arXiv:1606.06111v2 [q-fin.ST] 21 Jun 2018 hmi aydffrn as uhahgl heteroge- highly a Such ways. affect different can to many which subject factors, in each commercial them or are eco- political market geographical, curren- to nomic, the related different in e.g., The influences, traded multifarious relatively are [6]. are that analysis cies that for data accessible digital quan- easily large high-resolution of with availability of systems the tities is complex freedom other of fluctu- of degrees its many that in studying over world of advantage behavior the An ation in [5]. volume market of financial terms which largest (FOREX) the exchange de-centralized constitutes foreign the in is trade international composition heterogeneous highly near expected behavior [4]. universal transitions re- from phase can also heterogeneity deviation such may in that parts known sult is different connection It in differ. local elements greatly The the exhibit among may density dynamics. system For distinct complex a elements. qualitatively of individual components of het- of example, properties degree the large in a by erogeneity characterized com- often systems, interacting are of Such number ponents, large society. a general comprising and from uncover nature apart in to complex of occurring trying dynamics systems collective those the for underlying principles challenge di- poses large major phenomena The critical a under- non-equilibrium [3]. well in lacking seen yet still versity non- of is not universality transitions for are evidence equilibrium strong equilibrium Indeed, systems from [2]. stood socio-economic far fluctua- and are effort, biological that cen- considerable in previous despite behavior However, the tion in [1]. physics tury statistical path-breaking of a been achievement has transition phase a near versality eitosfo nvraiyi h utainbhvo of behavior fluctuation the in universality from Deviations rttpcleapeo ope ytmwt a with system complex a of example prototypical A uni- exhibit equilibrium at systems that discovery The ytmrva nrni rpriso opnns h case The components: of properties intrinsic reveal system h eeoeeu aueo opnnsi te ope sys identi complex other in in help components may of here nature presented heterogeneous that the to similar Approaches flu of tails univers putative heavy i a the the ( from properti in characterizing diverge the dynamics systematically exponents rate in currencies the exchange heterogeneity that the how the consider show of by we example masked systems, an such be As can signature systems. complex far-from-equilibrium iuiepoesswihw eaet h nicreae n anti-correlated the to relate ex we develop less the which ex the by processes of of exhibited diffusive those diversity economies, walks developed and random to output uncorrelated belonging to production contrast propertie capita in macroscopic per fundamental of to measures law” square “inverse ≃ dniyn eairta srltvl nain ne di under invariant relatively is that behavior Identifying h nttt fMteaia cecs I aps Tarama Campus, CIT Sciences, Mathematical of Institute The )o h xoet.W eaetedge fdvaino par a of deviation of degree the relate We exponents. the of 2) .INTRODUCTION I. bii hkaot,Suy awrnadStbr Sinha Sitabhra and Easwaran Soumya Chakraborty, Abhijit Dtd oebr8 2018) 8, November (Dated: urnymarket currency rbto o atclrcrec ol erltdto related economy. underlying be the could of currency dis- properties fluctuation intrinsic particular the some a of nature for the tribution that [16– suggests regime This not Levy-stable the - 19]. outside tails lie they such whether of characterizing even values exponents the power-law concerning cur- agreement the different little for the of is tails there some been heavy rencies, 15], has reported [14, indeed currencies investigations have which of earlier fluctua- several rates of of exchange distribution subject the the in Although tions such contrast, of In facts”. paucity 13]. relative “stylized [12, a re- law” have often cubic processes “inverse [8–11], macroeconomic stock the indices as heavy- market individual to equity the ferred in as, of well nature fluctuations as the prices, of to distributions relate these tailed of seen The robust has [7]. most universality markets of single suggestive of evidence accumulating is dynamics that the microeconomics with of domain concerned the that such Note in under- dynamics systems. towards self-organizing contribution the underlying important the of standing an case regu- be specific empirical will robust the larity any For establishing market, physics. using FOREX statistical explained micro- be of of potentially uni- tools may show independent which to phenomena details, scopic expected i.e., hetero- be features, a can of versal par- system components In the complex whether geneous ask physicists. can by we investigated that ticular, been composition relatively typically the homogeneous have to having contrast systems stark simpler a provides system neous ac,adteTelidxta esrstedvriyof diversity the measures that index perfor- Theil economic the fluctua- domestic the and to mance, the gross related two capita the for on per (GDP) viz., - depending product indicators, law” currencies macroeconomic different square key of “inverse behavior as tion to which - refer signature universal we putative a from deviation atic nti ae eso htteei nedasystem- a indeed is there that show we paper this In lfr soitdwt h einvalue median the with associated form al eetcniin sacalnigts in task challenging a is conditions fferent deooisso hrceitc fsub- of characteristics show economies ed tr ftecrepnigfluctuations. corresponding the of ature ftecrepnigeooy viz., economy, corresponding the of s tems. yn nain etrsosue by obscured features invariant fying hnert yaisfrcurrencies for dynamics rate change otpout.W loso that show also We products. port taindsrbtosfrdifferent for distributions ctuation trainlcrec akt We market. currency nternational so h opnnscomprising components the of es h xsec fasemi-invariant a of existence the i hna 013 India. 