ACTA PHYSICA POLONICA A No. 1 Vol. 138 (2020)

Proceedings of the 10th Polish Symposium on Physics in Economy and Social Sciences (FENS 2019)

Uncovering Hierarchical Structureof International FOREX Market by Using Similarity Metric

between Fluctuation Distributions of Currencies

A. Chakrabortya, S. Easwarana and S. Sinhaa,b,∗

aThe Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, IndiabHomi Bhabha National Institute, Anushaktinagar, Mumbai 400094, India

Doi: 10.12693/APhysPolA.138.105 ∗e-mail: sitabhra@imsc.res.in

The decentralized international market of currency trading is a prototypical complex system havinga highly heterogeneous composition. To understand the hierarchical structure relating the price move-ment of different currencies in the market, we have focused on quantifying the degree of similaritybetween the distributions of exchange rate fluctuations. For this purpose we use a metric constructedusing the Jensen-Shannon divergence between the normalized logarithmic return distributions of the dif-ferent currencies. This provides a novel method for revealing associations between currencies in termsof the statistical nature of their rate fluctuations, which is distinct from the conventional correlation-based methods. The resulting clusters are consistent with the nature of the underlying economies, butalso show striking divergences during periods of major international crises.

topics: Exchange rates; Fluctuation distributions, International currency market, Jensen–Shannondivergence

1. Introduction

One of the biggest challenges facing physiciststrying to understand natural and man-made com-plex systems is to uncover principles underlyingtheir collective dynamics. Socio-economic phenom-ena, in particular, provide examples of systems thatexhibit a high degree of heterogeneity in the dy-namical behavior exhibited by their components [1].Unlike the dynamical systems traditionally investi-gated by physicists, such as arrays of coupled os-cillators, coupling between the different parts ofsuch complex systems do not necessarily result inthe components displaying similar behavior (anal-ogous to synchronization in oscillator arrays [2]).Instead, autonomous agents in economic systemsmay employ various strategies which could mani-fest in different segments of the system exhibitingqualitatively distinct behavior [3]. This is observed,for instance, in a financial market whose collectivedynamics can be described in terms of the pricemovements of the numerous assets that are tradedin it. These movements are not independent of eachother as the components of this complex system“interact” through the actions of agents (rangingfrom individual investors to large financial institu-tions) buying and selling these assets. The statis-tical properties of the asset price fluctuations candiffer remarkably from one asset to another, whichcan be linked to intrinsic, as well as extrinsic fac-tors. To understand the mechanisms through whichsuch properties emerge, we first need to describe

accurately how the fluctuations of the different com-ponents are related to each other.

In this paper, we consider the largest financialmarket in the world, the international currency orforeign exchange (FOREX) market [4]. To pro-vide a comparison, the average daily turnover inthe FOREX market was 5.3 × 1012 USD in April2013 while the average daily turnover for NewYork Stock Exchange, the biggest stock market inthe world, in 2013 was 1.69×1011 USD. It is a decen-tralized global market for trading currencies whichplays a key role in the international monetary and fi-nancial system. Furthermore, it is among the freestand most competitive markets in the world. Theforeign exchange market is particularly importantto complex systems researchers because of the largevolumes of data it is able to provide for statisticalanalysis.

Currency exchange rates fluctuate because ofa number of reasons, including the balance of trade,interest rate, monetary policies, etc. As such itis very difficult to discern the individual role thatthese different factors play in influencing the pricefluctuations of a currency. Instead, we propose touncover the hierarchical structure of the network re-lating the different currencies in terms of their fluc-tuation behavior. We use a metric for measuringthe distance between pairs of exchange rate fluc-tuation distributions to cluster the currencies (andby extension, the economies to which they belong)into similarity groups. This provides a novel ap-proach to grouping components in complex systems

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according to the statistical characteristics of theirdynamical behavior, distinct from the widely em-ployed method of using cross-correlations betweenthe time-series [5–9], which have limitations [10, 11].We have also performed a temporally resolved anal-ysis of the nature of the distributions at differentperiods that shows strong disruption in the other-wise regular pattern of systematic variations dur-ing the global financial crisis of 2008, indicat-ing its deep-rooted nature affecting the real econ-omy [12, 13].

