The Intrafirm Complexity of Systemically Important Financial Institutions

In November, 2011, the Financial Stability Board, in collaboration with the International Monetary Fund, published a list of 29"systemically important financial institutions"(SIFIs). This designation reflects a concern that the failure of any one of them could have dramatic negative consequences for the global economy and is based on"their size, complexity, and systemic interconnectedness". While the characteristics of"size"and"systemic interconnectedness"have been the subject of a good deal of quantitative analysis, less attention has been paid to measures of a firm's"complexity."In this paper we take on the challenges of measuring the complexity of a financial institution and to that end explore the use of the structure of an individual firm's control hierarchy as a proxy for institutional complexity. The control hierarchy is a network representation of the institution and its subsidiaries. We show that this mathematical representation (and various associated metrics) provides a consistent way to compare the complexity of firms with often very disparate business models and as such may provide the foundation for determining a SIFI designation. By quantifying the level of complexity of a firm, our approach also may prove useful should firms need to reduce their level of complexity either in response to business or regulatory needs. Using a data set containing the control hierarchies of many of the designated SIFIs, we find that in the past two years, these firms have decreased their level of complexity, perhaps in response to regulatory requirements.


Introduction
The Financial Stability Board (FSB) describes a systemically important financial institution, or SIFI, as a financial institution "whose disorderly failure, because of their size, complexity and systemic interconnectedness, would cause significant disruption to the wider financial system and economic activity" (Financial Stability Board 2011). 1 Developed in the aftermath of the 2008 global financial crisis (hereafter the 'post-crisis period'), this characterization represents an expanded regulatory definition relative to earlier ones based primarily on size (e.g., the list of "mandatory banks" subject to the Basel II capital regulations, see 72 FR 69298, December 7, 2007). While size-based thresholds are appealing from a regulatory perspective in that they produce a dichotomous outcome, thereby being transparent and easy to understand and implement, they are overly simplistic in the presumption that risk can be evaluated via a single value. 2 Although historically regulatory emphasis has been on the risk of a given organization, the collapse of Lehman Brothers in September 2008 highlighted the extensive interconnectedness of the financial system and the importance of considering not just the risk to a single firm but the risk to the entire financial system, i.e., the risk to financial 1 The definition used by others, including the Basel Committee on Bank Supervision (BCBS, 2014), includes two additional characteristics, namely the lack of readily available substitutes or financial institution infrastructure, and global (cross-jurisdictional) activity. Since our focus in this paper is on complexity, we motivate our topic using the simpler FSB definition but highlight that we are aware of the other definitions as well. We also note that the BCBS has advocated the use of an indicator for G-SIB designation that is an equally-weighted average of exposures in each of the five categories. For the determinants of complexity, they list: (1) notional amount of OTC derivatives, (2) trading and AFS securities, (3) level 3 assets, each with a 1/15 (6 2/3%) weight. Scores are then distributed into buckets with a capital add-on associated with the different buckets. 2 In their survey of metrics for quantifying systemic risk, Bisias et al. (2012) refer to reliance on a single number to assess risk as a "Maginot Line Strategy". They argue that risk is multifaceted and that therefore multiple metrics are required to capture risk. stability. 3 Much of the research on interconnectedness has been formulated mathematically in terms of networks. The importance of such research for policymakers was highlighted in a May 10, 2013 speech by then-Fed Chairman Ben Bernanke: "Network analysis, yet another promising tool under active development, has the potential to help us better monitor the interconnectedness of financial institutions and markets" (Bernanke 2013).
Yet despite a large literature addressing the interrelationships among economic and financial network participants (e.g., Elsinger et al., 2006, Cohen-Cole et al., 2010, Haldane and May 2011, Adamic et al., 2012, Battiston et al., 2012, Billio et al., 2012, Hautsch et al., 2012, 2013, Kapadia et al., 2013, Squartini et al., 2013, Caccioli et al., 2014, there has been comparatively little development of metrics concerning the complexity of the individual firms that comprise the system -the other key attribute highlighted in the FSB's definition. Alessandri et al. (2015) point to the Lehman collapse as an example of how widespread financial instability resulted from the failure of a complex organization and go on to argue that a firm's complexity is a much more critical determinant of systemic importance than its size. Hampering the development of research related to a financial institution's complexity, in addition to data limitations (National Academy of Sciences 2010), is that "we lack a clear consensus on how to assess an entity's complexity" (Cetorelli, McAndrews, andTraina 2014, Cetorelli andGoldberg 2014). Failing any direct definition, one view of an individual firm's complexity comes from the lens of governance: "high complexity" would be interpreted as a corporate control structure rife with governance challenges for a firm's management, resulting in a lack of oversight that in turn poses significant operational, reputational, and balance sheet risk (Adams 2010, Vitali et al., 2011. This same complexity might present challenges to bank regulators, tasked with gathering information from a variety of sources to assess the systemic importance of the firm. This is particularly the case for the Federal Reserve, as it has the unique role of being the consolidated supervisor. 4 Organizational complexity contributes to the possibility that subsidiaries act in relative obscurity within the organization, thus hindering the regulators' ability to carry out effective consolidated supervision. In this context, complexity therefore poses risk to an accurate assessment of the organization's systemic importance and hence to financial stability. 5 One way to describe the organizational structure of a firm is via its control hierarchy (Vitali et al., 2011), consisting of a (parent) company and all of its subsidiaries, considered in its natural hierarchical and networked arrangement. This is a standard representation of the intraconnectedness of a firm, along the lines of Coase (1937) who described a firm as a "system of relationships". 6 While interfirm financial network complexity is a well-studied subject (see e.g., Arinaminpathy et al. 2012, Caballero and Simsek 2013, Haubrich and Lo 2013 and references therein, Roukny et al. 2013, Battiston et al. 2016a,b, Aldasoro and Alves 2018, Berndsen et al. 2018, Constantin et al. 2018, Roukny et al. 2018), many of the network methods in that literature are not applicable to the very specific 4 In providing guidance as to how to effectively carry out this role, the Bank Holding Company Supervision Manual (Board of Governors of the Federal Reserve System, 2017) notes that "The Federal Reserve's conduct of consolidated supervision is central to and dependent on the coordination with, and reliance on, the work of other relevant primary supervisors and functional regulators" (page 374, §2060.05.6.8). 5 The idea that imperfect or incomplete information reduces the ability to mitigate risk is discussed in Battiston and Martinez-Jaramillo (2018). 6 In this seminal paper, Coase (1937) uses this description to argue that a firm cannot increase in size indefinitely and notes that as a firm grows, the losses due to mistakes will increase. In a different context (law firm mergers), Briscoe and Tsai (2011) note that following a merger, there is a delay in value creation and coordination due to the time it takes for relationships to develop, a situation that is likely to be the case in the context of financial firms as well. A similar delay in identifying and mitigating systemic importance is likely to occur as a result of the need for supervisory coordination. kinds of network topologies that characterize majority-control hierarchies, which are by construction rooted directed trees (in the parlance of computer science).
Complex majority-control hierarchies present difficulties for regulators tasked with supervisory oversight (Blair and Kushmeider 2006, National Academy of Sciences 2010, Viñals et al. 2013), particularly the Federal Reserve in its role as consolidated supervisor. 7 Greater complexity (in terms of organizational structure and business activities) of an individual firm makes it harder for a consolidated supervisor to disentangle and understand the firm's structure and increases the likelihood that some parts of the firm's activities and interrelationships go unnoticed. In the case of large multinational organizations, a complexity measure related to oversight would naturally account for the burdens posed by coordinating over multiple national and regulatory environments. Therefore, the identification of metrics that enable comparison across firms that may have very different majority-control hierarchies is of critical importance to consolidated supervision in the post-2008 financial crisis environment. 8 The supervisory challenges we envision are likely what motivated the requirement that SIFIs indicate the number of "material" supervisors and regulators with whom they must interact in the resolution plans that they must file to comply with Title II of the Dodd-7 Throughout this paper, we will use the terms "supervisor" and "regulator" interchangeably, to refer to an entity responsible for assessing some aspect of risk in an organization. In the context of our discussion, these references relate to external (e.g., governmental) assessment but in principle our proposed metrics may also be useful for risk managers within a firm. 8 This point is echoed by Viñals et al. (2013), "Recent evidence from the crisis does not implicate specific bank business models as susceptible to greater risk of failure. Nevertheless, structural measures could be a useful complement to traditional prudential tools under certain conditions. Targeting them to reflect firmspecific risk profiles increases their effectiveness relative to the one-size-fits-all approach envisaged by the recent structural reform proposals, albeit the targeted approach requires firm political commitment and support for supervisors." Frank Act. 9 In our case, it is not just the number of different supervisors and regulators that is relevant but the way in which those supervisors and regulators have to interact with each other. Thus our paper provides a natural extension to the existing understanding of the coordination challenges supervisors face, much as network analysis has advanced knowledge regarding interconnectedness by noting that it is not only the number of connections that is important but the way in which those connections arise.
Our paper is most closely related to the work of Carmassi and Herring (2014) who note that institutional complexity "...impeded effective oversight by the authorities ex ante and greatly complicated crisis management and the resolution of institutions ex post." Using Bankscope to obtain data on "majority-owned subsidiaries for which the G-SIB is the ultimate owner with a minimum control path of 50.01%" (i.e., a control definition that is similar to the one used in this paper), Carmassi and Herring (2014) define complexity as the number of subsidiaries but emphasize, "This is a very simplistic indicator of corporate complexity, but it remains the only indicator that can be measured with any degree of accuracy and even that is far from perfect." We agree with both their premise and characterization of the existing metrics for assessing complexity and indeed, it is the dearth of metrics that motivates our investigation. This paper proposes network-based metrics to encode organizational complexity.
Ours is a novel approach that uses the innate network structure of a majority-control hierarchy. As we explain below, we see this representation, as well as the metrics we construct, as intimately related to the kinds of oversight/regulatory challenges that the Federal Reserve as consolidated supervisor might face. As such we hope that our metrics 9 Carmassi and Herring (2014) also note in Chapter 4 that the FSB identified the need to cooperate as one of five remaining challenges that regulators faced in order for resolution to proceed in an orderly manner. based on intra-firm complexity will be a useful addition to the more-commonly-studied inter-firm complexity (i.e., the interconnectedness across firms) metrics.
The metrics we propose also are intended to inform the Federal Reserve's judgment regarding the SIFI designation. The network encoding and associated metrics also admit the use of simulations as a means of assessing changes in complexity should a firm alter its business structure via a change in majority-control hierarchy. Such simulations could prove helpful for understanding the supervisory implications of altering a firm's majority-control hierarchy in the process of winding down a firm (such as in the case of the dismantling of Lehman Brothers), or in arranging a rapid acquisition, (e.g., in the cases of the JP Morgan Chase acquisitions of Bear Stearns and Washington Mutual, the Bank of America acquisition of Merrill Lynch, or the Wells Fargo acquisition of Wachovia). The goal would be to ensure continuity of consolidated supervision in the wake of a crisis, thereby fostering financial stability, a key component of the Federal Reserve's mandate.
Among the questions we address using network-based metrics are:  Do the SIFI institutions in our sample appear to be more complex?
 Do size rankings match complexity rankings that utilize country and SIC (Standard Industrial Classification) code information?  How has the complexity of the firms in our sample changed over time?
The paper proceeds as follows: Section 2 describes a small dataset that we will use to illustrate the type of insights that can be drawn using our proposed metrics. Section 3 introduces the metrics we employ. Section 4 provides a proof-of-concept by reporting results of our methods on the sample dataset. Section 5 contains additional discussion and Section 6 concludes.

