Introduction

Since the 2008 global financial crisis (GFC'08), there has been growing awareness of the interdependence of international financial markets and the systemic risk, i.e. the probability of a system-wide failure, resulting from their intrinsic entanglement1. It was shown how even supposedly minor distress of individual actors in a network of major financial institutions could lead to the annihilation of large economic value2. Conventional macroeconomic models, such as dynamic stochastic general equilibrium models3,4,5, which assume the existence of a stable equilibrium with disturbing shocks coming from outside the system, failed to predict, or even describe, the GFC'08.

According to the current understanding1,6,7,8,9,10, the proliferation of certain financial products prior to the GFC'08, such as collateralised mortgage obligations and credit default swaps (CDS), has led to a global network of strong interdependence of financial institutions with its geographical center in the United States (US). The crisis was then triggered by a relatively small shock inside the US mortgage market, which spread globally over this network11, leading to a dry-up of inter-bank lending. Subsequently, this resulted in the actual crisis, putting many systematically important institutions at risk2. While it is estimated that the US banking sector suffered losses of about 1.8 trillion USD9, the value of global financial assets declined by around 16 trillion USD12, not taking into account knock-on effects generated by this decline. Thus, the GFC'08, originating from a destabilising internal shock, which propagated over a complex network of strong interactions. This scenario is exactly in the realm of network science.

The study of large-scale economic networks has seen considerable progress in recent years, as demonstrated by13,14,15,16,17. However, much of the data relevant to the detailed investigation of global inter-institutional networks are deemed strategically important for the involved institutions and are not available. On the other hand, actual macroscopic dependencies among actors in international financial markets, such as the involvement of whole economies and the macroscopic interplay between different sectors of the global financial system may be probed by means of proxy networks, using aggregated flows/positions on the inter-economy level.

Motivated by this observation, we investigate two types of cross-border portfolio investment networks (PIN), namely equity- (E) and long-term debt (LD) securities, which represent major components of international capital markets. The respective networks are labelled LD-PIN and E-PIN, where nodes are given by individual countries and directed and weighted edges by consolidated investment positions, measured in USD, originating from residents/institutions in one country to residents/institutions in another18. Details about data and the used methodologies can be found in the sections Methodology and Discussion, while a presentation of the relevant concepts of graph theory, as well as additional information on both networks and results, are given in the Supplementary Information (SI).

The GFC'08 is clearly reflected in both PIN, where an overall reduction in investment positions (contraction) is observed, followed by a re-bounce as early as 2010. Our analysis reveals two early-warning indicators for potential financial crises. On the one hand, the algebraic connectivity of the E-PIN, interpreted as an indicator for the structural robustness of the global financial system and, as such, for the world-economy, drops sharply as early as 2005. This observation is associated with the emergence of a sparsely connected group of countries, particularly involving Middle Eastern countries, the United Kingdom (UK) and several high-degree off-shore financial centres (OFC), where we detect a general over-representation of OFC among countries associated with this structural instability. On the other hand, the edge density of the LD-PIN, as a network-proxy to measure the interdependence of financial markets, scales with the total market values of several over-the-counter-traded (OTC-traded) financial derivative products, which have been directly linked to the crisis, such as CDS19,20,9, but also equity-linked derivatives (ELD). Based on this, a simple phenomenological model is introduced, which allows for the description of the proliferation of such derivative products and, as such, for the detection of potentially high levels of market interdependency. We propose a dynamic monitoring mechanism which, taking the GFC'08 as a testing ground, generates clear warning signals between 6–12 months ahead of the crisis.

Results

General properties of PIN and the GFC'08

Both PIN are rapidly growing in terms of the numbers of nodes N and edges M in the years before the GFC'08, as can be seen in Fig. 1, where all quantities are shown with respect to their year-2002 values (dotted reference line). A summary of network statistics is given in Tab. S-2 of the SI. For both the E- and the LD-PIN, basic network parameters scale with major macroeconomic quantities. Taking into account the very different time resolutions of data points, the number of edges in the E-PIN (a) tracks quite well global stock market indices, such as the S&P Global 120021,22. The initial mismatch can be explained through the previous burst of the “dot-com bubble” and the following mild recession, which affected mainly the US23. In addition, the total trade volume between PIN countries scales roughly with the volume of the E-PIN, as shown in Fig. S-2-a in the SI. Since stock markets, which the E-PIN is inherently connected to and international trade are widely accepted measures for economic performance, the E-PIN offers a network-proxy for measuring the state of the world economy.

Figure 1
figure 1

Temporal Evolution of network key properties: number of nodes N, number of edges M and the resulting edge density ρ, of the two considered PIN relative to their corresponding year-2002 values (dotted reference line).

(a): E-PIN. The daily time series of the S&P Global 1200 stock index is given for reference. (b): LD-PIN. The temporal evolution of the gross-market value of equity-linked derivatives (GMV-ELD) and the total notional outstanding amount of credit default swaps (NOA-CDS, data not available prior to 2005) mirror the qualitative evolution of the percolation edge density .

Full size image

For the LD-PIN, the temporal evolution of the edge density at the minimal percolation edge threshold (see section Methodology) mirrors qualitatively the evolution of gross-market value (GMV) of OTC-traded ELD, as well as the notional outstanding amount (NOA) of CDS24, which is shown in panel (b) of Fig. 1.

