβ(C)= ρ(C[β,1]) – ρ(C[1-β,β]), (11)

where ρ denotes the Spearman rank correlation.

The choice of β is a more subtle problem. To avoid possible random fluctua-
tions, in Durante, Foscolo, Jaworski and Wang (2014, p. 4025; 2015, p. 219), 
it was proposed to measure the set of β s for which the index is positive. How-
ever, it is recommended to follow the approach proposed in Jaworski and Pitera 
(2014, p. 308), based on the paradigm that there is no contagion for Gaussian 
distributions, namely, for Gaussian copula ∆β is changing sign for 

 

Definition 2. 
Let C be a copula and   (0.5, 1). 
Δβ(C)= ρ(C[β,1]) – ρ(C[1-β,β]),        (11) 
 
where  denotes the Spearman rank correlation. 
The choice of  is a more subtle problem. To avoid possible random fluctuations, in Durante, Foscolo, 

Jaworski and Wang (2014, p. 4025; 2015, p. 219), it was proposed to measure the set of  s for which the 
index is positive. However, it is recommended to follow the approach proposed in Jaworski and Pitera 
(2014, p. 308), based on the paradigm that there is no contagion for Gaussian distributions, namely, for 
Gaussian copula ∆β is changing sign for 1 0.21   (compare Jaworski and Pitera (2014, p. 308)). It is 
positive for a smaller median part and negative for bigger. Thus we selected 0 .8  . To reduce the 
impact of the fluctuations, due to the random nature of our data, we apply the triangular average of empir-
ical estimates: 

 

’        (12) 
where 
∆̂β = ρ̂[β ,1] − ρ̂[1−β ,β]. 
            (13) 
Here ρ̂[β1 ,β2] denotes the Spearman rank correlation of truncated empirical data. 
 
Note that both the contagion index and its empirical estimate do not depend on marginal distribution 

of daily changes of bonds yields. They extract “the pure dependence”. The positive values indicate that the 
co-movements are stronger and more often, when the price of the first bond drops significantly. The nega-
tive values indicate that the movements in opposite direction are stronger and more often, when the price 
of the first bond drops significantly. In the first case we observe contagion, in the second divergence. 

 
Empirical analysis 
We took under the consideration Thomson/Reuters yield indices for bonds of different maturities is-

sued in USA, Germany, Japan, Australia, Poland, Italy and Portugal for the period of time from 2004-11-19 
to 2014-11-17. We compare the market behavior before and after the change of regime implied by the 
financial crisis. As a breaking point we select 2009-12-31. The choice is affected by empirical analysis of 
Eurozone bonds, where at the end of 2009 the final break is observed (see figure 4 for quotations refer-
ring to Germany and Portugal). 

In tables 1 and 2 we present values of contagion index ∆ for years 2004–2009 and 2010–2014, calcu-
lated for the daily changes of the Thomson/Reuters yield indices. Contagion or divergence goes from the 
indices in left column to indices in the first row. Note that after the change of regime, the values are quite 
different. On Figures from 5 to 16 we show the visualization of the data given in tables.  

The bar charts (figures 5 to 16) illustrate the results presented in tables 1 and 2. We may conclude, 
that bars going upward illustrate contagion effect, while bars directed downwards reflect the divergence. 
Basing on our sample, we can note that the divergence effect prevails over contagion. It mirrors the fact, 
there was no worldwide retreat from bond markets as a result of crisis. The threat of sovereign insolvency 
during the crises was limited only to selected countries. The sample includes safe haven countries (Ger-
many, US, Australia), emerging markets (Poland), troubled ones (Portugal, Italy) and Japan. As it was al-
ready mentioned, the sample has been selected heuristically. As our methodology complies with the port-
folio theory, we do not concentrate on fundamental explanation for the obtained results, but rather try to 
capture statistical relationship between the time series using defined measures. Generally speaking, there 
was some contagion before the crisis, although divergence prevailed. After outbreak of the crisis, we can 
see increase of contagion. Graphic comparison between pre-crisis and post-crisis observation may be seen 
at the diagrams (figure 17 and figure 18). We can see the rise of shock transmission referring to Far East 
Asia sovereign bonds (Japan, Australia) after the outbreak of the crisis. The most characteristic case out of 
our sample seems to be Japan. Before the crisis there was negative interdependency between Japanese 
and other countries’ sovereign bonds. At present, among our sample Japan produces more danger for an 
international bond markets in terms of contagion than Italy or Germany. Basing on our countries selec-
tion, we found out that there is more divergence among the EMU countries than between the EMU sover-
eigns and Japanese or the US bonds. The analysis gives us also an insight into the term structure of interest 
rates reaction following shock. We used the bonds representative for the short end-, middle section- and 
long end of each government yield curve. Before the crisis contagion effect affected predominantly long 

 
(compare Jaworski and Pitera (2014, p. 308)). It is positive for a smaller median 
part and negative for bigger. Thus we selected β = 0.8. To reduce the impact of 
the fluctuations, due to the random nature of our data, we apply the triangular 
average of empirical estimates:

 

Definition 2. 
Let C be a copula and   (0.5, 1). 
Δβ(C)= ρ(C[β,1]) – ρ(C[1-β,β]),        (11) 
 
where  denotes the Spearman rank correlation. 
The choice of  is a more subtle problem. To avoid possible random fluctuations, in Durante, Foscolo, 

Jaworski and Wang (2014, p. 4025; 2015, p. 219), it was proposed to measure the set of  s for which the 
index is positive. However, it is recommended to follow the approach proposed in Jaworski and Pitera 
(2014, p. 308), based on the paradigm that there is no contagion for Gaussian distributions, namely, for 
Gaussian copula ∆β is changing sign for 1 0.21   (compare Jaworski and Pitera (2014, p. 308)). It is 
positive for a smaller median part and negative for bigger. Thus we selected 0 .8  . To reduce the 
impact of the fluctuations, due to the random nature of our data, we apply the triangular average of empir-
ical estimates: 

 

’        (12) 
where 
∆̂β = ρ̂[β ,1] − ρ̂[1−β ,β]. 
            (13) 
Here ρ̂[β1 ,β2] denotes the Spearman rank correlation of truncated empirical data. 
 
