Risk Analysis of Conglomerates with Debt and Equity Links

Abstract

Conglomerates play an important role in the functioning of capital markets. Therefore, assessing their response to external shocks is a significant risk management challenge not only for conglomerate executives but also for investors and regulators alike. In this context, a conglomerate refers to a group of companies typically operating across different industries and interconnected through both equity and debt relationships. Essentially, a conglomerate functions as a financial network whose nodes are linked by two layers of reciprocal connections. This paper introduces an algorithm to evaluate a conglomerate’s response to external shocks. Additionally, it proposes a protocol based on five key metrics that collectively summarize the conglomerate’s overall resilience. These metrics offer two major advantages: they facilitate comparisons between the strengths of different conglomerates and help assess the effectiveness of various strategies, such as internal capital reallocations, aimed at enhancing a conglomerate’s resilience. The algorithm’s usefulness, including its ability to detect cascades or “second-wave” defaults, is demonstrated through two illustrative examples.

1. Introduction

Conglomerates play a critical role in the functioning of capital markets, and therefore, it is important to have effective tools to assess their resilience. The present study is concerned with conglomerates’ ability to withstand external shocks. Unfortunately, the single-value metrics (e.g., debt-to-equity ratios, short-term vs. long-term debt), which are often sufficient for evaluating individual firms, have proven inadequate to deal with conglomerates since they cannot account for their network structure. Thus, from a risk management viewpoint, conglomerates present a difficult challenge.

Moreover, the interconnected nature of conglomerates means that disruptions in one sector can have ripple effects across other industries, amplifying their economic impact. This situation is particularly pronounced in emerging markets, where conglomerates play a disproportionately larger role compared to their counterparts in developed economies. In short, the capacity to evaluate the resilience of conglomerates is also a problem with significant public policy and regulatory implications.

Conglomerate, in our study, refers to a group of companies that are inter-connected at both the equity and debt levels and operate, in principle, in different sectors of the economy. In essence, a conglomerate is a network whose nodes are linked via a two-level array of reciprocal dependencies. (A formal definition of a conglomerate is presented in Section 3.) This contrasts with the typical financial network (e.g., a banking system) in which the connections are in general due to interbank lending; thus, the firms interact only via debt-linked mechanisms, and there is only a one-level array of connections.

Previous research that focused on the response of financial networks to external shocks has dealt almost exclusively with networks defined by a one-level array of connections. The risk assessment of financial networks with a dual connectivity structure, such as conglomerates, is still a problem that lacks a widely accepted method of analysis. Furthermore, the need to have a protocol based on simple metrics, to compare the resilience of different conglomerates, and to explore the merits of competing risk-mitigating strategies remains unmet.

The motivation behind this study is twofold. First, our goal is to provide a practical method—an easy-to-implement simulation algorithm—to evaluate the response of a conglomerate to external shocks, and second, we seek to provide a suitable set of metrics to describe the resilience to shocks of a given conglomerate.

Our model is based on a set of linear equations that can be solved analytically and offers several advantages. It accounts explicitly for the post-default value of the debt of the firms of the conglomerate that have failed due to an external shock. It allows the identification of cascades (“second-wave” failures) and, in combination with the metrics we introduce, makes the assessment of different strategies aimed at improving the conglomerate strength very straightforward.

Having such tools is vital in the context of enterprise risk management (ERM), as the internal capital allocation within a conglomerate and its interconnectedness play key roles in mitigating the impact of external shocks (Boguth et al. 2021; Fraser et al. 2024). The tools we propose can also be used as diagnostic tools by both regulators and investors, not only to evaluate the resilience of a conglomerate but also to compare it with that of other conglomerates and to assess the effectiveness of various risk reduction strategies, if needed.

2. Literature Review

Conceptually, a conglomerate is a network comprising multiple agents (nodes) interconnected through reciprocal exposures (inter-dependencies). Prior to the subprime crisis of 2007/2008, only a few researchers examined the significance of systemic risk and how such risk could arise from the interconnectedness of financial agents. One of the pioneering studies in this area was conducted by Sheldon and Maurer (1998), who explored the Swiss interbank loan system and its response to defaults, offering one of the earliest frameworks for assessing risk within a financial network.

Subsequently, various authors examined different aspects of the issue, namely, the response of banking networks to external shocks. Notable studies include those by Eisenberg and Noe (2001), Freixas et al. (2000), Furfine (2003), Iori et al. (2008), van Lelyveld and Liedorp (2006), Nier et al. (2008), Upper and Worms (2004), and Wells (2004). Freixas et al. (2000), for example, explored the role of central banks and the importance of liquidity facilities, while Iori et al. (2008) analyzed the Italian interbank lending market. Eisenberg and Noe’s pioneering paper demonstrated that, under relatively mild conditions, there is always a unique clearing system for a network formed by reciprocal lending arrangements.

Following the subprime crisis, there was a significant surge in research focused on assessing risk in financial networks. Studies by Ahelegbey et al. (2021), Cifuentes et al. (2021), Georg (2013), and Elliott et al. (2014), for instance, examined how financial networks respond to external shocks, particularly those stemming from credit events. These works highlighted the critical role of cascades—identifying companies or nodes whose failures could trigger a chain of subsequent failures within the network.

Several papers have also explored the characteristics of financial networks and how their features influence their dynamic behavior. For instance, Haldane and May (2011) highlighted that prudential regulation has historically focused on individual banks, overlooking their interconnectivity. Martínez-Jaramillo et al. (2010) used the Mexican banking system as a case study, introducing a metric based on the conditional VaR to assess the system’s fragility. Their work distinguished between bank failures caused by an initial external shock and those resulting from the subsequent contagion process. Morrison et al. (2017) employed CDS spreads to identify transmission mechanisms. Other notable contributions include Avdjiev et al. (2019) and Veraart (2020), who addressed various aspects of the issue. Caccioli et al. (2018) offered a comprehensive review of the literature, summarizing significant advances in recent years.