2. Materials and Methods

In this study we have analyzed a data-set com-prising the daily exchange rates of N = 75 cur-rencies (see Table I) with respect to the US Dol-lar (USD), for the period October 23, 1995 toFebruary 10, 2016, corresponding to τ = 7416days. Specifically, we use the midpoint betweenthe bid and ask rates for 1 USD against a givencurrency. The source of the data is an onlinearchive of historical interbank market rates madeavailable by the Oanda Corporation [14] which pro-vides internet-based currency trading and informa-tion services. The interbank rate (also referred toas spot rate) is used for large volume transactionscarried out by banks and financial institutions, andis the exchange rate that is typically reported inthe media. From the online archive we have ob-tained the rate for each day in the period underconsideration, averaged over all quotations collectedwithin the previous 24 h period (terminating atmidnight UTC) from various frequently updatedsources around the world, including online currencytrading platforms, market data vendors, and con-tributing financial institutions. The USD is cho-sen as the numeraire as it is the preferred currencyfor most international transactions and continuesto remains the reserve currency of choice for manyeconomies [15, 16].

To see how the use of a particular numeraire, viz.,the USD, may have introduced bias in the results re-ported here, we have also carried out our analysis byexpressing the exchange rates of currencies in termsof Special Drawing Rights (SDR), a supplementaryinternational reserve asset created by the Interna-tional Monetary Fund (IMF), whose value is deter-mined by a basket of currencies [17]. The compo-sition of the basket is altered regularly to ensurethat it reflects the relative importance of differentcurrencies in international trade and finance. Dailyquotes for SDR in terms of USD is available fromthe IMF [18] for all days (except those on whichIMF is closed for business, which include weekends).We have used this to create a daily exchange ratetime-series with SDR as the numeraire, spanningthe 5210 days for which data is available in the pe-riod under study.

The 75 currencies considered in our study arechosen based on the exchange rate regime fol-lowed by them (see Table I). The information about

the regime type of each currency has been obtainedfrom the same online source from which we havecollected the daily rates [14], supplemented with in-formation from the website of another online foreignexchange tools and services company [19]. We haveexplicitly avoided any currency whose exchange ratewith respect to USD does not vary over time. Mostof the currencies chosen for our analysis are eitherfreely floating under the influence of market forcesor managed to an extent with no pre-determinedpath. The remainder, although pegged to USD orsome other important currency (such as EUR), doexhibit variation within a band which may be eitherfixed or can change over time. We have also classi-fied the economy of the countries to which these cur-rencies belong by using the Morgan Stanley CapitalInternational (MSCI) market classification frame-work [20]. Using multiple criteria including the sus-tainability of economic development, the number ofcompanies meeting certain size and liquidity crite-ria, ease of capital flow, as well as efficiency andstability of the institutional framework, MSCI clas-sifies the economies into three categories, viz., devel-oped, emerging and frontier markets (see Table I).An important point to note is that for the periodprior to January 1, 1999, when the EUR was firstintroduced, we have used in its stead the exchangerate for the ECU (European Currency Unit).

3. Results

Instead of analyzing the raw price information,the fluctuations in the exchange rates of 75 curren-cies (see Table I) with respect to the US Dollar overthe period 1995–2016 is measured. To make the re-sult independent of the unit of measurement, thevariation in the exchange rate Pi(t) of the i-th cur-rency (i = 1, . . . , N) at time t is quantified by itslogarithmic return defined over a time-interval ∆tas

Ri(t,∆t) = ln(Pi(t+ ∆t)

)− ln

(Pi(t)

).

Since our data comprises the daily exchange rates,we consider ∆t = 1 day. The standard devia-tion σ of the returns measures the intensity of fluc-tuations in the exchange rates (volatility) whichvaries for different currencies. Hence, to comparebetween the return distributions of different curren-cies, we normalize the returns of each currency i bysubtracting the mean value

〈Ri〉 =

T∑t=1

Ri(t)/T

and dividing by the standard deviation

σi(t) =

√√√√ 1

T − 2

∑t′ 6=t

[Ri(t′)− 〈Ri〉]2

(removing the self contribution from the measure ofvolatility), obtaining the normalized return,

ri(t) = (Ri(t)− 〈Ri〉)/σi(t).

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TABLE I

The currencies of developed (1–14), emerging (15–44) and frontier (45–75) economies considered in the study.The columns indicate the currency code along with the nature of the exchange rate regime (as obtained fromOanda and XE sites), the character of the economy (as categorized by MSCI) and the geographical region of thecorresponding countries.