Note that both the contagion index and its empirical estimate do not depend on marginal distribution 

of daily changes of bonds yields. They extract “the pure dependence”. The positive values indicate that the 
co-movements are stronger and more often, when the price of the first bond drops significantly. The nega-
tive values indicate that the movements in opposite direction are stronger and more often, when the price 
of the first bond drops significantly. In the first case we observe contagion, in the second divergence. 

 
Empirical analysis 
We took under the consideration Thomson/Reuters yield indices for bonds of different maturities is-

sued in USA, Germany, Japan, Australia, Poland, Italy and Portugal for the period of time from 2004-11-19 
to 2014-11-17. We compare the market behavior before and after the change of regime implied by the 
financial crisis. As a breaking point we select 2009-12-31. The choice is affected by empirical analysis of 
Eurozone bonds, where at the end of 2009 the final break is observed (see figure 4 for quotations refer-
ring to Germany and Portugal). 

In tables 1 and 2 we present values of contagion index ∆ for years 2004–2009 and 2010–2014, calcu-
lated for the daily changes of the Thomson/Reuters yield indices. Contagion or divergence goes from the 
indices in left column to indices in the first row. Note that after the change of regime, the values are quite 
different. On Figures from 5 to 16 we show the visualization of the data given in tables.  

The bar charts (figures 5 to 16) illustrate the results presented in tables 1 and 2. We may conclude, 
that bars going upward illustrate contagion effect, while bars directed downwards reflect the divergence. 
Basing on our sample, we can note that the divergence effect prevails over contagion. It mirrors the fact, 
there was no worldwide retreat from bond markets as a result of crisis. The threat of sovereign insolvency 
during the crises was limited only to selected countries. The sample includes safe haven countries (Ger-
many, US, Australia), emerging markets (Poland), troubled ones (Portugal, Italy) and Japan. As it was al-
ready mentioned, the sample has been selected heuristically. As our methodology complies with the port-
folio theory, we do not concentrate on fundamental explanation for the obtained results, but rather try to 
capture statistical relationship between the time series using defined measures. Generally speaking, there 
was some contagion before the crisis, although divergence prevailed. After outbreak of the crisis, we can 
see increase of contagion. Graphic comparison between pre-crisis and post-crisis observation may be seen 
at the diagrams (figure 17 and figure 18). We can see the rise of shock transmission referring to Far East 
Asia sovereign bonds (Japan, Australia) after the outbreak of the crisis. The most characteristic case out of 
our sample seems to be Japan. Before the crisis there was negative interdependency between Japanese 
and other countries’ sovereign bonds. At present, among our sample Japan produces more danger for an 
international bond markets in terms of contagion than Italy or Germany. Basing on our countries selec-
tion, we found out that there is more divergence among the EMU countries than between the EMU sover-
eigns and Japanese or the US bonds. The analysis gives us also an insight into the term structure of interest 
rates reaction following shock. We used the bonds representative for the short end-, middle section- and 
long end of each government yield curve. Before the crisis contagion effect affected predominantly long 

 (12)

where

 

Definition 2. 
Let C be a copula and   (0.5, 1). 
Δβ(C)= ρ(C[β,1]) – ρ(C[1-β,β]),        (11) 
 
where  denotes the Spearman rank correlation. 
The choice of  is a more subtle problem. To avoid possible random fluctuations, in Durante, Foscolo, 

Jaworski and Wang (2014, p. 4025; 2015, p. 219), it was proposed to measure the set of  s for which the 
index is positive. However, it is recommended to follow the approach proposed in Jaworski and Pitera 
(2014, p. 308), based on the paradigm that there is no contagion for Gaussian distributions, namely, for 
Gaussian copula ∆β is changing sign for 1 0.21   (compare Jaworski and Pitera (2014, p. 308)). It is 
positive for a smaller median part and negative for bigger. Thus we selected 0 .8  . To reduce the 
impact of the fluctuations, due to the random nature of our data, we apply the triangular average of empir-
ical estimates: 

 

’        (12) 
where 
∆̂β = ρ̂[β ,1] − ρ̂[1−β,β]. 
            (13) 
Here ρ̂[β1 ,β2] denotes the Spearman rank correlation of truncated empirical data. 
 
Note that both the contagion index and its empirical estimate do not depend on marginal distribution 

of daily changes of bonds yields. They extract “the pure dependence”. The positive values indicate that the 
co-movements are stronger and more often, when the price of the first bond drops significantly. The nega-
tive values indicate that the movements in opposite direction are stronger and more often, when the price 
of the first bond drops significantly. In the first case we observe contagion, in the second divergence. 

 
Empirical analysis 
We took under the consideration Thomson/Reuters yield indices for bonds of different maturities is-

sued in USA, Germany, Japan, Australia, Poland, Italy and Portugal for the period of time from 2004-11-19 
to 2014-11-17. We compare the market behavior before and after the change of regime implied by the 
financial crisis. As a breaking point we select 2009-12-31. The choice is affected by empirical analysis of 
Eurozone bonds, where at the end of 2009 the final break is observed (see figure 4 for quotations refer-
ring to Germany and Portugal). 