Data
To illustrate how our proposed methods might be used in practice, we use an anonymized data set provided to us by Kingland Systems 10 of twenty-nine large financial institutions that include 19 of the original 29 SIFIs and 10 other firms (5 non-SIFI banks and 5 insurance companies). See Appendix A for a complete list of the firms; Table A1 contains some basic descriptive statistics for the anonymized sample. In all subsequent analysis, the data are numbered in random order within group (i.e., SIFIs, non-SIFI banks, and insurance companies) to protect confidentiality. Additionally, because Kingland is not the only company that collects such data, our SIFI sample is restricted to those firms for which Kingland collects data; hence not all SIFIs are available for our analysis. The other 10 firms in our analysis (i.e., the 5 non-SIFI banks and 5 insurance companies) were chosen at random by Kingland based on some rudimentary guidelines (i.e., that the non-SIFI banks be from comparable SIC codes, that there be an equivalent number from the insurance sector for comparison and that the two sets be roughly comparable to each other in asset size) and provided to us for analysis. We emphasize that these data are intended to be illustrative of the potential use of our measures and that conclusive evidence will be left for future research on a more representative/complete sample.
For each firm, we obtain underlying data that encodes the control hierarchy. In this context "control" is defined by Kingland Systems as the parent controlling at least 50%+1 of the voting share for the child subsidiary. 11 As described above, this is the intra-institutional system of all parent-child majority-control relationships that stem from the "ultimate parent" (the firm of interest) through the ongoing process of creation, acquisition, and dissolution of subsidiaries by various entities in the institution. We do not know the names of the subsidiaries in the control hierarchies, only the parent-child linkages (as given by anonymized Kingland ID). In addition to the control relationships the dataset also contains the country of origin and four-digit SIC (Standard Industrial Classification) code of each subsidiary; we refer to these characteristics as 'labels' below. 12 We have data for the twenty-nine institutions at two distinct dates, May 26, 2011 andFebruary 25, 2013, spanning either side of the change (due to the Dodd-Frank Act) in the reporting threshold for consolidated entities.

Methods
We draw on techniques from the science of networks to analyze the organizational structure of these large market participants (see Newman 2010 for a basic reference).
Network analysis has already proven important for its ability to articulate complex interrelationships in financial networks, in particular, Roukney et al. (2018) highlight the importance of the network structure itself (i.e., the topology of the network) in the context of interconnected networks. Our research is related to their idea but we instead consider intrafirm networks; to date there has been little research on this aspect of financial detailed discussion of alternative sources of control hierarchy data and the limitations inherent in each. While our data has the advantage of being comprehensive in that it is also used by the firms for business and regulatory purposes, its proprietary nature is an obvious limitation. As the goal of our paper is to highlight the way in which metrics can be constructed from network methods to inform supervisory assessments, in particular the assessments of the Federal Reserve tasked with a responsibility as consolidated supervisor, we do not explore alternative definitions. 12 https://www.osha.gov/pls/imis/sic_manual.html stability. 13 Subsequent to the writing of this paper, Flood et al., 2017 similarly emphasize the importance of the topology of intrafirm networks for financial stability, particularly in times of crisis when rapid decision-making is critical (e.g., in assessing ease of resolution of a failing firm). The relevance of intrafirm networks for financial stability also stems from the microprudential approach to the determination of systemic importance.

