Sovereign debt crisis in the European Union: A minimum spanning tree approach

Physica A 391 (2012) 2046–2055

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Sovereign debt crisis in the European Union: A minimum spanning tree approach João Dias ∗ ISEG-Technical University of Lisbon and UECE, Rua do Quelhas, 6, 1200-781 Lisbon, Portugal

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Article history: Received 24 March 2011 Received in revised form 30 September 2011 Available online 17 November 2011 Keywords: Minimum spanning tree Sovereign debt Government bonds Crisis

abstract In the wake of the financial crisis, sovereign debt crisis has emerged and is severely affecting some countries in the European Union, threatening the viability of the euro and even the EU itself. This paper applies recent developments in econophysics, in particular the minimum spanning tree approach and the associate hierarchical tree, to analyze the asynchronization between the four most affected countries and other resilient countries in the euro area. For this purpose, daily government bond yield rates are used, covering the period from April 2007 to October 2010, thus including yield rates before, during and after the financial crises. The results show an increasing separation of the two groups of euro countries with the deepening of the government bond crisis. © 2011 Elsevier B.V. All rights reserved.

1. Introduction In the aftermath of the recent financial and economic crisis, many European Union (EU) member states, as well as countries in other regions, have significantly raised their budget deficits and public debts. One such example is the case of Greece, which recorded a government debt and budget deficit representing 126.8% and 15.4% of Gross Domestic Product (GDP) respectively in 2009. This unsustainable situation created difficulties in accessing international financial markets and a new crisis emerged, this time related to government bonds. After Greece, Ireland became the next country to draw on financial assistance from the EU and the International Monetary Fund (IMF), with Portugal following some months later. Spain also has been impacted by high government bond yield rates although its budget and debt problems are not of the same magnitude as those of the other three countries. Nevertheless, strong pressure has loomed over the euro area, given that other countries, such as Italy or Belgium, have also accumulated large public financial imbalances. There is now an increasingly widespread fear that the Euro might be in jeopardy, with even the European Union itself called into question as a project for economic and political integration in Europe, should this crisis not be contained. In this paper, sovereign debt crisis in the European Union is analyzed with tools developed and largely applied in the field of econophysics. The euro area is of particular concern and, thus, the main focus of the paper lies in the network topology of the eurozone members. The minimum spanning tree (MST) provides the main analytical approach and the dynamics of daily government bond yields are investigated using rolling windows of three months, from April 2007 through October 2010. Our methodology is similar to that used in a recently published paper in this journal [1], related to comovements in government bond markets over 1993–2008. Like in this latter paper, we also base our analysis on minimum spanning trees, hierarchical trees and use rolling windows. However, our subject is different, since we are interested in the analysis of the current sovereign debt crisis in the EU. Although we use 10-year government bold yield rates as well, the country

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0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.11.004

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Table 1 Country groups, abbreviations and group symbol. Abbrev.