In tables 1 and 2 we present values of contagion index ∆ for years 2004–2009 and 2010–2014, calcu-
lated for the daily changes of the Thomson/Reuters yield indices. Contagion or divergence goes from the 
indices in left column to indices in the first row. Note that after the change of regime, the values are quite 
different. On Figures from 5 to 16 we show the visualization of the data given in tables.  

The bar charts (figures 5 to 16) illustrate the results presented in tables 1 and 2. We may conclude, 
that bars going upward illustrate contagion effect, while bars directed downwards reflect the divergence. 
Basing on our sample, we can note that the divergence effect prevails over contagion. It mirrors the fact, 
there was no worldwide retreat from bond markets as a result of crisis. The threat of sovereign insolvency 
during the crises was limited only to selected countries. The sample includes safe haven countries (Ger-
many, US, Australia), emerging markets (Poland), troubled ones (Portugal, Italy) and Japan. As it was al-
ready mentioned, the sample has been selected heuristically. As our methodology complies with the port-
folio theory, we do not concentrate on fundamental explanation for the obtained results, but rather try to 
capture statistical relationship between the time series using defined measures. Generally speaking, there 
was some contagion before the crisis, although divergence prevailed. After outbreak of the crisis, we can 
see increase of contagion. Graphic comparison between pre-crisis and post-crisis observation may be seen 
at the diagrams (figure 17 and figure 18). We can see the rise of shock transmission referring to Far East 
Asia sovereign bonds (Japan, Australia) after the outbreak of the crisis. The most characteristic case out of 
our sample seems to be Japan. Before the crisis there was negative interdependency between Japanese 
and other countries’ sovereign bonds. At present, among our sample Japan produces more danger for an 
international bond markets in terms of contagion than Italy or Germany. Basing on our countries selec-
tion, we found out that there is more divergence among the EMU countries than between the EMU sover-
eigns and Japanese or the US bonds. The analysis gives us also an insight into the term structure of interest 
rates reaction following shock. We used the bonds representative for the short end-, middle section- and 
long end of each government yield curve. Before the crisis contagion effect affected predominantly long 

 (13)

Here 

 

Definition 2. 
Let C be a copula and   (0.5, 1). 
Δβ(C)= ρ(C[β,1]) – ρ(C[1-β,β]),        (11) 
 
where  denotes the Spearman rank correlation. 
The choice of  is a more subtle problem. To avoid possible random fluctuations, in Durante, Foscolo, 

Jaworski and Wang (2014, p. 4025; 2015, p. 219), it was proposed to measure the set of  s for which the 
index is positive. However, it is recommended to follow the approach proposed in Jaworski and Pitera 
(2014, p. 308), based on the paradigm that there is no contagion for Gaussian distributions, namely, for 
Gaussian copula ∆β is changing sign for 1 0.21   (compare Jaworski and Pitera (2014, p. 308)). It is 
positive for a smaller median part and negative for bigger. Thus we selected 0 .8  . To reduce the 
impact of the fluctuations, due to the random nature of our data, we apply the triangular average of empir-
ical estimates: 

 

’        (12) 
where 
∆̂β = ρ̂[β ,1] − ρ̂[1−β,β]. 
            (13) 
Here ρ̂[β1 ,β2] denotes the Spearman rank correlation of truncated empirical data. 
 
Note that both the contagion index and its empirical estimate do not depend on marginal distribution 

of daily changes of bonds yields. They extract “the pure dependence”. The positive values indicate that the 
co-movements are stronger and more often, when the price of the first bond drops significantly. The nega-
tive values indicate that the movements in opposite direction are stronger and more often, when the price 
of the first bond drops significantly. In the first case we observe contagion, in the second divergence. 

 
Empirical analysis 
We took under the consideration Thomson/Reuters yield indices for bonds of different maturities is-

sued in USA, Germany, Japan, Australia, Poland, Italy and Portugal for the period of time from 2004-11-19 
to 2014-11-17. We compare the market behavior before and after the change of regime implied by the 
financial crisis. As a breaking point we select 2009-12-31. The choice is affected by empirical analysis of 
Eurozone bonds, where at the end of 2009 the final break is observed (see figure 4 for quotations refer-
ring to Germany and Portugal). 

In tables 1 and 2 we present values of contagion index ∆ for years 2004–2009 and 2010–2014, calcu-
lated for the daily changes of the Thomson/Reuters yield indices. Contagion or divergence goes from the 
indices in left column to indices in the first row. Note that after the change of regime, the values are quite 
different. On Figures from 5 to 16 we show the visualization of the data given in tables.  