Definitions and Assumptions
Before stating our assumptions, we first need to define some standard network terms.
The networks describing the majority-control structures of the firms in our study are characterized by a rooted directed tree structure (see Figure 1). A tree is a network without loops. This type of network is composed of nodes and edges. It is directed if the edges (i.e., the links between the nodes) come with a direction. Note (as indicated in Figure 1) a rooted tree has a special node, the "root", also called the ultimate parent. In the trees of interest (representing majority-control hierarchies) all edges point away from the root. An edge pointing from node A (the "parent") to node B (a "child") encodes the fact that entity A has majority control over entity B (i.e., entity B is a subsidiary of entity A). Nodes that have no directed edges to other nodes (i.e. they do not have majority-control over any entities) are called leaves. The maximum number of nodes that a path from the root to any leaf would 13 In the Journal of Financial Stability special issue on Network Models, Stress Testing and Other Tools for Financial Stability Monitoring and Macroprudential Policy Design and Implementation, (April 2018), for example, none of the papers considered intrafirm network structure, suggesting that despite growing literature on the complexity of interconnected financial networks, there is still much work to be done in the area of intrafirm complexity. pass through is referred to as the depth of the tree. The number of children for a given node is called the degree of that node. 14 Figure 1. Basic network terminology and structures for rooted directed trees. This figure shows a particular example of a rooted directed tree. The root is node A, while B, D, and E are leaves or leaf nodes. Node C is neither the root nor a leaf and is sometimes called an internal node. Nodes B and C are children of the node A, which is the parent of these nodes. In addition, this is a regularly branching tree in which each node that has children has exactly two children. This tree has depth 2 (the distance of from the root node A to the farthest children down the tree, in this case either node D or node E) and a total of five nodes. If this were the tree corresponding to a control structure of a financial institution, then the "ultimate parent" would be node A and nodes B and C would be direct subsidiaries of A in which A still held a controlling interest, while D and E would denote subsidiaries of C in which C held a controlling interest.
We refer to the layout of nodes and edges corresponding to the majority control-hierarchy as the topology of the network. We make the following assumptions about this topology: A1. Information: Information flows more easily between nodes with the same label (i.e., the same country or industry). Therefore the number (proportion) of child nodes whose labels differ from their immediate parent is a proxy for the amount of coordination required to exchange information between those nodes and their immediate parent. The lower the proportion, the greater the ease of accessing needed information.

A2. Firm-level decision-making:
Coordination is critical to both short-term and long-term decision-making regarding a firm. It follows from A1 that coordination is easier within 14 In more general directed networks this is often referred to as the "out-degree" of the node, to distinguish from the "in-degree", that is, the number of immediate parents of a given node. However, in a rooted tree, all nodes except for the root have in-degree one, so throughout this paper we will use "degree" to refer to outdegree. of the firm for which it is responsible, due to the amount of coordination required. A more complex organizational structure renders the task more challenging because it hinders both: (a) communication, including the communication of risk, and (b) the coordination that is necessary for risk mitigation and management, as well as for resolution efforts (Cetorelli and Goldberg 2014, Alessandri et al. 2015, Flood et al. 2017), at both the firm and supervisory level. An implication of Assumptions A1-A3 is that it is not just the number of children emanating from a node that is important for consolidated supervision. Rather (Assumption A1) it is the ease by which a consolidated supervisor can access the information necessary in order to make his or her assessment. A subsidiary with children that all fall under the same supervisor or functional regulator is easier to assess than one with children that each fall under different country or SIC classifications because no coordination with other supervisors is required in the first case while the second requires coordination across a number of different supervisors. 16 The assumptions also imply that less breadth of expertise is required to supervise a subgroup that is totally within one's domain. 16 Cetorelli and Goldberg (2014, footnote 11) make a similar point in the context of the challenges a firm's management faces when a parent and its affiliate are located in different countries. They also mention the monitoring difficulty that organizational complexity might pose in their footnote 14.
The next three assumptions link financial stability to the coordination challenges associated with a firm's organizational structure: A4. Risk Implications. In the absence of coordination, opacity ensues, threatening financial stability by increasing the probability that a risk will either occur or go undetected for longer.
A5. Transmission of risk within the firm: Spillover of risk (i.e., from one country to another or one industry to another) within a firm is more likely to occur across heterogeneous linkages (i.e., edges that connect nodes with different labels) than across homogeneous linkages (i.e., edges that connect nodes with the same label) due to the above-mentioned coordination/information assumptions. Regulatory considerations create constraints on the arrangement of the nodes and edges in the rooted trees derived from the control hierarchy. For example, the Banking Act of 1933 (more commonly known as the Glass-Steagall Act, after its Senator and Congressman sponsors) required a separation between commercial and investment banks (as well as other restrictions). In a network theoretic framework, this would mean that under Glass-Steagall, commercial and investment banks would need to be in different subtrees in the control hierarchy (e.g., in Figure 1, investment banks would need to be on one side of the root and commercial banks on the other). Similarly, legal and tax incentives might drive patterns of country-incorporation, resulting in a tree structure where nodes associated with a specific business classification are also associated with a specific country. The 1999 passage of the Gramm-Leach-Bliley Act repealed many of the Glass-Steagall restrictions, fostering substantial growth-by-acquisition in the banking sector as banks diversified into new industries and countries (DeYoung et al. 2009). In tree terminology, the repeal would suggest that banking trees are less characterized by country-or SIC-specific subtrees but instead have become more jumbled. As the recent financial crisis unfolded, many viewed the repeal of Glass-Steagall as partially responsible, and calls to reenact it intensified, resulting in the 2010 Dodd-Frank Act. As a result, we might expect to see banking trees moving back toward their pre-1999 subtree separation (moving closer to a "perfect tree" in the terminology of the next section). In addition, the 2012 lowering of the threshold requiring subsidiaries to report on a consolidated basis (as stipulated in the 2010 Dodd-Frank Act) suggests we should see an increase in the number of nodes included in the majority-control hierarchies between 2011 and 2013. Therefore, the patterns of SIC and country codes (as node labels) in the majority-control hierarchy could help to reify these regulatory changes and should be of interest to banking supervisors.