Sl. no. Currency CodeExchange Rate Regime

(Oanda, XE)Market Type

(MSCI)Region

1 Canadian Dollar CAD floating developed Americas2 Danish Krone DKK pegged within horizontal band developed Europe3 Euro EUR floating developed Europe4 Great Britain Pound GBP floating developed Europe5 Iceland Krona ISK floating developed Europe6 Norwegian Kroner NOK floating developed Europe7 Swedish Krona SEK floating developed Europe8 Swiss Franc CHF floating developed Europe9 Israeli New Shekel ILS floating developed Middle East10 Australian Dollar AUD floating developed Asia-Pacific11 Hong Kong Dollar HKD fixed peg developed Asia-Pacific12 Japanese Yen JPY floating developed Asia-Pacific13 New Zealand Dollar NZD floating developed Asia-Pacific14 Singapore Dollar SGD floating developed Asia-Pacific15 Bolivian Boliviano BOB crawling peg emerging Americas16 Brazilian Real BRL floating emerging Americas17 Chilean Peso CLP floating emerging Americas18 Colombian Peso COP floating emerging Americas19 Dominican Republic Peso DOP floating emerging Americas20 Mexican Peso MXN floating emerging Americas21 Peruvian Nuevo Sol PEN floating emerging Americas22 Venezuelan Bolivar VEB fixed peg emerging Americas23 Albanian Lek ALL floating emerging Europe24 Czech Koruna CZK floating emerging Europe25 Hungarian Forint HUF pegged within horizontal band emerging Europe26 Polish Zloty PLN floating emerging Europe27 Russian Rouble RUB floating emerging Europe28 Turkish Lira TRY floating emerging Europe29 Algerian Dinar DZD floating emerging Africa30 Cape Verde Escudo CVE fixed peg emerging Africa31 Egyptian Pound EGP floating emerging Africa32 Ethiopian Birr ETB floating emerging Africa33 Mauritius Rupee MUR floating emerging Africa34 Moroccan Dirham MAD fixed peg emerging Africa35 South African Rand ZAR floating emerging Africa36 Tanzanian Shilling TZS floating emerging Africa37 Chinese Yuan Renminbi CNY fixed peg emerging Asia38 Indian Rupee INR floating emerging Asia39 Indonesian Rupiah IDR floating emerging Asia40 Papua New Guinea Kina PGK floating emerging Asia41 Philippine Peso PHP floating emerging Asia42 South Korean Won KRW floating emerging Asia43 Taiwan Dollar TWD floating emerging Asia44 Thai Baht THB floating emerging Asia

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TABLE I (cont.)

Sl. no. Currency CodeExchange Rate Regime

(Oanda, XE)Market Type

(MSCI)Region

45 Guatemalan Quetzal GTQ floating frontier Americas46 Honduran Lempira HNL crawling peg frontier Americas47 Jamaican Dollar JMD floating frontier Americas48 Paraguay Guarani PYG floating frontier Americas49 Trinidad Tobago Dollar TTD floating frontier Americas50 Croatian Kuna HRK floating frontier Europe51 Kazakhstan Tenge KZT floating frontier Europe52 Latvian Lats LVL fixed peg frontier Europe53 Botswana Pula BWP crawling peg frontier Africa54 Comoros Franc KMF fixed peg frontier Africa55 Gambian Dalasi GMD floating frontier Africa56 Ghanaian Cedi GHC floating frontier Africa57 Guinea Franc GNF fixed peg frontier Africa58 Kenyan Shilling KES floating frontier Africa59 Malawi Kwacha MWK floating frontier Africa60 Mauritanian Ouguiya MRO floating frontier Africa61 Mozambique Metical MZM floating frontier Africa62 Nigerian Naira NGN floating frontier Africa63 Sao Tome and Principe Dobra STD fixed peg frontier Africa64 Zambian Kwacha ZMK floating frontier Africa65 Jordanian Dinar JOD fixed peg frontier Middle East66 Kuwaiti Dinar KWD fixed peg frontier Middle East67 Syrian Pound SYP fixed peg frontier Middle East68 Brunei Dollar BND fixed peg frontier Asia69 Bangladeshi Taka BDT floating frontier Asia70 Cambodian Riel KHR floating frontier Asia71 Fiji Dollar FJD fixed peg frontier Asia72 Lao Kip LAK floating frontier Asia73 Pakistan Rupee PKR floating frontier Asia74 Samoan Tala WST fixed peg frontier Asia75 Sri Lankan Rupee LKR floating frontier Asia

3.1. Hierarchical clustering based on similarityof fluctuations distribution

We have investigated the inter-relation betweenthe different currencies by considering how similarthey are in terms of the nature of their fluctua-tions. For this we have measured the differencebetween the normalized logarithmic return distri-butions of each pair of currencies using a prob-ability distance metric, viz., the similarity dis-tance D(Pi, Pj) between a pair of return distri-butions Pi(r) and Pj(r) [21]. It is defined asthe square root of the Jensen-Shannon (JS) diver-gence [22], which in turn can be defined in terms ofthe Kullback-Leibler (KL) divergence for a pair ofprobability distributions Pi(x) and Pj(x) of a dis-crete random variable x:

KL(Pi, Pj) =∑x∈X

Pi(x) log

(Pi(x)

Pj(x)

).