The prices/values of financial derivatives are linked to an underlying asset, such as a stock, debt security or commodity, which by definition creates market interdependences, while their primarily uses are risk management (hedging) and speculation19,20,9,25. The value of ELD is derived from the price of some stock or a stock index. As such, ELD provide an indirect link between the two PIN, where it is believed that cross-asset hedges and capital structure arbitrage trades are the most important drivers26,20. Surprisingly, the inter-connectedness between equity and debt markets created in this way has so far not attracted any attention, but might complement the understanding of the GFC'08. One reason for this could be the large variety of ELD, such as single- or multi-stock and index forwards, options and swaps, where the macroscopic impact of individual products is difficult to assess.

Debt securities can be traded before the borrowed amount (principal) is repaid, i.e. before their maturity, or be used as base assets for financial derivative products such as CDS. These are credit derivative products, where the credit exposure from one or several third parties is passed from the buyer to the seller. This activity is expected to lead to additional inter-connectedness between a larger number of debtors and creditors. The general understanding is that CDS played an important role in transmitting the shock from the 2007-US subprime mortgage crisis, through large insurers, to the international banking system6,7,8,9,10,19. These observations make the LD-PIN a suitable network-proxy to measure the inter-connectedness and, consequently, the interdependence of financial markets, which have been identified as major factors contributing to financial crises27.

Note that NOA and GMV of financial derivative products are interpreted as two different risk measures. NOA describes the market value with respect to the underlying base assets (face value), such as mortgages, bonds or stocks. It may be taken as a measure for the overall market interconnectedness/interdependence, while it does not represent the actual amount at risk. GMV is the cost to replace existing contracts at the current market rate, which may be less coupled to the underlying assets than to the prevailing economic environment. It is seen to be a better (short-term) risk indicator than NOA.

Figure 2 illustrates structural properties of both PIN in terms of their cumulative distributions of node strength (a, b) and eigenvector centrality Cev (c, d). These two measures evaluate a country's importance within both networks from different points of view. Node strength, which is the sum of all in-coming and out-going investment positions, measures the size of a country in a network, while eigenvector centrality is a recursive measure, which accounts for a country's embedding into the weighted and directed network topology (see Section S-2 in the SI for more information on both measures).

Figure 2
figure 2

Cumulative node strength (a), (b) and eigenvector centrality (c), (d) distributions of the E- (a), (c) and the LD-PIN (b), (d), respectively.

Numerical values cover about the same ranges for both networks. In particular, node strengths span about five orders of magnitude with the largest values before the GFC'08. Strength distributions may be classified as “super-heavy-tailed”, since there is a (1)-probability for most values which indicates a strong hierarchical network structure. The US is by far the most central node in the E-PIN, with the highest strength and a dominating eigenvector centrality.

Full size image

An interesting feature of both PIN is the strong hierarchical structure, as depicted by their node strength distributions. These can be classified as being “super-heavy-tailed”, in the sense that node strengths span several orders of magnitude, while there is an (1)-probability for most values.

We see that the E-PIN is strongly dominated by the US and to a minor degree by the UK during all times, which is especially true for the eigenvector centrality (see Fig. 2-c). Its wave-like structure indicates a multi-layered topology with the US at the center. Taken together, such a configuration renders the E-PIN fragile against shocks originating from its most central nodes28.

Both measures are more homogeneously distributed in the LD-PIN (see Figs. 2-b and -d), where the most central nodes are Japan, France, the US and Germany. The proliferation of ELD, as shown in Fig. 1-b, is now interpreted as an additional (geographical) shift of weight from the LD-PIN to the E-PIN and towards the US. This can be understood as follows. Given the observed relation between ELD and the edge density of the LD-PIN, a high centrality of countries in Europe and Asia in the LD-PIN links them to the E-PIN, where the US is the dominating country. This configuration is expected to amplify shocks, as for example those from the US subprime mortgage crises of 2007.

Next, we will investigate how the GFC'08 is reflected in both PIN. PIN are large-scale economic structures. The total volume contained in all PIN together, which includes short-term debt (SD) securities, is of the same order in magnitude as world-GDP (gross-domestic product29, see blue line in Fig. 3-a), peaking at approximately 56% of world-GDP at the beginning of 2008. The crisis leads to an overall contraction of both networks, with the total volume reduced to less than 45% of world-GDP and a partly re-bounce to pre-crisis levels as early as 2010.

Figure 3
figure 3

Left: Temporal evolution of general PIN characteristics.

(a): Volumes (edge weight sums) in USD of the total PIN, the E- and the LD-PIN. For better comparability, all monetary values have been adjusted for changes in world-GDP (constant year-2013 values). The GFC'08 causes a contraction of both networks, which is especially pronounced in the E-PIN. (b): Fractions of total PIN volume from panel (a) contained in the E- and LD-PIN. One observes a constant shift in composition from the LD-PIN towards the E-PIN in the years before crisis. (c): Algebraic connectivity λ1 as a measure for network robustness against node/edge failure (removal). reaches an all-time low just before the GFC'08. and are given as an alternative reference scale to evaluate the stability of the E-PIN. (d): Cut depths of Fieder bi-sections of the E- and the LD-PIN. Numbers represent the fractions in percent of nodes contained in the corresponding smaller sections. While the LD-PIN is cut evenly without large variations in cut depth, the corresponding quantities of the E-PIN point to major topological changes before and during the crisis. Right (e): Comparison of potential macroeconomic reference variables (RV) for setting a dynamic warning threshold wth(t) for the gross-market value (GMV) of OTC-traded equity-linked derivatives (ELD): E- and LD-PIN volumes, world-GDP29, world trade volume (goods)31, global stock of foreign direct investment (FDI)32, global stock exchange trading volume29, foreign exchange market turnover volumes in Singapore and North America (US, Canada and Mexico, only available from October 2004 on). For a RV X, the temporal evolution of the quantity GMV-ELD/X is given relative to its year-2002 value, where nearest points in time have been matched. The shaded area indicates the shape a suitable RV should follow to generate a clear warning signal prior to the GFC'08. World-GDP (blue line) is seen to offer the best RV from the given sample.