A common element in the above-mentioned studies (which should not be taken as criticism) is that they have focused on financial networks—chiefly banking networks—that is, networks in which the nodes are linked only via reciprocal credit exposures. A good example is the study by Chen et al. (2020), which focused on the Chinese banking system. The case of conglomerates, that is, a network consisting of a group of companies, not necessarily financial entities, connected at the equity and/or debt level (that is, interconnected via cross-ownership and cross-lending) has received almost no attention.

Two notable exceptions are the works of Gourieroux et al. (2012) and Hauton and Héam (2015). These authors developed similar sets of 2 × N nonlinear equations (assuming a network or conglomerate consisting of N companies) to model such systems. However, despite the merits of these efforts, they overlooked a significant factor: that by incorporating market data to estimate the value of defaulted debt (a topic we address in detail later), these equations could be simplified considerably. Furthermore, their formulations did not allow the identification of cascades. Nevertheless, the merits of these studies lie in their pioneering attempts to conduct a quantitative analysis of how conglomerates respond to external shocks.

In summary, evaluating the resilience of conglomerates to external shocks, improving the capacity to assess their capital adequacy requirements, and addressing the potential risk to the broader economy in the event of their failure remain pressing concerns. For further discussion, see Bank of International Settlements (2012), Estrada and Sánchez-Aragón (2020), Gatzert and Schmeiser (2011), Henrard and Olieslagers (2004), and Ku and Whang (2020).

An important aspect of analyzing the response of financial networks to shocks is the development of effective metrics or protocols for comparing the resilience of different networks. The challenge arises from the need to simultaneously account for two key factors: the network’s topology (the structure of the links connecting the nodes) and the balance sheets of the companies within the conglomerate. Notice that each node in the network is associated with attributes that can vary widely in their characteristics, adding complexity to the analysis.

Metrics that focus solely on balance sheets, such as average debt-to-equity ratios, but ignore network connectivity have proven inadequate. Similarly, metrics based only on network connectivity, such as network centrality or node depth, but disregarding balance sheet structures, also fall short. A paper by Glasserman and Young (2016) provided a comprehensive overview of what they term Measures of Vulnerability and Contagion but concludes with a cautionary note: “As we have seen, the interaction between these factors and the network topology is quite complex and not fully understood even at a theoretical level.”

Thus, it is fair to say that neither practitioners nor academics have yet been able to propose widely accepted tools, protocols, or metrics for assessing and comparing the resilience of financial networks when their connections are defined solely by debt links. The complexity of the network obviously increases when equity-linked connections are introduced, which effectively adds a second layer of connectivity—an essential feature of conglomerates. This additional layer makes the challenge of evaluating resilience even more daunting.

In conclusion, the challenge of developing a practical risk management framework to assess the resilience of conglomerates to external shocks remains unresolved. Similarly, the need for effective metrics to compare the resilience of multiple conglomerates is still unmet. This issue is highlighted in a recent paper by Fraser et al. (2024), titled What’s Wrong with Enterprise Risk Management? where the authors argued for moving beyond static models and emphasized the importance of accounting for the dynamic nature of interconnected networks. The method presented in this paper represents an attempt to address both of these critical needs.

3. Problem Statement

Consider a conglomerate made up of N firms. For simplicity, but without loss generality, we assigned index 1 to the controlling entity (or the holding company); thus, the other components (“subsidiaries”) are represented by indices 2, 3, …, N.

In this study, the term “controlling entity” is really used for the expedience of notation rather than reflecting control in the conventional sense of the term. In a purist sense, a controlling entity can be interpreted to reflect at least 50% ownership of a company stock. Control, however, is a more subtle term since it can be achieved, for example, via holding a majority of stock with voting rights while holding a minor position in terms of cash flow rights. Or, alternatively, control can be attained through managerial and executive dominance. In any event, as it will become apparent shortly, the method we propose makes no assumptions regarding the percentage of equity that node 1 owns in each of the subsidiaries (nodes 2, 3, …, N).

We also assumed that each firm within the conglomerate has a balance sheet with the same basic (but quite general) structure, described as follows.

3.1. Assets

A = cash (and cash equivalents) + investments + physical assets;

B = equity investments in other firms within the conglomerate;

C = debt investments in other firms within the conglomerate.

Note: (i) investments, in A, refer to any financial interest in entities outside the conglomerate (e.g., stocks, bonds, mutual funds); and (ii) physical assets, in A, refer to factories, machinery, real estate, etc.

3.2. Liabilities

D = debt;

E = equity.

Note: (i) D captures both obligations to other firms within the conglomerate as well as obligations to outsiders, and (ii) E, again, is the sum of the equity positions held by firms within the conglomerate and outside investors.

3.3. Single-Firm Equilibrium Condition

Initially, that is, before an external shock is applied, the entire system is in a state of equilibrium. In short

$$A_k+B_k+C_k=D_k+E_k$$

(1)

for k = 1, 2, …, N.

Let α_k,i denote the fraction of the equity of firm i, owned by firm k. Hence,

$$B_k={\displaystyle \sum }_{i=1}^N{\alpha }_{k, i} E_{i }$$

(2)

Analogously, let us β_k,i denote the fraction of the debt issued by firm i, owned by firm k. Thus,

$$C_k={\displaystyle \sum }_{i=1}^N{\beta }_{k, i} D_{i }$$

(3)

Therefore, Equation (1) can be reorganized as

$$E_k=A_k-D_k +{{\displaystyle \sum }}_{i=1}^N{\alpha }_{k, i} E_{i } +{{\displaystyle \sum }}_{i=1}^N{\beta }_{k, i} D_{i }$$

(4)

for k = 1, 2, …, N.

