An analysis of network filtering methods to sovereign bond yields during COVID-19
Raymond Ka-Kay Pang, Oscar Granados, Harsh Chhajer, Erika Fille Legara
AAn analysis of network filtering methods to sovereign bond yieldsduring COVID-19
Raymond Ka-Kay Pang ∗ Oscar Granados † Harsh Chhajer ‡ Erika Fille Legara § September 29, 2020
Abstract
In this work, we investigate the impact of the COVID-19 pandemic on sovereign bond yieldsamongst European countries. We consider the temporal changes from financial correlations usingnetwork filtering methods. These methods consider a subset of links within the correlationmatrix, which gives rise to a network structure. We use sovereign bond yield data from 17European countries between the 2010 and 2020 period as an indicator of the economic healthof countries. We find that the average correlation between sovereign bonds within the COVID-19 period decreases, from the peak observed in the 2019-2020 period, where this trend is alsoreflected in all network filtering methods. We also find variations between the movements ofdifferent network filtering methods under various network measures.
Keywords : Sovereign bonds, Crisis, Financial correlations, Econophysics
The novel coronavirus disease 2019 (COVID-19) epidemic caused by SARS-CoV-2 began in Chinain December 2019 and rapidly spread around the world. The confirmed cases increased in differentcities of China, Japan, and South Korea in a few days of early January 2020, but spread globallywith new cases in Iran, Spain, and Italy within the middle of February. We focus on sovereignbonds during the COVID-19 period to highlight the extent to which the pandemic has influencedthe financial markets. In the last few years, bond yields across the Euro-zone were decreasing un-der a range of European Central Bank (ECB) interventions, and overall remained stable comparedwith the German Bund, a benchmark used for European sovereign bonds. These movements weredisrupted during the COVID-19 pandemic, which has affected the future trajectory of bond yieldsfrom highly impacted countries, e.g., Spain and Italy. However, in the last months, the Europeancentral banks intervened in financial and monetary markets to consolidate stability through anadequate supply of liquidity countering the possible margin calls and the risks of different markets ∗ London School of Economics and Political Science, Department of Mathematics, Houghton Street, London WC2A2AE, UK, Email: [email protected]. † Department of Economics and International Trade, Universidad Jorge Tadeo Lozano, Bogot´a, Colombia, Email:[email protected] ‡ Centre for Biosystems Science and Engineering, Indian Institute of Science, Bangalore 560012, India, Email:[email protected] § Aboitiz School of Innovation, Technology, and Entrepreneurship, Asian Institute of Management, Makati, Philip-pines, Email: [email protected] a r X i v : . [ q -f i n . S T ] S e p nd payment systems. These interventions played a specific role in sovereign bonds because, on theone side, supported the stability of financial markets and, on the other side, supported the gov-ernments’ financial stability and developed a global reference interest rate scheme. Understandinghow correlations now differ and similarities observed in previous financial events are important indealing with the future economic effects of COVID-19.We consider an analysis of sovereign bonds by using network filtering methods, which is partof a growing literature within the area of econophysics [29, 44, 30, 28, 17]. The advantages inusing filtering methods is the extraction of a network type structure from the financial correlationsbetween sovereign bonds, which allows the properties of centrality and clustering to be considered.In consequence, the correlation-based networks and hierarchical clustering methodologies allow usto understand the nature of financial markets and some features of sovereign bonds. It is not clearwhich approach should be used in analyzing sovereign bond yields, and so within this paper, weimplement various filtering methods to the sovereign bond yield data and compare the resultingstructure of different networks.Our analysis shows that over the last decade, the mean correlation peaks in October 2019 andthen decreases during the 2020 period, when COVID-19 is most active in Europe. These dynamicsare reflected across all network filtering methods and represent the wide impact of COVID-19towards the spectrum of correlations, compared to previous financial events. We consider thenetwork centrality of sovereign bonds within the COVID-19 period, which remains consistent withprevious years. These trends are distinctive between filtering methods and stem from the nature ofcorrelations