Loan maturity aggregation in interbank lending networks obscures mesoscale structure and economic functions

Since the 2007-2009 financial crisis, substantial academic effort has been dedicated to improving our understanding of interbank lending networks (ILNs). Because of data limitations or by choice, the literature largely lacks multiple loan maturities. We employ a complete interbank loan contract dataset to investigate whether maturity details are informative of the network structure. Applying the layered stochastic block model of Peixoto (2015) and other tools from network science on a time series of bilateral loans with multiple maturity layers in the Russian ILN, we find that collapsing all such layers consistently obscures mesoscale structure. The optimal maturity granularity lies between completely collapsing and completely separating the maturity layers and depends on the development phase of the interbank market, with a more developed market requiring more layers for optimal description. Closer inspection of the inferred maturity bins associated with the optimal maturity granularity reveals specific economic functions, from liquidity intermediation to financing. Collapsing a network with multiple underlying maturity layers or extracting one such layer, common in economic research, is therefore not only an incomplete representation of the ILN's mesoscale structure, but also conceals existing economic functions. This holds important insights and opportunities for theoretical and empirical studies on interbank market functioning, contagion, stability, and on the desirable level of regulatory data disclosure.


A Maturity layer analysis
Here we give a detailed description and network analysis of the Russian interbank lending network to compare to other interbank networks used in research. We divide the Russian interbank loan network {G α } by maturity class α into separate layers consisting only of the loans with loan maturity inside each maturity class. (Note that in the main text we use the symbol l instead of α.) We evaluate the properties per maturity layer (often also referred to as layer for simplicity) to show the differences between them. We denote the eight maturity layers as: G <1d , G 2−7d , G 8−30d , G 31−90d , G 91−180d , G 0.5−1y , G 1−3y , G >3y .

Activity and volume
One expects that the number of banks participating in a given loan maturity layer decreases as the loan maturity lengthens. Table 1 shows that this is approximately correct. More importantly, one sees that the number of active banks becomes very small for G 1−3y and G >3y . We define an active bank as a bank that has lent or borrowed at least once in a given time window and maturity layer. We will always only consider active banks to deal with the death and birth of banks in the Russian interbank market over the course of time 1 . For the layers with long loan maturity, this results in an exceedingly high average lending activity per bank, comparing to layers with shorter loan maturity. It seems that G >3y is occupied by a small club of banks that trade relatively intense.  Table 2 and the top panel of this Table element-wise. Care must be taken when interpreting; this is not indicative of an 'average' bank since the degree distributions are heavy-tailed (as will be discussed below). Also note that the columns '1998' and '2004' are biased because they are incomplete in the data (respectively 5 and 10 months missing months).

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simply as the number of loans recorded during a certain period. A crucial observation is that the lending activity sharply decreases for longer maturities, which is also reported for other interbank markets in 2 . We see that the overnight segment (G <1d ) is the most active, together with G 2−7d . The impacts of the crises in 1998 and 2004 have had clear impact on the lending activity, except for the longer maturities; these seem relatively unaffected.  -90d  49  231  530  662  964  1,182  844  91-180d  16  38  62  99  159  233  201  0.5-1y  13  20  26  49  81  158  190  1-3y  10  17  23  17  16  76  84  >3y  11  12  15  31  235  143  89   Table 3 lists the total loan volumes (i.e. loan sizes) by loan maturity layer and year. We observe that the relative importance of each maturity segment, as measured by the total volume of loans traded within it, follows the ranking of the loan maturity lengths. The loan volumes are log-normally distributed 3 , especially for shorter loan maturities.

Maturity layer activity of banks
A node i is defined as active on layer α if it has at least one connection on this layer, i.e. its degree k α i > 0. In symbols, The total activity B i measures the number of layers the node participates in, i.e. B i = ∑ α b α i 4 . Figure 1 shows the distribution of B i on monthly time scales. We see that the average total activity grows steadily as the network develops, finally settling around at a value of about three. Almost no banks make use of more than six maturity layers on a monthly basis. The distribution of the total activity is quite broad and relatively unpeaked, especially during the stable phase of the network. This has been reported as a typical quality of real-world multiplex network 5 . The distribution of the total activity throughout time.
average activity with respect to Figure 1. For a given month, the area between the thick lines is proportional to the relative number of active banks with a certain B i value, which is indicated on the right. The dashed lines make up the moving coordinate system of B i , in relation to which the average total activity indicated by the thick purple line must be understood. For example, B i is about 2 (3) for month 1 (75). The summer of 2004 crisis is mirrored by the drop in B i during the last months.