Full size image

Both effects are stronger in the E-PIN, which experiences a faster growth and larger contraction, than the LD-PIN before and during the crisis. About 80% of the total contraction during the crisis result from the E-PIN, which shrinks by about 47% (or 10trillion USD) from 2008–2009. Note, at this point, that the contraction of both PIN already captures a large part of the decrease in value of global financial assets resulting from the GFC'0812.

The higher variability of the E-PIN during the GFC'08 is attributed to the volatile nature of its links as compared to those of the LD-PIN. This is due to the fact that investment in equity is generally riskier than investment in long-term debt. Equity securities can be readily sold in a crisis, additionally experiencing dramatic changes in value (here, edge weight). This is not the case for long-term debt securities, where links are by definition more durable with fixed weights, since future returns on investment are generally determined at the time of issuance. Moreover, there has been a reduction of liquidity in the debt markets30, as a consequence of the crisis, which prevented the LD-PIN from contracting substantially.

Besides an overall expansion of both networks, we observe a gradual shift from debt markets to equity markets in the years before the GFC'0812, which is reflected in a changing composition of the total PIN, as seen in Fig. 3-b, where the temporal evolution of the respective fractions of volume of the total PIN (blue line in Fig. 3-a) contained in the E- and LD-PIN are shown. This is interpreted as a growing fragility of the global financial system because the E-PIN is more susceptible to financial crises than the LD-PIN. We remark that the strong anti-correlation between E- and LD-PIN volumes is expected because they represent the majority of cross-border portfolio investment18.

A simple network measure for the structural robustness of the E-PIN against edge or node failures (removals) is the algebraic connectivity (see Section S-2 in the SI for a detailed description), where a zero value means the decomposition of the E-PIN into two disconnected components. drops sharply in the beginning of 2005 and reaches an all-time low in the beginning of 2007 (Fig. 3-c), pointing to a structural fragility of the E-PIN prior to the GFC'08. This is an intriguing fact from the network perspective, because both PIN are dense with edge densities ρ > 0.3, average in-/out-degrees well above 20 and a minimal edge weight of eth = 52 million USD during all times (see Tab. S-2 in the SI).

We have seen that the build-up of the GFC'08 can be well described under the PIN framework, confirming the conventional understanding, yet providing new insights from the network point of view. Two rather simple early-warning indicators for potential financial crises emerge from this analysis: The algebraic connectivity of the E-PIN, as a measure for structural robustness and the edge density of the LD-PIN, as a measure for the level of interdependence within the global financial system.

Indicator I: the algebraic connectivity of the E-PIN

A (hypothetical) decomposition of the E-PIN is expected to lead to a global economic crisis, since it disconnects international equity markets. The precise value of depends on the numbers of nodes and edges and the network topology, rendering values for different times not directly comparable to each other. However, as explained in Section S-2 in the SI, for most practical applications with , one is an effective upper bound and, as such, the interval [0, 1] offers a quasi-absolute scale. An alternative scale is set by the value of the next-largest eigenvalue of the normalised Laplacian λ2 ≥ λ1, where one can consider the difference of both values Δλ = λ2 − λ1, given that λ2 is approximately constant and smaller than one. Looking at Fig. 3-c, where and ΔλE are given for reference, both scales point to an increased structural fragility of the E-PIN before and during the GFC'08, which we will investigate more closely now. Two fundamental questions here concern the properties of this instability, namely if it affects the whole network and to what extend and which countries are most-related to it.

To address the first issue, we consider the graph bi-section, given by the ordering of nodes according to the signatures of the entries of the eigenvector associated with (Fiedler vector), which we call Fiedler bi-section33,34. As stated in Section S-2 in the SI, this is expected to result in a good graph bi-section, in terms of a low cut ratio R(s+, s−) = scut/(|s+| · |s−|), when is close to zero.

To evaluate the quality of Fiedler bi-sections of both PIN, we compare them with different types of random bi-sections, where the nodes of a PIN are randomly divided into two groups of equal size, random sizes of 1 to N − 1 nodes or sizes of the corresponding Fiedler bi-section (labelled Fiedler-like). The results of this exercise are summarised in Section S-3 in the SI, where we use the cut depth Dcut of a bi-section of weighted graphs, which relates the average weight of edges between the two partitions of a bi-section to the average edge weight of the whole network. A value larger than one indicates that two partitions are connected by stronger-than-average edges, while the opposite is true for a value smaller than one. We see that Fiedler bi-sections, which are far outside the given 99%-confidence intervals, are significantly different from any random bi-section of the E- as well as the LD-PIN.

Thus, given the moderately low values of observed in the years before the GFC'08, the Fiedler vector can be used to detect the fault lines in the E-PIN. The details of this bi-section are summarised in Tab. S-3 in the SI, where the nodes contained in the smaller sections and their in- and out-degrees (numbers in parentheses), as well as the corresponding fractions of nodes for all years, are given. Note that such a bi-section is always possible, while one expects an approximately equal partition of a graph for large values of λ1, since this will, on average, minimise the cut ratio in the absence of a particular λ1-instability. This is exactly what was observed for the E-PIN at the beginning and the end of the observation period and for the LD-PIN at all times.

We use the parameter triple for the description of the observed structural instability of the E-PIN prior and during the GFC'08. Considering Fig. 3-d, where the percentage of nodes is given by the red numbers, one sees that the drop of at the beginning of 2005 can be explained by the emergence of a sparsely connected group of countries, which is expressed through the small cut depth at that time.