A few observations:

i.

clearly, α_k,k = 0, for k = 1, …, N since no firm can own its own equity;

ii.

also, α_2,1 = α_3,1 = … = α_N,1 = 0 since firms 2, 3, …, N (subsidiaries) are assumed not to have any participation in the controlling entity (firm 1);

iii.

note that regarding α_k,1 + α_k,2 + … + α_k,N (for k = 1, …, N), we can only claim that this sum is ≥0 and can certainly be larger than 1; and

iv.

we claim that α_1,k + α_2,k + … + α_N,k ≤ 1 (for k = 1, …, N); the equality holds when there are no outside investors in the equity of firm k.

Analogous observations are valid for the β coefficients.

3.4. Matrix Formulation

The set of N equations described by the single-firm equilibrium conditions, e.g., Equation (4), can be written in matrix form as

$$\left(\left[I\right]-\left[\alpha \right]\right) \mathit{E}=\mathit{A}+\left(\left[\beta \right]-\left[I\right]\right) \mathit{D}$$

(5)

where the [ ] denotes a matrix, and the boldface is reserved for vectors; [I] is the identity matrix.

3.5. External Shocks

An external shock could occur due to a number of situations. First, there is fraud. Think of the Wirecard case in which suddenly $2 billion went missing from its balance sheet (Jo et al. 2021; Nicola 2022). Second, some financial investments can abruptly lose value. For example, during the subprime crisis, many AAA-rated structured products, in a short time, went from being valued at par to single digits (Fender and Scheicher 2009). And finally, acts of God (e.g., earthquakes, hurricanes), despite the existence of insurance protection, can impair physical assets and cause big monetary losses. All these cases can be represented by a sudden reduction in the value of A_q. In short, a shock can be modeled as

$$A_{q }\left(post shock value\right)=\left(1-\delta \right)\times A_{q }\left(pre shock value\right)$$

(6)

The problem, then, is to assess the response of the conglomerate (an N-node network) to an external shock, assuming that we know the balance sheet of each firm—represented by the vectors A, B, C, D, and E—and the matrices [α] and [β]. Additionally, we assume that the networks in initially in a state of equilibrium.

4. Simulation Algorithm

Broadly speaking, the simulation algorithm proceeds as follows: We assume that, initially, the network is in a state of equilibrium; that is, Equation (4) is satisfied at each node. Then, (1) with the new (post shock) value of A, (defined by q and δ, see Equation (6)), we recalculate the new value of E; (2) any firm whose new equity value is equal or less than zero (bankrupted) is removed from the network and a defaulted value is assigned to its debt; and (3) we continue recalculating the value of E (and removing the firms that have failed) until the system reaches a new equilibrium.

More specifically, the approach outlined above can be implemented in two different but totally equivalent ways, depending on whether we follow a single-equation (firm-by-firm) approach or a matrix formulation. For the sake of completion, we describe each approach in detail.

4.1. Firm-by-Firm Approach

[1]

Let A be the new (post shock) vector.

[2]

For k = 1, 2, …, N proceed as follows:

Using Equation (4), solve for E_k.

If the new value of E_k is such that E_k > 0 it means that the firm has survived the shock (although it could well happen that the new E_k value reflects a reduction compared to the initial value of the equity). Update vector E.

If the new value is such that E_k ≤ 0, it means that firm k has gone into bankruptcy. Thus, E_k is set permanently to zero; also, D_k is re-set at D_k = μ D_k, its defaulted value (clearly, 0 < μ < 1). Thus, vectors E and D are updated.

[1] The loop above ([2]) is repeated until the vector E reaches a stable value (no additional firms collapse).

If a firm collapses due to a shock to firm q (whether or not it is firm q itself that collapses), we call this a primitive failure. When a firm collapses as a result of a primitive failure—meaning a failure triggered by a previously failed firm—we refer to these subsequent failures as cascades or “second-wave” failures.

4.2. Matrix Formulation Approach

[1]

Let A be the new (post shock) vector.

[2]

Solve Equation (5) to calculate the new E.

If all E components are positive, it means the system has reached a state of equilibrium, and we stop. If at least one component of E is ≤ 0; say E_m, it means that company m has failed (is bankrupted) and its equity is now 0. Also, the value of its debt is re-set, D_m = μ D_m.

Now, a new (reduced) version of Equation (5) needs to be solved again to calculate the new vector E (in which the m component has been removed). In short, the $\left(\left[\mathrm{I}\right]-\left[\alpha \right]\right)$ matrix is now reduced in size since the m column and m row have been removed; the m component is also removed from the vector A; and finally, the m row is removed from the $\left(\left[\beta \right] - \left[\mathrm{I}\right]\right)$ matrix. See Appendix A for a more schematic description. Again, the failures that occur after solving the original (N × N) system of equations are designated as primitive. Subsequent failures (those identified by solving these reduced systems of equations) are designated as cascades. This process is continued (each time solving a reduced version of the previous system) until the solution to the linear system yields a vector with only positive values.