600113, Chennai ni, iua urnyfo uhan such from currency ticular eeoeeu complex heterogeneous a fteinternational the of 2 exports of the corresponding countries (see data descrip- maining currencies, a few are pegged to USD or some tion for details). Thus, several underdeveloped (frontier) other important currency (such as EUR), but with some economies exhibit currency fluctuations whose distribu- variation within a band (which may either be fixed or tions appear to be of a Levy-stable nature, while those of moving in time). Note that as the EUR was introduced most developed economies fall outside this regime. The in January 1, 1999, i.e., within the time interval con- median value of the exponents quantifying the heavy- sidered by us, we have used the exchange rate for the tailed nature of the cumulative fluctuation distributions ECU (European Currency Unit) for the period October for all the currencies occur close to 2, i.e., at the bound- 23, 1995 to December 31, 1998. ary of the Levy-stable regime. Our study demonstrates To ensure that the observed differences in the nature how robust empirical regularities in complex systems can of the fluctuation distributions of currencies is not just be uncovered when they are masked by the intrinsic het- a trivial outcome of the different exchange rate regimes, erogeneity among the individual components. We have we have performed a two-sample Kolmogorov-Smirnov also characterized the distinct nature of the exchange rate test [24] with the null hypothesis that the pegged and dynamics of different currencies by considering their self- floating currencies are sampled from the same continu- similar scaling behavior. Our analysis reveals that while ous distribution. A measured p-value of 0.39 indicates currencies of developed economies follow uncorrelated that the null hypothesis cannot be rejected at 5% level random walks, those of emerging and frontier economies of significance. We also carried out a Wilcoxon rank sum exhibit sub-diffusive (or mean-reverting) dynamics. test [25] with the null hypothesis that both pegged and floating currencies are sampled from continuous distri- butions with equal medians. We obtained a p-value of II. DATA DESCRIPTION 0.27, again indicating that there is not enough evidence to reject the null hypothesis at 5% level of significance. The data-set we have analyzed comprises the daily ex- We thus conclude that the distinct behavior of the cur- change rates with respect to the US Dollar (USD) of rencies in terms of the distribution of their exchange rate N = 75 currencies (see Table I) for the period October returns cannot be simply explained away as being related 23, 1995 to April 30, 2012, corresponding to τ = 6035 to their pegged or floating nature. days. The rate we use is the midpoint value, i.e., the av- In order to explore whether the nature of the fluctua- erage of the bid and ask rates for 1 USD against a given tion distribution of a particular currency could be related currency. The data is obtained from a publicly accessible to the characteristics of the underlying economy, the archive of historical interbank market rates maintained countries to which these currencies belong are grouped by the Oanda corporation, an online currency conversion into three categories, viz., developed, emerging and fron- site [20] that is used by major corporations, tax author- tier markets, as per the Morgan Stanley Capital Interna- ities and auditing firms worldwide. The interbank (or tional (MSCI) market classification framework [26]. This spot) rate for a currency is the official rate quoted in the is done on the basis of several criteria such as, the sus- media and that apply to large transactions of 106 USD tainability of economic development, number of compa- or higher (typically taking place between banks and fi- nies meeting certain size and liquidity criteria, ease of nancial institutions). For each day, the site records an capital flow, as well as, efficiency and stability of the in- average value that is calculated over all rates collected stitutional framework. over a 24 hour period from frequently updated sources To make more explicit the connection between devia- in the global foreign exchange market, including online tion from universality and the heterogeneity of the con- currency trading platforms, leading market data vendors, stituents of the FOREX market, we have examined in de- and contributing financial institutions. We have chosen tail certain macro-economic factors characterizing a na- USD as the base currency for the exchange rate as it tional economy for the role they may play in determining is the preferred currency for most international transac- the nature of the fluctuation dynamics of a currency. In tions and remains the reserve currency of choice for most particular, we find that a prominent role is played by (a) economies [21, 22]. We have verified that using other the GDP per capita g, as well as, (b) the Theil index T base currencies lead to qualitatively similar fluctuation of export products, which we define below. distributions for exchange rates. The GDP per capita of a country is obtained by divid- The choice of currencies used in our study is mainly ing the annual economic output, i.e., the aggregate value dictated by the exchange rate regime (see Table I), which of all final goods and services produced in it during a is obtained from the site [20] where we collected the ex- year, by the total population. It is one of the primary in- change rates data and supplemented by information from dicators of the economic performance of a country, with the site of another online FOREX services company [23]. higher GDP per capita indicating a higher standard of In particular, we have not considered currencies whose living for the people living in it [27]. The annual GDP exchange rate with respect to USD is constant over time. per capita of the countries whose currencies have been in- Most of the currencies in our database are floating, either cluded in our study are obtained from publicly accessible freely under the influence of market forces or managed to data available in the website of the International Mone- an extent with no pre-determined path. Among the re- tary Fund (IMF) [28]. We have averaged the data over 3