Country

Euro

Group

AT

Austria

1999

E5, E7

BE

Belgium

1999

E7

CZ

Czech Republic

–

NE6

DK

Denmark

–

NE6

FI

Finland

1999

E5, E7

FR

France

1999

E3, E5, E7

DE

Germany

1999

E3, E5, E7

GR

Greece

2001

G4

HU

Hungary

–

NE6

IE

Ireland

1999

G4

IT

Italy

1999

E7

NL

Netherlands

1999

E3, E5, E7

PL

Poland

–

NE6

PT

Portugal

1999

G4

SK

Slovakia

2009

–

SI

Slovenia

2007

–

ES

Spain

1999

G4

SE

Sweden

–

NE6

UK

United Kingdom

–

NE6

Symbol

composition (only EU countries), period under analysis (last four years) and even frequency of data (daily values) are quite distinct. Besides, we use a larger set of measures in the rolling windows. In addition to this Introduction, the paper is structured as follows. Section 2 briefly describes the data used. Section 3 explains the methodology adopted and Section 4 presents the results obtained. Finally, Section 5 draws the main conclusions. 2. Data characterization We analyze daily yield rates on 10-year government bonds for nineteen EU countries. Thirteen of them belong to the euro area: Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Portugal, Slovakia, Slovenia and Spain. However, Slovakia and Slovenia only recently became members of the eurozone. The six non-Euro countries are the Czech Republic, Denmark, Hungary, Poland, Sweden and the United Kingdom. Table 1 lists all the countries and groups in the sample, their abbreviations and the year they adopted the euro. The data for all countries corresponds to the Thomson Reuters Government Bond Indices and are end-of-day 10-year government bond yields as calculated by Datastream. All data are valued in local currencies and were obtained from this last database. The sample covers the last four years, from the beginning of April 2007 to the end of October 2010, with a total of 933 observations. This corresponds to distinct phases in the current economic and financial crisis, although we do not pretend here to provide any rigorous characterization of it. Accordingly, the sample is split into three periods. The first (P1) includes the observations from April 2007 up to the end of August 2008 and largely corresponds to pre-crisis data, even though the last months already show significant declines in stock markets all over the world. Period two (P2) covers the most critical financial crash from September 2008 up to December 2009. Finally, the last period (P3), from January to October of 2010, shows some economic and financial recovery in many countries but exposes deep budget problems in others and gives rise to a new crisis, this time related to government bonds. Fig. 1 illustrates this periodization with the evolution of the S&P 500 index (S&P500) and 10-year government bond yield rates for Greece (Greece_Y10). In the first period, in spite of a substantial reduction in the stock market index, no clear trend is observed in the case of bond yield rates, whose values ranged from 4.21 to 5.31. During the second period, the crash also generated some concerns with government bonds and yield rates jumped to a maximum of 6.18. However, the recovery in stock markets was accompanied by some pressure relieve on government bond yield rates during most of the second half of 2009. These rates exploded in the third period, from a minimum of 5.55 to a maximum of 12.27 and this peak was followed by some decline in the S&P500. On average, yield rates have been decreasing from the first to the last period. Fig. 2 shows the evolution of the average and standard deviation computed over the 19 countries. Average values were reduced by around ten percent between the first

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J. Dias / Physica A 391 (2012) 2046–2055 Table 2 Government bond yield rate averages. Country or group

All

P1

P2

P3

AT BE CZ DK FI FR DE GR HU IE IT NL PL PT SK SI ES SE UK

3.98 4.04 4.55 3.82 3.85 3.84 3.58 5.75 8.11 4.87 4.37 3.84 5.92 4.56 4.53 4.81 4.22 3.55 4.15

4.40 4.45 4.56 4.39 4.37 4.37 4.24 4.64 7.44 4.47 4.61 4.36 5.83 4.54 4.61 4.91 4.41 4.22 4.89

4.01 4.00 4.89 3.74 3.82 3.74 3.37 5.12 9.26 5.05 4.37 3.79 6.11 4.24 4.78 4.95 4.08 3.28 3.72

3.23 3.41 3.97 2.96 3.01 3.11 2.79 8.68 7.39 5.25 3.96 3.01 5.77 5.12 3.97 4.43 4.14 2.86 3.57

EU

4.54

4.72

4.54

4.24

E3 E5 E7 G4 NE6

3.75 3.82 3.93 4.85 5.02

4.32 4.35 4.40 4.52 5.22

3.63 3.75 3.87 4.62 5.17

2.97 3.03 3.22 5.80 4.42

Fig. 1. S&P 500 index (right scale) and bond yield rates for Greece (left scale).

and the third period. However, disparities between countries increased, revealing strong contrasts within the EU in 2010. In fact, the values of standard deviations increased from near one to more than two, expressing large differences of countries like Greece and Germany or Hungary and Sweden. Table 2 shows the evolution in detail for each country and some country groups. The group of six non-euro countries reduced yield rates, particularly from period two to period three, although there is a clear difference between Poland and Hungary and the other members of this group. But the most striking contrast is between the two main groups of the euro area. While in the group of Greece, Ireland, Portugal and Spain (G4) yield rates increased by almost one third, the opposite occurred in the other countries of the euro zone (E7), with a global reduction of around one third. As a result, the G4 group has a value eighty percent above the value of the E7 group, even taking into account that Spain has lower rates than Greece, Ireland and Portugal. In fact, only these last three countries, exactly those currently under external financial assistance, increased average yield rates between the first and the last period.

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Fig. 2. Average and dispersion of government bond yields in the EU.