The limitations of the KL divergence, viz., that itis asymmetric and is, moreover, undefined when ei-ther Pi or Pj is zero for any value of x ∈ X, isovercome by the JS divergence which is defined as:

JS(Pi, Pj) =1

2KL(Pi, P ) +

1

2KL(Pj , P ),

where P = (Pi + Pj)/2. As returns are con-tinuous variables, in order to calculate the diver-gences between their distributions we have dis-cretized the values using a binning procedure (in-volving intervals of width ∆r ∼ 0.05). The samebinning procedure has been also used for the anal-ysis described later in which the data is split intofour non-overlapping segments belonging to differ-ent time periods. Note that, the related general-ized JS measure has been used earlier to measurethe similarity of tick frequency spectrograms for dif-ferent currency exchange rates [23].

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The matrix of similarity distances D between allpair of currencies is used for clustering them in a hi-erarchical manner. Note that this approach is dis-tinct from earlier attempts of hierarchical classifi-cation which use synchronous cross-correlation be-tween exchange rate fluctuations in order to iden-tify clusters of related currencies (see, e.g., [24–26]).Given a set of nodes to be clustered and a matrixspecifying the distances between them, the methodof hierarchical clustering [27] involves (i) initiallyconsidering each node as a cluster, (ii) mergingthe pair of clusters which have the shortest distancebetween them, (iii) re-computing the distance be-tween all clusters, and repeating the steps (ii) and(iii) until all nodes are merged into a single cluster.Clustering methods can differ in the way the inter-cluster distance is calculated in step (iii). If this dis-tance is taken as the maximum of the pairwise dis-tances between members of one cluster to membersof the other cluster, it is known as complete-linkageclustering. On the other hand, in the single-linkageor nearest neighbor clustering, the minimum ofthe distance between any member of one cluster toany member of the other cluster is chosen. Average-linkage clustering, as the name implies, considersthe mean of the pairwise distances between mem-bers of the two clusters. Note that, the hierarchi-cal clustering obtained using the complete-linkagemethod will be same as one obtained using a thresh-old distance to define membership of a cluster, whilethat constructed using the single-linkage method isidentical to the minimal spanning tree [28].

We have shown the hierarchical clustering (us-ing complete-linkage clustering) of the different cur-rencies considered in this study in Fig. 1 usinga polar dendrogram representation. We note thatthe technique divides the currencies at the coarsestscale into two groups, the smaller of which is al-most exclusively composed of currencies from fron-tier economies (the sole exception being VEB whichbelongs to an emerging economy) that are char-acterized by large fluctuations. Although some ofthese currencies (e.g., KWD, TTD and VEB) be-long to countries having high GDP per capita, theytypically also have a high Theil index [29] indicatingthat their foreign trade is dominated by the exportof a few key products (e.g., crude oil in the case ofKuwait, Trinidad & Tobago and Venezuela). Theircurrencies are therefore potentially highly suscepti-ble to fluctuations in the worldwide demand.

Focusing now on the larger group, we observethat it is further divided broadly into two sub-groups, one of which is dominated by relativelystable currencies from developed and emergingeconomies, with only a single frontier economy be-ing represented (viz., LVL) which has a relativelyhigher GDP per capita than the other frontiereconomies. The other subgroup is composed of cur-rencies from emerging, as well as frontier economies(with the exception of HKD and ISK which belongto developed economies). The occurrence of these

Fig. 1. Polar dendrogram representation obtainedby hierarchical clustering of different currenciesin terms of the statistical distance between theirfluctuation distributions. A similarity distance Dobtained from the Jensen-Shannon divergence be-tween the corresponding normalized logarithmic re-turn distributions of a pair of currencies has beenused as the clustering metric. The currencies havebeen clustered using complete linkage algorithmand the height of a branch measures the linkagefunction d, i.e., the distance between two clus-ters. Using a threshold of dth = 0.47, the largestnumber of distinct clusters (viz., 22 clusters repre-sented by the different colored branches of the den-drogram, black branches indicating isolated nodes)can be identified. The largest cluster comprisesonly currencies of developed economies with the ex-ception of CZK and HUF (belonging to emergingeconomies). Currencies are distinguished accord-ing to the average annual GDP per capita 〈g〉 ofthe corresponding economy (represented by fontsize, which scales logarithmically with 〈g〉) andthe geographical region to which they belong (rep-resented by font color, viz., black: Americas, red:Europe, blue: Middle East, magenta: Asia-Pacific,green: Africa and brown: Asia).