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Density Real-world interbank networks are typically sparse, meaning that in directed and undirected views of the network only a small fraction of all possible edges exist. One can then ask how the edges are distributed among the banks. To this end, we define the degree of a node. For undirected networks, this is the number of edges connected with a given node, i.e. the number of the bank's counterparties. For directed networks, the in-degree (out-degree) of a node is the number of incoming (outgoing) edges.
The in-degree (out-degree) of a bank is then simply the number of loans borrowed (lent) by it.
The above seen differences in lending and bank activity per loan maturity layer is reflected in the directed density of the layers, shown in Fig. 2. All layers can be considered sparse, except for G 1−3y and G >3y , which have respectively moderate and an extremely high density 6 . We also observe that the density in almost all layers grows steadily as the network develops, pointing to an increasing interconnectedness with increasing development of the market 7 .

Degree distribution
Interbank networks, like many real-world networks 8 , exhibit heavy-tailed degree distributions. In a nutshell, this means that few nodes have many links and many nodes have few links. In general, many typical distributions are heavy-tailed -in fact, they abound in descriptions of natural events like avalanches, earthquakes, turbulent flow and rainfall 9, 10 ; another example in economics are the non-Gaussian return distributions in financial markets.
Early work in the previous decade tended to postulate power laws for the degree distributions. Networks whose degree distributions follow a power law (at least asymptotically) are called scale-free networks. However, while agreeing on the heavy-tail character, recent literature has cast serious doubt on the idea that power laws are the best candidate for the degree

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distributions, and thus on the scale-free character of interbank networks. In our case, power laws have been decisively rejected as best fit candidates for the heavy-tailed degree distributions of the <1d and 2-7d loan maturity by Vandermarliere et al.
We plot the heavy-tailed degree distributions with complementary cumulative distribution functions (ccdfs) 10 with doubly logarithmic scales. If we denote the degree and its distribution by k and p(k) respectively, the ccdf is given by where cdf(k) is the usual cumulative distribution function.
provide the best overall fit for the bulk+tail multi-directed in-and out-degree for monthly and yearly time windows. In Eq. 3, d is the in-degree or out-degree, C a normalization constant and λ , β distribution parameters. The stretched exponential can be understood as a Weibull distribution, with β being the shape parameter, and 1/λ the scale parameter. If 0 < β < 1, the distribution has a fat tail, with a smaller β putting more weight towards smaller d. 1/λ widens the distribution, the mean of f being proportional to it.
We fitted f to the degree distributions to see if the result by Vandermarliere et al. (2015) 3 can be extended to longer maturities. The conclusions are identical for the in-and out-degrees. First, the stretched exponential fit does seem to describe the time-aggregated degree distributions of G <1d , G 2−7d and G 8−30d . Then the leap in residual sum-of-squares suggests that G 31−90d and longer maturities can not be satisfactorily described by the stretched exponential. This is confirmed by visual inspection. While (parts of) the bulk distributions of the longer loan maturities are reasonably well captured by f , the fits systematically underestimate the ccdf in the tail; put differently, they underestimate for a given large degree d how many nodes exists with an even larger degree d .
To investigate whether frequent lenders are also frequent borrowers, we first rank the banks with regard to in-and out-degree, then we calculate Kendall's coefficient of concordance W , which measures the rate of agreement 0 < W < 1 between two rankings, per loan maturity layer. The significant results together with the joint degree distributions are shown in Fig. 5. We conclude that lending and borrowing are highly correlated for G <1d , G 2−7d , and G 8−30d ; in other words, the dominant lenders are likely to be dominant borrowers, and vice versa. This correspondence breaks down markedly for the layers with longer loan maturity, with orderly decreasing W but insignificant test results. This can also be seen in the shapes of the contour plots in Indegree distribution for each maturity layer, aggregated over time.