4.3. General Considerations

Some observations are in order:

(a)

It is worth noting that the firm-by-firm approach is akin to the Gauss–Seidel method for solving linear equations, whereas the matrix approach relies on directly solving a sequence of decreasing-size linear systems. In essence, both approaches rely on an iterative procedure which just manifests itself in a different manner;

(b)

It might be tempting to believe that assigning the defaulted debt a fraction (μ) of its original value is rather arbitrary. That sentiment would be misguided. There are plenty of data regarding the value of defaulted corporate debt, roughly a value between 30% and 70%, depending on whether we are dealing with subordinated unsecured positions or senior secured loans (Gunter and Kraemer 2023). Making this assumption has the advantage that leaves us with a system of linear equations without sacrificing accuracy. In fact, assuming such value for the defaulted debt, since it captures all the frictions involved in a bankruptcy proceeding, is far more realistic than leaving the defaulted debt as an unknown to be determined by simple arithmetic (that is, matching assets and liabilities) in the post-default scenario;

(c)

Note that the transpose of [α] is such that the sum of the entries in any row is less or equal to 1 (the strict inequality holds if there are outside investors). In short,${\left(\left[\mathrm{I}\right]-\left[\alpha \right]\right)}^T$ is a diagonally dominant matrix. And, in the case where there are outside investors, it is strictly diagonally dominant. Thus, recalling that a matrix and its transpose have the same determinant, we can claim that whenever ${\left(\left[\mathrm{I}\right]-\left[\alpha \right]\right)}^T$ is strictly diagonally dominant, then $\left(\left[\mathrm{I}\right]-\left[\alpha \right]\right)$ is non-singular, which gives assurance that Equation (5) has a unique solution. This is a sufficient but not necessary condition for having a unique solution. On the other hand, in cases in which ${\left(\left[\mathrm{I}\right]-\left[\alpha \right]\right)}^T$ is just diagonally dominant (but not strictly diagonally dominant) since the diagonal has only 1’s and most off-diagonal entries are either zero or less than 1 (in absolute value), it is highly likely that $\left(\left[\mathrm{I}\right]-\left[\alpha \right]\right)$ will also be non-singular and stable,

(d)

Finally, it is important to realize that the framework proposed is flexible enough to accommodate a number of network structures. For example, this formulation places no restrictions on what we can describe as “outside” connections. In other words, investors external to the conglomerate can hold equity and/or debt positions in any of the subsidiaries (nodes, 2, 3, …, N). This is important since conglomerates often have joint ventures with entities that are not part of the conglomerate. Additionally, whether the equity in node 1 (controlling entity) is held by private investors or floated is irrelevant. These considerations are crucial from a risk management viewpoint since, in a real situation, it is necessary not only to examine the response of the conglomerate to a shock under its current configuration but also to examine its response after introducing potential re-arrangement to its structure.

5. Examples of Application

The following two examples showcase some of the analyses that can be carried out with the proposed algorithm.

5.1. Example 1

Consider a seven-firm conglomerate whose balance sheets are described in

Table 1. Seven-firm conglomerate: the balance sheet of each firm (firm 1 is the controlling entity).

	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7
Assets
A	70	12	11	13	13	33	19
B	14	1	3	4	3	0	12
C	16	1	9	4	2	0	6
Total	100	14	23	21	18	33	37
Liabilities
D	90	7	16	7	11	14	30
E	10	7	7	14	7	19	7
Total	100	14	23	21	18	33	37
Leverage	10.0	2.0	3.3	1.5	2.6	1.7	5.3

Note: Leverage is defined as the firm’s total liabilities divided by its equity (e.g., (D + E)/E); A = cash (and cash equivalents) + investments + physical assets; B = equity investments in other firms within the conglomerate; C = debt investments in other firms within the conglomerate; D = debt; and E = equity.

(firm 1 is the controlling entity). Clearly, their capital structures are quite heterogeneous, and firm 1, in terms of market value (as is often the case), dominates.

Table 2. Seven-firm conglomerate: [α] and [β] matrices.

[α]
||	1	2	3	4	5	6	7	Total
1	0.0%	14.3%	42.9%	14.3%	57.1%	15.8%	14.3%	1.59
2	0.0%	0.0%	14.3%	0.0%	0.0%	0.0%	0.0%	0.14
3	0.0%	0.0%	0.0%	0.0%	28.6%	0.0%	14.3%	0.43
4	0.0%	57.1%	0.0%	0.0%	0.0%	0.0%	0.0%	0.57
5	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	42.9%	0.43
6	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.00
7	0.0%	0.0%	0.0%	7.1%	0.0%	57.9%	0.0%	0.65
Total	0.00	0.71	0.57	0.21	0.86	0.74	0.71
[β]
||	1	2	3	4	5	6	7	Total
1	0.0%	0.0%	68.8%	0.0%	0.0%	35.7%	0.0%	1.04
2	0.0%	0.0%	0.0%	0.0%	0.0%	7.1%	0.0%	0.07
3	0.0%	0.0%	0.0%	28.6%	0.0%	0.0%	23.3%	0.52
4	0.0%	0.0%	0.0%	0.0%	27.3%	0.0%	3.3%	0.31
5	0.0%	28.6%	0.0%	0.0%	0.0%	0.0%	0.0%	0.29
6	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.00
7	0.0%	0.0%	0.0%	0.0%	0.0%	42.9%	0.0%	0.43
Total	0.00	0.29	0.69	0.29	0.27	0.86	0.27

Note: Non-zero entries in bold. If the sum of the entries in a column is less than one, it implies that a fraction of the equity or debt of that firm is owned by investors outside the conglomerate.

shows the corresponding [α] and [β] matrices. Notice that for each column k of [α], (k = 1, …, 7), the sum of the entries is less than 1. In other words, there are outside-the-conglomerate investors who hold at least a fraction of the equity of each firm. This, in turn, means that the ${\left(\left[\mathrm{I}\right]-\left[\alpha \right]\right)}^T$ matrix (and any subsequent reduced versions of it) is strictly diagonally dominant and, hence, non-singular.

Figure 1 provides a graphical representation of the network’s connectivity. As previously mentioned, unlike banking networks, the connectivity here operates through two channels: debt and equity links. For example, firm 4 controls 57.1% of the equity in firm 2, while firm 3 owns 23.3% of the debt issued by firm 7. The [β] matrix also indicates that external investors hold a portion of the debt issued by each firm in the conglomerate (with the sum of the entries in each column being strictly less than 1). Additionally, we assume μ = 50%, a value consistent with recent data reported by S&P on loans and bonds (Gunter and Kraemer 2023).