This picture is changing dramatically during the GFC'08, when the cut depth rises substantially in the beginning of 2009, while recovers partially. This is explained trough a very peculiar network configuration, where the US and UK, which are the most central nodes in the E-PIN (see Fig. 2-c), enter the much smaller section (see Tab. S-3 in the SI), indicating considerable fault lines within the E-PIN during the crises.

Next, we identify the nodes which can be related strongest to the emergence of the fragility of the E-PIN at the beginning of 2005. Many of the high-degree nodes in during 2005–2008 are classified as OFC, defined as “a country or jurisdiction that provides financial services to non-residents on a scale that is incommensurate with the size and the financing of its domestic economy”35 (page 7), such as the Bahamas, the Cayman Islands or Guernsey. On average, OFC make up around one third of nodes in the E-PIN, while lists of OFC according to35 and found in the E-PIN are given in Tab. S-1 in the SI. This finding raises the question if a tight integration of OFC into the E-PIN can be associated with the observed structural instability.

To answer this question, we remove multiple groups of 1–10 nodes and observe the resulting changes in for 2005–2008. The new E-PIN is defined as the largest strongly-connected component after that node removal. It is reasonable to assume that a node or a group of nodes are relevant for the formation of the observed -instability if removing it leads to a considerable increase in to a value larger than 0.5: either a clear jump of to a value around 0.7, or the same qualitative picture as shown in Fig. 3-c, is observed. Such instability lifting between 2005–2008 can be partial or complete, i.e. during some years or the whole period. The only two countries, which completely lift when being removed, are Bahrain and Kuwait. This suggests that the observed structural instability is centred around these two Middle Eastern countries, which are both not highly central to the E-PIN in terms of their total degrees or strengths. One may therefore conclude that the observed -instability is not of systemic relevance. Note, however, that this is a static picture, where it is not at all clear how a full or partial break-up of a graph, caused by the removal of some edges or nodes, affects the dynamics in the network, where scenarios such as cascading failures are imaginable. Furthermore, the modern financial system is a global one and the situation turns out to be more complex when considering groups of countries, as is expected from the network perspective.

For pairs of countries, Bermuda and Guernsey are the only combination which can partially lift , bar pairs involving Bahrain or Kuwait. For triples, we find two groups made of the UK, Jersey and either Bermuda or Egypt, which partially lift the instability; again, bar those involving previously found single countries or pairs. For larger groups, the picture turns out to be more complex. Note that an exhaustive check of all possible combinations of groups of sizes n = 4–10 is computationally infeasible because of the super-exponentially growing number of possible groups. We therefore use a statistical two-step breadth-first search algorithm to detect combinations of countries that are involved in the formation of the -instability. After removing Bahrain and Kuwait from the pool of possible selections, we perform the following computations:

  • Breadth: For n ∈ {4, …, 10}, randomly draw 104 samples and separately calculate between 2005 and 2008. Next, order all countries according to the frequency that they have been involved in groups lifting for at least one year.

  • Depth: For n ∈ {4, …, 10}, draw again 104 samples and check the -instability, but now half (round-up) of nodes of each sample is drawn from the corresponding ten most-frequent countries from step one and the other half is drawn from the remaining pool. This procedure is based on the observation that approximately the ten most-frequent countries show a relative frequency in their numbers of occurrences higher than what one expects from a pure random selection. Again, order all countries according to their frequencies of being found in groups lifting for at least one year.

Since this search routine is non-deterministic for n = 4–10, we perform five full rounds and average the results over all outcomes. The final results are summarised in Tab. 1, where the average probability of finding a combination of nodes, whose removal leads to a jump in , is given by pλ. Its step-like behaviour for rising values of n is caused by the rounding rule of step two. We should point out that there are no considerable fluctuations in the overall search results between different rounds, which is especially true for large n, such as n = 9 and n = 10, with mean-over-standard-deviation ratios of pλ of 0.017 and 0.008, respectively.

Table 1 List of countries most-associated with the -instability for groups of sizes n = 1-10. A country or group of countries is associated with the -instability if its removal from the E-PIN leads to a partial or complete lifting of over a value of 0.5 during the period 2005-2008. The cases n = 1,2,3 have been treated exhaustively. Bahrain and Kuwait are seen to be the only single countries whose removal lifts , while the picture turns out to be more complex when considering larger groups. For n = 4-10, a two-step breadth-first search algorithm has been used, excluding Bahrain and Kuwait. Results have been averaged over five full rounds. The average probability of finding a combination of nodes, whose removal leads to a jump in , is given by pλ. Its step-like behaviour for rising values of n is caused by the rounding rule used in step two of the search routine. The 15 most-frequently found countries are shown. The group consisting of Bermuda, Guernsey, United Kingdom, Jersey and Egypt is persistently associated with the -instability, suggesting that these countries are most central for its formation. estimates the average representation of off-shore financial centres (OFC) among all nodes found by the search routine. A value larger than one, which is observed persistently, means a relative over-representation of OFC
Full size table

The countries mostly associated with the -instability are shown in the right column. For n > 3, the 15 most-frequently found countries are ordered according to their frequencies of occurrence. There is a group of OFC (Bermuda, Guernsey and Jersey), the UK and Egypt, which are found persistently and are considered central to the observed instability. For less-frequently associated nodes, there are several countries which show up repeatedly, but in different positions. These changes of macroscopic ordering are attributed to network effects, in the sense that the removal of some nodes can only lead to a rise in when contained in certain compounds. It is most difficult, i.e. there are the fewest combinations found to lift , in the beginning of 2008, suggesting that the instability is most deeply rooted in the network structure of the E-PIN at the onset of the GFC'08. In addition, any combination able of lifting at that time involves the group of nodes Guernsey, Egypt, Luxembourg and the UK, which is particularly interesting because Luxembourg is not frequently occurring, but highly central to the E-PIN. Note, furthermore, that the three countries mostly associated with the drop in in 2005, Bahrain, Bermuda and Kuwait, are found to be related to a similar phenomenon within the SD-PIN (see Section S-9 in the SI), which underpins the obtained results and further demonstrates the applicability of the used concepts.