Figure 3.
In-degree distribution with attempts to fit the bulk+tail with stretched exponentials, drawn as piecewise linear functions connecting each predicted value. The correspondence with this distribution seems to break down for layers with loan maturity longer than 8-30d.
for maturity longer than 31-90d. G 1−3y and G >3y even display modest anti-correlation for the degree tails, meaning that the most frequent lenders (borrowers) area unlikely to be among the most frequent borrowers (lenders).
1e − 1) respectively, with all standard errors below 1%.  Figure 5. The time-aggregated joint degree distribution for each layer, drawn as density contour plots with almost transparent data points. For the first four maturity bins, a highly significant Kendall W was obtained and put in the top left corner of the panel. Note that all scales are equal for easy comparison.

Average shortest path length
The average shortest path length (denoted from here on as just 'average path length') D is defined for the largest connected component of the undirected view of interbank networks. It indicates the typical distance between two randomly chosen nodes, i.e. the smallest number of edges needed to reach one node from the other. Small-world networks are characterized by high clustering and small average path length. Both should be compared to their mean values in an ensemble of random networks with the same number of nodes and edge 11 .
Most studies find that D is small for interbank networks, which indicates compact network structure, but not all conclude that they are small-worlds. Hubs tend to lower the average path length; scale-free networks are ultra-small-worlds. D is a measure of the typical length of intermediation chains that are taking place among the market participants (at least in the largest connected component). Longer intermediation chains arise when D is large, which effectively contribute to slowing down the market transactions between participants and consequently harming the liquidity allocation between financial institutions.
In contrast, when D is small, the information between the market participants flows quickly in the network, giving rise to a well-functioning liquidity allocation in the market 12 .
In the case of the Russian interbank network, the average path length peaks at times when the network is in crisis, and decreases gradually when the network is maturing. The same patterns have been reported in other empirical literature, such as in 13 .  Table 1 on page 2. The relative size of the largest weakly connected can be found in Fig. 7.

Clustering coefficient
Clustering coefficients measure to which extent banks form triangles. Put differently, they measure the tendency of connected nodes to have common neighbors in undirected views of interbank networks.
According to Bargigli et al. 2 , an inverse relationship between the degree and the clustering of a node is observed quite commonly. In the core-periphery picture, low clustering values of core nodes indicate that they essentially behave as star centers. A star graph exhibits zero clustering, as the periphery nodes are unconnected amongst themselves. Consequently, deviations from the star graph, which has idealized core-periphery structure, can be probed by measuring the local clustering coefficients of the periphery nodes. One can identify these heuristically by simply considering the nodes with low degree, so the aforementioned inverse relationship hints that the core-periphery structure may exhibit considerable complexity. A second and more robust implication is that the clustering coefficient of the complete interbank network is dominated by the clustering of low-degree nodes if this inverse relationship is observed together with heavy-tailed degree distributions.
High clustering has obvious implications for systemic risk 3 and therefore clustering coefficients are of interest to the interbank network literature. Table 1 in the main text shows, that at least a few studies claim opposite 'typical' values for clustering coefficients. This may be due to the fact that the clustering tends to increase with longer time windows 2 , or that the coefficients may not have been compared to the mean clustering coefficient obtained for random networks of the same size and number of edges.
The local clustering coefficient 8 of a node i is defined in function of triangle motifs in undirected networks: where a triple on i means an unordered pair of nodes connected via i, and possibly connected directly. If that is the case, the triple is counted as a triangle, ending up in the nominator of Eq. 4 as well. Thus C i expresses the degree of connectedness among the neighbors of i 14 . The (global) clustering coefficient of an undirected (N, E) network is then simply the average of the local coefficients: To assess whether an observed network possesses a non-random clustering structure and thus a significant clustering coefficient C 0 , we generate n random networks of the same size, i.e. the same number of nodes and edges, having clustering coefficients . . ,C n }. Then we can calculate the z-score of the observed C 0 as Large z indicates a significant value of C 0 , the sign indicating more (+) or less (−) clustering than a random network of the 13/29 same size.
We have calculated clustering coefficients and z-scores for the undirected view of the loan maturity layers using yearly time windows which are displayed in Fig. 9. A clear pattern can be seen: the clustering varies from significantly high to moderate (in order G <1d , G 2−7d , and G 8−30d ), via moderately low to insignificant (in order G 31−90d , G 91−180d , and G 0.5−1y ), to significantly low (G 1−3y and G >3y ). Since all layers exhibit heavy-tailed degree distributions, the sum in Eq. 5 is dominated by the (local) clustering coefficients of banks with small degree. We can use C to proxy the clustering of the periphery banks Within that approximation, one sees that the core-periphery structure expected in the short maturity layers deviates considerably from the idealized star network, at least when looked at with yearly resolution. Notwithstanding the strong presence of intermediation found earlier, the lower-tier banks still trade extensively with each other, indicating that contagion risk is not located entirely in the high-tier banks.
In contrast, the two longest maturity layers, where source-sink structures are expected, exhibit considerable star-like structure, low degree nodes and periphery banks being equivalent in most cases. In addition to the low clustering, we recall the modest anticorrelation between in-and outdegree; these observations lead us to believe that G 1−3y and G >3y behave as a sort of source-sink star layers. These layers have star-like hub structure, but hardly any intermediation occurs in the hubs. They act simply as sources and sinks, generating and dissipating excess liquidity in the interbank network. Of course, one could argue that for loans with maturities of at least one year, no intermediation is possible within the scope of one year. Looking at Fig. 5, which completely aggregates time, however, we see that the banks with the largest degrees tend to be either sinks or sources, especially in G >3y . Thus the tendency of the hubs to be either sink or source, but not an intermediary, holds for time scales longer than one year. The next question is whether the star centers in the source-sink star layers are connected, or rather at the center of disconnected components. Disconnected components are plentiful compared to the number of active banks per year (see Fig. 8 and Table 1), and less than half of the latter participate in the largest (weakly) connected component, as Fig. 7 shows.
Furthermore, we report that in the two source-sink star layers the average shortest path length is between 2 and 3 (see Fig. 6, which shows the average path length for G 1−3y and G >3y on a monthly basis as being typically 2), which, together with the proven existence of hubs, indicates a compact star-like structure. In a nutshell, the source-sink star layers G 1−3y and G >3y are composed in general of many disconnected 'islands', of which the largest exhibit an almost perfect star-like structure around a sink or source hub.
All results are highly significant, except for G 91−180d and G 0.5−1y : these layers do not possess any structural clustering structure on their own. As always, the clustering in the aggregated network resembles mainly the first two layers. We further note that clustering is most volatile in times of crisis, and increases during the growing phase of the network 3 .
14/29 Figure 9. Time series of clustering coefficients for the loan maturity layers and the aggregated network, together with a ribbon centered around C rand of total width 2 × sd(C rand ). Following the methodology in 2 , a total of 100 Erdős-Rényi networks were generated to test the significance of the observed C's.