The bar charts (figures 5 to 16) illustrate the results presented in tables 1 and 2. We may conclude, 
that bars going upward illustrate contagion effect, while bars directed downwards reflect the divergence. 
Basing on our sample, we can note that the divergence effect prevails over contagion. It mirrors the fact, 
there was no worldwide retreat from bond markets as a result of crisis. The threat of sovereign insolvency 
during the crises was limited only to selected countries. The sample includes safe haven countries (Ger-
many, US, Australia), emerging markets (Poland), troubled ones (Portugal, Italy) and Japan. As it was al-
ready mentioned, the sample has been selected heuristically. As our methodology complies with the port-
folio theory, we do not concentrate on fundamental explanation for the obtained results, but rather try to 
capture statistical relationship between the time series using defined measures. Generally speaking, there 
was some contagion before the crisis, although divergence prevailed. After outbreak of the crisis, we can 
see increase of contagion. Graphic comparison between pre-crisis and post-crisis observation may be seen 
at the diagrams (figure 17 and figure 18). We can see the rise of shock transmission referring to Far East 
Asia sovereign bonds (Japan, Australia) after the outbreak of the crisis. The most characteristic case out of 
our sample seems to be Japan. Before the crisis there was negative interdependency between Japanese 
and other countries’ sovereign bonds. At present, among our sample Japan produces more danger for an 
international bond markets in terms of contagion than Italy or Germany. Basing on our countries selec-
tion, we found out that there is more divergence among the EMU countries than between the EMU sover-
eigns and Japanese or the US bonds. The analysis gives us also an insight into the term structure of interest 
rates reaction following shock. We used the bonds representative for the short end-, middle section- and 
long end of each government yield curve. Before the crisis contagion effect affected predominantly long 

 denotes the Spearman rank correlation of truncated empirical data.

Note that both the contagion index and its empirical estimate do not de-
pend on marginal distribution of daily changes of bonds yields. They extract 
“the pure dependence”. The positive values indicate that the co-movements are 



 ConTagion and divergence on sovereign bond markeTs 51

stronger and more often, when the price of the first bond drops significant-
ly. The negative values indicate that the movements in opposite direction are 
stronger and more often, when the price of the first bond drops significantly. In 
the first case we observe contagion, in the second divergence.

Empirical analysis

We took under the consideration Thomson/Reuters yield indices for bonds of 
different maturities issued in USA, Germany, Japan, Australia, Poland, Italy and 
Portugal for the period of time from 2004-11-19 to 2014-11-17. We compare the 
market behavior before and after the change of regime implied by the financial 
crisis. As a breaking point we select 2009-12-31. The choice is affected by em-
pirical analysis of Eurozone bonds, where at the end of 2009 the final break is 
observed (see figure 4 for quotations referring to Germany and Portugal).

In tables 1 and 2 we present values of contagion index ∆ for years 2004– 
–2009 and 2010–2014, calculated for the daily changes of the Thomson/Reu-
ters yield indices. Contagion or divergence goes from the indices in left column 
to indices in the first row. Note that after the change of regime, the values are 
quite different. On Figures from 5 to 16 we show the visualization of the data 
given in tables. 

The bar charts (figures 5 to 16) illustrate the results presented in tables 
1 and 2. We may conclude, that bars going upward illustrate contagion effect, 
while bars directed downwards reflect the divergence. Basing on our sample, 
we can note that the divergence effect prevails over contagion. It mirrors the 
fact, there was no worldwide retreat from bond markets as a result of crisis. 
The threat of sovereign insolvency during the crises was limited only to se-
lected countries. The sample includes safe haven countries (Germany, US, Aus-
tralia), emerging markets (Poland), troubled ones (Portugal, Italy) and Japan. 
As it was already mentioned, the sample has been selected heuristically. As our 
methodology complies with the portfolio theory, we do not concentrate on fun-
damental explanation for the obtained results, but rather try to capture statis-
tical relationship between the time series using defined measures. Generally 
speaking, there was some contagion before the crisis, although divergence pre-
vailed. After outbreak of the crisis, we can see increase of contagion. Graphic 
comparison between pre-crisis and post-crisis observation may be seen at the 
diagrams (figure 17 and figure 18). We can see the rise of shock transmission 
referring to Far East Asia sovereign bonds (Japan, Australia) after the outbreak 



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki52

of the crisis. The most characteristic case out of our sample seems to be Japan. 
Before the crisis there was negative interdependency between Japanese and 
other countries’ sovereign bonds. At present, among our sample Japan produc-
es more danger for an international bond markets in terms of contagion than 
Italy or Germany. Basing on our countries selection, we found out that there is 
more divergence among the EMU countries than between the EMU sovereigns 
and Japanese or the US bonds. The analysis gives us also an insight into the 
term structure of interest rates reaction following shock. We used the bonds 
representative for the short end-, middle section- and long end of each govern-
ment yield curve. Before the crisis contagion effect affected predominantly 
long term bonds. After the Lehman collapse we may observe the reaction along 
the whole government yield curve.

The applied contagion index enables intra-country contagion analysis. Con-
tagion between bonds of a single issuer representing extreme ends of the gov-
ernment yield curve is tantamount to yield curve twist movement. Divergence 
within the bonds of the same issuer with different maturity results in paral-
lel shift. If the bond yield at the middle of yield curve diverge relative to oth-
er yields, it means that we face the yield curve butterfly movement (results in 
change of the yield curve convexity). Of course the parallel shift dominates. 
However when it comes to Polish government debt, investors tend to shift mon-
ey between short term and long term bonds and the yield curve finally twists.

There are some more interesting observations: When we look at Fig-
ures 9–10 we may observe that divergence effect between Polish, German and 
the US markets in years 2004–2009 changed into contagion effect in period 
2010–2012. The same effect can be seen for Japanese market (figures 11–12). 
The divergence effect between Japanese and German bonds changed into con-
vergence effect. What regards German market, we observe strong divergence 
towards Italy and Japan. This was to disappear later on while the divergence 
towards Australia market turned into contagion. There was, in turn strong di-
vergence form Australian to German, Italian and Polish markets in years 2004– 
–2009 which was to disappear in years 2010–2014.