To quantify the concentration of countries classified as OFC among the nodes associated with the instability of the E-PIN, we estimate the average representation of OFC among all nodes found by step two of the above search routine. We define the quotient

where , x ∈ {λ, E}, is the averaged relative frequency of OFC between 2005–2008 in the groups associated with the -instability and in the E-PIN, respectively. A value of greater than one, which is observed consistently, means a statistical over-representation of OFC among the countries associated with the -instability before the GFC'08. One observes a decrease of , with rising n which is expected because OFC represent the minority of nodes.

In our opinion, this consistent over-representation of OFC suggests the necessity of a further investigation of the central role of individual countries and groups of countries, such as OFC, regarding the stability of the global financial system, for which methodologies from network science, as presented in this work, could prove useful.

Two remarks are due at this point. First, even though we observed a structural destabilisation of the E-PIN, as measured by , as early as 2005, a note of caution should be made, when directly relating this finding to the GFC'08. It is not clear, if and how this instability played a role during the crisis, where the main question concerns the reconciliation of the static structural picture with a dynamic event. A low value of might, however, be associated with a potentially increased susceptibility of the financial system towards shocks.

Second, the algebraic connectivity of the LD-PIN, , might as well serve as an indicator for potential financial crises because a separation of large parts of international debt markets is equally expected to lead to drastic consequences for the global financial system. In the current analysis, the temporal evolution of does not give reasons for major concerns. However, from the beginning of data taking, its value is constantly decreasing, still remaining at a relatively high level, till it suddenly rises in the beginning of 2006. A possible explanation for this behaviour, which will not be discussed further, is the rise of the Target Nominal Federal Funds Rate of the US Federal Reserve System36. This coincides with a temporary slow-down in the increase of the number of edges in the LD-PIN (see Fig. 1-b) and a halt in its expansion (see Fig. 3-a) at that time.

Indicator II: the edge density of the LD-PIN

Our second network indicator for potential financial crises is the edge density of the LD-PIN at the minimal percolation edge threshold million USD. As shown in Fig. 1-b, its temporal evolution can be used to describe the magnitudes of NOA-CDS and GMV-ELD24. It can be seen that the evolution of all three quantities share some prominent features, while the curves for NOA-CDS and GMV-ELD lag somehow behind. The fact that the scale of the rise and fall of the values of derivatives is much larger than that of can be explained by several factors. First, financial derivatives can be re-bundled to higher-order products, inflating the total value with respect to the underlying base assets, resulting in a global derivative market (notional amounts) which is at least one order in magnitude larger than the total volume in PIN24 or world-GDP29. Furthermore, it is known that small changes in (edge) density can cause abrupt changes in global properties of a wide range of systems (phase transition), including many complex networks28,37,38.

Consequently, there must be non-linear effects present. We introduce a simple phenomenological non-linear short-term memory model (NLSMM), which is based on the assumption that total market values of financial derivatives scale non-linearly with the percolation edge density of the LD-PIN, , while certain memory (hysteresis) effects can be observed.

For the total market value VD (tn) (NOA or GMV) of some derivative VD at time tn, with respect to a reference value Vr = VD (tr), we write

Here, tn− 1 denotes the data point before tn, which is the previous year for a 1-year period of the CPIS data18; γ1 and γ2 are scaling exponents, where a value different from one implies non-linearity; ar is a scaling factor, accounting for the arbitrarily chosen reference value Vr.

We consider different lead/lag time shifts of Δt = +12, +6, 0, −6, −12 months, according to the 6-months period of derivative data24, between and NOA-CDS/GMV-ELD, when performing the fits to determine the scaling exponents γ1,2 (see section Methodology for details on the fitting procedure.). Positive values of Δt (lead) indicate that changes in cause changes in NOA-CDS/GMV-ELD, while negative values (lag) indicate that changes in are caused by changes in NOA-CDS/GMV-ELD. We find (Δt)CDS = 6 months and (Δt)ELD = 0 months. As expected, there is a tendency towards a lead of with respect to NOA-CDS/GMV-ELD, while a higher resolution in both data sets18,24 is expected to considerably clarify these relations. Note furthermore that positive values of Δt enhance the applicability of the NLSMM (2) for the indication of financial crises, since potentially high levels of market interdependence can be spotted earlier.

The results for both fits, using (2), are shown in Figs. 4-a and -b with reference values of 2005 and 2002, respectively. One can see that the NLSMM achieves a good description of the proliferation of CDS and ELD in most cases and that the agreement is especially good for the years around the GFC'08 between 2007–2009. The mismatch for the year-2005 NOA-CDS point may either be considered as an outlier, or be explained through the coinciding start of the CDS statistics data taking24, which fell into a time of large market growth.

Figure 4
figure 4

Fit of the non-linear short-term memory model (NLSMM, Eq. 2) for the phenomenological description of the proliferation of OTC-traded financial derivative products, where values are given in relative magnitudes with respect to a reference year.