Bank size mixing as degree size mixing
Many studies point to disassortative mixing with respect to the bank size, meaning that small banks trade mainly with large banks and vice versa. Given the core-periphery picture in these studies, one expects that the core banks are the large banks.
Indeed, it is found that total bank assets are significant in explaining core membership 15 . As bank size correlates with total degree, we would also expect disassortative mixing of the banks with respect to the (total) degrees, i.e. high (low) degree nodes tend to be connected to low (high) degree nodes 8 . In fact several studies report this as an additional stylized fact 2, 16 . Figure 10 shows the assortativity with respect to the total node degrees on yearly basis, which is also called the total degree correlation. The layers with short maturity show a clear preference for the low-degree nodes to attach to the high-degree ones.
This preference weakens when we look at longer maturity layers. The assortativity or the long maturity layers is pushed up by the large number of disconnected clusters, in most cases simply isolated pairs of trading banks. The disassortative degree mixing in the larger clusters is present, caused by their star-like structure. As the number of unconnected components grows drastically, we would expect that the same mechanism is behind the high assortativity for G 91−180d and especially for G 0.   Figure 10. The degree assortativity measures to which extent nodes with a given degree associate preferentially with other nodes of similar degree -see Equation (2) in 17 for the formal definition. In this case the type of degree considered is the total degree, i.e. the sum of the in-and outdegree. A yearly resolution was chosen because monthly aggregation suffered from bad statistics starting from G 91−180d .

Summary
A summary of the results of this maturity layer analysis can be found in Table 1 Figure 11. The degree assortativity for the total degrees throughout the years. The degree assortativity is also called degree correlation 8 .
report the findings here as is and only make claims about function and structure in the main text where steps have been taken to gather statistical evidence for both the construction of the layers as the occurrence of structure in such layers.