towards economic factors e.g., positive correlations show a stable trend in the individualcentrality, compared with the volatile trends for negative correlations, where central nodes withinthese networks are less integrated in the Euro-area. Although there is a change in the magnitude ofcorrelations, the overall structure relative to the central node is maintained within the COVID-19period.Previous studies have used different methods to analyze historic correlations as random matrixtheory to identify the distribution of eigenvalues concerning financial correlations [27, 39, 23],the approaches from information theory in exploring the uncertainty within the financial system[20, 12], multilayer network methods [1, 7, 46, 24, 18, 40], and filtering methods. Several authorshave used network filtering methods to explain financial structures [31, 37], hierarchy and networksin financial markets [50], relations between financial markets and real economy [34], volatility [51],interest rates [33], stock markets [21, 52, 53, 2], future markets [8] or topological dynamics [45] tolist a few. Also, the comparison of filtering methods to market data has been used for financialinstruments. Birch, et al [10] consider a comparison of filtering methods of the DAX30 stocks.Musmeci, et al [35] propose a multiplex visual network approach and consider data of multiplestock indexes. Kukreti, et al [26] use the S&P500 market data and incorporate entropy measureswith a range of network filtering methods. Aste, et al [5] apply a comparison of network filteringmethods on the US equity market data and assess the dynamics using network measures.In order to evaluate the European sovereign bonds, based on filtering methods, this work isorganized as follows. In Section 2, we describe the network filtering methods and present thedata sets with some preliminary empirical analyses. We apply in Section 3 the filtering methodsto sovereign bond yields and analyze the trend of financial correlations over the last decade andconsider aspects of the network topology. We construct plots in Section 4 representing the COVID-19 period for all methods and analyze the clustering between countries. In Section 5, we discussthe results and future directions. 2 Materials and methods
We introduce a range of network filtering methods and consider a framework as in [31] for sovereignbond yields. We define n ∈ N to be the number of sovereign bonds and bond yields Y i ( t ) of the i th sovereign bond at time-t, where i ∈ { , ..., n } . The correlation coefficients r ij ( t ) ∈ [ − ,
1] aredefined using Pearson correlation as: r ij = (cid:104) Y i Y j (cid:105) − (cid:104) Y i (cid:105)(cid:104) Y j (cid:105) (cid:114)(cid:0) (cid:104) Y i (cid:105) − (cid:104) Y i (cid:105) (cid:1) (cid:16) (cid:104) Y j (cid:105) − (cid:104) Y j (cid:105) (cid:17) , (1)with (cid:104)·(cid:105) denoting the average of yield values. The notion of distance d ij ∈ [0 ,
2] considers thevalues of the entries r ij of the correlation matrix R ∈ [ − , n × n , with d ij = (cid:112) − r ij ). A distanceof d ij = 0 represents perfectly positive correlations and d ij = 2 represents bonds with negativecorrelations. The network filtering methods are then applied to the distance matrix D ∈ [0 , n × n ,where a subset of links (or edges) are chosen under each filtering method. The set of edges isindicated by { ( i, j ) ∈ E ( t ) : nodes i and j are connected } at time-t, defined for each filteringmethod.We define the time frames of financial correlations as X for the set of observations, with n different columns and T rows. From the set of observations X , we consider windows of length 120,which is equal to six months of data values. We then displace δ windows by 10 data points, whichis equal to two weeks of data values, and discard previous observations until all data points areused. By displacing the data in this way, we can examine a time series trend between each window X .We verify the statistical reliability of correlations by using a non-parametric bootstrapping ap-proach as in Efron [15], which is used in Tumminello, et al [48, 49]. We randomly choose rows equalin number to the window length T , allowing repeated rows to be chosen. We compute the corre-lation matrix for this window X ∗ m and repeat the procedure until m samples are generated, whichis chosen at 10,000. The error between data points described in Efron [15] is equal to (1 − ρ ) /T ,where highly positive and negative correlated values ρ have the smallest errors. The minimum spanning tree (MST) method is a widely known approach which has been used withincurrency markets [22], stocks markets [42, 43] and sovereign bond yields [13]. The MST from Table1 considers the smallest edges and prioritizes connections of high correlation to form a connectedand undirected tree network. This approach can be constructed from a greedy type algorithm e.g.Kruskal’s and Prim’s algorithm and satisfies the properties of subdominant ultrametric distancei.e, d ij ≤ max { d ik , d kj } ∀ i, j, k ∈ { , ..., n } .A maximum spanning tree (MaST) constructs a connected and undirected tree network with n − n − n − etwork FilteringMethods Number oflinks (edges) Reference Description Minimum SpanningTree (MST) n − n nodes which minimizesthe total edge weight.Maximum SpanningTree (MaST) n − n nodes which maximizesthe total edge weight.Asset Graph (AG) n − n − n −