3. Methodology In order to analyze the behavior of N government bond yields, we use the framework introduced by Mantegna [2] and Mantegna and Stanley [3] for the analysis of financial markets. This approach has been successfully applied in many distinct areas, including the analysis of currencies [4–6] and international prices [7]. Let Yi (t ) be the yield rate of government bond i at time t. We begin by constructing the Pearson correlation coefficient between the daily bond yields of two countries i and j: rij = 

Yi2

⟨

⟨Yi Yj ⟩ − ⟨Yi ⟩ ⟨Yj ⟩  , ⟩ − ⟨Yi ⟩2 ⟨Yj2 ⟩ − ⟨Yj ⟩2

where ⟨·⟩ is the average over a given period. From the matrix of correlation coefficients R, we compute a symmetric (N × N ) matrix of distances D with elements, dij =

 

2 1 − rij ,



which vary between 0 (the two bond yield rates are perfectly correlated) and 2 (the rates are perfectly anti-correlated). From this distance matrix, we can obtain the minimum spanning tree (MST) by applying, for example, Kruskal’s algorithm [8,9]. In constructing the MST, we lose some information since we move from a matrix of N (N − 1)/2 correlation coefficients to the information contained in the N − 1 edges of the tree. However, this reduction allows us to easily extract the most relevant data characteristics. Connected  with  this MST is a particular subdominant ultrametric space [10] in ⟩

⟩

⟩

which distance fulfills the condition dij ≤ max dik , dij , which is a stronger condition than the usual triangular inequality

dij ≤ dik + dkj . The minimum spanning tree allows us to obtain a taxonomy for the N yield topological space and this MST and the associated ultrametric distance make it possible to get a hierarchical tree representation of the synchronization of the N yields. This is obtained by using the single linkage cluster analysis (SLCA), but other different algorithms are also available [11], for example the average cluster analysis (ALCA). Since ALCA gives usually a better representation of the hierarchical structure, we will use here ALCA instead of SLCA. The empirical part will explore the resulting graphs of the MST and the hierarchical tree. For the purpose of comparing the evolution of the yield rates for different countries or groups of countries, it is useful to proceed with analysis for different time periods. In our case, the data are analyzed for the entire time-span set out and for the three different periods we consider here. However, to gain a more informative picture about the evolution of the yields, the data are also analyzed using a rolling window constructed as follows. Beginning with a first set of T observations we apply our framework to this data set. Next, let us displace our window by moving it to include the next δ T observations and exclude the first δ T data points. We now proceed with the analysis for this new window and we go on with new windows constructed in the same way, until all the available data have been exhausted. In this paper, we use windows of a length equal to 60 days, that is, about three months of data. For the displacement, we apply a step length of 5 days (a week of data). In order to quantify the statistical reliability of the links of MST and hierarchical trees a non parametric bootstrap technique is applied to the data. The numbers appearing in Figs. 3a to 6a and 3b to 6b quantify this reliability (bootstrap

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Fig. 3a. MST for April 2007 through October 2010.

Fig. 3b. Hierarchical tree for April 2007 through October 2010.