The outcome of the research process and conclusions

The latest research puts a lot of attention on contagion effect. However, there is 
a kind of a variety among contagion effect definitions. It is important to stress, 
that the contagion does not simply equal strong interdependency: the conta-



 ConTagion and divergence on sovereign bond markeTs 53

gion require change of interdependency level (significant rise) in times of con-
traction. In this aspect classic portfolio theory may lead to catastrophic risk 
underestimation as it captures normal market conditions, i.e. it considers the 
value range for random variables in the middle of their distributions, thus ig-
noring extreme events. What is more, not only the extreme events probability 
but also the linear as well as non-linear interdependency increase must be tak-
en into account. 

There is a robust methodology on modeling the relationship between mul-
tidimensional random variables with special focus on contagion effect. The pa-
per presents and somehow places these techniques in some sort of order with 
a following approach forged by Durante and Jaworski (2010, p. 551), Jaworski 
and Pitera (2014, p. 303), Durante and Foscolo (2013, p. 319) and Durante et al. 
(2014, p. 4023; 2015, p. 217; 2017, p. 1). The copula theory is worth the atten-
tion, as it captures non-linear contagion effects, so demanded by the practition-
ers (Embrechts, 2009, p. 639).

The article proposes definition and measure for bond divergence effect, 
which opposite to contagion effect. Therefore our research puts an equal ac-
cent on both contagion and divergence effect on bond sovereign bond markets 
as we see the latter effect somehow overlooked in literature and research.

We conducted our calculations basing on sovereign bonds representative 
for safe haven countries (Germany, US, Australia), emerging markets (Poland), 
troubled ones (Portugal, Italy) and Japan. 

We found some contagion before the crisis, although divergence prevailed. 
After the outbreak of recent crisis contagion increased. Prior to that, contagion 
affected predominantly long term bonds. After Lehman collapse we may ob-
serve contagion along the whole government yield curve.

Paradoxically, judging from our sample analysis, one can find an increase of 
divergence between the EMU countries. This may serve as another evidence for 
the thesis that introduction of the European Monetary Union contributed more 
to divergence than convergence in bond yields. Germany and Australia play the 
role of safe havens in our sample. During contraction times, Japanese investors 
retreat from bond markets shifting their money to Australian and to some ex-
tend into US-sovereign debt. 

The advantage of the proposed method is that there is no need of subjective 
subdivision of the time axle into the calm and crisis periods. Instead, one just 
compares the size of changes which means it is enough to look at the distribu-
tion of empirical values.



[54]

Figure 2. Scaterplot of 1000 points simulated from the copula of X and Y  
from the toy model

 

Figure 2. Scaterplot of 1000 points simulated from the copula of X and Y from the toy model 
 

 
 
Source: own calculations. 
 
Figure 3. Scaterplot of 1000 points simulated from the copula of Y and Z from the toy model 
 

 
 
Source: own calculations. 

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

S o u r c e : own calculations.

Figure 3. Scaterplot of 1000 points simulated from the copula of Y and Z  
from the toy model

 

Figure 2. Scaterplot of 1000 points simulated from the copula of X and Y from the toy model 
 

 
 
Source: own calculations. 
 
Figure 3. Scaterplot of 1000 points simulated from the copula of Y and Z from the toy model 
 

 
 
Source: own calculations. 

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

S o u r c e : own calculations.



[55]

Figure 4. Yields of German 3 year bonds and Portugal 5 year bonds

 

 
Figure 4. Yields of German 3 year bonds and Portugal 5 year bonds 
 

 
 
Source: own calculations based on Thomson/Reuters data. 
 
Table 1. Contagion index ∆ for years 2004–2009 
 

∆ G1 G G I1 I5 I3 Pl Pl PlGERMA- 0 - - - - - 0 0 0 0GERMA- - 0 - - - - - - -GERMA- - - 0 - - - - - 0Italy 10Y - - - 0 - - 0 1 0 0Italy 5Y - - - - 0 - - - -Italy 3Y - - - - - 0 - - -Poland - - - - - - 0 - 0Poland - - - - - - - 0 0Poland - - 0 0 0 0 - - 0Japan 0 0 0 0 0 0 0 0 - 0Japan 5Y - - - - - 0 - - 0Japan 3Y - - - - - - - - -USA 10Y - - - - - - - - -USA 5Yg - - - - - - - - -USA 2Y 0 0 0 0 0 0 0 - - -Portugal - - - - - - - - -AUS- - - - - - - - - -AUS- - - - - - - - - -AUS- - - - - - - - - - 
 

∆ J1 J5 J3 U1 U U Pt A1 A AGERMA- - - - - - - - 0 0 0 0GERMA- - - - - - - - - - -GERMA- - - - - - - - - - -Italy 10Y - - - - - - - - - -Italy 5Y - - - - - - - - - -Italy 3Y - - - - - - - - - -Poland 0 - - 0 - - - 0 0 0 0Poland 0 0 0 - - - - 0 0 0 0Poland 0 - - - - - 0 0 0 - -Japan 0 0 0 0 0 0 0 0 0 0 0 0Japan 5Y - 0 - - - - - - - -Japan 3Y - - 0 - - - - - - -USA 10Y 0 0 0 0 - - - 0 1 0 0USA 5Y - 0 0 - 0 - - 0 0 0 0USA 2Y - - 0 - - 0 0 0 0 0 0Portugal - - - - - - 0 - - -AUS- - - - 0 0 0 - - 0 - -AUS- - - - 0 0 0 - - - 0 -AUS- - - - 0 0 0 - - - - 0 

S o u r c e : own calculations based on Thomson/Reuters data.