latter two in this subgroup can be related to the se-vere financial crises encountered by these economiesat different times, viz., the 2003 SARS crisis inthe case of Hong Kong and the 2008 banking col-lapse in the case of Iceland, during the period un-der consideration. The largest number of similarityclusters into which the currencies can be grouped isobtained for a threshold value of dth = 0.47. Mostof the currencies belonging to developed economiesare seen to occur in the largest cluster consisting of12 currencies, indicating that these economies havea relatively similar high degree of stability for theircurrency exchange rates. As expected, almost all ofthem have high GDP per capita. The only membersof this highly stable currency cluster which are not

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The 100 years anniversary of the Polish Physical Society — the APPA Originators

in the developed category (viz., CZK and HUF) be-long to countries that are members of the EuropeanUnion, and whose economies are therefore highlyintegrated with the other members of this cluster.The second largest cluster comprising 5 currenciesalso appear to bring together economies that havesimilar GDP per capita even though they vary intheir market classification from developed (SGD)to emerging (MXN, PLN and ZAR) and frontier(LVL). The remaining clusters are mostly pairs (orat most triplets) of currencies.

To ensure that the hierarchical organization ev-ident from the dendrogram is not too sensitivelydependent on the specific numeraire we have used,in Fig. 2 we have compared dendrograms con-structed using exchange rates expressed in termsof SDR and USD. As the daily quotes for SDR isavailable only for a subset of the dates for whichwe have information about the rates expressed inUSD, for a fair comparison we have constructedthe two dendrograms using the set of dates com-mon between the two classes of data. As is evident,the clusters of currencies observed in the two hi-erarchical structures are qualitatively similar. Theresemblance between the two dendrograms can bequantified by using the cophenetic distance cor-relation coefficient (CDCC). This is computed bymeasuring the cophenetic distance between pairsof elements, which is defined as the height ofthe branch where two elements become mem-bers of the same cluster in the dendrogram, andthen obtaining the correlation between the corre-sponding distances in the two dendrograms [30–32].The value of CDCC between the dendrogramsshown in panels (a) and (b) of Fig. 2 is foundto be 0.52, suggesting a high degree of similaritybetween them.

As the shape of a distribution can be described byits moments [33], to delve deeper into whether anyspecific property of the fluctuation distribution isresponsible for the hierarchical clustering reportedabove, we have investigated the relation betweenthe lowest order moments of the distributions forthe different currencies. As we consider normal-ized returns, the first and second moments (cor-responding to mean and variance) of all the dis-tributions are identical. Furthermore, as most ofthe return distributions are approximately symmet-ric (large deviations being shown only by NGN,DZD, MWK, PGK, VEB, MZM, TRY, FJD andSTD), the distributions also do not markedly dif-fer in terms of their skewness (related to the thirdmoment). Thus, we focus on the kurtosis, mea-sured in terms of the fourth moment, which quanti-fies the probability of occurrence of extreme valuesreflected in the heavy-tailed nature of the distri-bution. Considering higher order moments is notlikely to provide any further information as it isknown that specifying a finite number of momentsof a distribution only determines its tail but notits bulk [34].

Fig. 2. Comparison between hierarchical cluster-ing of currencies according to the statistical dis-tance between corresponding fluctuation distribu-tions when the exchange rates are expressed interms of different numeraires, viz., (a) SDR and(b) USD. As in Fig. 1, clustering is performed us-ing complete linkage algorithm. In (a), using athreshold of dth = 0.78, 5 distinct clusters, repre-sented by the different colored branches of the den-drogram (with black branches indicating isolatednodes), can be identified. Note that the SDR dataused for constructing the dendrogram is availableonly for 5210 days during the period October 23,1995 to February 10, 2016. Using this same setof dates, we have constructed in (b) a dendrogramfrom the data in which currency exchange rates areexpressed in terms of USD, for comparison with (a).For a threshold of dth = 1.30, we observe 5 distinctclusters in this case also. The cophenetic distancecorrelation coefficient between the two dendrogramsis 0.52, suggesting a high degree of similarity be-tween them.