Statistical Description of Tree Heterogeneity
This section of the paper delineates some metrics to quantify several important characteristics of the institutions' majority-control hierarchy trees that relate to the ease of Electronic copy available at: https://ssrn.com/abstract=2604166 coordination that are highlighted in Assumptions A1-A6. Despite a high degree of heterogeneity in the firms' structures, these metrics provide a basis for comparison.
A fundamental quantitative descriptor for any network (tree or not) is its degree distribution, describing the probability distribution associated with the network's set of degrees (i.e., the function d(i) that records the fraction of nodes with i children). Just as the degree distribution of a tree describing a firm's reporting lines might be used to characterize the spans of control (Urwick 1956) of its management, the degree distribution of a firm's majority-control hierarchy analogously might be used to describe a consolidated supervisor's span of control in assessing the firm's systemic importance.
For this reason, we therefore introduce a new metric of complexity that we see as related to supervisory challenges derived from the need to coordinate oversight efforts across a variety of jurisdictions and agencies. It follows from the assumptions that the simplest supervisory structure a firm can have is a "perfect" supervisory tree, where all nodes of a certain label are clustered (in the language of Flood et al. 2017, this equates to the number of maximally homogeneous subgroups -subtrees that consist of nodes of a single label -being equal to the number of labels).
Assumption A3 forms the baseline for our analysis. In assessing the complexity of any given firm, we derive a measure reflecting the distance of the majority-control hierarchy tree from this "ideal" supervisory or perfect tree structure. 18 18 That is, we begin by defining a "perfect tree" as one in which all nodes belong to the same (country or 2digit SIC) classification. In other contexts, the phrase "perfect tree" refers to a tree with the same number of directed edges emanating from each node. It is important to recognize that our definition differs from that one.
We illustrate how this can be done using our Kingland data sample. For each firm, we consider a "perfect" supervisory baseline to be one with topology equal to the firm in question, where beginning with the nodes at level 2 each child (subsidiary) has the same label (either country or 2-digit SIC classification) as its immediate parent, --i.e., we take as given the firm's heterogeneity at level 1, reflecting its decisions regarding the distribution of business or geographic lines to each of the immediate subsidiaries (children) of the ultimate parent. By imposing that all edges beyond level 1 join nodes with similar labels, this process will produce a tree with (country or SIC) distributions that may be quite different from the firm's actual majority-control hierarchy. In our framework, a perfect tree is comprised of perfect groupings, that is, country-specific (or SIC-specific) groupings that do not involve other countries (SIC codes). Of course in reality none of the firms we consider will have an exact perfect tree structure (i.e., one where there are no edges connecting two nodes with different labels). To this end we introduce the notion of perfect tree similarity.
For each firm, we compare the proximity of their actual control hierarchy to their perfect supervisory baseline via a perfect tree statistic, in each case computing the fraction of nodes with the same label (i.e., country or 2-digit SIC code) as their immediate parent.
The statistic is therefore bounded between zero and one. Mathematically, let E denote the number of firms in our sample and Ni the total number of subsidiaries in firm i, i.e., the count of all nodes that are in firm i's majority-control structure). We index all nodes in the firm with {0, 1,…,Ni}, where we always assign the root node index 0. Also let si,j ϵ {1,…,S} be the label (i.e., the country or 2-digit SIC code) associated with subsidiary j of firm i, and pa(j) the parent node of that subsidiary j. Finally let δ(٠) = 1 if the expression within the parentheses is true and zero otherwise. Then the perfect tree statistic for firm i is given by: Electronic copy available at: https://ssrn.com/abstract=2604166 1 , , 1 As noted above, this is just the fraction of non-root nodes in firm i whose parents have the same label as the node itself.
Note that in a perfect tree (i.e., a tree in which the perfect tree statistic is equal to one), removal of a node and all subtrees below it will not change the value of the statistic. 19 In contrast, a value of zero means that the subsidiaries below depth 1 are always different in character from their immediate parent (with respect to a given characteristic -country or SIC code) and therefore would require coordination among supervisors across all countries (industries) in which the firm operates. Thus, to the extent that a firm's tree structure is closer to a perfect supervisory tree, we reason that it is easier for both the firm and the supervisor to track/monitor the activities of its subsidiaries or to achieve resolution should the firm fail. In practice we recognize that the perfect tree characterization is too rigid and the costs of a fully-segmented structure may far outweigh the benefits of globalization and cross-border banking, but we believe it may inform consolidated supervision by providing a regulatory ideal against which real world instances can be compared. We refer to the comparison between the perfect supervisory baseline and the firm's actual majority-control hierarchy via their associated perfect tree statistics as the perfect tree similarity statistic.
The closer a firm is to its own perfect supervisory tree, the easier it will be for the consolidated supervisor to satisfy the mandate of the Bank Holding Company Act (to aggregate information from supervisors in different jurisdictions or industries evaluating the portion of the firm for which they are responsible) in order to form an assessment of the consolidated organization and without need for additional supervisory review. Firms can be evaluated according to their own internal structure and the level of complexity they exhibit as a result of that structure can be compared, thus informing the allocation of supervisory resources.

Comparing complexity across heterogeneous firms
Assumptions A1-A6 lead us to the following: "By comparing the firm's actual organizational structure to a hypothetical perfect tree structure, we can draw inferences about the challenges a firm and/or its supervisors would experience were one of its subsidiaries to experience deterioration." It additionally follows from assumption A5 that the greater number of perfect groupings in a tree, the less likely that difficulty faced by a subsidiary in one country (industry) will spill over into subsidiaries in other countries (and the easier it will be to 'ring-fence' the risk in a resolution situation). 20 The closer a structure is to a perfect tree, the easier it is for a supervisor to evaluate its risk. This is because the supervisor has direct access to information from a higher proportion of subsidiaries in the firm. Similarly, the farther away from a perfect tree, the more coordination is required for information to transmit to the consolidated supervisor.
We expect firms to have moved closer to a perfect tree structure between 2011 and 2013. 20 A resolution framework where each national authority has responsibility for the resolution of banks that fall under its jurisdiction is referred to as the "subsidiarization" model in of Carmassi and Herring (2014); in their context, our country and industry perfect tree statistics could inform the geographic subsidiarization and subsidiarization across lines of business that they propose.
As noted above, Assumptions A1-A6 imply that organizational structures that are closer to a "perfect tree" will be easier for a supervisor to evaluate. However, simply calculating the proportion of edges in the perfect tree that connects nodes of differing labels will not provide a satisfactory metric because different firms have different business models (and hence the distribution of their country and SIC classifications will differ) and thus some firms more than others will have structures and distributions that lend themselves naturally to be closer to a perfect tree. Such a comparison would give rise to the "apples-tooranges" and aforementioned one-size-fits-all criticisms that banks often level at regulations, a reference to the idea that by their nature, regulations need to be general and hence don't adequately account for the fact that firms have different business models.
Recognizing these criticisms, we therefore need to find a way to determine how unusual a specific firm's organizational structure is, by comparing it to all the other structures that might arise, given the specific distribution of that firm's characteristics. Specifically, we take a firm's tree topology as fixed and bootstrap different labelings of the nodes (country or 2-digit SIC code) according to the firm-specific empirical label transition probabilities.
Concretely, for firm i, we first set the label of the root node, si,0, to its observed (actual) value. We then traverse the tree following the edges through each node, sampling the label of each node according to the empirical conditional distribution: We continue this process recursively for the child nodes of the root node and their children until we reach the leaf nodes. When j is a non-leaf node (i.e., a node with children), we restrict s to be from the subset of labels associated with the non-leaf nodes in the observed tree; similarly, when j is a leaf node (i.e., a node with no children), we restrict s to be from the subset of labels associated with the leaf nodes in the observed tree. 21 When drawing labels for the leaf nodes we add a small correction factor to the empirical probability, 1/|N|, where |N| is the number of nodes in the tree, to account for the fact that conditional on the simulated parent, drawing from the leaf-only empirical distribution might result in labels that have a 0% conditional probability. This correction factor admits the possibility that another label (i.e., different from the observed one) might be assigned to the leaf; hence a greater variety of tree labelings can be explored.
The number of bootstrap replications that we use for each firm varies in order to achieve For each replication, the perfect tree similarity statistic is computed.
The replications therefore allow us to generate a distribution of possible structures for each firm, conditional on its overall topology (organizational structure), and to compute a range of summary statistics from these distributions. Our approach allows a supervisor to compare complexity across firms that have different country and industry profiles, while still holding the tree topology and distribution of countries/industries fixed for each firm.
In particular, from the replications we can compute a percentile ranking for each firm, representing the proportion of simulations that produced a perfect tree similarity statistic that was lower (higher) than the one given by the firm's observed structure.