2) [32] A planar filtered graph under an as-signed objective function.Table 1: List of network filtering methods.for the given threshold and therefore the connection of unconnected nodes is unknown, relative toconnected components.The triangulated maximal filtering graph (TMFG) constructs a network of 3( n −
2) fixed edgesfor n nodes, similar to the planar maximal filtered graph (PMFG) [47], which has been used toanalyze US stock trends [35]. The algorithm initially chooses a clique of 4 nodes, where edgesare then added sequentially, in order to optimize the objective function e.g., the total edge weightof the network, until all nodes are connected. This approach is non-greedy in choosing edges andincorporates the formation of cliques within the network structure. A TMFG is also an approximatesolution to the weighted planar maximal graph problem, and is computationally faster than thePMFG. The resulting network includes more information about the correlation matrix comparedwith spanning tree approaches, while still maintaining a level of sparsity between nodes. The European sovereign debt has evolved in the last ten years, with some situations affecting theconvergence between bond yields. After the 2008 crisis, European countries experienced a financialstress situation starting in 2010 that affected bond yields, thus the investors saw an excessiveamount of sovereign debt and demanded higher interest rates in low economic growth situationsand high fiscal deficit levels. During 2010-2012, several European countries suffered downgrades intheir bond ratings to junk status that affected investors’ trust and fears of sovereign risk contagionresulting, in some cases, a differential of over 1,000 basis points in several sovereign bonds. Afterthe introduction of austerity measures in GIIPS countries, the bond markets returned to normalityin 2015.The 2012 European debt crisis particularly revealed spillover effects between different sovereignbonds, which have been studied using various time series models e.g. VAR [11, 4] and GARCH[6]. The results showed that Portugal, Greece, and Ireland have a greater domestic effect, Italyand Spain contributed to the spillover effects to other European bond markets and a core groupof ABFN (Austria, Belgium, France, and Netherlands) countries had a lower contribution to thespillover effects, with some of the least impacted countries residing outside of the Euro zone.During the sovereign debt crisis, public indebtedness increased after Greece had to correct thepublic finance falsified data, and other countries created schemes to solve their public financeproblems, especially, bank bailouts. In consequence, the average debt-to-GDP ratio across the4 ountry Mean Variance Skewness Kurtosis Jarque-Bera p-value (JB) ( < . × − )Austria 1.38 1.38 0.55 2.06 217.59 0.00Belgium 1.67 1.93 0.60 2.07 239.53 0.00Czech Republic 1.93 1.34 0.54 2.25 177.64 0.00France 1.48 1.30 0.39 1.88 195.06 0.00Germany 1.02 1.07 0.58 2.42 173.68 0.00Greece 9.57 50.04 1.79 6.37 2500.95 0.00Hungary 4.79 4.56 0.50 1.85 239.06 0.00Iceland 5.86 1.37 -0.89 3.60 368.35 0.00Ireland 2.97 9.17 1.20 3.55 626.33 0.00Italy 3.07 2.28 0.55 2.25 183.10 0.00Netherlands 1.24 1.21 0.50 2.13 181.50 0.00Poland 3.80 1.67 0.46 2.16 160.86 0.00Portugal 4.50 11.67 1.11 3.51 534.11 0.00Romania 4.98 2.18 0.68 2.55 211.98 0.00Spain 2.82 3.47 0.49 1.86 235.85 0.00Switzerland 0.37 0.57 0.58 2.38 178.27 0.00UK 1.89 0.83 0.52 2.65 124.63 0.00Table 2: Summary statistics of the 10Y sovereign bond yield data of 17 European countries fromJanuary 2010 to June 2020.Euro-zone countries rose from 72% in 2006 to 119.5% in 2014, as well as the increase in sovereigncredit risk [3, 9]. After the Fiscal Compact Treaty went into effect at the start of 2013, which definedthat fiscal principles had to be embedded in the national legislation of each country that signed thetreaty, the yield of sovereign bonds started a correction, although some investors and institutionspushed for financial and monetary authorities to introduce an additional