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 from Oanda and XE sites), the character of the economy (as categorized by MSCI), the geographical region, the average GDP per capita (provided by IMF) and the mean Theil index (calculated from data available from MIT OEC) for the corresponding countries.

Sl.no. Currency Code ExchangeRateRegime MarketType Region hgi in USD hT i (Oanda,XE) (MSCI) (IMF) (MIT) 1 Canadian Dollar CAD Floating Developed Americas 32561.46 1.95 2 DanishKrone DKK Peggedwithinhorizontalband Developed Europe 44617.1 1.49 3 Euro EUR Floating Developed Europe 28200.99 - 4 Great Britain Pound GBP Floating Developed Europe 32126.2 1.54 5 IcelandKrona ISK Floating Developed Europe 39213.54 3.69 6 NorwegianKroner NOK Floating Developed Europe 59286.29 3.45 7 SwedishKrona SEK Floating Developed Europe 39571.51 1.63 8 Swiss Franc CHF Floating Developed Europe 52059.39 1.96 9 IsraeliNewShekel ILS Floating Developed MiddleEast 22478.26 2.64 10 Australian Dollar AUD Floating Developed Asia-Pacific 35251.16 2.38 11 HongKongDollar HKD Fixedpeg Developed Asia-Pacific 27406.74 1.98 12 Japanese Yen JPY Floating Developed Asia-Pacific 36942.47 1.95 13 New Zealand Dollar NZD Floating Developed Asia-Pacific 23459.35 2.14 14 Singapore Dollar SGD Floating Developed Asia-Pacific 30538.39 2.65 15 BolivianBoliviano BOB Crawlingpeg Emerging Americas 1287.16 3.65 16 Brazilian Real BRL Floating Emerging Americas 6254.18 1.93 17 Chilean Peso CLP Floating Emerging Americas 7563.51 3.23 18 Colombian Peso COP Floating Emerging Americas 3864.52 3.01 19 DominicanRepublicPeso DOP Floating Emerging Americas 3509.27 2.84 20 MexicanPeso MXN Floating Emerging Americas 7556.32 2.15 21 PeruvianNuevoSol PEN Floating Emerging Americas 3243.22 2.99 22 VenezuelanBolivar VEB Fixedpeg Emerging Americas 6302.1 4.85 23 Albanian Lek ALL Floating Emerging Europe 2319.21 2.77 24 Czech Koruna CZK Floating Emerging Europe 11701.17 1.44 25 HungarianForint HUF Peggedwithinhorizontalband Emerging Europe 9151.13 1.87 26 Polish Zloty PLN Floating Emerging Europe 7866.73 1.41 27 Russian Rouble RUB Floating Emerging Europe 5791.06 3.23 28 Turkish Lira TRY Floating Emerging Europe 6451.81 1.58 29 AlgerianDinar DZD Floating Emerging Africa 2890.28 5.17 30 CapeVerdeEscudo CVE Fixedpeg Emerging Africa 2130.76 3.71 31 EgyptianPound EGP Floating Emerging Africa 1727.67 2.73 32 EthiopianBirr ETB Floating Emerging Africa 208.91 4.33 33 MauritiusRupee MUR Floating Emerging Africa 5432.83 3.39 34 MoroccanDirham MAD Fixedpeg Emerging Africa 1997.64 2.54 35 SouthAfricanRand ZAR Floating Emerging Africa 4751.66 2.14 36 TanzanianShilling TZS Floating Emerging Africa 361.18 3.17 37 ChineseYuanRenminbi CNY Fixedpeg Emerging Asia 2173.96 1.55 38 IndianRupee INR Floating Emerging Asia 774.57 1.74 39 IndonesianRupiah IDR Floating Emerging Asia 1630.89 1.99 40 PapuaNewGuineaKina PGK Floating Emerging Asia 1014.91 4.34 41 PhilippinePeso PHP Floating Emerging Asia 1440.56 3.05 42 South Korean Won KRW Floating Emerging Asia 15655 2.11 43 Taiwan Dollar TWD Floating Emerging Asia 15707.7 - 44 Thai Baht THB Floating Emerging Asia 3194.12 1.78 45 Guatemalan Quetzal GTQ Floating Frontier Americas 2134.53 2.54 46 HonduranLempira HNL Crawlingpeg Frontier Americas 1380.32 3.23 47 Jamaican Dollar JMD Floating Frontier Americas 4042.29 4.25 48 Paraguay Guarani PYG Floating Frontier Americas 1892.51 3.81 49 Trinidad Tobago Dollar TTD Floating Frontier Americas 12983.73 4.21 50 Croatian Kuna HRK Floating Frontier Europe 9166.72 1.75 51 Kazakhstan Tenge KZT Floating Frontier Europe 4399.18 3.97 52 Latvian Lats LVL Fixed peg Frontier Europe 6912.26 2.35 53 Botswana Pula BWP Crawling peg Frontier Africa 5447.23 5.45 54 Comoros Franc KMF Fixed peg Frontier Africa 609.64 5.05 55 Gambian Dalasi GMD Floating Frontier Africa 513.84 4.27 56 Ghanaian Cedi GHC Floating Frontier Africa 871.99 4.11 57 GuineaFranc GNF Fixedpeg Frontier Africa 419.31 5.03 58 KenyanShilling KES Floating Frontier Africa 586 2.95 59 Malawi Kwacha MWK Floating Frontier Africa 226.69 4.5 60 MauritanianOuguiya MRO Floating Frontier Africa 752.88 4.96 61 MozambiqueMetical MZM Floating Frontier Africa 330.54 4.28 62 Nigerian Naira NGN Floating Frontier Africa 782.49 6.02 63 SaoTomeandPrincipeDobra STD Fixedpeg Frontier Africa 864.42 4.15 64 Zambian Kwacha ZMK Floating Frontier Africa 677.53 4.67 65 JordanianDinar JOD Fixedpeg Frontier MiddleEast 2617.85 3 66 KuwaitiDinar KWD Fixedpeg Frontier MiddleEast 25554.56 5.49 67 SyrianPound SYP Fixedpeg Frontier MiddleEast 1732.47 4.21 68 Brunei Dollar BND Fixed peg Frontier Asia 23516.1 5.45 69 Bangladeshi Taka BDT Floating Frontier Asia 436.09 3.63 70 Cambodian Riel KHR Floating Frontier Asia 498.85 3.85 71 FijiDollar FJD Fixedpeg Frontier Asia 3052.58 3.37 72 Lao Kip LAK Floating Frontier Asia 577.82 3.66 73 Pakistan Rupee PKR Floating Frontier Asia 756.7 2.87 74 Samoan Tala WST Fixed peg Frontier Asia 2215.73 4.63 75 Sri Lankan Rupee LKR Floating Frontier Asia 1438.09 2.65 4

5 a a c 6 6 0 r ( t ) −5 5 5 20 b −

γ 4 4 0 r ( t )

−20 3 3 08/03/1997 16/04/2001 10/11/2006 19/12/2010 Time

1 2 2 10 c SEK INR 2 3 4 5 6 0 0.5 γ 0 P ( ) 10 TTD b − )

