Who Possesses Whom in Terms of the Global Ownership Network

Abstract

The concentration of wealth has become a momentous and contentious global issue as the world has grown more tightly connected through economic relations. The objective of this chapter is to address the question of who possesses whom through a web of stock ownership relations on a global scale. We construct annual ownership networks annually on the basis of a comprehensive database that includes virtually all of the world’s listed firms and their market capitalizations. The networks thus constructed are first decomposed into bow-tie components with the giant strongly-connected components making up a core that mainly consists of Japanese listed firms. The Helmholtz–Hodge decomposition, which allows flow structures to be decomposed into a directed network of hierarchical and circular flow components, is then applied to the strongly-connected components in order to resolve multilateral cross-holding relationships embedded within them. The Helmholtz–Hodge potential quantitatively illuminates the hierarchical structure of ownership networks. In addition to the direct owners of the listed firms, their ultimate owners are traceable through the hierarchical ownership flow. We discuss this issue from a wide variety of viewpoints, including the hierarchical positions of countries, the distribution of corporate equity holdings within them, cross-ownership between them, and the progressive dominance of institutional investors based in the United States.

Appendices

Appendix 1

Here, we describe the detailed steps that we took to construct the global ownership networks utilized in the present study from the Global Equity Ownership data from Thomson Reuters. In fact, the construction process was not an easy task.

The raw data consists of the following data files:

• Historical Holdings File

• Security File (History)

• Owner File (History)

• Owner Issue Map File.

These data files contain a wide range of ownership information for the period 1997–2017. The Holdings File matches security IDs with owner IDs and can thereby enable an owner’s portfolio to be constructed or the owners of a given security described. The Security File is a list that links a security ID with its issuer name, issuer ID, and issuing country ID. Moreover, the Owner File is a list that links the owner ID with the owner Name and country ID.

We created network data for each year using the security owner and issuer as nodes and their ownership relations as links. Data for each year was generated using the latest reporting date for each security up to the geven year. However, as different IDs are assigned to the same company for the owner ID and issuer ID, the company (such as a listed company), which can be both the owner and issuer, becomes multiple nodes, and in order to unify these, it is necessary to establish one-to-one correspondence between the owner ID and issuer ID for those who are not only owners but also issuers. However, the mapping list provided by Thomson Reuters that links the owner ID to the issuer ID is far from complete. Therefore, we used the owner name, issuer name, country ID, and issuing country ID to complement the mapping list with name identification. The procedure for creating the new mapping list was as follows:

• 1. Anything that existed in the mapping list provided to Thomson Reuters would be used with priority.

• 2. Words with notational fluctuations were unified into one word (for example, “corp” and “corporation” were unified into “co”).

• 3. Names were compared word by word for issuers and owners coming from the same country. Those without country information were compared with companies in all countries.

• 4. If a word matched in the owner name and issuer Name, a score that was inversely proportional to the frequency of the word was given (for example, the words “Industry” and “Insurance” have a low score due to their high frequency, and a high score if a unique name such as “Toyota” matches).

• 5. The value obtained by dividing the sum of the scores of the matching words by the sum of the scores of all of the unique words of both was defined as the similarity, and a pair with a similarity of 0.9 or more was obtained as a matched pair. When being matched with multiples, only the highest one was taken.

In this way, we obtained a new mapping list that mapped 12,117 by Thomson Reuters and 11,758 by our own name identification scheme. As was demonstrated in Sect. 2, the global ownership database constructed in this way encompassed virtually all of the world’s listed firms.