decision that permittedthem to include sovereign bonds in their portfolios. The rate interest policy of the European CentralBank helped to consolidate the trust in this kind of asset; the bonds confirmed their adjustmentespecially Germany, France, Spain, during the fourth quarter of 2013, while countries like Greeceand Italy started in 2014 with variations of over 500 basis points during the following months. By2015, all European bonds increased their yields as a result of an adjustment of the market rally of2014.We analyze the sovereign bond yield data for the following countries Austria (AUT), Belgium(BEL), Czech Republic (CZE), France (FRA), Germany (DEU), Greece (GRC), Hungary (HUN),Iceland (ISL), Ireland (IRL), Italy (ITA), Netherlands (NLD), Poland (POL), Portugal (PRT),Romania (ROU), Spain (ESP), Switzerland (CHE), and the UK (GBR). Out of 17 Europeancountries, seven countries are outside the Euro-zone (Czech Republic, Hungary, Iceland, Poland,Romania, Switzerland, and the UK) and three are not within the EU (Iceland, Switzerland, and theUK). Four of the listed countries are part of the G7 and G20 economic groups (Germany, France,Italy and the UK). We consider sovereign bond yields with a 10 year maturity between January2010 and June 2020. This data is taken from the financial news platform . In total, there are2,491 data values for each country with an average of 240 data points within 1 year.Table 2 provides summary statistics of the 10Y bond yield data. The results show Greek yieldsto have the highest values across all statistical measures compared with other countries yields, We compute the correlation matrix for each window X with a displacement of δ between windows,and consider the mean and variance for the correlation matrix. We define the mean correlation r ( t )given the correlations r ij for n sovereign bonds r ( t ) = 2 n ( n − (cid:88) i Variance of correlation Figure 1: The left plot represents the mean and variance of the correlation matrix with windowsof length 120 and δ = 10 days. u ( t ) = 2 n ( n − (cid:88) i 95 in Oct 2019. Thissuggests that a COVID-19 impact was a continuation on the decrease of the mean correlation, andthroughout the punitive lock down measures introduced by the majority of European countries in6eb-Mar 2020. The decreases in mean correlation are also observed within the in the 2012 periodduring the European debt crisis, in which several European countries received EU-IMF bailouts tocope with government debt and in 2016, under a combination of political events within the UK andthe increased debt accumulation by Italian banks. The variance u ( t ) also follows a trend similarto the mean correlation, with the smallest variance of 0 . 002 in October 2019. Within 2020, thevariance increases between sovereign bonds and reflects the differences between the correlations oflow and high yield. We consider the normalized network length L ( t ), which is introduced in Onnela, et al [36] as thenormalized tree length. We define the measure as the normalized network length, as this measure isconsidered for AG and TMFG non-tree networks. The network length is a measure of the mean linkweights on the subset of links E ( t ), which are present within the filtered network on the distancematrix at time-t L ( t ) = 1 { ( i, j ) ∈ E ( t ) } (cid:88) ( i,j ) ∈ E ( t ) d ij ( t ) , (4)with the variance V ( t ) defined on network links Normalized network length Variance of network length MSTTMFGMaSTAG Figure 2: The plots represent the normalized and variance of the network length for MST, TMFG,MaST and AG networks, with windows of length 120 and δ = 10 days. V ( t ) = 1 { ( i, j ) ∈ E ( t ) } (cid:88) ( i,j ) ∈ E ( t ) ( d ij ( t ) − L ( t )) . (5)7he plots in Figure 2 represent the mean and variance of the network length. As each filteringmethod considers a subset of weighted links, the normalized length L ( t ) is monotonic between allmethods and decreases with the increased proportion of positive correlated links within the network.We highlight the movements in the normalized network length during the COVID-19 period, whichis reflected across all filtering methods. This movement is observed within 2016, but only towardsa subset of correlations, in which the network length of the MaST and TMFG increases comparedwith the MST and AG. The relative difference between the normalized networks lengths is leastevident in periods of low variance; this is observed in the 2019-2020 period, where the differencebetween all methods decreases.We find the variance is highest within the TMFG and lowest with the AG approach. Theincreased inclusion of links with a higher reliability error in the TMFG increases the variance,particularly within