+ 0.5 γ Developed −1 P ( r ) 10 P ( Emerging 0 Frontier −2 2 3 4 5 6 10 γ + −3 6 d 6 e 10 + −10 −5 0 5 10 − γ r 5 γ 5 4 4 3 3 Exponent FIG. 1: (color online). Heavy tailed behavior in the dis- Exponent tribution of currency exchange rate fluctuations. The 2 2 time-series of normalized log returns r(t) for currencies of de- 1 1 0 2 4 0 2 4 veloped economies, e.g., SEK (a), shows relatively lower am- 10 10 10 10 10 10 Kurtosis α Kurtosis α plitude variations compared to that of currencies of frontier 4 4 economies, e.g., TTD (b), in general (note the different scales in the ordinate of the two panels). However, the probability density functions of r for all currencies show a heavy-tailed FIG. 2: (color online). (a-c) Deviation from universality nature, shown in (c) for currencies from a developed econ- for exchange rate fluctuations. The probability distribu- omy, SEK (black, circles), an emerging economy, INR (red, tion of the power law exponents γ+ (b) and γ− (c) obtained squares), and a frontier economy, TTD (blue, triangles). For by maximum likelihood estimation (MLE) for the positive comparison, the standard normal distribution is shown using and negative tails, respectively, of the individual return dis- a solid curve. tributions for the 75 currencies, show a peak around 3 with median values of 3.11 (for γ+) and 3.28 (for γ−). Error bars indicate the uncertainty in the estimated values and are ob- the 18 year period (1995-2012) considered in our study tained by a non-parametric bootstrap technique. Points lying to obtain the mean GDP per capita hgi. closer to the diagonal (γ+ = γ−, indicated by a broken line) in (a) imply a higher degree of symmetry in the distribution The Theil index measures the diversity of the ex- of r for the corresponding currency, i.e., positive and negative port products of a country [29] and is defined as T = fluctuations of similar magnitude are equally probable. The 1 M xi xi M i=1( x¯ ln x¯ ), where xi is the total value (in USD) heavy-tailed nature of the distributions characterized by the of theP i-th export product of a country,x ¯ is the average tail-exponents correspond closely to their peakedness mea- value of all export products and M is the total number of sured using the kurtosis α4, as shown by the scatter plot different products that are exported. A high value of T between (d) α4 and γ+ and (e) α4 and γ− for the curren- corresponds to large heterogeneity in the values of the dif- cies. The best log-linear fits, indicated by broken lines, cor- −β± ferent exported products, indicating that a few products respond to α4 = exp[(γ±/A±) ] with A+ = 5.8, β+ = 2.4 dominate the export trade. By contrast, low T implies (d) and A− = 5.6, β− = 2.8 (e). The Pearson correlation coefficient between log(log(α4 )) and log(γ±) are ρ = −0.67 that a country has a highly diversified portfolio of export (p = 10−11) for (d) and ρ = −0.59 (p = 10−8) for (e). Differ- products and therefore, relatively protected from the va- ent symbols and colors are used to indicate currencies from garies of fluctuations in the demand for any single prod- developed (black, circles), emerging (red, squares) and fron- uct. To compute the Theil index we have used the an- tier (blue, triangles) economies, while symbol size is propor- nual export product data of different countries available tional to log(hgi) of the corresponding countries. from the Observatory of Economic Complexity (OEC) at MIT [30]. We have used the four digit level of the Standard International Trade Classification for catego- III. RESULTS rizing different products which corresponds to M = 777 distinct export products in the data set. We have aver- aged the annual Theil indices over the period 1995-2012 We have measured the fluctuations in the exchange to obtain the mean Theil index hT i for each country. rates of 75 currencies (see data description for details) 5 with respect to the US Dollar over the period 1995-2012. each other for the different currencies. Currencies that To ensure that the result is independent of the unit of occur closer to the diagonal line γ+ = γ− have similar measurement, we have quantified the variation in the nature of upward and downward exchange rate move- exchange rate Pi(t) of the i-th currency (i = 1,...,N) ments. However, currencies which occur much above the at time t by its logarithmic return defined over a time- diagonal (i.e., γ+ <γ−) will tend to have a higher prob- interval ∆t as Ri(t, ∆t) = ln Pi(t + ∆t) − ln Pi(t). As ability of extreme positive returns compared to negative explained in the data description, our data comprises ones, while those below the diagonal are more likely to daily exchange rates and we therefore consider ∆t = 1 exhibit very large negative returns. We note in passing day. Different currencies can vary in terms of the in- that the skewness depends, to some extent, on the state tensity of fluctuations in their exchange rates (volatil- of the economy of the country to which a currency be- ity) as can be measured by the standard deviation σ of longs, with return distributions of developed economies the returns. Thus, to compare the return distributions being the least asymmetric in general, having mean skew- of the different currencies, we normalize the returns of ness 0.52 ± 1.28, while those of emerging and frontier each currency i by subtracting the mean value hRii = economies are relatively much higher, being 6.54 ± 15.24 τ−1 Σt=1 Ri(t)/(τ − 1) and dividing by the standard devi- and 6.60 ± 18.04, respectively.