Fig. 22

Total bilateral capitalization transaction, with the sum of market capitalization flowing in both directions for each link, in the illustrative network on the basis of [29]; Figure S4 in Appendix S1 of this paper, where the intrinsic value of each node is assumed to be one, i.e., $\boldsymbol{v} = (1, 1, 1, 1, 1, 1)^{\text {T}}$

Appendix 2

We applied the Helmholtz–Hodge decomposition to the illustrative example given by [29], in Fig. S4 in Appendix S1 of their paper. Figure 22 depicts the extent to which nodes were coupled through total bilateral capitalization transactions; that is, sum of market capitalization flows in both directions for each link, leading to an undirected network. In Fig. 23, on the other hand, we show the balance flow network obtained by calculating the net market capitalization flow for each link, which is a directed network. The solution for the Helmholtz–Hodge potential $\boldsymbol{\phi }=(\phi _{1},\phi _{2},\cdots ,\phi _{6})^{\text {T}}$ in the balance flow network is given by

$$\begin{aligned} \boldsymbol{\phi }=\left( \begin{array}{c}1.993 \\ 0.993 \\ 0.663 \\ 0.945 \\ 1.000 \\ 0.000\end{array}\right) , \end{aligned}$$

(18)

where the potential of the sixth node is set to zero in order to fix the arbitrariness in the potential by constant and $G_{ij}$ is assumed to be the total capitalization transaction between nodes i and j. From the results for the Helmholtz–Hodge potential, Eq. (18), the gradient flow component and associated loop flow component in each link can be derived using Eqs. (2) and (3). Figures 24 and 25 show the resulting gradient and circular flow networks, respectively. The hierarchical structure hidden in the original network appears in the gradient flow network.

Fig. 23

Balance flow and the subtraction of market capitalization flows in both directions for each link, in the illustrative network on the basis of [29]: see Fig. S4 in Appendix S1 of their paper

Fig. 24

Gradient flow component for the network in Fig. 23

Fig. 25

Loop flow component for the network in Fig. 23

The network value $v_{i}^{\text {net}}$ of node i in an ownership network is defined as the sum of its intrinsic value $v_{i}$ and portfolio value $p_{i}^{\text {net}}$ integrated through ownership relations⁶:

$$\begin{aligned} v_{i}^{\text {net}}=v_{i}+p_{i}^{\text {net}}. \end{aligned}$$

(19)

The network portfolio value $p_{i}^{\text {net}}$ is given by:

$$\begin{aligned} p_{i}^{\text {net}}=\sum _{j}W_{ij}v_{j}^{\text {net}}, \end{aligned}$$

(20)

where $W_{ij}$ denotes the ownership share of node i for node j, that is, the share of the equity capital of node j possessed by node i, and is further decomposed into the direct and indirect terms as

$$\begin{aligned} p_{i}^{\text {net}}=p_{i}^{\text {dir}}+p_{i}^{\text {ind}}. \end{aligned}$$

(21)

with

$$\begin{aligned} p_{i}^{\text {dir}}=\sum _{j}W_{ij}v_{i}, \end{aligned}$$

(22)

and

$$\begin{aligned} p_{i}^{\text {ind}}=\sum _{j}W_{ij}p_{j}^{\text {net}}. \end{aligned}$$

(23)

We note that $p_{i}^{\text {net}}$ is expressible in terms of $v_{i}$ as:

$$\begin{aligned} p_{i}^{\text {net}}=\sum _{j}\widetilde{W}_{ij}v_{j}, \end{aligned}$$

(24)

where $\widetilde{W}_{ij}$ is given by

$$\begin{aligned} \widetilde{W}=(1-W)^{-1}W=W+W^2+W^3+\cdots , \end{aligned}$$

(25)

in a matrix form. If the expansion, Eq. (25), for $\tilde{W}$ is truncated at the leading order, Eq. (24) is reduced to Eq. (22). We thus see that $\widetilde{W}_{ij}$ plays a role in the effective ownership share of node i for node j, which is connected to node i, even in an indirect way. In the presence of loops, however, the network effects are overemphasized by multiple counting. In the worst case that is, when there is a closed loop without any root node attached to it in the network, Eq. (24) gives rise to a divergent result. This is the first drawback of the original input-output theory [13] initially pointed out by [29].