One mostly observes a good agreement between data and the NLSMM. Setting a warning threshold wth, which has been set as a multiple of world-GDP (gray dashed line), potentially dangerous levels of interdependency (red dots) can be detected, where the red dashed line indicates the time of first warning. For positive time shifts Δt, such a warning signal is generated before the actual fitted values reach wth, as is the case for NOA-CDS (a) and NOA-UAD (c). For all shown derivatives, the NLSMM in combination with the specified warning thresholds wth, produces warning signals at the beginning of 2007. It is particularly interesting that the NLSMM describes the “hidden” variable NOA-UAD well24,39,40. (a): NOA-CDS, with γ1 = 11.0, γ2 = 6.6, Δt = 6 months, pr = 0.92, wth = 0.56 world-GDP. (b): GMV-ELD, with γ1 = 7.3, γ2 = 8.0, Δt = 0, pr = 0.95, wth = 0.014 world-GDP. (c): NOA-UAD, with unallocated derivatives, γ1 = 5.9, γ2 = 6.0, Δt = 6 months, pr = 0.95, wth = 0.75 world-GDP.

Full size image

For both fits, the scaling exponents γ1,2 are much larger than one, showing that the relations between and the market values of financial derivative products are highly non-linear. Note that changes in the reference point Vr mainly change the coefficient ar, while the scaling exponents γ1,2 stay approximately constant. We define the ratio m ≡ γ2/γ1, which indicates the time span over which changes in the LD-PIN and derivatives markets are coupled, where a value greater than one means that past-year values of contribute stronger in (2) than present-year values. It might serve as a measure for “market memory”. Here, , indicating that changes in have a longer lasting effect on GMV-ELD than on NOA-CDS, which means, on the other hand, that NOA-CDS reacts faster to changes in the LD-PIN.

A way to make use of the NLSMM for the quantitative indication of systemic risk and, as such, potential financial crises, is to set a warning threshold wth in terms of a maximal level for the market value of a certain product, which one deems still safe. One such possibility is a dynamic threshold, i.e. a threshold which allows for market changes which a certain derivative product may be coupled to, such as a macroeconomic reference variable (RV). If VRV (t) denotes the value of an RV at time t, we require

where VD is the market value of some derivative product (NOA or GMV) and is a multiplier, which sets the maximal proportion between VD and VRV. This is a dynamic threshold, in the sense that, if VRV increases, VD is also allowed to increase. However, if VRV decreases, such that VD comes close to wth or if VD exceeds the threshold, VD must be reduced by some mechanism. A regulatory approach, based on monetary incentives to enforce a certain “equilibrium level” , could be to imposed an RV-progressive transaction or holding tax on financial derivatives, which makes especially speculative trading unattractive, as soon as the level of derivative D comes close to .

In Fig. 3-e, we investigate eight macroeconomic quantities with respect to their usability as RV for GMV-ELD, where the relative magnitude of Cref ≡ VELD/VRV is plotted with reference to their year-2002 values (foreign exchange market turnover volumes for North America, FX vol. (N.A.), are only available from October 2004 on).

We indicate the shape which a suitable reference curves Cref should approximately follow (shaded area in Fig. 3-e), to deliver a clear warning signal before the GFC'08, i.e. to be able to set wth appropriately. We identify four potential RV, namely world-GDP29, global trade volume of goods (world trade vol.31,), global stock of foreign direct investment (world-FDI stock32,) and the foreign exchange market turnover volume in Singapore (FX vol. (SG)41,), while we suggest to exclude the latter because it only covers a regional market. In our opinion, world-GDP offers the best RV for the case at hand. In addition, GDP is among the most-frequently referred-to indicators to quantify economic activity and, as such, represents a well-established baseline which the OTC-derivatives market may be compared to.

The value of does not need to be set precisely, in the sense that equivalent warning signals will be generated within a comfortably large range of values. Taking world-GDP as RV, can be set within the ranges of 0.53 – 0.85 and 0.012 – 0.018 for warning signals in the beginning of 2007 for NOA-CDS and GMV-ELD, respectively, while the actual values have been set to 0.56 and 0.014 in Figs. 4-a and -b. This might provide enough time to prevent the worst consequences of the insetting downturn. To further investigate the relation between the LD-PIN and the OTC-derivatives market and the applicability of the proposed methodology, we tested the NLSMM (2) for the description of all major classes of financial derivatives and their first subcategories, given in24. The results are summarised in Tab. 2. One can see that the NLSMM gives mostly good descriptions for CDS and ELD, while those for foreign exchange (FXD) and interest rate derivatives (IRD) are generally poor. Interestingly, NOA of unallocated derivatives (UAD, see Fig. 4-c), which stems from the difference of reporting institutions between24 and39,40, is described very well, offering a tool to indirectly measure this large, but “hidden” variable.