17/29 B OG inference with non-contiguous binning
Here we present the results of the OG inference where non-contiguous binning was allowed in Fig. 12 (as opposed to the contiguous binning in the main text). This allows for partitions where short maturity loans are merged with long maturity loans.
As explained in the main text, the creation of a new bin in the OG needs to be warranted by enough statistical evidence for it to be necessary; i.e. that it will contain a lending pattern that is significantly different from the lending patterns in the other bins. In those cases where very long maturity loans are merged with short maturity loans, we suspect that there is hardly any actual structure (and thus not enough statistical weight) in the long maturity layer due to its sparcity. It then makes sense that such layer is merged with the densest layer available since this merging will cause the least amount of new connections to be introduced relative to the other, less dense, bins.
Comparing Fig.3 (a) and (b) in the main text to Fig. 12 (a) and (b) shows the results to be qualitatively the same.

D Activity characteristics per OGB
We investigate the OGBs and their typical characteristics. Fig. 14 shows the distribution of the log group strengths across time and per OGB. We discern three types of group strength: outstrength, instrength and internal strength. The group outstrength in a certain month of bank group b is the total amount lent by all banks belonging to b to banks belonging to another group b = b.
In symbols, The two other strength types and the constraint that the maturity layer l must be in a given OGB are obtained likewise from this definition. We see that the groups for OGBs 2, 3, 4 show definite "anti-community characteristics", i.e. they hardly lend or borrow internally. There is also an interesting asymmetry between group instrength and group outstrength in OGB 4. Looking at the group strength modes, each OGB is seen to function on its own magnitude scale (see Fig. 5 in the main text).

20/29 E Other percentiles for instrength/strength of important banks
We look at the lending behaviour of individual banks. As in the main text we again use the bank strength to single out the "important" banks. We show that the conclusions from the main text hold for different bank strength cutoffs. Each row of subplots in Fig. 15 uses a different cutoff, respectively with the bank strength (s) unlimited, limited to the top 10%, top 5%, and the top 1%. At monthly time scales the important banks in "short" bins tend to both lend and borrow equal amounts of money, while the important banks in the "long" bins tend to either lend or borrow.  log10(count) Figure 15. The time-integrated distribution of the strength and the instrength/strength ratio of the monthly "important banks" for the four OGBs. As in Fig. 4(b) in the main text we gauge a bank's importance by its strength: For a given month a bank is deemed important if its strength lies in the top X, with X varying per row of subplots (the top X% is defined by the (1 − X)th percentile). With growing OGB index, the important banks increasingly tend to either lend or borrow. Banks in OGB 1 and OGB 2 tend to balance lending and borrowing. This suggests that the economic function of the important banks in the various OGBs changes from financial intermediation (short maturity bins) to financing (long maturity bins) on a monthly time scale.

21/29 F The trust crisis
The it just fuelled the hysteria on the interbank market. Rumours about the identities of these ten banks started to spread rapidly.
Soon several inconsistent blacklists were circulating in the banking community as bankers tried to guess which banks were officially suspected of money laundering. Anecdotal evidence suggests that banks were actively helping to spread the rumour by removing themselves from the list and adding competitors in an attempt to escape the carnage. By consequence, The union of all the blacklists circulating in the banking community expanded in a few days to include dozens of banks, including several market leaders.
In the presence of total uncertainty about the quality of their counterparties, banks began to reduce limits on each other, which reverberated into an acute liquidity drought on the interbank market 21 16 short term long term Montagna and Kok (2016) 25 short term long term

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H Term structure of interest rates and loan volumes in the interbank loan market