value) and they represent the fraction of replicas preserving each link in the MST and each internal node in hierarchical tree. The number of replicas used in each period was 10,000. Developed by Efron in 1979 [12], the bootstrap method has been recently proposed as a technique to measure reliability of links of minimum spanning trees and hierarchical trees obtained with financial data [13,14]. Refs. [13–15,6] give details on this technique and illustrate its empirical application in a similar context. 4. Results The MST for the 2007–2010 period shows the G4 group separated in the outer limits of the tree, but directly linked to the euro group E7 (Fig. 3a). The recent eastern members of the EU form a different branch. The Netherlands occupies a more central position than Germany and this characteristic is persistent throughout the different periods. In Ref. [1], it was already emphasized that France is more central than Germany in the period 1993–2008 for 13 EU government bond markets. In our case, the countries composing the E3 group (Netherlands, Germany and France) always remain linked together, a characteristic already appearing in Ref. [1] as well. Another interesting feature of the tree, also observed in different periods, is the fact that non-euro member states Denmark, Sweden and the UK are closely connected to the euro core E3. The observation of the hierarchical tree (Fig. 3b) shows that the E7 group, together with these three countries, form a separate cluster linked by a lower distance than all the other countries. Again, the eastern countries appear in a different cluster. Greece, Portugal and Ireland also are isolated in a different group, but Spain is not directly linked to them. Note that, since we are using ALCA, the fact that Austria and Belgium (or Hungary and Poland) are directly linked in the HT does not imply a direct connection between these countries in the MST. The bootstrap values obtained for the MST are very high, in most cases equal to one or near one. This means that most links along the MST are very reliable, remaining from replica to replica in our bootstrap experiment. Only the link between Slovenia and Austria has a much lower reliability measure. A similar pattern of high reliability of the internal nodes of the hierarchical tree is also, in general, observed. For the first period, that is, prior to the emergence of the financial crisis, all the euro area countries appear more consistently linked together, without any clear separation into particular groups (Fig. 4a). This is also observed through the hierarchical tree which shows that, in fact, the eurozone members form a cluster with the lowest distances in the tree (Fig. 4b). In contrast, the five eastern members of the EU form another cluster with the highest distances. The average distance (along the MST) for the former group is 0.27 and for the latter group is 0.71. The average of correlation coefficients

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Fig. 4a. MST for April 2007 through August 2008.

Fig. 4b. Hierarchical tree for April 2007 through August 2008.

between all possible pairs of the eleven eurozone members is 0.94, showing a high yield bond synchronization involving this group of countries. The average of the correlation coefficients between each of these eleven countries and each of the eastern countries is much lower, only 0.44. In the second period, as the financial crisis peaked, we already observe a first deterioration in consistency within the euro area. Greece and Ireland are separated off from the other eurozone countries in the MST (Fig. 5a) and, together with Hungary, they form a cluster linked at a high distance to the other countries in the hierarchical tree (Fig. 5b). The countries of the euro core E3 remain linked together in the MST and these links are highly reliable, as are the links of Belgium, Italy or Finland with this group. Spain is also directly connected to the E3 group, but Portugal already appears near Greece and Ireland. Yet, it still remains connected to the other euro countries, both in the MST and in the hierarchical tree. In fact, the dendrogram in Fig. 5b shows that Portugal connects to the cluster of Ireland and Greece at a high distance. This movement of segmentation in the eurozone is further extended in the final period, that is, during the sovereign debt crisis. Now, Greece, Ireland, Portugal and even Spain emerge isolated in the MST (Fig. 6a) from the euro core E3 (and E5). In the hierarchical tree (Fig. 6b), the three southern European countries plus Ireland form a clearly separate group, although Spain is only linked to the others at a greater distance. In this case, even Italy already appears directly connected to this group in the MST and joins a separate group (with the Czech Republic, Poland and Hungary) linked to the others at higher distances in the hierarchical tree. Now, the average of the correlation coefficients between all pairs of countries of the euro area (Slovenia and Slovakia excluded) is very low, only 0.08 and it is also low (0.27) for the correlation coefficients between each of these eleven countries and each of the eastern EU members. However, there is a very strong synchronization between Germany, France, Netherlands, Austria and Finland, given that the correlation coefficient between any two of these countries is at least 0.98. It is interesting to compare the behavior of the G4 group and other groups, in particular those of the euro area. Table 3 portrays similar behavior of the G4 and E3 (E5 or E7) groups in the period before the financial crisis. The synchronization decreased in the second period and there was a clear anti-correlation between the two groups in the debt crisis period. We now turn to the analysis of some characteristics of the MST and their evolution over time. A first useful tool for this purpose is the normalized tree length introduced by Onnela et al. [16] as a simple measure of the state of the market. This is

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Fig. 5a. MST for September 2008 through December 2009.

Fig. 5b. Hierarchical tree for September 2008 through December 2009.

Fig. 6a. MST for January 2010 through October 2010.

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Fig. 6b. Hierarchical tree for January 2010 through October 2010. Table 3 Mean of correlation coefficients between countries of G4 and countries of other groups.