On the basis of the gradient flow network, which is free from loops, we can calculate the network value for each node as:

$$\begin{aligned} \boldsymbol{v}^{\text {net}}_{\text {HH}}=\left( \begin{array}{c}2.857 \\ 1.919 \\ 1.000 \\ 1.000 \\ 3.143 \\ 1.000\end{array}\right) . \end{aligned}$$

(26)

The first drawback of the original theory was later rectified by [12, 29]. A comparison with the optimal result on the basis of [29],

$$\begin{aligned} \boldsymbol{v}^{\text {net}}_{\text {VGB}}=\left( \begin{array}{c}1.500 \\ 5.000 \\ 4.378 \\ 4.667 \\ 4.714 \\ 1.000\end{array}\right) , \end{aligned}$$

(27)

shows that the network values of the nodes in the SCC as given in Eq. (27) are appreciably larger than the corresponding results in Eq. (26). This is because there still exist inner feedback loops in the calculation of [29]. According to our new formulation based on the Helmholtz–Hodge decomposition, the fifth node is a root node (ultimate owner), as well as the first node in the illustrative network because they have no incoming links to the gradient flow network. As it should be, the sum of their network values in Eq. (26) amounts to the total market value, i.e., the sum of the intrinsic value of all of the nodes, $6=\sum _{i}v_{i}$ in the present example. However, such a sum rule does not hold in Eq. (27). Moreover we can see that the total market value is almost evenly divided by the two root nodes; the fifth node is slightly more influential than the first node, however. This clearly demonstrates that the new method does not suffer from the second drawback encountered by input-output theory [13], which leads to the exclusive possession of the entire intrinsic value of the nodes by the first node, with a single root node in the initial network.

We can therefore see the new formulation based on the Helmholtz–Hodge decomposition, which has a sound mathematical foundation, and is superior to the method of [29].

Appendix 3

The Herfindahl–Hirschman index (HHI) was devised [53] as a tool to measure the extent to which an industrial market is occupied by a small number of firms. It is defined as:

$$\begin{aligned} HHI=\sum _{i=1}^{N} s_{i}^2, \end{aligned}$$

(28)

where $s_{i}$ is the market share of firm i with $\sum _{i=1}^{N} s_{i}=1$ and N is the number of firms in the market. Here, we use the HHI to measure the concentration of corporate equity holdings by stock owners through direct or ultimate ownership relations. The HHI takes 1/N when corporate equity holdings are equally distributed over the owners, whereas it maintains unity when they are held by a single owner.

Fig. 26

Annual change in the Herfindahl–Hirschman index (HHI), measuring the degree of concentration in the distribution of corporate equity holdings by owners

Figure 26 shows the annual change in HHI for the three major categories of stock owners, private corporations, and holding companies, as well as strategic individuals and family members, and institutional investors, ordered from left to right in the four countries/regions of the United States, Japan, the EU and China, ordered from top to bottom. The broken and solid lines represent the results based on direct and ultimate ownership relations, respectively. Note that the inverse of the HHI, plotted on the vertical axis, is normalized by the number N of firms in each category.

The temporal change in the HHI for the institutional investors sector in the United States highly contrasts with that for the private corporations and holding companies sector in China; there is no appreciable difference observed between the results based on the direct and ultimate ownership relations in either of these two countries. The former confirms that the distribution of corporate equity holdings is steadily ever more greatly concentrated in the large funds based in the United States. On the other hand, the latter shows that the economic development of China continuously diversifies corporate equity holdings among private corporations and holding companies in the country.

We also note that the HHI for the private corporations and holding companies sector in the United States, Japan, and the EU, and for the institutional investor sector in Japan, significantly depends on which ownership relations we pay attention to, whether direct or ultimate. In 2007, the degree of concentration in the distribution of corporate equity holdings within the private corporations and holding companies sector increased in all three countries when the direct ownership relations were replaced by the ultimate ones. This tendency holds across the entire period in the United States and EU. On the other hand, the crisis in 2008–2009 prompted by the collapse of Lehman Brothers turned over the situation in Japan, with corporate equity holdings becoming more diversified among ultimate owners in the sector in the wake of the financial shock. The distribution of corporate equity holdings within the institutional investors sector in Japan was also more concentrated for the ultimate owners than the direct ones throughout the entire period.