1 ′ ′ 2 The distribution of the exponents characterizing the ation σi(t) = q τ−2 Σt 6=t[Ri(t ) − hRii] (removing the power-law nature of the exchange-rate returns shown in self contribution from the measure of volatility), obtain- Figs. 2 (b-c) peaks around 3 for both the positive and neg- ing the normalized return, r (t) = (R (t) − hR i)/σ (t). i i i i ative tails. As a probability distribution function with a power law characterized by exponent value γ ≃ 3 implies that the corresponding CCDF also has a power-law form A. The “inverse square law” of the distribution of but with exponent value α = γ − 1 ≃ 2 [33], this result fluctuations for currency exchange rates suggests an “inverse square law” governing the nature of fluctuations in the currency market in contrast to the As can be seen from Fig. 1 (a-b), the returns quantify- “inverse cubic law” that has been proposed as governing ing the fluctuations in the exchange rate of currencies can the price and index fluctuations in several financial mar- appear extremely different even though they have been kets [8–13]. However, as is the case here, such a “law” normalized by their volatilities. The temporal variation is only manifested on the average, as the return distribu- of r(t) for SEK [shown in Fig. 1 (a)], the currency of a tions for individual assets can have quite distinct expo- developed economy, is mostly bounded between a narrow nents [11]. Here, we observe that the different currencies interval around 0 with the fluctuations never exceeding can have exponents as low as 2 and as high as 6. More- 6 standard deviations from the mean value. By contrast, over, there appears to be a strong correlation between the Fig. 1 (b) shows that TTD, belonging to a frontier econ- nature of the tail and the state of the underlying econ- omy, frequently exhibits extremely large fluctuations that omy to which the currency belongs. Thus, developed can occasionally exceed even 20 standard deviations - an economy currencies tend to have the largest exponents, event extremely unlikely to have been observed had the while most of the lowest values of exponents belong to distribution been of a Gaussian nature [31]. These ob- currencies from the frontier economies. This suggests an servations suggest that the distributions of the exchange intriguing relation between the nature of currency fluc- rate fluctuations have long tails and that different cur- tuations and the state of the underlying economy, that rencies may have significantly different nature of heavy- could possibly be quantified by one or more macroeco- tailed behavior. As shown in Fig. 1 (c), where the dis- nomic indicators. This theme is explored in detail below. tributions of r for SEK, TTD and an emerging economy currency, INR, is displayed, this is indeed the case. In order to verify that the nature of fluctuations in The nature of the tails of the return distributions is exchange rates does not change drastically depending on established quantitatively by fitting them to a power-law the specific choice of base currency, we have re-calculated decay for the probability distribution having the func- the exponents γ characterizing return distributions of tional form P (r) ∼ r−γ through maximum likelihood different currencies that are obtained using each of the estimation (MLE) [32]. Uncertainty in estimating the 75 currencies as the base. Fig. 3 shows that for all optimal value of γ is calculated by performing MLE of base currencies used in our study, the exponents γ+ exponents from 100 surrogate data-sets for each currency. are distributed about mean values that fluctuate around These are constructed by random sampling with replace- hhγ+ii ∼ 3 (a similar behavior is seen for the exponents ment from the original return time-series data [32]. While of the negative returns distributions, γ−). This suggests both the positive and negative returns show heavy tails, that the inverse square law form for the heavy tails of re- we note that the exponents characterizing them need not turn distributions is valid on average relatively indepen- be identical for a currency, such that the corresponding dent of the base currency used to calculate the exchange return distribution is asymmetric or skewed. The scatter rates. plot in Fig. 2 (a) shows how the positive and negative The character of the heavy tails of the returns r is tail exponents, γ+ and γ− respectively, are related to closely related to the peaked nature of the distribution 6

values (corresponding to the positive and negative tails) for each return distribution, we shall henceforth focus on the single kurtosis value that characterizes the distribu- tion.

B. Deviation from universality related to macroeconomic factors

Given the variation in the nature of fluctuation distri- bution of different currencies from a single universal form, we ask whether the deviations are systematic in nature. Note that, the currencies belong to countries having very diverse economies, that trade in a variety of products & services with other countries and which may have con- trasting economic performances. An intuitive approach would be to relate the differences in the return distribu- FIG. 3: (color online). Distribution of fluctuations for tions with metrics which capture important aspects of currency exchange rates with respect to different base the economies as a whole. Fig. 4 shows that there is in- currencies display an “inverse square law” on average. deed a significant correlation between the kurtosis of the (a) The ensemble of distributions of power-law exponents γ+ return distributions for the currencies and two macroe- for the positive return distributions of 75 currencies calculated conomic indicators of the underlying economies, viz., the with respect to each of 75 base currencies that are arranged mean GDP per capita, hgi, and the mean Theil index, according to the mean GDP per capita hgi of the correspond- hT i, that describe the overall prosperity and the diver- ing economy (of the base currency). (b) The mean values sity of export products, respectively (see data description of the exponents γ+ (circles) obtained using each of the base for details). currencies are almost all clustered around the value of 3, indi- cating an “inverse square law” behavior of the heavy tails of Fig. 4 (a) shows that the scatter of kurtosis α4 against return distributions that is relatively stable against the choice hgi can be approximately fit by a power law of the form: −2.2 of different bases for measuring the fluctuation. Error bars α4 ∼ hgi . The Pearson correlation coefficient be- indicated represent the standard deviation in the estimated tween the logarithms of the two quantities is ρ = −0.55 −7 values of γ+ for different currencies for a given base currency. (p = 10 ). Thus, in general, currencies of countries hav- The broken line represents the grand average (hhγ+ii = 3.21) ing higher GDP per capita tend to be more stable, in the of the values for the the exponent γ+, taken over all currencies sense of having low probability of extremely large fluctu- and bases. ations. However, there are exceptions where currencies exhibit high kurtosis even when they belong to countries with high GDP per capita (e.g., HKD and ISK which are that can be quantified by its kurtosis which is defined as indicated in the figure). In these cases, the peakedness of 4 4 α4 = E(r − µ) /σ , where E() is the expectation while the distribution may reflect underlying economic crises, µ and σ are the mean and standard deviation, respec- e.g., the 2008 Icelandic financial crisis in the case of ISK tively, of r. Fig. 2 (d-e) shows the relation between the and the 2003 SARS crisis for HKD. Furthermore, we ob- kurtosis and the exponents for the tails of the return dis- serve that currencies belonging to high GDP per capita tributions of the different currencies. The fitted curve economies that are dependent on international trade of shown qualitatively follows the theoretical relation be- a few key resources - such as, crude oil - also exhibit tween the two which can be derived by assuming that high kurtosis (e.g., KWD and BND). This suggests a de- the distribution is Pareto, i.e., follows a power law (al- pendence of the nature of the fluctuation distribution on though for such a situation, the kurtosis is finite only the diversity of their exports, which is indeed shown in for exponent values γ > 5). We observe that the rela- Fig. 4 (b). The dependence of the kurtosis on T (which tion between the exponents and kurtosis suggested by is a measure of the variegated nature of trade) of the the scatter plots can be approximately fit by the func- corresponding economy is approximately described by a −β± 9.1 tion α4 ∼ exp[(γ±/A±) ] with β+ = 2.4, A+ = 5.8 power-law relation: α4 ∼ hT i . The Pearson correlation for the positive tail and β− =2.8, A− =5.6 for the neg- coefficient between the logarithms of the two quantities ative tail [Fig. 2 (d) and (e),respectively]. The strong is ρ = 0.53 (p = 10−6). This implies that, in general, correlation between the peakedness of the distribution currencies of countries having low hT i, i.e., having well- and the character of the heavy tails can be quantified by diversified export profile, tend to be more stable. the Pearson correlation coefficients between log(γ±) and Note that the fluctuations of the currencies depend on −11 log(log(α4)), viz., ρ = −0.67 (p = 10 ) for the posi- both of these above macroeconomic factors, and the dif- tive returns and ρ = −0.59 (p = 10−8) for the negative ferences in their nature cannot be explained exclusively returns. Thus, instead of using two different exponent by any one of them. It is therefore meaningful to per- 7