E3 E5 E7 NE6

All

P1

P2

P3

−0.277 −0.279 −0.230 −0.174

0.896 0.913 0.933 0.616

0.355 0.437 0.480 0.208

−0.721 −0.716 −0.607 −0.508

Fig. 7. Normalized tree length (ntl). Rolling windows with a length of 60 days and a step length of 5 days.

defined as the mean of distances along the N − 1 edges of the MST, that is,

∑ ntl =

dij

N −1

.

Since we are interested in the behavior of a particular group (G4) and its comparison with another group (E3), it would be useful to introduce a similar measure for a particular set of nodes (vertices) in the MST. For this purpose, and given a group of countries of interest, we define the group mean distance length (gmdl) as the mean of distances of the edges whose vertices are countries belonging to our group. Any edge only enters once, even when both vertices are member of the group. In Fig. 7, the evolution of ntl is shown using a rolling window of 60 days and step length of 5 days. These means of distances increased from an average of 0.38 in the first period to 0.5 in the second, but with only a small increase up to 0.55 in the third period. The group of Germany, France and the Netherlands had a very low gmdl value in period one (0.21), increased to 0.39 in the second period but remained at this level (0.4) in the last period. The group of Greece, Ireland, Portugal and Spain started out with a similar score to the E3 group (0.24) in the first period, rising to 0.48 in P2, but not sharply different from the E3 score. However, they clearly diverged from the E3 group in period three (Fig. 8) as the average of the gmdl indicator surged to 0.66 in this last period. The evolution of the MST may be further analyzed using other relevant tree characteristics. We consider three additional measures here. The first is the tree diameter, i.e. the maximum distance between two nodes in the tree [17]. Using moving windows as defined above, Fig. 9 shows this measure did not change significantly from one period to another. Our second tool is the radius of the MST, which measures the maximum distance from the center of the tree to the other nodes of the tree [17]. Fig. 9 also shows the evolution of the radius and it is clear that this measure is also more or less constant, on average, throughout the period under analysis. Our third tool to help characterize the tree topology is the mean occupation layer (mol)

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Fig. 8. Group mean distance length (gmdl): G4 and E3. Rolling windows with a length of 60 days and a step length of 5 days.

Fig. 9. Diameter, radius and mol for the set of 19 EU countries. Rolling windows with a length of 60 days and a step length of 5 days.

proposed by Onnela et al. [18], useful to measure the spread of nodes on the MST. Designating a node that occupies a central position on the tree as the central vertex, the level of any node is the number of edges separating it from this central vertex. Hence, the mean occupation layer is the average of levels of the different nodes. That is, for each window, we obtain the MST and compute, N ∑

mol =

lev(vi )

i =1

N

,

where lev(vi ) is the level of node i and N is the number of nodes. Here, we obtained the central vertex using Onnela’s [18] second definition, that is, the node with the highest sum of correlation coefficients between it and the nodes that are directly connected to it (incident edges). The evolution of this indicator shows similar behavior to the diameter and radius, thus revealing no tendency to increase or decrease between 2007 and 2010. Of particular interest to us, in terms of the MST topological properties, is the positioning of the G4 group in relation to the ‘‘core’’ E3 or E5 euro countries. Fig. 10 portrays the evolution in the mean distance (in terms of number of edges) on the MST separating G4 from E3 and E5. There is an increasing distance between the two eurozone groups. In period 1, the number of edges separating the G4 countries from the E3 group was, on average, 3.2. This rose to 4.1 in the second period and doubled to 6 in period 3. A comparison with the E5 groups gives similar results. Therefore, the contrast observed between these two groups when we analyzed the MST results incorporating all the data for each period is not due to a particular division of our data and is clearly confirmed by our analysis of the evolution of the MST over 2007–2010. 5. Conclusions Following the recent financial and economic crisis, a new budgetary and debt turbulence has emerged in Europe, and some countries of the euro zone are being particularly afflicted by this new crisis. Greece, Ireland, Portugal and even Spain are now at the center of this turbulence. This is a serious problem not only for the countries concerned but also for the euro as a currency adopted by a large number of European Union member states. The very future of the European Union may be at stake should the euro collapse. The level of this danger rises with the time taken to bring the crises under control, as contagion to other countries would have dramatic consequences. For example, in addition to this group of four countries (not equally impacted, in particular Spain has thus far remained in a better position to access international financial markets than the other three countries), Italy and Belgium are becoming the object of some concern, and certainly other countries would follow should the situation in this group deteriorate. Our analysis of the evolution of government market bond yields for nineteen EU countries, covering daily rates for the period from April 2007 up to October 2010, has provided a coherent topological description of the current government bond