Two Examples
As explained in Section 2, the majority-control hierarchy imposes a tree structure on the data, driven by consolidated supervision considerations. That is, for the purposes of consolidation, as we navigate through a chain of subsidiaries, we should never loop back to any of the intermediate (or parent) entities. For illustrative purposes, it is useful to select a firm with a small number of nodes to demonstrate the information in the tree layouts further. The control hierarchy of SIFI S11, color-coded by distance from the ultimate parent, by depth, country, and both 1-digit and 2-digit SIC classifications (classifications are available at www.secinfo.com/$/SEC/SIC.asp?Division=I). A consistent layout is used for all four representations, for comparability. Node size is proportional to distance to the ultimate parent, with larger nodes corresponding to closer distance.  have an immediate parent that is also incorporated in Japan, suggesting that it will be relatively easy for a Japanese supervisor to obtain information at all levels of control of the firm since there are only two entities beyond the immediate reach of the supervisor. 24 Figure 2c illustrates S11's control hierarchy labeled by 1-digit SIC code. At the 1digit level, this firm is fairly homogeneous, with 31 of the children of the ultimate parent operating in the same industry. In addition to financial services, this firm has control over one subsidiary in SIC area 3 (roughly construction and equipment), and 11 in services (areas 7 and 8); for the most part the services are concentrated in one subtree (to the left of the diagram). With most of the tree falling into the same SIC classification, it is evident that a financial services supervisor would be able to assess most of the firm's activities without having to rely on coordination with supervisors from other industries. In addition, since the subsidiaries that fall under SIC classification 7 are concentrated in one subtree, the diagram 24 The tree structure with most subsidiaries being controlled by a Japanese subsidiary may reflect a keiretsu arrangement (Berglof and Perotti, 1994). We thank a referee for bringing this to our attention. Cetorelli and Goldberg (2014) similarly note that using their metrics and definition of complexity, Japanese banking organizations appear to be less complex. highlights a single link that might warrant additional scrutiny or that might be severed should either the firm or its supervisor wish to reduce the range of the firm's business activities. An additional implication is that should this link (between the subtree under SIC classification 7 and its parent node with SIC classification 6) be severed, either due to deacquisition or the subtree falling below the majority-control threshold, there would be little change to the burden of the consolidated supervisor. Figure 2d analogously illustrates S11's control hierarchy labeled by 2-digit SIC code.
For comparison, consider Figure 3 in which we see the same kinds of snapshots (at the same date) for a much larger firm (S16): this firm has 1778 nodes, corresponding to 32 countries and 100 4-digit SIC codes, with a tree depth of 5. We provide three representations color-coded according to distance from the root, country of origin, and 1digit SIC code, respectively. Figure 3a shows that the majority of nodes are at depth 2.
Specifically, there are 299 nodes at depth 1, 1186 nodes at depth 2, 188 nodes at depth 3, 24 nodes at depth 4 and 80 nodes at depth 5. Figure 3b illustrates that in addition to the parent company being located in the US (the largest circle in the center), the majority of S16's subsidiaries are also located in the US. In addition, some of those US subsidiaries themselves have quite elaborate majority-control hierarchies, judging from the large clusters at the top of the figure, as well as the ones on the far left, far right and bottom of the figure, that are almost exclusively comprised of US subsidiaries. There are also a large number of subsidiaries located in the UK (green), Germany (yellow), and Spain (light blue), with the UK and Spanish subsidiaries having a number of children that are located in the same country as their immediate parent. Figure 3c shows that while many The control hierarchy of SIFI S16, color-coded by distance from the ultimate parent, by depth, country, and 1-digit SIC classification. A consistent layout is used for all three representations, for comparability. Node size is proportional to distance to the ultimate parent, with larger nodes corresponding to closer distance.
of S16's subsidiaries are in the same 1-digit SIC classification as the parent company (classification 6 -Finance, Insurance, and Real Estate), the company also has a fairly diversified range of subsidiaries with the second largest 1-digit classification in the wholesale and retail trade sectors (SIC code 5).
Taken together, Figures 2 and 3 illustrate that this view of complexity is multidimensional, that is, a firm that has a complex SIC classification structure may not have a complex country structure and vice versa. 25 Identification of these complexity dimensions provides a way to evaluate and compare sometimes very disparate firms and may inform allocation of supervisory resources. In addition, network information can be used to develop metrics with which to compare complexity across firms.

Results
In this section we present several results describing the tree topology of the firms in our sample; discussion of the implications of these results for our three main questions of interest is reserved for the next section. The main results are given in the following subsections, corresponding to different metrics: 1) Degree statistics 2) Markov statistics (parent-child transitions)

3) Perfect tree similarity statistics
We then use these metrics to illustrate the manner in which supervisors might draw inferences about the firms and explore whether there are differences between the coordination efforts required for different types of institutions. Finally, we consider whether our metrics provide additional information beyond the size delineation that has traditionally been used to classify institutions that warrant greater regulatory oversight due to their systemic importance. Note that while in this paper we focus our discussion exclusively on the overall tree topology of the firms in our sample, a similar analysis can be readily conducted on any subtrees of interest, for example to inform ring-fencing decisions that supervisors might need to make once a risk has been identified.

Overall degree statistics.
One way to characterize the organizational structure of the firms is via the hierarchy distribution, that is, the proportion of nodes at each level of the tree hierarchy. To aid our understanding of how tree hierarchies might be used in the context of large financial institutions, it is useful to first consider how different organizational structures correspond to different hierarchy distributions. For example, an institution with a very flat (i.e., "entrepreneurial") management structure would have a large proportion of nodes at level one and relatively few branches extending from those nodes. In contrast, an institution that concentrates its decision-making among only a few senior managers who are then held accountable for large portions of the firm would have a larger proportion of nodes at lower levels of the tree. Such a diffuse tree might also be found among organizations that have experienced significant growth by acquisition, such as many financial institutions in the decade preceding the 2008 financial crisis, where the tree of an acquired complex organization may have been grafted to the tree of the acquiring parent somewhere below the highest level, creating a very hierarchical structure of great depth (a "bureaucratic" structure). Firms also might be arranged along geographical ("divisional") or industry ("functional") lines (Armour and Teece 1978). 26 The hierarchy distribution for our sample of firms is summarized in Figure 4, for 2011 and 2013 separately. Note that there is substantial variation across the firms. For example, while in 2011 more than one-third of the firms have more than half of their nodes at the first level of the tree hierarchy, others have relatively few nodes branching from the ultimate parent and instead have a large concentration of nodes farther down the tree (e.g., S13 and B1). None of the firms with the deepest trees (i.e., more than six levels) have node concentration at the first level of the tree hierarchy, indicating a flatter or more diffuse organizational structure. Note firm I4 has a tree structure that spreads out at each level in 2011. Across all firms in our sample, roughly one-third of the nodes are in each of the first two levels, another 22% in the third level, and only 10% at deeper levels in the tree hierarchy.
In contrast, by 2013, the tree hierarchies of the firms in our sample deepened substantially. For example, only half as many (five) firms now have more than half their nodes at the first level, while 14 have less than 10% of their nodes at the first level. In addition, 11 firms now have more than seven levels while just two years earlier, none did.
Across all firms in the sample, by 2013 roughly 25% of the nodes were at deeper than the third level. Thus, from the perspective of consolidated supervision, the challenges associated with assessing these firms increased dramatically, with many entities in the organization being much farther removed from the ultimate (root) parent. 26 An earlier version of our paper (Lumsdaine, et al., 2016) discussed the similarity of the firms' degree distributions to power law distributions (distributions that have the form x -r for some r>1. Power law distributions suggest certain patterns of growth in the network and also have implications for its robustness and stability. 1.0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 B1  B2  B3  B4  B5  I1  I2  I3  I4  I5  1.0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 B1  B2  B3  B4  B5  I1  I2  I3  I4  I5  . For each firm, we plot the fraction of nodes at each level in its control hierarchy (i.e., the distance from the root), for 2011 and 2013. For ease of comparability between 2011 and 2013, for firms with more than nine levels in 2013, the fraction of nodes beyond level 9 are aggregated into the ">10" distance. The fraction of nodes at each level for the entire sample of firms we consider is shown in the right-most column ("avg").