4 form a multi-linear regression of α4 as a function of both 10 GDP per capita and Theil index using an equation of the a b form: log(α4) = b0 + b1 log(hgi)+b2 log(hT i), where the constants b (= 6.74),b (= −0.48) and b (= 1.69) are the 3 0 1 2 10 ISK best-fit regression coefficients. The coefficient of deter- 2 mination R , which measures how well the data fits the 4 HKD −8 α statistical model, is found to be 0.39 (p ≃ 10 ). This in- 2 dicates that together the macroeconomic factors of GDP 10 per capita (related to the overall economic performance) Kurtosis and Theil index (related to the international trade of the 1 country) explain over 39% of the variation between the 10 nature of the return distributions of the different curren- cies. 0 One of the assumptions of multi-linear regression anal- 10 2 3 4 5 0 1 ysis is that the explanatory variables [viz., log(hgi) and 10 10 10 10 10 10 Mean GDP per capita 〈 g 〉 (USD) Mean Theil Index 〈 T 〉 log(hT i)] are not highly correlated with each other. Thus we need to explicitly test for the absence of significant collinearity, i.e., linear dependence of one explanatory FIG. 4: (color online). Variation of the kurtosis α4 variable on the other variables. A commonly used in- of exchange rate fluctuation distributions of differ- dicator of collinearity is the variance inflation factor ent currencies with (a) annual GDP per capita, hgi (V IF ) [34]. When the variation of a specific explanatory (in USD) and (b) annual Theil index of the export variable (referred to as a predictor) is largely explained products, hT i, for the corresponding countries, aver- by a linear combination of the other predictors, VIF for aged over the period 1995-2012. The Pearson correla- that predictor is correspondingly large. Complete ab- tion coefficient between log(hgi) and log(α4) is ρ = −0.55 − sence of collinearity corresponds to the case V IF = 1 (p = 10 7), the best-fit functional relation between the two −2.2 and the inflation is measured relative to this reference being α4 ∼ hgi . Currencies of developed economies that value. V IF have been shown to correspond to the diag- are outliers from this general trend, viz., ISK and HKD that onal elements of the inverse of the matrix of correlations have high kurtosis despite having high GDP per capita, are between the predictors [34] and using this method we explicitly indicated in (a). A similar analysis shows that the Pearson correlation coefficient between log(hT i) and log(α4) obtain V IF = 1.28 for both the macroeconomic factors is ρ = 0.53 (p = 10−6), with the best-fit functional rela- considered by us. As commonly collinearity is consid- 9.1 tion being α4 ∼ hT i . Different symbols are used to in- ered to be a cause for concern only if VIF values are dicate currencies from developed (black, circles), emerging higher than 5, GDP per capita and Theil index can be (red, squares) and frontier (blue, triangles) economies, while reasonably treated as independent explanatory variables symbol size is proportional to log(hgi) of the corresponding in our analysis. We have also investigated the possible countries. Error bars represent the standard deviation of the dependence of the nature of the fluctuation distribution annual values of g and T over the period 1995-2012 for the on other economic factors, such as the foreign direct in- countries corresponding to each currency. vestment (FDI) net inflow, but none of these appear to be independent of the two factors considered above. to white noise, while γDFA > 1/2 (< 1/2) implies that To investigate the reason for the strong relation be- the time-series is correlated (anti-correlated). As seen tween the kurtosis of the return distribution for a cur- from Fig. 5 (a), the DFA exponents of currencies for most rency and the corresponding underlying macroeconomic developed economies - which also have the lowest kurtosis factors, we need to delve deeper into the nature of the - are close to 0.5, indicating that these currencies are fol- dynamics of the exchange rate fluctuations. For this we lowing uncorrelated random walk [36]. In contrast, cur- first look into the self-similar scaling behavior of the time- rencies of the emerging and frontier economies, possess- series of exchange rate of a currency P (t) using the de- ing higher values of kurtosis, typically have γDFA < 0.5 trended fluctuation analysis (DFA) technique suitable for indicating sub-diffusive dynamics. analyzing non-stationary processes with long-range mem- To understand the reason for this sub-diffusive behav- ory [35]. Here, a time-series is de-trended over differ- ior we have analyzed the exchange rate time-series using ent temporal windows of sizes s using least-square fitting the variance ratio (VR) test. This technique, based on with a linear function. The residual fluctuations F (s) of the ratio of variance estimates for the returns calculated the resulting sequence, measured in terms of the stan- using different temporal lags, is often used to find how γDF A dard deviation, is seen to scale as F (s) ∼ s , where close a given time-series is to a random walk [37]. For a γDFA is referred to as the DFA exponent. The numerical sequence of log returns {Rt}, the variance ratio for a lag value of this exponent (lying between 0 and 1) provides l is defined as: information about the nature of the fractional Brownian τ k−1 2 motion undertaken by the system. For γDFA ≃ 1/2, the k=l( t=k−l Rt − lµR) V R(l)= P2 P , (1) process is said to be equivalent to a random walk subject σRl(τ − l + 1)(1 − [l/τ]) 8