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Fig. 3. Polar dendrogram representation obtainedby hierarchical clustering of different currencies onthe basis of how similar they are in exhibiting ex-treme values in their exchange rates, i.e., in terms ofthe heavy-tailed nature of the fluctuation distribu-tions. The relative difference between the kurtosisof corresponding normalized logarithmic return dis-tributions of a pair of currencies has been used asthe clustering metric. As in Fig. 1, clustering isperformed using complete linkage algorithm. Us-ing a threshold of dth = 0.08, the largest num-ber of distinct clusters (viz., 24, represented bythe different colored branches of the dendrogram,with black branches indicating isolated nodes) canbe identified. Unlike the clusters obtained usingJensen-Shannon divergence shown in Fig. 1, all clus-ters here are relatively small. The relative dis-similarity between the clusters seen here and thatin Fig. 1 is quantitatively indicated by the valueof the cophenetic distance correlation coefficientbetween the two dendrograms being 0.2. How-ever, most of the currencies belonging to developedeconomies do seem to occur close to each other, withthe exceptions of CHF, HKD and ISK (resultingfrom rare instances of extremely large deviations intheir exchange rates). As in Fig. 1, currencies aredistinguished according to the average annual GDPper capita 〈g〉 of the corresponding economy (rep-resented by font size, which scales logarithmicallywith 〈g〉) and the geographical region to which theybelong (represented by font color, viz., black: Amer-icas, red: Europe, blue: Middle East, magenta:Asia-Pacific, green: Africa and brown: Asia).

For each pair of currencies i, j, we have measuredthe relative difference between the kurtosis α4 oftheir fluctuation distributions as

δ(αi4, α

j4) =

|αi4 − α

j4|

〈αi4, α

j4〉,

where 〈. . .〉 represents the mean. The matrixof these differences is then used to hierarchicallycluster the currencies using the complete-linkage

method. Figure 3 shows the results of the cluster-ing using a polar dendrogram representation. Thereare clearly discernible differences with the struc-ture shown in Fig. 1, indicating that there are as-pects of the distribution other than kurtosis (andhence, the behavior at the tails) which are respon-sible for the similarity in the nature of fluctuationsof the currencies belonging to a cluster. The differ-ences can partly be ascribed to the fact that the kur-tosis is strongly affected by rare (or even single in-stances of) extreme events. For example, this ispossibly the explanation for the adjacency of cur-rencies DZD, NGN and MWK seen in Fig. 3, asall three experienced a large deviation in their ex-change rates on November 7, 1995. This has beenexplicitly verified by reconstructing the dendrogramafter discarding this specific extreme return valueand noting that the three currencies no longer oc-cur in close proximity.

We also note that this single extreme eventstrongly affects the synchronous cross-correlationsbetween the currency movements over the entire pe-riod under study, resulting in the three currenciesbeing identified as a prominent cluster moving inde-pendently of other currencies [9]. This is apparentfrom the dendrogram shown in Fig. 4 which rep-resents relations between the currencies in termsof the degree of synchrony in their fluctuations.It underlines one of the weaknesses of using cross-correlations between asset price fluctuations for in-ferring significant dynamical relationships betweenthe assets, as a lone outlier may be sufficient to sig-nificantly bias the results obtained.

Another example of a rare event influencingthe kurtosis is the occurrence of CHF (belonging toa developed economy) far from the neighborhood ofthe other currencies of developed economies. Thiscan be traced to a rapid rise in its exchange rateimmediately following the depegging of CHF fromEUR in January 2015. If the clustering is per-formed by considering only the time period pre-ceding this event, we note that CHF occurs alongwith the other currencies belonging to economieshaving similar character. We do note, however,that, in general, currencies belonging to similareconomies tend to be grouped in neighboring loca-tions in the dendrogram, which could be related tothe fact that the kurtosis of the fluctuation distri-bution of a currency is related to the overall pros-perity, as measured by the gross domestic productper capita, of the corresponding economy [35].

3.2. Temporal evolution of the system properties

In the analysis presented above we have consid-ered the entire temporal duration which our data-set spans. However, as the world economy under-went significant changes during this period, mostnotably, the global financial crisis of 2008, it isof interest to see how the hierarchical structure ofthe currency market has evolved over time. Forthis purpose we divide the data-set into four equal

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Fig. 4. Polar dendrogram representation of the hi-erarchical clustering of currencies using syn-chronous cross-correlations between the time-seriesof their exchange rate fluctuations. Distance be-tween each pair of currencies i, j is computed fromthe Pearson correlation Cij between their normal-ized returns over the period under study as dij =√