Aggregation Across a Sample of Firms: Markov Statistics (parent-child similarities)
As noted above in the discussion of both degree and hierarchy distributions, from a supervisory perspective it is not just the number (or proportion) of child nodes that emanate from a parent node that matters but also the similarity between the parent and child. 27 In this section we therefore ask the following: Given that a node has a particular label s, what is the conditional ('birth') probability P(s|s) that the node below it also has label s, not just for a given firm but across a sample of firms? Birth probabilities are one simple measure of a type of "homophily" in a network, or the predilection for a node (in this case a parent) to be connected to another node of the same kind. 28 In the context of the rooted directed tree networks we consider, the Markov similarity statistic P(s|s) for label s is computed as: Note the resemblance between the second sum and the formula for the perfect tree similarity statistic. In the case of the Markov similarity statistic, however, the sum is computed per unique value of the label s over all firms in the sample. A higher Markov similarity statistic indicates greater within-label (country or industry) linkage (as opposed to across-label linkage) and hence may suggest a greater likelihood of spillover to other firms with the same label. From a financial stability perspective, if we were able to observe the entire network of firms, such a metric provides a measure of the degree of coordination burden involved in a particular dimension (e.g., with respect to that country) across the entire sample and hence may be useful in identifying potential contagion effects should a firm with a particular label (e.g., a specific country) start to exhibit signs of stress. Such a statistic may help alert supervisors of high Markov-similarity countries or industries to the 27 Although for expositional purposes much of our discussion has focused on the ease of a supervisor to "look below" in examining entities that are lower down the tree, our focus on the similarity between parent and child nodes stems from the view that consolidated reporting and risk management will be easier when, for example, a child has the same legal, accounting, tax, or supervisory framework as its parent. To illustrate, in Table 1 we compute the Markov similarity statistics for each country, using all firms in our sample, for both 2011 and 2013. The probabilities vary dramatically for different countries. In 2011, Canada has the highest probability, with P(Canada | Canada) = 0.97; in contrast Switzerland has the lowest, with P(Switzerland | Switzerland) = 0.11. Part of the reason for this variation is differences in country frequency; that is, even under random assignment a country that has more nodes in the network has a greater likelihood of being paired with its own country than does a country that has fewer nodes in the network. By comparing the probabilities in 2011 to those in 2013, however, we see that for most countries, the Markov similarity statistic increases, with many countries above 0.9. This suggests that across the network of 29 firms there has been a shift to consolidate subsidiaries from a given country under parents from the same country, consistent with our prediction based on Assumptions A1-A6.

Complexity and Changing Structure -Perfect Tree Similarity Statistics
In this section we document the fluidity of majority-control hierarchies by comparing the perfect tree statistics in 2011 and 2013. As is the case for many complex  Table 3. This table summarizes the "in" transition statistics with respect to country labels. That is, for any country A consider all the subsidiaries incorporated in A. Here we list the proportion of the children of such subsidiaries that are also incorporated in A (P(A|A)) for any country where in 2011 the in-country probability is neither zero or one, along with some additional countries from 2013.
systems, the structure of the majority-control hierarchy is a response to a variety of endogenous and exogenous forces. A significant component of the latter comes from regulatory frameworks, which can include pressures that come from tax and corporate legal structures as well as supervisory restrictions on activities. For example, in December 2012, the lowering of the threshold for consolidated supervision as a result of the Dodd-Frank Act of 2010 went into effect. Because our two data snapshots straddle this event, we can use our sample to illustrate how our metrics might be used to study or evaluate changes in policy.  . The columns labeled "Perfect" refer to the perfect treesimilarity statistic equal to the number of nodes that have the same country designation as their immediate parent divided by the total number of nodes in the firm's tree. In a perfect tree, deletion of a node and its children will not change the value of the statistic. From a supervisory perspective, this means that supervisory oversight follows firm organizational lines. A number closer to one signifies greater proximity to perfect. The columns "Mean", "Stdev" and "Quantile" are summary statistics computed over all replications of the simulated data, where each firm's tree structure is taken as given and the node identifiers are randomly drawn according to the conditional probability distribution of the firm's observed tree. See section 3.2 for details. shuffled to achieve proximity to a perfect tree than others. For this reason, it is useful also to consider the moments and/or quantiles generated using the perfect tree simulations. For example, a comparison of S9 and S11 in 2011 reveals that although S9 has a perfect tree statistic that is close to one, the arrangement of its subsidiaries is actually far from perfect (falling in the lowest third of the simulated distribution). In contrast, S11 has a much lower perfect tree statistic (0.279) but is in the 100% quantile compared to other control hierarchies that could result from its given country distribution, meaning that the simulations produced no better way to arrange the subsidiaries for this firm (i.e., in a manner that would result in less coordination burden among supervisors) than the observed arrangement.

Country Perfect Tree Similarity Statistics
In 2011, nine of the 29 firms had country structures that were significantly different from their corresponding perfect tree, given their topology (that is, a test of the null hypothesis that the firm's perfect tree similarity statistic is equal to one is rejected). In addition, 25 of 29 firms had country distributions that differed from a structure where all subsidiaries differ from their immediate parent (that is, a test of the null hypothesis that the firm's perfect tree similarity statistic is equal to zero is also rejected), suggesting most firms have majority-control structure that follows geographical lines. By 2013, only one firm had a country structure that was significantly different from its corresponding perfect tree and for all firms, the null hypothesis that the firm's perfect tree similarity statistic is equal to zero was rejected, corroborating the results of the Markov statistics of a shift toward geographical consolidation. Table 3 summarizes the 2-digit SIC-level perfect tree simulations for each firm and both dates, again grouped by firm type. 29 Similar to the country tree statistics, while in 11 of the 29 firms we fail to reject the null hypothesis that the firms have 2-digit SIC structures where all subsidiaries differ from their immediate parent in 2011 (perfect tree statistic is equal to zero), by 2013 the hypothesis is rejected for all of the firms' structures (at the 95% level of confidence). The results document a shift toward a perfect tree configuration; all except one firm's perfect tree similarity statistic (S19) increased and the number of firms with SIC perfect tree-similarity significantly less than one fell from 20 to 10. Nonetheless, in many cases the change in 2-digit SIC structure did not actually render the firms less complex, as the associated quantile declined for 14 of 29 firms.