The following analyses were performed with both approaches (firm-by-firm and matrix formulation) and yielded, as expected, the same results.

Table 3. Minimum shock size to bankrupt each firm in the seven-firm conglomerate.

Minimum Shock to Bankrupt	Shock to:
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7
Firm 1	0.2	-	-	-	-	0.6	-
Firm 2	-	0.6	-	-	-	-	-
Firm 3	-	-	0.7	-	-	-	-
Firm 4	-	-	-	-	-	-	-
Firm 5	-	-	-	-	0.6	-	-
Firm 6	-	-	-	-	-	0.6	-
Firm 7	-	-	-	-	-	0.4	0.4

Note: The firms to which the shocks are applied are listed horizontally (top row). Each vertical column identifies the minimum shock such that it will cause the failure of the firm listed on the furthest left-side column. For example, a shock of size 0.4 applied to firm 7 will cause the collapse of this firm; a shock of size 0.4 applied to firm 6 will cause the failure of firm 7; increasing the size of this shock to 0.6 will also cause the failure of firms 1 and 6 (firm 7 has already failed).

shows the result of applying shocks of different sizes to each of the firms individually. The firms to which the shocks are applied are listed horizontally (top row). Each vertical column identifies the minimum shock such that it will cause the failure of the firm listed on the furthest left-side column. To be clear, a shock of size 0.4 applied to firm 7 caused the collapse of this firm. A shock of size 0.4 applied to firm 6 will also cause the failure of firm 7. Whereas, a shock of size 0.6 applied to firm 6, in addition, caused the failure of firms 1 and 6.

Table 4. Type of failures after applying shocks of different sizes to firm 6.

Shock Size	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7
0	0	0	0	0	0	0	0
0.1	0	0	0	0	0	0	0
0.2	0	0	0	0	0	0	0
0.3	0	0	0	0	0	0	0
0.4	0	0	0	0	0	0	1
0.5	0	0	0	0	0	0	1
0.6	2	0	0	0	0	1	1
0.7	2	0	0	0	0	1	1
0.8	2	0	0	0	0	1	1
0.9	2	0	0	0	0	1	1
1	2	0	0	0	0	1	1

Note: 1 refers to a primitive failure; 2 refers to a failure due to a cascade effect.

presents the results of applying shocks of varying magnitudes (δ = 0, 0.1, …, 1) to firm 6. The smallest shock that triggers a bankruptcy occurs at δ = 0.4, causing the initial failure (primitive) of firm 7. When δ reaches 0.6, firm 6 also fails (primitive failure), and these failures subsequently lead to the collapse (cascade) of firm 1. This outcome is consistent with the data in Table 3, specifically under the column for firm 6. Notably, increasing the shock beyond 0.6 has no further effect, as firm 6 has already failed and been removed from the network.

Interestingly, the first firm to fail is not the one that received the shock (firm 6) but firm 7. Moreover, firm 1, with a leverage level of 10, collapses after firms 6 and 7, whose leverage levels are significantly lower, at 1.7 and 5.3, respectively. This highlights the limitations of relying solely on leverage to predict resilience—network connectivity plays a crucial role. The choice of firm 6 as the recipient of the shocks is arbitrary; it only serves to demonstrate the usefulness of this type of analysis. Similar approaches can be applied to other firms or groups of firms to shed light on different aspects of the network dynamics.

Figure 2 shows graphically and, in more detail, what Table 4 indicates, namely, how the equity of each firm decreases as δ (the size of the shock) grows. For δ = 0.4, only one firm shows zero equity (firm 7). δ = 0.6 shows three firms with zero equity (1, 6, and 7). As mentioned before, after firm 6 has failed, the system is indifferent to increasing the size of the shock, which is consistent with the stable equity values observed for δ > 0.6 (see Table 4). This figure also allows us to identify which firms (nodes), even if they have not failed, might have suffered a debilitating impact. For instance, the graph indicates that firm 3, despite the fact that it has survived the shocks applied to firm 6, has seen its equity reduced in value from 7 to 2. And thus, it is now in a more vulnerable position.

Table 5. Minimum shock size, applied to each firm, needed to bankrupt the controlling entity (firm 1).

Shock to	Shock Size
Firm 1	0.2
Firm 2	-
Firm 3	-
Firm 4	-
Firm 5	-
Firm 6	0.6
Firm 7	-

shows the minimum shock size applied to different firms that will cause the failure of the controlling entity (firm 1), regardless of the type of failure. Again, this information is consistent with the information presented in Table 3, i.e., a 0.6 shock applied to firm 6 triggers the failure of the controlling entity (firm 1). And no matter how big the shocks to firms 2, …, 5, and 7, they do not trigger the collapse of firm 1.

Table 6. Minimum shock size to bankrupt each firm in the seven-firm conglomerate based on different values of μ (recovery value of defaulted debt).