Table 1. Contagion index ∆ for years 2004–2009

∆ G10Y G5Y G3Y I10Y I5Y I3Y Pl10Y Pl5Y Pl3Y

GERMANY 10Y 0 -0.06 -0.11 -0.28 -0.17 -0.16 0.02 0.07 0.08

GERMANY 5Y -0.04 0 -0.04 -0.24 -0.1 -0.13 -0.08 -0.05 -0.01

GERMANY 3Y -0.13 -0.09 0 -0.21 -0.1 -0.14 -0.04 -0.08 0.02

Italy 10Y -0.16 -0.17 -0.15 0 -0.09 -0.08 0.13 0.14 0.06

Italy 5Y -0.15 -0.12 -0.13 -0.17 0 -0.07 -0.11 -0.05 -0.02

Italy 3Y -0.2 -0.17 -0.16 -0.23 -0.15 0 -0.21 -0.15 -0.14

Poland 10Y -0.09 -0.12 -0.1 -0.04 -0.08 -0.08 0 -0.05 0.03

Poland 5Y -0.11 -0.1 -0.07 -0.08 -0.02 -0.02 -0.07 0 0.06

Poland 3Y -0.01 -0.03 0 0.01 0.08 0.05 -0.03 -0.02 0

Japan 10Y 0.01 0.02 0.02 0.04 0.06 0.04 0 -0.04 0

Japan 5Y -0.07 -0.08 -0.07 -0.02 -0.01 0 -0.02 -0.04 0.01

Japan 3Y -0.12 -0.12 -0.1 -0.09 -0.04 -0.04 -0.08 -0.07 -0.03



[56][56]

∆ G10Y G5Y G3Y I10Y I5Y I3Y Pl10Y Pl5Y Pl3Y

USA 10Y -0.12 -0.09 -0.09 -0.16 -0.12 -0.13 -0.13 -0.21 -0.13

USA 5Yg -0.08 -0.04 -0.02 -0.07 -0.03 -0.04 -0.13 -0.18 -0.06

USA 2Y 0.01 0.03 0.05 0.04 0.1 0.07 -0.07 -0.08 -0.01

Portugal 5Y -0.15 -0.13 -0.1 -0.18 -0.11 -0.11 -0.05 -0.05 -0.04

AUSTRALIA 10Y -0.06 -0.08 -0.09 -0.13 -0.08 -0.05 -0.09 -0.1 -0.05

AUSTRALIA 5Y -0.12 -0.14 -0.16 -0.16 -0.16 -0.15 -0.08 -0.1 -0.06

AUSTRALIA 3Y -0.12 -0.16 -0.17 -0.15 -0.17 -0.16 -0.07 -0.1 -0.08

∆ J10Y J5Y J3Y U10Y U5Y U2Y Pt5Y A10Y A5Y A3Y

GERMANY 10Y -0.08 -0.1 -0.12 -0.05 -0.1 -0.1 -0.14 0.05 0.02 0.01

GERMANY 5Y -0.14 -0.15 -0.13 -0.07 -0.07 -0.06 -0.08 -0.08 -0.1 -0.1

GERMANY 3Y -0.15 -0.13 -0.1 -0.06 -0.03 -0.01 -0.06 -0.08 -0.11 -0.11

Italy 10Y -0.05 -0.09 -0.1 -0.08 -0.07 -0.06 -0.11 -0.1 -0.06 -0.04

Italy 5Y -0.1 -0.09 -0.11 -0.05 -0.03 -0.04 -0.12 -0.05 -0.08 -0.08

Italy 3Y -0.15 -0.13 -0.13 -0.03 -0.01 -0.03 -0.16 -0.08 -0.12 -0.12

Poland 10Y 0.04 -0.01 -0.06 0 -0.04 -0.03 -0.09 0.07 0.04 0.03

Poland 5Y 0.05 0.04 0.03 -0.11 -0.14 -0.09 -0.08 0.06 0.02 0

Poland 3Y 0.01 -0.04 -0.02 -0.04 -0.04 -0.02 0.03 0.04 -0.02 -0.04

Japan 10Y 0 0.03 0 0.03 0.02 0.01 0.05 0.04 0.06 0.08

Japan 5Y -0.03 0 -0.01 -0.1 -0.1 -0.12 -0.03 -0.02 -0.03 -0.02

Japan 3Y -0.17 -0.16 0 -0.06 -0.07 -0.07 -0.08 -0.05 -0.06 -0.03

USA 10Y 0.03 0.06 0.05 0 -0.08 -0.2 -0.16 0.11 0.11 0.11

USA 5Y -0.05 0.03 0.04 -0.05 0 -0.13 -0.05 0.05 0.08 0.07

USA 2Y -0.11 -0.06 0 -0.16 -0.08 0 0.06 0.02 0.04 0.04

Portugal 5Y -0.13 -0.1 -0.1 -0.04 -0.01 -0.04 0 -0.07 -0.08 -0.09

AUSTRALIA 10Y -0.16 -0.13 -0.13 0.09 0.01 -0.09 -0.12 0 -0.07 -0.12

AUSTRALIA 5Y -0.09 -0.13 -0.15 0.05 0.02 -0.09 -0.18 -0.13 0 -0.01

AUSTRALIA 3Y -0.09 -0.11 -0.14 0.07 0.04 -0.05 -0.18 -0.19 -0.02 0

S o u r c e : own calculations.