the 2014-2017 period. The variance of the MST on average is higher comparedwith the MaST, but when considering only the highest correlated links in the AG, the variancedecreases. We define the degree centrality for the node of maximum degree C ( t ) at time- t . This measureconsiders the number of direct links C ( t ) = max i ∈{ ,...,n } n (cid:88) j ∈ E ( t ) d ij > . (6)The mean occupation layer η ( t ) (MOL) introduced in Onnela, et al [36] is a measure of thecentrality of the network, relative to the central node υ ( t ). We define lev i ( t ) as the level of the node,which is the distance of the node relative to υ ( t ), where the central node and nodes unconnectedrelative to the central node have a level value of 0 η ( t ) = 1 n n (cid:88) i =1 lev i ( υ ( t )) . (7)We use the betweenness centrality to define the central node υ ( t ) for the MOL. Introduced inFreeman [16], the betweenness B ( t ) considers the number of shortest paths σ ij ( k ) between i and j which pass through the node k , relative to the total number of shortest paths σ ij between i and j ,where i (cid:54) = j (cid:54) = k B k ( t ) = (cid:88) i (cid:54) = k (cid:88) j (cid:54) = k,j (cid:54) = i σ ij ( k ) σ ij . (8)Within the MST, the degree centrality ranges between 3 to 5 for Euro-zone countries. The trendwithin the MST remains stable, where the central node under degree centrality is associated withmultiple sovereign bonds e.g., Netherlands 19%, Portugal 10% and Belgium 9% across all periods.The MaST has the highest variation, with a centralized network structure in some periods e.g., C ( t )of 16, forming a star shaped network structure. This is usually associated with Greece, Iceland andHungary, which are identified as the central node 55% of the time. The degree centrality on averageis naturally highest with the TMFG, under a higher network density, where the central nodes areidentified as Hungary and Romania sovereign bonds, similar to the MaST. The AG identifies theNetherlands and Belgium within the degree centrality, under a higher proportion of 25% and 13%compared with the MST. 8 012 2014 2016 2018 2020 2012 2014 2016 2018 202046810121416 Degree centrality Mean occupation layer Figure 3: The plots represent the degree centrality and mean occupation layer for MST, TMFG,MaST and AG networks, with windows of length 120 and δ = 10 days.Within Figure 3, the MOL on average is smallest for the AG, because of the 0 level values fromunconnected nodes, in which an unconnected node is present within 94% of considered windows.We find that the nodes within the TMFG are closest within the network, where the central node isdirectly or indirectly connected for all nodes, with an average path length of 1 . We analyze the temporal changes of sovereign bond yields between October 2019 and June 2020.The associated link weights on each filtering method for window X are the proportions in whichthe link appears within the correlation matrix, under the statistical reliability, across all samples m for the randomly sampled windows X ∗ m .Under the MST, Austria has the highest degree centrality of 4. The network also exhibits clustersbetween southern European countries connected by Spain, and the UK towards Polish and Germansovereign bond yields. Within the network, there is a connection between all ABFN countries, butcountries within this group also facilitate the connecting component within GIIPS countries, whereBelgium is connected with Spain and Irish sovereign bonds. The UK and eastern European countriesremain on the periphery, with ABFN countries occupying the core of the network structure.For the MaST in Figure 4, there exists a high degree centrality for Polish sovereign bonds betweenwestern European countries e.g., France and Netherlands. This contrasts to the observed regionalhub structure within the MST, with the existence of several sovereign bonds with high degreecentrality in the network. The UK remains within the periphery of the MaST structure when9onsidering anti-correlations, and shows UK bond yields fluctuate less with movements of otherEuropean bonds compared with previous years. This is also observed for sovereign bonds for othercountries with non-Euro currencies such as Czech Republic, Hungary, and Iceland. Figure 4: The plots represent the minimum and maximum spanning trees for the October 2019 -June 2020 period. The link weights represent the proportion in which the link is identified in thenetwork filtering method across all samples. Figure 5: The plots represent the triangulated filtering maximal graph (left) and asset graph (right)for the October 2019 - June 2020 period. The link weights represent the proportion in which thelink is identified in the network