µ = 0.75
Minimum Shock to Bankrupt	Shock to
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7
Firm 1	0.2	-	-	-	-	-	-
Firm 2	-	0.6	-	-	-	-	-
Firm 3	-	-	0.7	-	-	-	-
Firm 4	-	-	-	-	-	-	-
Firm 5	-	-	-	-	0.6	-	-
Firm 6	-	-	-	-	-	0.6	-
Firm 7	-	-	-	-	-	0.4	0.4
µ = 0.5
Minimum Shock to Bankrupt	Shock to
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7
Firm 1	0.2	-	-	-	-	0.6	-
Firm 2	-	0.6	-	-	-	-	-
Firm 3	-	-	0.7	-	-	-	-
Firm 4	-	-	-	-	-	-	-
Firm 5	-	-	-	-	0.6	-	-
Firm 6	-	-	-	-	-	0.6	-
Firm 7	-	-	-	-	-	0.4	0.4
µ = 0.25
Minimum Shock to Bankrupt	Shock to
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7
Firm 1	0.2	-	0.7	-	-	0.4	0.4
Firm 2	-	0.6	-	-	-	-	-
Firm 3	-	-	0.7	-	-	0.4	0.4
Firm 4	-	-	-	-	-	-	-
Firm 5	-	-	-	-	0.6	-	-
Firm 6	-	-	-	-	-	0.6	-
Firm 7	-	-	-	-	-	0.4	0.4

shows the effects of assuming different values for the defaulted debt. The base case (μ = 50%), which is shown in the center panel, coincides with the information shown in Table 3. As expected, a higher value of μ (75%) results in fewer failures, whereas a lower μ (25%) is associated with more failures. What is more interesting, however, is observing that changes in the value of μ alter the dynamics of the network’s response, that is, which firms fail, the sequence in which they fail, and the presence of cascades.

5.2. Example 2

Consider now the nine-firm conglomerate whose balance sheets are described in

Table 7. Nine-firm conglomerate: the balance sheet of each firm (firm 1 is the controlling entity).

	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7	Firm 8	Firm 9
Assets
A	319	156	176	322	45	137	307	177	282
B	394	99	87	15	231	102	33	120	77
C	351	14	46	40	94	22	20	14	19
Total	1065	270	310	377	370	260	359	311	378
Liabilities
D	665	140	190	180	150	179	160	200	200
E	400	130	120	197	220	81	199	111	178
Total	1065	270	310	377	370	260	359	311	378
Leverage	2.7	2.1	2.6	1.9	1.7	3.2	1.8	2.8	2.1

Note: Leverage is defined as the firm’s total liabilities divided by its equity (e.g., (D + E)/E); A = cash (and cash equivalents) + investments + physical assets; B = equity investments in other firms within the conglomerate; C = debt investments in other firms within the conglomerate; D = debt; and E = equity.

. Once again, firm 1 represents the controlling entity, and it dominates the rest of the firms in terms of market value.

Table 8. Nine-firm conglomerate: [α] and [β] matrices.

[α]
||	1	2	3	4	5	6	7	8	9	Total
1	0.0%	30.0%	30.0%	43.0%	44.0%	40.0%	15.0%	20.0%	30.0%	2.52
2	0.0%	0.0%	50.0%	20.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.70
3	0.0%	18.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	36.0%	0.54
4	0.0%	0.0%	0.0%	0.0%	7.0%	0.0%	0.0%	0.0%	0.0%	0.07
5	0.0%	0.0%	0.0%	30.0%	0.0%	10.0%	82.0%	0.0%	0.0%	1.22
6	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	66.0%	16.0%	0.82
7	0.0%	0.0%	0.0%	0.0%	0.0%	27.0%	0.0%	10.0%	0.0%	0.37
8	0.0%	0.0%	0.0%	0.0%	47.0%	20.0%	0.0%	0.0%	0.0%	0.67
9	0.0%	48.0%	12.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.60
Total	0.00	0.96	0.92	0.93	0.98	0.97	0.97	0.96	0.82
[β]
||	1	2	3	4	5	6	7	8	9	Total
1	0.0%	30.0%	24.0%	31.0%	32.0%	38.0%	20.0%	20.0%	10.0%	2.05
2	0.0%	0.0%	0.0%	8.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.08
3	0.0%	0.0%	0.0%	9.0%	0.0%	0.0%	0.0%	15.0%	0.0%	0.24
4	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	25.0%	0.0%	0.0%	0.25
5	0.0%	0.0%	0.0%	0.0%	0.0%	15.0%	42.0%	0.0%	0.0%	0.57
6	0.0%	0.0%	3.0%	0.0%	0.0%	0.0%	10.0%	0.0%	0.0%	0.13
7	0.0%	0.0%	4.0%	0.0%	0.0%	0.0%	0.0%	0.0%	6.0%	0.10
8	0.0%	8.0%	0.0%	0.0%	2.0%	0.0%	0.0%	0.0%	0.0%	0.10
9	0.0%	5.0%	0.0%	0.0%	0.0%	1.0%	0.0%	5.0%	0.0%	0.11
Total	0.00	0.43	0.31	0.48	0.34	0.54	0.97	0.40	0.16

Note: Non-zero entries in bold. If the sum of the entries in a column is less than one, it implies that a fraction of the equity or debt of that firm is owned by investors outside the conglomerate.

shows the corresponding [α] and [β] matrices. Unlike the previous case, the fraction of outside-the-conglomerate equity investors is very small (the average value of the sum of [α]-columns is 0.94); that is, on average, only 6% of each subsidiary is owned by outsiders. Figure 3 shows graphically the conglomerate connectivity. We also assume μ = 50%.

Table 9. Minimum shock size to bankrupt each firm in the nine-firm conglomerate.

Minimum Shock to Bankrupt	Shock to
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7	Firm 8	Firm 9
Firm 1	-	-	-	-	-	-	-	-	-
Firm 2	-	0.7	-	-	-	-	-	-	-
Firm 3	-	-	0.6	-	-	-	-	-	-
Firm 4	-	-	-	0.6	-	-	-	-	-
Firm 5	-	-	-	-	-	-	0.6	-	-
Firm 6	-	-	-	-	-	0.5	0.6	-	-
Firm 7	-	-	-	-	-	-	0.6	-	-
Firm 8	-	-	-	-	-	-	0.6	0.5	-
Firm 9	-	-	-	-	-	-	-	-	0.6

Note: The firms to which the shocks are applied are listed horizontally (top row). Each vertical column identifies the minimum shock such that it will cause the failure of the firm listed on the furthest left-side column. For example, a shock of size 0.6 applied to firm 9 will cause the collapse of this firm; a shock of size 0.6 applied to firm 7 will cause not only the failure of firm 7 but also the failures of firms 5, 6, and 8.

shows the outcomes of applying shocks of varying sizes to all firms. The fact that firm 1 (controlling entity) and firm 5 do not fail regardless of the size of the shock can easily be explained by noting that in both cases, A is less than E. In the case of firm 7, a shock of 0.6 not only knocks out this firm but also triggers the failures of firms 5, 6, and 8 (cascades), a situation which is shown more clearly in

Table 10. Type of failures after applying shocks of different sizes to firm 7.