Table 1. Contagion index ∆ for years 2004–2009



[57][57]

Table 2. Contagion index ∆ for years 2010–2014

∆ G10Y G5Y G3Y I10Y I5Y I3Y Pl10Y Pl5Y Pl3Y

GERMANY 10Y 0 -0.13 -0.12 -0.06 -0.07 -0.01 -0.12 -0.16 -0.11

GERMANY 5Y -0.19 0 -0.11 -0.02 -0.01 0.03 -0.07 -0.12 -0.07

GERMANY 3Y -0.26 -0.18 0 0.01 0.01 0.04 -0.11 -0.17 -0.1

Italy 10Y -0.3 -0.28 -0.27 0 0.01 0.03 0 0.03 0.02

Italy 5Y -0.25 -0.24 -0.23 -0.04 0 -0.02 0 0 0

Italy 3Y -0.27 -0.27 -0.28 -0.07 -0.08 0 -0.01 0.02 -0.02

Poland 10Y 0.12 0.14 0.1 -0.05 -0.03 -0.04 0 0.05 0.06

Poland 5Y 0.11 0.09 0.03 -0.01 -0.02 -0.04 0.04 0 -0.01

Poland 3Y 0.04 0.03 0 -0.07 -0.02 -0.04 -0.09 -0.1 0

Japan 10Y 0.07 0.11 0.12 -0.01 0.05 0.02 0.12 0.1 0.09

Japan 5Y 0.01 0.06 0.09 0.06 0.05 0.05 0.09 0.1 0.04

Japan 3Y 0.02 -0.02 -0.01 0.16 0.16 0.09 0.1 0.1 0.04

USA 10Y 0.04 0.06 0.08 0.18 0.13 0.1 0.01 0.01 0.03

USA 5Y -0.17 -0.11 -0.06 0.22 0.21 0.18 -0.02 -0.05 0.02

USA 2Y -0.22 -0.12 -0.08 0.2 0.22 0.22 0 -0.03 0.02

Portugal 5Y -0.16 -0.17 -0.16 -0.23 -0.21 -0.2 -0.2 -0.22 -0.17

AUSTRALIA 10Y -0.06 -0.05 -0.02 -0.05 -0.02 -0.05 0.01 0.06 0.07

AUSTRALIA 5Y -0.06 -0.02 0.06 0 0 -0.01 0.05 0.06 0.06

AUSTRALIA 3Y -0.06 -0.03 0.03 0 0.01 -0.01 0.03 0.03 0.04

∆ J10Y J5Y J3Y U10Y U5Y U2Y Pt5Y A10Y A5Y A3Y

GERMANY 10Y -0.06 -0.06 0.06 -0.05 -0.11 -0.08 0.04 -0.01 0 0

GERMANY 5Y 0.01 -0.01 0.07 -0.11 -0.15 -0.11 -0.03 0.15 0.16 0.16

GERMANY 3Y -0.03 -0.03 0.07 -0.12 -0.14 -0.12 0.01 0.1 0.11 0.12

Italy 10Y -0.11 -0.04 0.02 -0.18 -0.12 -0.03 -0.16 -0.05 -0.06 -0.05

Italy 5Y 0.02 0.02 0.14 -0.23 -0.2 -0.14 -0.26 -0.12 -0.13 -0.13

Italy 3Y 0.02 -0.02 0.03 -0.2 -0.15 -0.08 -0.25 -0.12 -0.15 -0.16

Poland 10Y -0.02 0.02 0.01 0.04 0.03 0.05 -0.03 0 0 0

Poland 5Y -0.04 -0.05 -0.08 0 0.02 0.08 0.01 0.03 0.04 0.05



[58]

∆ J10Y J5Y J3Y U10Y U5Y U2Y Pt5Y A10Y A5Y A3Y

Poland 3Y 0 -0.01 -0.03 -0.04 -0.02 -0.02 -0.08 -0.01 0 0.01

Japan 10Y 0 -0.01 0.07 0.05 0.09 0.12 -0.02 -0.07 -0.09 -0.08

Japan 5Y 0.06 0 0.24 -0.06 -0.05 -0.04 0.08 -0.12 -0.15 -0.16

Japan 3Y 0.11 0.12 0 0 0 0.01 0.15 -0.04 -0.09 -0.1

USA 10Y 0.09 -0.01 0.06 0 -0.08 -0.08 0.01 0 -0.02 -0.04

USA 5Y 0.07 0.02 0 -0.14 0 -0.05 -0.01 -0.01 -0.04 -0.03

USA 2Y 0.01 0.01 -0.07 -0.14 -0.07 0 0.04 -0.01 0.03 0.04

Portugal 5Y 0.01 0.02 -0.01 -0.02 0 0.09 0 -0.08 -0.06 -0.07

AUSTRALIA 10Y -0.07 -0.12 -0.02 -0.05 -0.09 -0.02 0.06 0 -0.12 -0.17

AUSTRALIA 5Y -0.09 -0.08 -0.04 -0.03 -0.07 -0.04 0.11 -0.19 0 -0.04

AUSTRALIA 3Y -0.15 -0.14 -0.09 -0.01 -0.06 -0.03 0.1 -0.25 -0.06 0

S o u r c e : own calculations.

Figure 5. Contagion/divergence from German to other markets,  
years 2004–2009

 

Source: own calculations. 
 