SIC Perfect Tree Similarity Statistics
An interesting example from this table is S18. From Table A1, we know that this firm experienced a modest (13%) reduction in nodes between 2011 and 2013, despite adding five additional countries to its majority-control hierarchy. Over the same period, it reduced its number of SIC categories by more than 50% and doubled its degree depth.
As noted above in Section 4.2, the increased degree depth is an indication of a shift toward a more bureaucratic organizational structure. Commensurate with this change, the firm's country perfect tree statistic increased both in level and quantile (see Table 4), indicating that the new control hierarchy was closer to a segmented structure, where children mimic the country of their immediate parent. This shift to a more Table 3. SIC Perfect Tree Similarity Statistics Notes to Table 5: Summary statistics describing the regularity of a firm's SIC control hierarchy for two periods in time, May 26, 2011 (left panel) and February 25, 2013 (right panel). The columns labeled "Perfect" refer to the perfect treesimilarity statistic equal to the number of nodes that have the same SIC designation as their immediate parent divided by the total number of nodes in the firm's tree. In a perfect tree, deletion of a node and its children will not change the value of the statistic. From a supervisory perspective, this means that supervisory oversight follows firm organizational lines. A number closer to one signifies greater proximity to perfect. The columns "Mean", "Stdev" and "Quantile" are summary statistics computed over all replications of the simulated data, where each firm's tree structure is taken as given and the node identifiers are randomly drawn according to the conditional probability distribution of the firm's observed tree. See section 3.2 for details. percentile in its proximity to a perfect tree.

Additional Thoughts
As noted in Section 3, complexity in our framework is a multidimensional concept. It is evident from a comparison of Tables 2 and 3 that while in many cases inference regarding a firm's complexity may be similar in the two dimensions we consider (i.e., country and SIC), in other cases the two dimensions may conflict, for example with a firm having a fairly complex country-label topology and a not-so-complex SIC-label topology, or vice versa. 30 In principle, such conflict can be addressed by redefining (and expanding) the label space to accommodate all possible label pairs (i.e., all country-SIC combinations). This framework also can be used to consider additional types of labels beyond just country and SIC classifications (e.g., by activity), were such information available. More generally one can define labels via n-tuples, where n represents the number of properties that describe each node. In practice, however, such a definitional expansion soon becomes computationally prohibitive.

Discussion
In this section we provide some thoughts to illustrate how results such as ours might be used in policy discussion and the implications of our findings for both the determination of systemic importance and financial stability. We emphasize that due to our limited sample, the remarks are offered as a proof-of-concept rather than intended to be definitive.

Are the SIFIs in our sample really more complex from a supervisory perspective?
In Table A1, we see that in 2011 SIFIs had tree structures with more than three times as many nodes, higher degree depth, greater geographical reach and more than double the amount of 4-digit SIC variation in their subsidiaries than both non-SIFI banks and insurance companies. In addition, data from Bloomberg® on total consolidated assets of all firms in our sample reveal that on average the SIFIs had $1.82tr in total consolidated assets as compared to $0.72tr for the non-SIFI banks and $0.61tr for the insurance companies in our dataset. So do our complexity measures tell a regulator anything more about the appropriateness of a systemic importance designation than can be gleaned from simply looking at the size of an institution?
Note that in asking this question, we do not mean to downplay the relevance of the other metrics (e.g., number of nodes, tree depth, etc.) or to suggest that our perfect tree similarity statistic is the only important property (or even the most important property) of a majority-control hierarchy network. To reiterate, we consider the term "complexity" from a supervisory perspective regarding the topology of the firm, namely: (1) how difficult is it for a consolidated supervisor to access the information it needs to assess the firm, (2) how likely are they to identify a problem with the firm if it exists, and (3) how easily can a problem be course it remains to be seen whether our findings also appear in a larger, representative sample.

Is size a sufficient statistic?
In the aftermath of the 2008 financial crisis, calls to end "too big to fail," the so-called practice of bailing out the largest, most systemically important financial institutions, have intensified. Most often, the concept of too big to fail implies a firm whose size is larger than a specified threshold. Yet size is but one of the criteria mentioned in the SIFI definition. 31 Consistent with the idea that size and systemic importance are synonymous, regulators typically delineate a size threshold to identify firms that pose considerable risk to the global financial system. Despite the ease of implementation, a size-based threshold is in many ways unsatisfactory, precisely because it does not take into account the level of complexity of a firm's business activities. To quantify this point more generally, we computed the Pearson rank correlation between total consolidated assets and both country-and SICperfect tree similarity statistics. 32 The rank correlation was -0.32 and -0.36, respectively.
In contrast, the rank correlation between size and the number of nodes was 0.58, reflecting the fact that firms with more total consolidated assets generally have more subsidiaries.
Taken together, these numbers highlight the fact that the link between asset size, number of subsidiaries, and the number of countries and industries in which a firm operates is not 31 There are a variety of definitions of size that arise in the banking and finance literature. The most common in recent banking regulations (e.g., Basel II, Basel III, Dodd-Frank) is specified in terms of total consolidated assets. Other definitions might include market capitalization, number of distinct entities, number of employees. Generally speaking, however, in the "too big to fail" context, size is usually considered in financial (e.g., dollar) terms, rather than in terms of features of organizational structure. 32 We report results using rank-based statistics in order to preserve confidentiality of the firms. The results are qualitatively similar using ln(assets). exact; that is, the perfect tree statistics are measuring something different than just the number of nodes. This point is also illustrated in Figure 5, comparing the ranks of the firms' size to the ranks of their country (left chart) and SIC (right chart) perfect tree similarity statistics. In particular, if size and the perfect-tree statistics measured the same thing, we would expect the points in this figure to lie along the 45-degree line. Instead there is a slight negative relationship, particularly among the SIFI firms; the larger firms are less complex from a supervisory standpoint, with a larger proportion of trees being selfcontained (where the child node is in the same country as its immediate parent).
To see how such information might be used in practice, we offer two examples: (1) Firms S4 and S5 have very similar asset-size, yet in 2013 S4 has three times as many subsidiaries (nodes) and activities in 50% more industries than S5. In contrast, it is only active in half as many countries. Hence the supervisory challenges associated with these firms may be very different, S4 requiring coordination among many more industry regulators and S5 requiring substantial coordination among different country supervisors. Yet both their country and SIC perfect tree-similarity statistics are similar and close to one. And while their country perfect tree similarity statistics would place them both at the 100% quantile (i.e., closer to a perfect tree than all simulated firms with identical tree structure), S4's SIC statistic, while higher in value than S5's, is below the 33 rd percentile (by comparison, S5's is just below the 54 th percentile). Taken together, these statistics would alert regulators to the SIC dimension of S4's business.
Note from Table A1 that the absolute number of 4-digit SIC activities of S4 is not particularly unusual: there are 12 firms that have and SIC (bottom chart) tree similarity ranks, for all 29 firms in our sample, shown separately for SIFIs, non-SIFI banks, and insurance companies. A higher rank indicates greater complexity and/or larger size. While it is evident that the non-SIFI banks and insurance companies are smaller in size than the SIFIs, there is also evidence that the insurance companies have country and SIC perfect tree similarity statistics that are farther from a perfect tree (i.e., they are more intermingled and hence potentially more difficult to supervise/regulate). In contrast, the non-SIFI banks are similar to their SIFI counterparts in that some have tree structures that are close to perfect and others far away. Data on size as of December 31, 2012 are obtained from Bloomberg®.
subsidiaries in a greater number of industries. Yet there is only one firm (S2) that has a 2digit SIC perfect tree similarity statistic with a lower quantile score.
(2) Another comparison of interest is between S6 and S12. These two firms are very similar in their perfect tree similarity statistics (both country and SIC), as well as the quantiles to which these statistics correspond. Both firms' SIC quantiles are near the median of the simulated distribution for their corresponding tree structures and hence might warrant additional supervision. Yet S6 is nearly 50% bigger in asset size and has more than four times the number of subsidiaries (nodes) than S12. A size-only threshold would potentially miss the complexity of S12.
Finally, Cohen and Lou (2012) use a Herfindahl-type measure to assess how complicated a firm is, computed as the sum of squared percentages of each business type (e.g., 0.4 2 +0.3 2 + 0.3 2 if there are three types that comprise 40%, 30%, and 30% of the firm's overall business) but do not take into account the topology of the firm's organizational structure.
The closer their measure is to one, the less complicated a firm is. To compare our work to this important paper, using our 2011 data we analogously compute indices that measure the country and SIC concentration of our firms. The correlation between these indices and our perfect tree statistics is 0.83 (country) and 0.72 (SIC). Yet when we consider the rank correlation between the indices and the perfect tree statistic quantiles, there is a negative relationship, providing further evidence that our complexity metric is detecting something different than business concentration. 33