The global categorical yield curve
The yield curve is obtained after "averaging" interest rates r over time and by maturity class. The averaging technique chosen is the median due to extreme outliers which disproportionately affect the mean: the standard deviation of the complete r population (including outliers) is σ ≈ 40%, yet 99% of the population is contained in [0, σ ]. Within that interval the standard deviation is about 7%, which is a more sensible measure of the interest rate dispersion. By using the median, we avoid choosing any cutoff.
Yield curves are usually considered with continuous maturities ranging from one day to several years, but the available data only records maturity classes. Although methods exist to estimate the continuous yield curve from discrete data 26 , we will only consider interest rates (and interests etc.) per category, i.e. per maturity class.
The solid line in the top panel of Figure 16 exhibits a typical stylized fact of the yield curve. It is upward sloping and has a convex shape, except for the >3y term, i.e. maturities three years or longer. The unusual steepness of the curve in Figure   16 is an artifact caused by the term categories; as they progress, they bucket a growing amount of maturities, so a continuous yield curve would be horizontally stretched with respect to the categorical yield curve. The upward slope is usually explained by classical expectations theory (ET). According to ET, interbank lending rates dynamics are determined by the structure 25/29 of liquidity supply and demand 27 ; the long-term interest rate is an average of expected future short-term rates, plus a term premium that increases with longer terms to compensate risk-averse lenders for the interest risk, which arises for lenders from fluctuating interest rates with respect to the base deposit policy rate. One can also extend ET by including separate premiums for liquidity risk (selling loans on the secondary market tends to be harder as their maturities lengthen) and default risk, also called counterparty risk, which is in theory governed by the credit rating of the borrower. Normally the short-end (long-end) of the yield curve is dominated by the liquidity (default) risk; both are considered components of interbank lending risk 28,29 . Each risk contributes to the upward slope of the yield curve given normal market conditions. The interest per average ruble is also displayed in the top panel of Figure 16. Loans with longer maturities are the most profitable for lenders. The drop for the >3y term could point to the fact that these contracts may be made on more amicable (flexible) terms; this would imply lower perceived interbank risk which could explain the negative slope between the 1-3y and >3y categories. To illustrate the dynamics that underlie the global yield curve, Figure 17(a) plots the categorical yield curve for each of the last 16 months in the data. In this period the by now mature interbank network deals with the trust crises which are explained in more detail in Appendix F. Figure 17(b) contains the (yield) spread, which we define as the difference in average yield between "long" maturities and "short" maturities. We define these maturities as two classes: the "long maturity" class consists of maturity classes (1-3y, >3y) and the "short maturity" class consists of maturity classes (<1d, 2-7d, 8-30d). The general shape

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of the curve is not sensitive to the definition of the "long maturity" class; the "short maturity" class is motivated in the paper, while the "long maturity class" is based on Basel recommendations and is also used in 2, 16 .
In normal times, interbank markets are among the most liquid in the financial sector: banks prefer to lend out excess cash since the central bank's interest rate on excess reserves is smaller than rates available in interbank markets 30 . During trust crises, the perceived default risk grows, which inflates interest rates according to ET. Riskier banks, i.e. banks at risk of being in financial distress, exert an externality on safer banks who subsidize their liquidity 30 . If the crisis gets worse, this externality on safer banks is so costly that they leave the unsecured market, and liquidity rich banks may prefer to hoard liquidity instead of lending it out to an adverse selection of borrowers; the interbank lending market dries up.
We see that this mechanism is indeed captured by the spread curve in the bottom panel of Figure 17: low spread seems to indicate abnormal market conditions with low liquidity because the short term interest rates increase quite relatively fast. Note in the upper panel that the long term interest rates stay almost constant during the crises and drop slightly during the recovery in between (roughly from January until April 2004). This suggests that we cannot make an analogy with the typical inverted yield curves of e.g. treasury securities that are associated with (predicting) recessions. Indeed, according to ET these are yield curves with negative slopes because investors have poor expectations of future interest rates. In contrast, we see for our data that the short term interest rates rise quickly during the trust crisis while the long term rates hardly change.

Characteristics of loan volumes and interests
The lower panel of Figure 16 shows the average interest and volume per term. The loan volumes are log-normally distributed 3 and this holds remarkably well for the interest too, especially for shorter terms. Because of the considerable variance on a linear scale, these averages may be interpreted only as rough order of magnitude estimates. With this in mind, Figure 16 shows that interest rate and volume are roughly negatively correlated: except for the bump at 0.5-1y and the case >3y, the volumes decrease as the interest rates increase. This can also be seen in the slowly varying average interest for the first five term categories. Table 3 lists the total loan volumes by term and per year. As stated there (p. 3), we observe that the relative importance of each term segment, as measured by the total volume of loans traded within it, follows the ranking of the loan terms. This, together with the typical volumes and interest rates in Figure 16 and the lending activity in Table 2 (p. 3), supports the conclusion that the Russian interbank network exhibits a distinct hierarchy with respect to the loan maturities, which we could summarize by saying that banks lend greater volumes at lower interest rates more often for shorter loan terms.