Table 2 Fit results between the percolation edge density of the LD-PIN and notional outstanding amounts (NOA) and gross-market values (GMV) of all major classes of OTC-traded financial derivatives and their first subcategories24, using the NLSMM (2) (scaling factor ar and exponents γ1,2 for a fit of VD(t)/Vr). The fraction/multiple of world-GDP of NOA/GMV as of the middle of 2008 is given for reference for each class/subcategory, where the mean between the 2008- and 2009-values has been taken for world-GDP. The quantity m ≡ γ2/γ1 indicates the time span over which changes in the LD-PIN and the derivative market are coupled (“market memory”), where a value greater than one means that past-year values of contribute stronger in (2) than present-year values. Δt (in months) is the best-fit lead/lag time shift between and NOA/GMV of a certain derivative class, where positive values indicate a lead of . We use the Pearson product correlation coefficient pr between the best-fit and market values of derivative products as a goodness-of-fit criterion, where we accept a fit if pr ≥ 0.9. We say that a certain product may be described by the NLSMM if 0.9 > pr ≥ 0.85 (conditional acceptance) and reject the fit if pr < 0.85. The major classes are credit default swaps (CDS), foreign exchange derivatives (FXD), interest rate derivatives (IRD), equity-linked derivatives (ELD) and commodity-linked derivatives (CLD). The first sub-categories are single- and multi- name instruments SNI and MNI for CDS, respectively and forwards and swaps (F&S), swaps (SWP), options (OPT) and forwards (FWD) for the other classes. CLD additionally include gold derivatives (GLD) and others commodities (OTH). Unallocated derivatives (UAD) are values which are not covered in24, but are included in39,40. Gross-credit exposure (GCE) measures the positive net-value of contracts, after mutual obligations have been set off (netting). Fit results for NOA of exchange-traded derivatives (EXD) are given for reference, where one sees that future contracts (FUT) can also be described by the NLSMM. The NLSMM, Eq. 2, is seen to be especially suitable for the description of the proliferation of CDS (NOA and GMV), which is the financial derivative product which has most-frequently been related to the GFC'08
Full size table

An additional class of derivatives where the NLSMM offers decent descriptive power and which saw a strong rise in popularity in the years before the GFC'08, for both hedging and speculation, are commodity-linked derivatives (CLD). Here, the strategy to hedge equity and bond risks using CLD, creating large amounts of additional interdependencies in global financial markets, failed in face of the crisis42.

One might ask if the edge density of the E-PIN ρE contributes additional information to the analysis similar to that provided by . Note, at this point, that there is an underlying reason for choosing to describe the proliferation of financial derivative products; namely, LD-securities can be seen as “durable” base assets for or be linked to various derivatives, which is generally not the case for E-securities. We may, however, infer some information from the evolution of ρE. We observe two major decreases of ρE between the years from 2003 to 2004 and 2005 to 2006 (see Fig. 1-a), where the number of edges is approximately increasing in the same manner as the number of nodes, consequently leading to a reduction of the edge density (see Section S-2 in the SI). These two events might be associated with the pick-up of equity investment after the 2001 recession23 and the slowdown of growth of the LD-PIN (see previous subsection), offering an incentive for investors to switch from debt to equity securities.

In summary, both early-warning indicators for financial crises are intrinsically tied to the network perspective, where the global financial system is treated as a complex multi-layered network consisting of strongly-interacting components. Especially, in view of the recent financial crisis, which conventional models have not been able to foresee, or even to describe properly, this offers a completely new perspective to macroeconomics, where the topology of a complex system, such as the international financial architecture, is taken into account explicitly. In our opinion, the GFC'08 is an excellent learning ground for the development of new methodologies aiming at the prevention of future large-scale economic downturns. We finally stress that one of the biggest obstacles for the development of such tools is the scarcity of openly available high-quality data, which is especially daunting in the dawning age of big data.

Discussion

We have investigated the applicability of the presented early-warning indicators for financial crises by looking at the robustness of the results against changes in the edge threshold eth, which has previously been set according to the percolation properties of the LD-PIN (see Section Methodology). It turns out that results from the E-PIN are highly insensitive to the choice of eth, while results stemming from the LD-PIN are robust against variations of eth in a conveniently large window of up to 30 million USD. To probe the threshold dependence of the results, we rise eth in a step-wise fashion from 1 million to 1 billion USD, considering a total of 500 values, which are equally-spaced on a logarithmic scale. The results from this exercise are shown in Fig. S-4 in the SI.

The threshold dependences of the network sizes, in terms of the numbers of nodes of the E- and LD-PIN, are shown in panels (a) and (b), respectively, where a near constant value of million USD is identified above which the LD-PIN (b) disintegrates rapidly.

The eth-dependence of the edge density ρLD of the LD-PIN, which we have taken as a network-proxy to measure the interdependence of financial markets, is shown in panel (d) of Fig. S-4. The detailed evolution of ρLD over time is rather sensitive to the choice of eth, while most fitting results for the description of financial derivatives, when using (2), do not change within a range of eth = 25–55 million USD (shaded regions in panels (b) and (d) of Fig. S-4), where the number of nodes of the LD-PIN is approximately constant. This window of possible values is considered comfortably large for practical implementations of the proposed methodologies. There are, however, slight differences of the goodness-of-fit for different derivatives. The proliferation of CDS is generally described better at larger threshold within the stated region, while one achieves better description of ELD and CLD for lower values of eth.

The threshold dependence of the algebraic connectivity of the E-PIN, as a measure for robustness of the global financial system and as such for the world economy, is shown in panel (c) of Fig. S-4. The main features of the temporal evolution of , such as its sharp drop in 2005 and its persistent low value until the GFC'08, are largely threshold-independent. No qualitative changes are observed for values of eth between 1 to 100 million USD, which makes the algebraic connectivity of the E-PIN a robust indicator for practical implementation in the current case. We remark that the actual details of such a -instability should be carefully investigated, when detected, as we have done in this work. Its large value for high thresholds eth stems from the induced removal of nodes, resulting in a more robust core-component.

Considering the anti-correlation of volume fractions of the total PIN shown in Fig. 2-b, one may also take an alternative approach: a joint threshold of, say, eth = 110 million USD, for both PIN together, where particular values used for the E- and LD-PIN are determined on a yearly basis according to their respective fractions of the total PIN. This has been done. It turns out that results do not change considerably, which is the reason for choosing a simpler single value for both networks at all times.