2(1− Cij), following Ref. [36]. As in Fig. 1, clus-tering is performed using complete linkage algo-rithm. For a threshold of dth = 1.4, 13 distinct clus-ters represented by the different colored branchesof the dendrogram (with black branches indicatingisolated nodes) can be identified. Note the clus-ter of three currencies (comprising DZD, NGN andMWK) at the top. Their apparently strong associ-ation arises from a single extreme fluctuation eventthat occurred on November 7, 1995.

non-overlapping periods comprising 1853 days, cor-responding to Period I: Oct 23, 1995–Nov 18, 2000,Period II: Nov 19, 2000–Dec 16, 2005, Period III:Dec 17, 2005–Jan 13, 2011 and Period IV: Jan 14,2011–Feb 10, 2016. Note that Period III corre-sponds to the crisis of the global economy span-ning 2007–2009 that also affected the FOREX mar-ket [37], allowing us to contrast the structure ofthe currency market in the pre-crisis era (PeriodsI and II) with the situation that prevailed duringthe crisis, as well as, post-crisis (Period IV). Foreach of these periods, we carry out the same proce-dures as outlined earlier for the entire data-set.

The hierarchical clustering of the currencies showstriking changes over time when they are con-structed separately for each of the four intervalsmentioned above (Fig. 5). The degree of simi-larity between the clusters seen in the four peri-ods can be quantitatively indicated by the cophe-netic distance correlation coefficient calculated forpairs of successive periods.The corresponding val-ues, viz., CDCC(I,II) = 0.23, CDCC(II,III) = 0.12and CDCC(III,IV) = 0.46, suggest that a major dis-ruption took place in the hierarchical structure inthe period immediately leading upto the crisis.

As seen for the data-set covering the entire pe-riod, the method classifies the currencies in the pre-crisis era broadly into two groups with one groupcomprising the relatively stable currencies of de-veloped and emerging economies. However, duringand following the crisis, the broad division sepa-rates only a small group of frontier economy cur-rencies (with the exception of VEB in Periods IIIand IV, and ISK in Period IV) from the rest. Whilethe developed economies largely remain members ofthe same group even after the crisis, the relative po-sition of most currencies show large changes. Evenamong the developed economy currencies, we ob-serve that ISK moves away from its peers follow-ing the crisis. This is possibly related to the Ice-landic crisis of 2008–2010 that saw a complete col-lapse of its financial system [38, 39]. The severity ofthe global crisis of 2008 is also reflected in the move-ment of GBP away from the neighborhood of cur-rencies belonging to the other developed economiesin Period III, although it subsequently returned inPeriod IV to the group in which it belonged.

4. Discussion

In this work, we have used the Jensen-Shannondivergence, that measures the difference betweentwo distributions, in a novel method for clusteringcurrencies into similarity groups based on the sta-tistical behavior of their fluctuations. In princi-ple, one can use other definitions for the distancebetween probability distributions, such as the to-tal variation distance and the Bhattacharyya dis-tance [40]. In the case of the international currencymarket this approach is particularly apt, comparedto correlation-based methods, for reconstructingthe hierarchical network relating the movementsof different currencies. This is because trading indifferent currencies is distributed across the globe,with activity in different locations (focused aroundspecific sets of currencies) peaking at different timesbecause of the corresponding time-zones. This isquite distinct from the situation for a stock marketlocated in a specific geographical location, wheretrading in all assets begins and ends at the sametime, which allows one to compute the synchronouscorrelations between the movements of the differ-ent assets. In contrast, when information abouttrading in different currencies is pooled togetherover a given 24 hour period, one may be comparingthe movement of a particular currency at the be-ginning of the trading day in the location wheretrading in it is most intense with that of anothercurrency at the close of the trading day in a dif-ferent location which focuses on trading in the lat-ter. This makes it essentially impossible to obtaina truly synchronized correlation matrix for the cur-rency market, unlike the situation in the stock mar-ket. Thus, the network of relations between curren-cies inferred from a correlation-based approach islikely to have inaccuracies. The technique based