Table 2. Contagion index ∆ for years 2010-2014 
 

∆ G1 G G I1 I5 I3 Pl Pl PlGERMA- 0 - - - - - - - -GERMA- - 0 - - - 0 - - -GERMA- - - 0 0 0 0 - - -Italy 10Y - - - 0 0 0 0 0 0Italy 5Y - - - - 0 - 0 0 0Italy 3Y - - - - - 0 - 0 -Poland 0 1 0 0 - - - 0 0 0Poland 0 1 0 0 - - - 0 0 0 -Poland 0 0 0 0 - - - - - 0Japan 0 0 0 0 - 0 0 0 1 0 0Japan 5Y 0 0 0 0 0 0 0 0 0 0 0Japan 3Y 0 0 - - 0 0 0 0 1 0 0USA 10Y 0 0 0 0 0 0 0 0 0 0 0USA 5Y - - - 0 0 0 - - 0USA 2Y - - - 0 0 0 0 - 0Portugal - - - - - - - - -AUS- - - - - - - 0 0 0 0AUS- - - 0 0 0 - 0 0 0 0AUS- - - 0 0 0 - 0 0 0 0 
 

∆ J1 J5 J3 U1 U U Pt A1 A AGERMA- - - 0 - - - 0 - 0 0GERMA- 0 - 0 - - - - 0 0 0GERMA- - - 0 - - - 9( 0 0 0Italy 10Y - - 0 - - - - - - -Italy 5Y 0 0 0 - - - - - - -Italy 3Y 0 - 0 - - - - - - -Poland - 0 0 0 0 0 0 - 0 0 0Poland - - - 0 0 0 0 0 0 0Poland 0 - - - - - - - 0 0Japan 0 - 0 0 0 0 0 - - - -Japan 5Y 0 0 0 - - - 0 - - -Japan 3Y 0 0 0 0 0 0 0 - - -USA 10Y 0 - 0 0 - - 0 0 - -USA 5Y 0 0 0 - 0 - - - - -USA 2Y 0 0 - - - 0 0 - 0 0Portugal 0 0 - - 0 0 0 - - -AUS- - - - - - - 0 0 - -AUS- - - - - - - 0 - 0 -AUS- - - - - - - 0 - - 0 
Source: own calculations. 
 
Figure 5. Contagion/divergence from German to other markets, years 2004-2009 
 

 

Source: own calculations. 
 
 
 
 
 

S o u r c e : own calculations.

Table 2. Contagion index ∆ for years 2010–2014



[59]

Figure 6. Contagion/divergence from German to other markets,  
years 2010–2014

 

Figure 6. Contagion/divergence from German to other markets, years 2010-2014 
 

 

 
Source: own calculations. 
 
Figure 7. Contagion/divergence from Italian and Portugal to other markets, years 2004-2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.

Figure 7. Contagion/divergence from Italian and Portugal to other markets,  
years 2004–2009

 

Figure 6. Contagion/divergence from German to other markets, years 2010-2014 
 

 

 
Source: own calculations. 
 
Figure 7. Contagion/divergence from Italian and Portugal to other markets, years 2004-2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.



[60]

Figure 8. Contagion/divergence from Italian and Portugal to other markets,  
years 2010–2014

 

Figure 8. Contagion/divergence from Italian and Portugal to other markets, years 2010-2014 
 

 

 
Source: own calculations. 
 
Figure 9. Contagion/divergence from Polish to other markets, years 2004-2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.

Figure 9. Contagion/divergence from Polish to other markets, years 2004–2009

 

Figure 8. Contagion/divergence from Italian and Portugal to other markets, years 2010-2014 
 

 

 
Source: own calculations. 
 
Figure 9. Contagion/divergence from Polish to other markets, years 2004-2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.



[61]

Figure 10. Contagion/divergence from Polish to other markets, years 2010–2014

 

Figure 10. Contagion/divergence from Polish to other markets, years 2010–2014 
 

 

 
Source: own calculations. 
 
Figure 11. Contagion/divergence from Japan to other markets, years 2004–2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.

Figure 11. Contagion/divergence from Japan to other markets, years 2004–2009

 

Figure 10. Contagion/divergence from Polish to other markets, years 2010–2014 
 

 

 
Source: own calculations. 
 
Figure 11. Contagion/divergence from Japan to other markets, years 2004–2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.



[62]

Figure 12. Contagion/divergence from Japan to other markets, years 2010–2014

 

Figure 12. Contagion/divergence from Japan to other markets, years 2010–2014 
 

 

 
Source: own calculations. 
 
Figure 13. Contagion/divergence from USA to other markets, years 2004–2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.

Figure 13. Contagion/divergence from USA to other markets, years 2004–2009

 

Figure 12. Contagion/divergence from Japan to other markets, years 2010–2014 
 

 

 
Source: own calculations. 
 
Figure 13. Contagion/divergence from USA to other markets, years 2004–2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.



[63]

Figure 14. Contagion/divergence from USA to other markets, years 2010–2014

 

Figure 14. Contagion/divergence from USA to other markets, years 2010–2014 
 

 

 
Source: own calculations. 
 
Figure 15. Contagion/divergence from Australian to other markets, years 2004–2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.

Figure 15. Contagion/divergence from Australian to other markets, years 2004–2009

 

Figure 14. Contagion/divergence from USA to other markets, years 2010–2014 
 

 

 
Source: own calculations. 
 
Figure 15. Contagion/divergence from Australian to other markets, years 2004–2009 
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.



[64]

Figure 16. Contagion/divergence from Australian to other markets, years 2010–2014 

 

 
Figure 16. Contagion/divergence from Australian to other markets, years 2010–2014  
 

 

 
Source: own calculations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

S o u r c e : own calculations.



[65]

Figure 17. Contagion diagram 2004–2014

 

 
 
Source: own calculations. 
 
Figure 18. Contagion diagram 2009–2014 

IT

GER 

POR 

AUS

POL 

USA 

J

S o u r c e : own calculations.



Piotr Jaworski, Kamil Liberadzki, Marcin Liberadzki66

Figure 18. Contagion diagram 2009–2014

 

 
 
Source: own calculations. 
 
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