Shock Size	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7	Firm 8	Firm 9
0	0	0	0	0	0	0	0	0	0
0.1	0	0	0	0	0	0	0	0	0
0.2	0	0	0	0	0	0	0	0	0
0.3	0	0	0	0	0	0	0	0	0
0.4	0	0	0	0	0	0	0	0	0
0.5	0	0	0	0	0	0	0	0	0
0.6	0	0	0	0	2	2	1	2	0
0.7	0	0	0	0	2	2	1	2	0
0.8	0	0	0	0	2	2	1	2	0
0.9	0	0	0	0	2	2	1	2	0
1	0	0	0	0	2	2	1	2	0

Note: 1 refers to a primitive failure; 2 refers to a failure due to a cascade effect.

. Also note that firm 5, despite the fact that it has the lowest leverage ratio among all the subsidiaries (1.7), and it did not receive the shock directly, failed. Again, as in the first example, a reminder that single metrics help little to predict resilience.

Figure 4, analogous to Figure 2, depicts the evolution of equity values as a function of shock size. Once firm 7 has collapsed, increasing δ has no further effects.

Finally,

Table 11. Minimum shock size to bankrupt each firm in the nine-firm conglomerate based on different values of μ (recovery value of defaulted debt).

µ = 0.75
Minimum Shock to Bankrupt	Shock to
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7	Firm 8	Firm 9
Firm 1	-	-	-	-	-	-	-	-	-
Firm 2	-	0.7	-	-	-	-	-	-	-
Firm 3	-	-	0.6	-	-	-	-	-	-
Firm 4	-	-	-	0.6	-	-	-	-	-
Firm 5	-	-	-	-	-	-	-	-	-
Firm 6	-	-	-	-	-	0.5	-	-	-
Firm 7	-	-	-	-	-	-	0.6	-	-
Firm 8	-	-	-	-	-	-	-	0.5	-
Firm 9	-	-	-	-	-	-	-	-	0.6
µ = 0.5
Minimum Shock to Bankrupt	Shock to
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7	Firm 8	Firm 9
Firm 1	-	-	-	-	-	-	-	-	-
Firm 2	-	0.7	-	-	-	-	-	-	-
Firm 3	-	-	0.6	-	-	-	-	-	-
Firm 4	-	-	-	0.6	-	-	-	-	-
Firm 5	-	-	-	-	-	-	0.6	-	-
Firm 6	-	-	-	-	-	0.5	0.6	-	-
Firm 7	-	-	-	-	-	-	0.6	-	-
Firm 8	-	-	-	-	-	-	0.6	0.5	-
Firm 9	-	-	-	-	-	-	-	-	0.6
µ = 0.25
Minimum Shock to Bankrupt	Shock to
	Firm 1	Firm 2	Firm 3	Firm 4	Firm 5	Firm 6	Firm 7	Firm 8	Firm 9
Firm 1	-	-	-	-	-	-	-	-	-
Firm 2	-	0.7	-	-	-	-	-	-	-
Firm 3	-	-	0.6	-	-	-	-	-	-
Firm 4	-	-	-	0.6	-	-	-	-	-
Firm 5	-	-	-	-	-	-	0.6	-	-
Firm 6	-	-	-	-	-	0.5	0.6	-	-
Firm 7	-	-	-	-	-	-	0.6	-	-
Firm 8	-	-	-	-	-	-	0.6	0.5	-
Firm 9	-	-	-	-	-	-	-	-	0.6

shows the impact of assuming different values for the defaulted debt. (Note that the base case, μ = 50%, which is shown in the center panel, coincides with the information shown in Table 9.) In contrast with the previous example, lowering the recovery rate does not change the patterns detected in the baseline scenario (μ = 50%), whereas increasing the value of μ to 75% completely eliminates the cascades. This shows that the sensitivity of the network response to changes in the value of μ is very network-specific. In fact, unlike the previous case, the controlling firm never fails, regardless of the value of μ.

5.3. A Useful Protocol and Metrics

Notwithstanding the usefulness of the analyses shown above, as mentioned previously, a perennial challenge when studying financial networks is the lack of a common standard to compare and/or describe their resilience and attributes.

For instance, the most basic (naïve) metric refers to the percentage of non-zero off-diagonal entries in [α] and [β]. These values are, respectively, 31% and 21% in the first example and 33% and 32% in the second example. Unfortunately, not much can be said based on this coarse indicator of connectivity. On the other hand, investigating whether a network is irreducible or not (in the sense that every node can be reached from every other node) or specifying centrality measures based on the Bonacich condition and the eigenvalues of the adjacent matrix (Bonacich 1972), is unlikely to be of any help to a risk manager, investor, or regulator. Or actually to anyone hoping to obtain any insight regarding the way a conglomerate might respond to external shocks.

With that as background, our aim here is to propose an easy-to-implement protocol that can be useful for assessing and comparing the resilience of conglomerates. In essence, we seek to define a number of figures of merit, which, taken together, can provide an intuitive—but accurate—assessment of the conglomerate’s strengths or weaknesses.

To this end, we proceeded as follows:

i.