Methods

Network set-up

Data for different types of cross-border portfolio investment networks (PIN) for eleven consecutive years, 2002–2012 (beginning of the year), come from the Coordinated Portfolio Investment Survey (CPIS)18, conducted by the International Monetary Fund (IMF). Portfolio investment is an indirect investment, defined here as a cross-border transactions or positions of equity or debt securities, excluding direct investment, reserve assets, financial derivatives and bank loans43. The CPIS includes aggregated data for end-of-the-year positions in USD from 78 reporting creditor countries for three types of securities: equity (E), long- (LD) and short-term (SD) debt. E-securities, such as shares, stocks, participations or similar documents indicating ownership and LD-securities, such as bonds, debentures and notes with a maturity of more than one year, make up for the majority in terms of total volume, approximately 94%, as seen in Fig. 3-a and Tab. S-2 in the SI. We therefore concentrate on these two types. A summary of results on the SD-PIN and how the proposed methodologies can be used for its analysis is given in the SI.

In the networks constructed from the CPIS data, nodes are represented by individual countries and weighted directed edges by aggregated portfolio investment positions between them. In terms of the weight matrix W (see Section S-2 in the SI), an investment position from country i in country j is given by wij.

To allow for a better comparability of results over time, all monetary values (investment positions, derivatives statistics, trade flows and market turnover volumes) have been adjusted for changes in world-GDP, using the GDP deflator29 (constant 2013-values).

The LD-PIN is seen to have an approximately constant percolation edge threshold with an minimum value of million USD (see Fig. S-4-b in the SI). This is the edge weight above which the LD-PIN rapidly disintegrates, when consecutively removing edges of larger weights, as measured by the size of its largest strongly-connected component. There are two notable exceptions to this behaviour. Namely, before and during the GFC'08, in 2007 and 2008, the percolation point rises by about a factor of three, which is an interesting observation in its own right. The graph connectivity at the edge threshold , where edges are deleted, i.e. wij ≡ 0, is expected to contribute dominantly to the global properties of the weighted network, while still allowing for a good comparability of results over time.

Data between non-reporting countries, as well as liabilities between any two countries, are missing from the CPIS18, which gives the resulting networks a “sea urchin-like” topology. Nevertheless, the present data include mutual investment positions for all major economies, bar the Mainland of China, where cross-border capital flows are highly restricted and, as such, would not contribute significantly44,45.

To account for both the percolation properties of the LD-PIN and the incompleteness of data, we define a PIN as the largest strongly-connected component, after applying the minimum percolation edge threshold million USD.

There is no such general percolation point for the E-PIN. However, for some years, a rather low value below 10 million USD can be found, while for other instances, a more continuous disintegration with a rising edge threshold is observed, as shown in Fig. S-4-a in the SI. This is most notable for the year 2007, where the largest number of nodes for low values of eth is observed, which already indicates a relatively fragile expansion of the E-PIN prior to the GFC'08. We note that it may come to negative positions in the CPIS data under certain conditions, as explained in the CPIS guide18. Such positions are indeed observed, making a total number of nodes larger than the 78 reporting countries possible.

For reasons of comparability and simplicity, the same edge threshold is used for the E-PIN in this investigation, where the network volume is of similar magnitude. Final PIN volumes, after extracting the largest strongly connected components, are around 95% of the initial amounts, i.e. we make use of the great majority of data after the above-described clean-ups.

Model fit

We use a least-square method to fit the percolation edge density of the LD-PIN to the data from the OTC-derivatives statistics24, when implementing the non-linear short-term memory model (NLSMM, Eq. 2). The best-fit lead/lag time shifts between both quantities, where we considered values of Δt ∈ {+12, +6, 0, −6, −12} months, are obtained by minimising the normalised squared-difference ||VD − fit(VD, Δt)||2/||VD||2, where ||·|| is the Euclidean norm of a vector containing values from different time instances, VD is the market value of financial derivative D, in either notional outstanding amounts (NOA) or gross-market values (GMV) and fit(VD, Δt) is the best-fit of (2) for a given Δt. Overall differences in the average values of derivatives for different time windows are accounted for in this way.

The Pearson product correlation coefficient pr46 has been used as a goodness-of-fit criterion, where we accept the best-fit fit(VD, Δt) if pr(VD, fit(VD, Δt)) ≥ 0.9, which turns out to provide overally-good results. We conditionally accept a fit if 0.85 ≤ pr(VD, fit(VD, Δt)) < 0.9, while we reject a fit if pr(VD, fit(VD, Δt)) < 0.85.

Note that PIN and derivatives data have different time resolutions of 12 and 6 months, respectively. To be able to make full use of all available data, we interpolate to obtain the same resolution for both data sets. This is justified on the ground that is a slowly-changing variable compared to all derivatives data. Alternatively, dropping one half of the derivatives data is seen to lead to less accurate results, in the sense that more fits are actually accepted due to the induced smoothing-effects.

As can be seen from Tab. 2, one may obtain one negative scaling exponent (mostly γ1) for the best-fit solution in some cases. Depending on the absolute value, as compared to the other exponent, this means that the corresponding contribution is suppressed and the fit is effectively achieved through one parameter only. Note, however, that fit results are not satisfactory in all of these cases.

Data for credit default swaps (CDS) are only available starting from the end of 2004 onward. To be able to make use of all data points for all times, we consider intervals of varying lengths for different values of Δt, which introduces a bias towards lags of , when selecting the best-fit. Obtaining persistent lead relations for all given CDS statistics, strengthens the results, which does not change when considering intervals of equal lengths by dropping part of the data.