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Fig. 5. Temporal evolution of the hierarchical clustering of different currencies according to the similarity oftheir exchange rate fluctuation distributions. Dendrogram representations of the clusters constructed followingthe same procedure as for Fig. 1 are shown for four different time intervals: (a) Period I: Oct 23, 1995-Nov18, 2000, (b) Period II: Nov 19, 2000-Dec 16, 2005, (c) Period III: Dec 17, 2005-Jan 13, 2011, and (d) PeriodIV: Jan 14, 2011-Feb 10, 2016. The threshold distances used for obtaining the largest number of clusters ineach period are dth = 0.80, 0.90, 0.70 and 0.65, which yield 19, 16, 17 and 16 clusters, respectively. Note thatthe statistical distance between the currencies shrink in the latter two periods following the major economiccrisis that occurred in 2008. The cluster of currencies belonging to developed economies seen in the earlierperiods is also seen to be disrupted to an extent. The degree of similarity between the different dendrogramsis quantified by the cophenetic distance correlation coefficient as follows: CDCC(I,II) = 0.23, CDCC(II,III)= 0.12 and CDCC(III,IV) = 0.46. As in Fig. 1, currencies are distinguished according to the average annualGDP per capita 〈g〉 of the corresponding economy (represented by font size, which scales logarithmically with〈g〉) and the geographical region to which they belong (represented by font color, viz., black: Americas, red:Europe, blue: Middle East, magenta: Asia-Pacific, green: Africa and brown: Asia).

on measuring the similarity between fluctuation dis-tributions of currencies that is described here doesnot have the limitation of requiring the informationfor the different assets to be acquired in a synchro-nized manner. It is therefore more likely to providea correct description of the underlying hierarchicalstructure of the market.

Comparing the hierarchical clustering organiza-tion in the currency market across different peri-ods can provide us with an indication of the in-tensity of specific disruptive events in the economy

of different nations. For instance, we observe thatISK and HKD have changed their position relativeto other currencies in the dendrograms represent-ing hierarchical clustering of the currencies at dif-ferent eras (Fig. 5). ISK lies close to the cluster ofdeveloped economy currencies in the first two peri-ods considered, but neighbors emerging and frontiereconomy currencies in the later periods. This helpsus to connect the atypical characteristics shown bythe currency with the effects of the major finan-cial crisis that affected Iceland in the latter era.

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Triggered by the default of all three major privately-owned commercial banks in Iceland in 2008, the cri-sis resulted in the first systemic collapse in any ad-vanced economy [41]. A sharp drop in the value ofISK followed, with exchange transactions halted forweeks and the value of stocks in the financial marketcollapsing. The crisis led to a severe economic de-pression in Iceland lasting from 2008–2010. By con-trast, HKD appears close to other developed econ-omy currencies in Period I, but in the neighborhoodof emerging and frontier currencies in subsequentperiods. This again helps us to link the unusual be-havior of HKD with the crisis triggered by the SARSepidemic of 2003 affecting mainland China, Taiwanand large parts of Southeast Asia, that caused ex-tensive economic damage to Hong Kong with unem-ployment hitting a record high [42]. For the HongKong currency and banking system that had sur-vived the Asian Financial Crisis of 1997–1998 [43],the epidemic was an unexpected shock [44], with anet capital outflow observed during the persistentphase of the disease [45]. In addition, the domi-nance of the service sector in the Hong Kong econ-omy meant that the reduction in contact followingthe epidemic outbreak had a large negative impacton the GDP [45]. Thus, the deviation in the behav-ior of specific currencies from that expected becauseof the macro-economic characteristics can be tracedto particular disruptive events that specifically af-fected them.

5. Conclusions

To conclude, we have quantitatively measuredthe degree of similarity between different curren-cies by applying a distance metric on the distri-bution of their fluctuations. This has allowed usto uncover the hierarchical structure relating move-ments of different currencies in the internationalFOREX market, in particular, showing that cur-rencies belonging to economies of similar naturetend to cluster together during normal periods inthe market. More importantly, economic disrup-tions in a particular country tend to displace it fromits usual position in the hierarchy. The severe im-pact of the global financial crisis of 2008 is reflectedin the large-scale disruption of the arrangementof the clusters that were in existence in pre-crisistimes. Thus, considering the temporal dimension inour analysis allows us to relate particularly strongeconomic shocks to changes in the relative posi-tions of currencies in their hierarchical clustering.We suggest that using statistical distance betweendistributions characterizing the dynamical states ofthe components of a complex system, to reconstructthe hierarchical arrangement of the network of inter-relations between them, can complement existingcorrelation-based methods. Indeed, in situationswhere information about the different componentscannot be obtained simultaneously (thereby pre-cluding the construction of a synchronous cross-correlation matrix), the technique proposed here

may provide the means for accurately construct-ing the network of interactions between the com-ponents.

Acknowledgments

We thank Anindya S. Chakrabarti, TanmayMitra and V. Sasidevan for helpful suggestions.We gratefully acknowledge the assistance of UdayKovur in the preliminary stages of this work. Thiswork was supported in part by the IMSc Econo-physics (XII Plan) Project and by the IMSc Centreof Excellence in Complex Systems & Data Science,both funded by the Department of Atomic Energy,Government of India.

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