Apply to each firm of the conglomerate (sequentially, not simultaneously) a set of shocks ranging from δ = 0 (no shock) to δ = 1. This is undertaken using a discrete set of increments (Δδ). Thus, the total number of shock scenarios is M = 1/Δδ. And since we have N firms, S = M × N is the total number of firm-level shock scenarios;

ii.

Count the number of scenarios in which

(a)

One or several firms experience failure (bankruptcy), N1;

(b)

The firm receiving the shock goes into bankruptcy, N2;

(c)

The controlling firm (1) fails, N3;

(d)

A firm that does not receive the shock directly goes into bankruptcy, N4; and

(e)

There are failures due to cascades, N5.

iii.

We define π_i (for i = 1, …, 5), as π_i = N_i/S. Clearly, the π’s can only take values between 0 and 1;

iv.

We claim that π₁, π₂, and π₃ offer a general assessment of the conglomerate resilience. In other words, lower values are associated with higher levels of resilience. On the other hand, π₄ and π₅ provide a general view of the “degree of connectivity”, namely, the ability of the network to propagate the effects of a shock, and higher values suggest more likelihood of propagating shocks.

Table 12. Value of the different metrics (π₁, …, π₅), for the two conglomerates.

Metric	Seven-Firms	Nine-Firms
π1	48.86%	36.44%
π2	46.00%	36.44%
π3	18.57%	0.00%
π4	9.14%	4.89%
π5	6.29%	4.89%

summarizes the values of these metrics for the two networks already described, based on a simulation using M = 50. The first and most important observation is that all metrics were higher in the case of the seven-firm conglomerate (Example 1). This suggests that the nine-firm conglomerate (Example 2) is more resilient. All in all, the second conglomerate was not only stronger (lower first three metrics, π₁, π₂, and π₃) but also less likely to propagate the effects of an external shock (lower last two metrics, π₄ and π₅). The most salient difference is that the controlling entity in the case of the nine-firm conglomerate always survived (π₃ = 0). Obviously, it would be naïve to conclude that the relative strength of the second conglomerate was the result of having more nodes. The difference in strength between the two networks is clearly the result of the combined interaction of both node connectivity and balance sheet structure.

Another interesting difference refers to π₄ and π₅. They were identical in the nine-firm case, whereas in the seven-firm case, π₅ < π₄. The interpretation is clear: in the case of the seven-firm example, there were certain shocks that triggered a failure of both the firm receiving the shock and the firm that did not receive the shock. Table 4 shows such an example. For instance, a 0.6-size shock applied to firm 6 triggered a primitive failure of firms 6 and 7. This explains why π₄ was higher than π₅. On the other hand, in the nine-firm conglomerate, the only way a firm not receiving a shock failed was due to a cascade. Thus, π₄ = π₅ always (in other words, a shock never triggered a primitive failure of a firm other than the firm receiving the shock).

6. Conclusions

The usefulness of the algorithm was demonstrated through two realistic examples. These examples showed how, in combination with appropriate scenario analyses, the method could identify which firms (nodes) in a conglomerate are most vulnerable to external shocks, whether these shocks might lead to the failure of the controlling entity, and the likelihood of triggering cascades within the conglomerate.

The proposed protocol, along with the associated metrics, provides an effective framework for assessing a conglomerate’s resilience. To our knowledge, this is the first effort to summarize the dynamics of a financial network with a few interpretable parameters that capture both balance sheet characteristics and connectivity matrices. This approach makes the comparison of different conglomerates’ relative strengths quite straightforward.

The algorithm, when used with the protocol, also serves as a diagnostic tool. It can help regulators and stakeholders (e.g., risk managers, investors) evaluate the advantages and disadvantages of modifying the conglomerate’s structure. For instance, if a weakness is identified, such as the potential for a subsidiary’s failure to trigger a cascade, these tools can assess whether it is better to increase the subsidiary’s equity, limit its intra-conglomerate debt exposure, or, in more serious cases, reorganize its balance sheet. The five metrics provide a clear way to evaluate potential risk-mitigation strategies.

The assumption of a fixed value for defaulted debt (μ in this paper’s notation) is significant. While it may seem appealing to calculate the “true” post-default value of debt by balancing assets and liabilities, this overlooks the legal and regulatory factors that influence debt value after a default. Using a fixed value supported by available data is more realistic. Moreover, this assumption simplifies the equilibrium equations considerably, making them linear, unlike the more complex nonlinear approach of Gourieroux et al. (2012), which complicates identifying cascades by skipping intermediate steps. Our step-by-step method explicitly reveals these intermediate stages, allowing for a clearer identification of cascades.

The examples also highlighted the importance of sensitivity analyses. Variations in key variables (e.g., the value of μ, the entries in [α] and [β], or balance sheet items) can significantly affect network dynamics.

It might be argued that the examples are “theoretical” and lack practical relevance, but such a criticism would be misplaced. These examples were intended as proof-of-concept illustrations of the algorithm’s usefulness in modeling the behavior of financial networks. Furthermore, accessing detailed inter-link data of real conglomerates is often impossible without regulatory power or direct involvement in the conglomerate. For private firms, these data are rarely available, and public companies only provide consolidated reports without specifics on interconnections. This is why researchers, including ourselves, often use hypothetical examples to showcase their methodologies (e.g., Cabrales et al. 2017; Gatzert and Schmeiser 2011; Glasserman and Young 2016; Hauton and Héam 2015).

Finally, this framework opens the door to more advanced analyses. For example, one could explore scenarios where the impact of external shocks is absorbed by both debt and equity in predetermined proportions rather than just the equity. Additionally, a time dimension could be incorporated, allowing for the shock to unfold over multiple time steps (t = 0, Δt, 2Δt, 3Δt, etc.), which would enable capturing network evolution, and the corresponding bankruptcies in a more detailed fashion. We leave these interesting topics for future study.