dynamics of these currencies as arising from the anti- 0.6 a correlated nature of their successive fluctuations which 0.4 prevents excursions far from the average value. Thus, when we consider the time-series of all currencies after DFA γ 0.2 normalizing their variance, the fluctuations of the emerg- ing and frontier economy currencies mostly remain in HKD ISK 0 the neighborhood of the average value with rare, occa- sional deviations that are very large compared to devel- b 1 oped economy currencies. This accounts for the much heavier tails of the return distributions of the former and the corresponding high value of kurtosis. It is intriguing VR 0.5 to consider whether the difference in the nature of the movement of exchange rates of the currencies could be HKD ISK 0 possibly related to the role played by speculation in the 0 1 2 3 4 10 10 10 10 10 trading of these currencies [38]. We also note that these Kurtosis α 4 results are in broad agreement with the fact that effi- cient markets follow uncorrelated random walks and the notion that the markets of developed economies are far FIG. 5: (color online). Variation of (a) the long-range more efficient than those of emerging and frontier ones. auto-correlation scaling exponent γDF A obtained us- A temporally resolved analysis of the nature of the dis- ing detrended fluctuation analysis of the exchange tributions at different periods shows strong disruption rate time series, and (b) the variance ratio (V R) of of the otherwise regular pattern of systematic deviation the exchange rate fluctuations calculated using lag l during the severe crisis of 2008-09, indicating its deep- (= 10 days), with the kurtosis α4 of the normalized logarithmic return distributions of different curren- rooted nature affecting the real economy. cies. Different symbols are used to indicate currencies from developed (black, circles), emerging (red, squares) and fron- tier (blue, triangles) economies, while symbol size is propor- C. Temporal evolution of system properties tional to log(hgi) of the corresponding countries. The bro- ken lines in (a) and (b) indicate the values of γDF A(= 0.5) and V R(= 1) corresponding to an uncorrelated random walk. In the analysis presented above we have considered the Currencies of developed economies that are outliers, viz., ISK entire temporal duration which our data-set spans. How- and HKD that have much higher kurtosis than others in the ever, as the world economy underwent significant changes group, are explicitly indicated. during this period, most notably, the global financial cri- sis of 2008, it is of interest to see how the properties we investigate have evolved with time. For this purpose 2 2 where µR = hRti and σR = h(Rt − µR) i are the mean we divide the data-set into three equal non-overlapping and variance of the {Rt} sequence. An uncorrelated ran- periods each comprising 2011 days, corresponding to Pe- dom walk is characterized by a VR value close to 1. If riod I: Oct 23, 1995 - Apr 25, 2001, Period II: Apr 26, VR > 1, it indicates mean aversion in the time-series, 2001 - Oct 28, 2006 and Period III: Oct 29, 2006 - Apr i.e., the variable has a tendency to follow a trend where 30, 2012. Note that the last period corresponds to the successive changes are in the same direction. In contrast, crisis of the global economy spanning 2007-2009. For VR< 1 suggests a mean-reverting series where changes each of these, we carry out the same procedures as de- in a given direction are likely to be followed by changes in scribed earlier in the context of the the entire data-set. the opposite direction preventing the system from mov- As seen from Fig. 6, the behavior in the first two inter- ing very far from its mean value. Fig. 5 (b) shows the VR vals appear to be quite similar in terms of the various values for different currencies, calculated using lag l = 10 properties that have been measured, but large deviations days, as a function of their kurtosis. Consistent with the are seen in the third interval. This is apparent both for DFA results reported above, it is seen that for currencies the relation between kurtosis and mean GDP per capita of developed economies the VR is close to 1, indicating [Fig. 6 (a-c)], as well as that between kurtosis and mean uncorrelated Brownian diffusion as the nature of their Theil index (figure not shown). The dependence of the exchange rate dynamics. However, for most frontier and nature of the fluctuation distribution on the properties a few emerging economy currencies, the VR value is sub- of the underlying economy seem to have weakened in Pe- stantially smaller than 1, implying that their trajectories riod III. For example, while there is significant strong have a mean-reverting nature. As in Fig. 4, we note that negative correlation between log(hgi) and log(α4) for the HKD and ISK appear as outliers in Fig. 5 in that, al- first two intervals, viz., ρ = −0.60 (p = 10−8) and −0.57 though belonging to the group of countries having high (p = 10−8), respectively, it decreases to only ρ = −0.28 GDP per capita, they share the characteristics shown by (p = 10−2) for the third interval. Furthermore, the first most emerging and frontier economies. two intervals show a 1/hgi2 dependence of the kurtosis We can now understand the sub-diffusive nature of the α4, same as that seen for the entire period that we have 9