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Fig. 10. Mean distance (number of edges) of G4 in relation to E3 and E5. Rolling windows with a length of 60 days and a step length of 5 days.

crisis in the European Union, clearly revealing the asynchronization in the evolution of bond yields in the euro zone and the formation of two distinct and increasingly separate blocks in this area. This was achieved by applying the MST approach both for three different periods and by using a rolling window in order to observe the evolution of some relevant aspects of the MST, over a period extensive enough to differentiate between three distinct situations: before the financial crisis, during the financial crisis and during the sovereign debt crisis. While Germany, France and Netherlands always appear strongly linked together throughout the different periods (the E3 core), Greece, Ireland, Portugal and Spain have progressively joined in an isolated cluster and separated off from the E3 core both in the MST and the hierarchical tree. The situation of the sovereign debt crisis in the euro area has been recently deteriorating, threatening also other countries in the area. This shows clearly the lack of a permanent regulatory mechanism in the eurozone to prevent and solve this type of problems but also some political fragilities at the EU level to consolidate the European integration. Acknowledgments This work has benefited from partial financial support from the Fundação para a Ciência e a Tecnologia-FCT, under the Multi-annual Funding Project of UECE, ISEG, Technical University of Lisbon. Helpful comments and suggestions from two anonymous referees substantially contributed to improve the paper and are gratefully acknowledged. References [1] C.G. Gilmore, B.M. Lucey, M.W. Boscia, Comovements in government bond markets: a minimum spanning tree analysis, Physica A 389 (2010) 4875–4886. [2] R.N. Mantegna, Hierarchical structure in financial markets, The European Physical Journal B 11 (1) (1999) 193–197. [3] R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics, Cambridge University Press, United Kingdom, 2000. [4] T. Mizuno, H. Takayasu, M. Takayasu, Correlation networks among currencies, Physica A 364 (2006) 336–342. [5] W. Jang, J. Lee, W. Chang, Currency crisis and the evolution of foreign exchange market: evidence from minimum spanning tree, Physica A 390 (2011) 707–718. [6] M. Keskin, B. Deviren, Y. Kocakaplan, Topology of the correlation networks among major currencies using hierarchical structure methods, Physica A 390 (2011) 719–730. [7] R.J. Hill, Comparing price levels across countries using minimum-spanning trees, The Review of Economics and Statistics 81 (1) (1999) 135–142. [8] J.B. Kruskal, On the shortest spanning subtree of a graph and the travelling salesman problem, Proceedings of the American Mathematical Society 7 (1) (1956) 48–50. [9] T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, third ed., The MIT Press, Cambridge, Massachusetts, 2009. [10] R. Rammal, G. Toulouse, M.A. Virasoro, Ultrametricity for physicists, Reviews of Modern Physics 58 (3) (1986) 765–788. [11] M.R. Anderberg, Cluster Analysis for Applications, Academic Press, New York, 1973. [12] B. Efron, Bootstrap methods: another look at the Jacknife, The Annals of Statistics 7 (1) (1979) 1–26. [13] M. Tumminello, F. Lillo, R.N. Mantegna, Hierarchically nested factor model from multivariate data, Europhysics Letters 78 (2007) 30006. [14] M. Tumminello, C. Coronnello, F. Lillo, S. Miccichè, R. Mantegna, Spanning trees and bootstrap reliability estimation in correlation-based networks, International Journal of Bifurcation and Chaos 17 (7) (2007) 2319–2329. [15] M. Tumminello, F. Lillo, R.N. Mantegna, Correlations, hierarchies, and networks in financial markets, Journal of Economic Behavior & Organization 75 (2010) 40–58. [16] J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertész, A. Kanto, Dynamics of market correlations: taxonomy and portfolio analysis, Physical Review E 68 (2003) 1–12. [17] P. Hage, F. Harary, Eccentricity and centrality in networks, Social Networks 17 (1995) 57–63. [18] J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertész, Dynamic asset trees and black monday, Physica A 324 (2003) 247–252.