Has complexity of the firms in our sample changed over time?
A comparison of the left and right blocks of columns in Table 2   In contrast, while most firms' SIC structure moved more closely to a perfect tree in 2013 (i.e., the number of firms with SIC perfect tree-similarity statistic significantly less than one fell by 50%, from 20 to 10), the associated quantiles declined in almost half the firms, indicating that the change in SIC structure did not commensurately reduce the firms' complexity along this dimension. financial crisis consolidation of the banking sector has led to greater jurisdictional complexity; for most firms, coordination with respect to both supervisory oversight and possible wind-down or paring of assets would be easier now, given their organizational structure, than in 2011.

Conclusions
The 2008 financial crisis highlighted the risks that large, multinational, complex, interconnected banks pose. Since then, debate concerning which firms warrant a SIFI designation has led to a large amount of research into the web of interconnections inherent in financial transaction networks and the associated risk to the global financial system. There has been comparatively less emphasis on the risks that a firms' internal complexity might pose, particularly the coordination challenges for supervisors tasked with evaluating a firm's riskiness on a consolidated basis. As noted by Carmassi and Herring (2014): "[Also,] opacity of organizational structures impedes regulatory oversight. If regulators do not have a clear understanding of how lines of business map into legal entities and how the legal entities interact with each other, they cannot perform effective prudential supervision, nor can they implement an orderly resolution" "Information tends to flow more freely within a regulatory organization than between organizations and more freely among regulatory institutions within a country than across national borders." In this paper, we propose using a firm's control hierarchy as a proxy for such supervisory challenges. By defining complexity as a function of the firm's tree topology we demonstrate that complexity and size are not synonymous and thus warrant distinct mention in the FSB's SIFI definition. We additionally propose using a perfect tree similarity statistic to quantify the ease of supervision in a number of dimensions: (1) a consolidated supervisor's need to rely on coordination or information from supervisors from other countries, (2) the implications of severing the link to a subsidiary, and (3) the ability to assess how to wind down a firm through paring of subsidiaries with minimal risk. 34 By comparing data from 2011 and 2013, we find that on average the ease of supervision along these dimensions improved. While our data sample prevents definitive conclusions regarding this improvement, one possible reason is the 2012 lowering of the threshold under which subsidiaries must submit to consolidated supervision. Although purely speculative at this stage, the pattern that we have observed in our data, namely of dramatic increases in the number of nodes and decreases in the level of complexity as defined by our metrics, is consistent with what we would expect from such a change in threshold. No doubt other explanations are possible. In addition, our results suggest that contrary to conventional 34 Although we have emphasized the potential value of our metrics to the supervisory/regulatory community, it is also worth noting that our metrics additionally might be useful as measures of operational risk. The Risk Management Association on their website (http://www.rmahq.org/operational-risk/?gmssopc=1, accessed 8/12/16) notes, "The definition of operational risk is the risk of loss resulting from inadequate or failed internal processes, people, and systems, or from external events, but is better viewed as the risk arising from the execution of an institution's business functions." We thank several people at various conferences and seminars where an early version of this paper was presented for mentioning this connection to operational risk.
wisdom, some of the SIFI-designated institutions may not pose greater supervisory challenge since their control hierarchy more closely resembles a perfect tree than some of the other firms in our sample. Specifically, we find little difference between the SIFI and non-SIFI banks in our sample. In contrast, the insurance companies in our sample are more complex according to the perfect tree similarity statistics, despite being smaller in size, having fewer subsidiaries, and being less geographically or industry-diverse than the banks.
We emphasize, however, that our sample is highly selected and that generalizability of these findings remains a topic for further research.
In closing we stress the importance of further development of metrics aimed at measuring complexity as it relates to the challenges of supervisory oversight and coordination. As a key aspect of the FSB's definition of systemic importance, viable measures of complexity are critical to ensuring financial stability. In this paper, we make the argument that the firms' topology may provide insights into the challenges that external supervisors are known to face. In our view, the metrics proposed in this paper represent just the beginning of the discussion on this topic. We emphasize that, distinct from size and interconnectedness, much work on complexity remains to be done.
on, the work of other relevant primary supervisors and functional regulators". "Effective consolidated supervision requires strong, cooperative relationships between the Federal Reserve and relevant primary supervisors and functional regulators." (page 62). "Information sharing among domestic and foreign supervisors, consistent with applicable law and the jurisdiction of each supervisor, is essential to ensure that a banking organization's global activities are supervised on a consolidated basis." (page 63). "For example, more independent Federal Reserve work typically will be required to assess consolidated asset quality or earnings for large complex BHCs with significant nonbank activities that are not functionally regulated. However, where all material holding company assets are concentrated in a single depository institution subsidiary, a minimal level of incremental Federal Reserve efforts typically will be required to assess consolidated asset quality and earnings." (page 84). "The Federal Reserve's conduct of consolidated supervision is central to and dependent on the coordination with, and reliance on, the work of other relevant primary supervisors and functional regulators." (page 374) "Many large complex institutions have interstate operations that expand with the continuation of mergers and acquisitions. In this environment, close cooperation with the other federal and state banking agencies is critical." (page 563).
The next quotes note the challenges for large multinational BHCs and link these challenges to macroprudential supervision.

5.
"Financial trouble in one part of an organization can spread rapidly to other parts of the organization; moreover, large BHCs increasingly operate and manage their businesses on an integrated basis across corporate boundaries. Risks that cross legal entities or that are managed on a consolidated basis cannot be monitored properly through supervision directed at any one of the legal entity subsidiaries within the overall organization." (page 61) "There are, however, areas of focus for the Federal Reserve that are unique to a holding company's international operations. For example, some host-country legal and regulatory structures and supervisory approaches are fundamentally different from those in the United States. As a result, the banking organization often must devote additional resources to maintain expertise in local regulatory requirements. In some instances, privacy concerns have led to limits on the information a BHC's foreign office may share with its parent company, thereby limiting the parent company's ability to exercise consolidated risk management on a global basis. Additionally, while considerable progress has been made to strengthen supervisory crossborder cooperation and information sharing, the Federal Reserve and other U.S. supervisors have, at times, faced challenges in accessing information on a bank's or BHC's foreign operations or in carrying out examinations of cross-border or foreign activities." (page 89) "The Federal Reserve aims to reduce systemic risk…Supervision carried out under this framework will support a variety of macroprudential supervisory approaches beyond those already discussed including…(f) Enhancing international coordination with foreign counterparts, including national supervisors and international bodies such as the Basel Committee on Bank Supervision, the Financial Stability Board, and the Senior Supervisors Group. These activities focus on enhancing oversight of internationally active financial firms and markets and on minimizing the opportunities for firms to take advantage of weaker or inconsistent regulations." (page 597)