filtering method across all samples.We find nodes within the TMFG to have the highest degree in Iceland at 13 and Poland at10. Although the MST is embedded within the TMFG network structure, a high resemblanceis observed to links from the MaST, where 69% of links which are present within the MaST arecommon in both networks. There is also the associated degree centrality of the MaST, whichis observed within the TMFG connected nodes. Under the TMFG, nodes have a higher degreeconnectivity when considering an increased number of links, this is the case for the UK, which10as 9 links compared with other spanning tree approaches. The AG exhibits three connectedcomponents between western European countries, southern European countries and the UK witheastern European countries. These unconnected nodes within the AG are associated with non-Euroadopting countries, with the remaining countries connected in an individual component. By solelyconsidering the most positive correlations, we observe the formation of 3-cliques between countries,which is prevalent within the western European group of 6 nodes.The average statistical reliability is highest at 0 . 92 within the MaST and AG, 0 . 89 for the MSTand 0 . 82 for the TMFG. Under the TMFG, the increased inclusion of links with a lower magnitudein correlations decreases the reliability in link values. Other filtering approaches which consider asmaller subset can still result in low reliability values between some nodes e.g. Austria and Romaniaat 0 . 51 in the MST, Germany and Netherlands at 0 . 47 in AG.Under various constraints, we observe a commonality between sovereign bonds across networkfiltering methods. We find for tree networks, that Euro-area countries have a high degree central-ity and countries with non-Euro currencies e.g. Czech Republic and the UK are predominatelylocated within the periphery of the network. This is further observed within the AG, where cliquesare formed between GIIPS and ABFN countries, which is distinctive during the COVID-19 periodcompared with previous years. The anti-correlations within the MaST inform the trends of thenegative correlations between eastern European countries and other European countries. By con-sidering the TMFG with an increased number of links for positive correlations, we find similaritieswith the MaST degree centrality. As a response to the COVID-19 pandemic, most countries implemented various socio-economicpolicies and business restrictions almost simultaneously. An immediate consequence was an increasein yield rates for these nations. The resulting upward co-movement and upward movements inother yield rates explain the decrease in the mean correlation in bond dynamics, coinciding withthe pandemic outbreak. Thus, understanding the dynamics of financial instruments in the Euroarea is important to assess the increased economic strain from events seen in the last decade.In this paper, we consider the movements of European sovereign bond yields for network filteringmethods, where we particularly focus on the COVID-19 period. We find that the impact of COVID-19 decreased the mean correlation, which was reflected within the normalized network length ofall filtering methods. The network topology remained consistent with previous years, in whichthe trends between approaches were distinctive, where ABFN countries were central nodes whenconsidering positive correlations and Eastern European countries within negatively correlated typenetworks. We identified the network structures of filtering methods within the COVID-19 period,which showed three main clusters within positive correlated networks, and a centralization towardsEastern European countries of negative correlation. These networks represent a level of fragmen-tation within the correlation trends which is reemerging between groups, based on the economicimpact of COVID-19 on sovereign bonds.However, depending on the terms of each bond, the European bond market reacted positivelyafter central banks (e.g., Bank of England, European Central Bank, Swiss National Bank) increasedtheir financial programs directed to alleviating the financial pressure on markets and to providingfinancial liquidity to issuers. Namely, the bond purchase programs had aimed to consolidatedmarket recovery and help to displacing investors toward other financial assets. As a result, pricesrecovered and remain close to the highs of the 2020 second quarter but not at the same level beforeMarch’s stress situation, especially in 10Y bonds. Additionally, if liquidity provided by Central11anks starts to drop off, the market dynamics could adjust to economic performance and not itsfinancial performance. 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