3 a b c different in the third - in part because the VR for the

104 developed and some emerging economies have adopted α 2 10 values > 1 (i.e., exhibiting mean aversion) in this last in- terval, while earlier they were close to 1 (i.e., similar to a 1 10 random walk). While Periods I and II had their share of Kurtosis 0 10 economic booms and busts, it is instructive to note that 2 3 4 5 2 3 4 5 2 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 the 2008 crisis was severe enough to disrupt systemic fea- Mean GDP per capita 〈 g 〉 ( USD ) tures that were otherwise maintained over time. d e f 1.5

1 IV. DISCUSSION AND CONCLUSION VR 0.5 The work we report here underscores the importance of 0 1 2 3 1 2 3 1 2 3 studying economic systems, especially financial markets, 10 10 10 10 10 10 10 10 10 Kurtosis α for gaining an understanding of the collective dynamics 4 of heterogeneous complex systems. At the largest scale, such a system encompasses the entire world where the FIG. 6: (color online). Temporal evolution of the statis- relevant entities are the different national economies in- tical properties of exchange rate fluctuation distribu- teracting with each other through international trade and tions of different currencies. The variation of (a-c) the the foreign exchange market. The far-from-equilibrium kurtosis α4 of the distributions with annual GDP per capita, behavior of this highly heterogeneous complex system has g (in USD) and that of (d-f) the variance ratio (VR) of the been investigated here by focusing on the fluctuations of different normalized fluctuations time series with kurtosis α4, exchange rates of the respective currencies. Understand- are shown for three different periods, viz., Period I: Oct 23, ing the overall features of this dynamics is crucially im- 1995 - Apr 25, 2001 (a & d), Period II: Apr 26, 2001 - Oct 28, portant in view of the human and social cost associated 2006 (b & e) and Period III: Oct 29, 2006 - Apr 30, 2012 (c with large-scale disruptions in the system, as was seen & f), which divide the duration under study into three equal, during the recent 2008 world-wide economic crisis. non-overlapping segments. The GDP per capita of the differ- ent countries for each period are obtained by averaging the Our results suggest a putative invariant signature in annual values over the corresponding periods. The Pearson the dynamics of exchange rates, possibly the first such correlation coefficients between log(hgi) and log(α4) for the seen in macroeconomic phenomena. This is in contrast −8 three periods are ρI = −0.60 (p-value = 10 ), ρII = −0.57 to microeconomic systems like individual financial mar- −8 −2 (p-value = 10 ) and ρIII = −0.28 (p-value = 10 ). For the kets where robust stylized facts such as the “inverse cubic first two periods, the best-fit functional relation between the law” has been established for some time. The “inverse 2 two is α4 ∼ 1/hgi , while for the third period, the dependence square law” that we report here also has a fundamental of α4 on hgi shows a strong deviation from the inverse square distinction in that distributions characterized by CCDF relation seen in the other two periods. Comparing the vari- exponents α ≤ 2 belong to the Levy-stable regime. By ance ratio values for the three different periods show a higher contrast, the logarithmic return distributions of equities degree of mean aversion in the third period. Period III, during and indices of financial markets that have exponent val- which the major economic crisis of 2008-09 occurred, is distin- guished by large deviation from the trends seen in the other ues around 3 are expected to converge to a Gaussian form two periods. Different symbols are used to indicate curren- at longer time scales [13, 39]. It suggests that extreme cies from developed (black, circles), emerging (red, squares) events corresponding to sudden large changes in exchange and frontier (blue, triangles) economies, while symbol size is rates, in particular for currencies belonging to emerging proportional to log(hgi) of the corresponding countries. and frontier economies, should be expected far more often compared to other financial markets. The “inverse square law” has recently been also reported in at least one other reported above. However, this is not true for the last market, viz., that of Bitcoins in the initial period follow- interval where the best fit for the dependence of α4 on ing its inception [40]. We note that agent-based modeling hgi shows a strong deviation from the behavior seen in of markets suggest that such a distribution can arise if other periods. Similarly, we have found significant high market players are relatively homogeneous in their risk correlation between log(hT i) and log(α4), corresponding propensity [41, 42]. to Pearson coefficients ρ =0.50 (p = 10−6) and ρ =0.46 To conclude, the results of our study help in reveal- (p = 10−5), respectively, for the first two intervals. In ing a hidden pattern indicative of relative invariance in a contrast, for the third interval we observe a relatively highly heterogeneous complex system, viz., the FOREX smaller correlation ρ = 0.35 (p = 10−2). In addition, market. The robust empirical feature that we identify the relation between the variance ratio and the kurto- here is a power law characterizing the heavy-tailed na- sis of the returns [Fig. 6 (d-f)], as well as that between ture of the fluctuation distributions of exchange rates for the DFA exponent and the kurtosis (figure not shown), different currencies. The systematic deviation of indi- are seen to be similar in the first two intervals but very vidual currencies from the universal form (the “inverse 10 square law”), quantified in terms of their kurtosis mea- lar approaches may be used for identifying invariances in suring the peakedness of the return distributions, can be other biological and socio-economic systems. linked to metrics of the economic performance and de- gree of diversification of export products of the respective countries. By doing detrended fluctuation analysis, the distinct behavior of currencies corresponding to devel- Acknowledgments oped, emerging and frontier markets can be linked to the different scaling behaviors of the random walks under- We thank Anindya S. Chakrabarti, Tanmay Mitra and taken by these currencies. Our work shows how robust V. Sasidevan for helpful suggestions. We gratefully ac- empirical regularities among the components of a com- knowledge the assistance of Uday Kovur in the prelimi- plex system can be uncovered even when the system is nary stages of this work. This work was supported in part characterized by a large number of heterogeneous inter- by IMSc Econophysics (XII Plan) Project funded by the acting elements exhibiting distinct local dynamics. Simi- Department of Atomic Energy, Government of India.

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