2.1 Balance sheet and payment equilibrium

An interbank system is considered here, which consists of several banks that are linked by interbank lending via unsecured debt contracts signed at the initial period. To better clarify the risk contagion process, we start with the balance sheet of a stylized commercial bank within the interbank system. As illustrated in Fig 1, bank i’s total assets A_i are composed of liquidities c_i, securities s_i, interbank loans l_i and other assets p_i; thus, the following equation holds: (1)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Balance sheet of a stylized commercial bank.

https://doi.org/10.1371/journal.pone.0200209.g001

Meanwhile, bank i’s total liabilities H_i consist of shareholder equity e_i, deposits d_i, interbank borrowing b_i and other liabilities q_i: (2)

In fact, the risk in interbank payment systems can originate from different factors; for example, [21] discussed how a shock to deposits could lead to the bank’s default on interbank borrowing or even part of its retail deposits, and [22] considered that the uncertain returns on banks’ securities and other assets were likely to cause a system risk. Unlike the abovementioned studies, this paper does not distinguish the reasons that cause the risk, but rather focuses on the bank’s default induced by any possible initial shocks. Moreover, we further assume that each bank embedded in the interbank system have borrowing or loans at least with the other one bank, namely b_i > 0 and l_i > 0, because the banks without any interbank lending and borrowing are isolated nodes and should not be considered into the interbank system.

Note that the liabilities displayed in the balance sheet can be divided into the senior type and the junior type, and therefore they should be repaid in a different order when shocks occur. Here, the deposits d_i have seniority relative to the bank’s other liabilities; in other words, the available liquidities of bank i should repay d_i first and then b_i as well as the other liabilities. Furthermore, the other liabilities generally contain long-term borrowing, bonds and sub debts that really exist as a commercial bank’s liabilities as displayed in Fig 1. Although interbank borrowings and other liabilities mentioned above both belong to the junior liabilities, interbank borrowings are always short-term and other liabilities are often long-term. Thus, we consider that interbank borrowings should be repaid firstly and immediately compared to other liabilities.

Let al_i denote the total available liquidities of bank i, and then regardless of the reasons that cause the shocks, two cases may exist: (1) if al_i < d_i, bank i is called complete default. In this case, the total available liquidities of bank i are not enough to pay its senior liabilities; (2) if d_i ≤ al_i < d_i + b_i, bank i is called part default, which means that, in this case, senior liabilities can be paid in full and the junior creditors are repaid in part. Furthermore, let x_ij (j ≠ i) denote the amount of money borrowed by bank j from bank i. When bank j is in part default, the amount of money repaid by bank j to bank i (denoted as y_ij) is in proportion to their contract x_ij. In mathematics, we have (3)

Summarizing the two cases above, y_ij can be further expressed as (4) where x_ij implies the network structure imbedded in the interbank payments, and therefore their interdependence may induce a cascade of defaults when one or more banks default. Then, Eq (4) will be useful in determining the payment equilibrium because part default actually exists when financial contagion occurs in interbank payments.

Furthermore, because rapid liquation is costly in most cases, bank i can only recover a fraction η_i < 1 of the securities s_i and other assets p_i, where the defined fraction η_i is defined as the discount factor of bank i. According to the expression of repaid money shown in Eq (4), the total available liquidities of bank i can be expressed as (5)

Here, ∑_jy_ij denotes the realised payments made by all the other banks.

To sum up, Eq (4) is a rule that a bank returns the interbank borrowings when facing default, and Eq (5) measures the total available liquidity of a bank. By considering Eqs (4) and (5) together, it seems that the fix-point method can directly be adopted by using the framework of [1], but we do not adopt it in our model setting because the fix-point method ignores the time process of risk contagion to some extent and potentially assumes that all the banks can make the optimal decisions at the same time when the default risk appears in the system. Parallel to these studies related to DebtRank [23,24], this paper also pays attention to the dynamic process of risk contagion and provides the dynamic mechanism as displayed in the next subsection. Besides, this paper defines and validates several risk distances without needing all the banks make the optimal decisions at the same time, and therefore the method of this paper is quite different with fix-point method.

2.2 Mechanism of risk contagion

In order to make clear the mechanism of risk contagion captured by this paper, Fig 2 provides a visual and simple representation, where the interbank system consists of three banks with different bank sizes and these banks are linked by their lending-borrowing relationship. Here, the process of risk contagion can be roughly divided into three successive phases.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Sketch map of risk contagion.

https://doi.org/10.1371/journal.pone.0200209.g002

As Phase 1 displays, one of the banks suffers a sufficiently large negative shock so as to completely default, and thus the other banks’ loans to this bank become unrecoverable. As a result, the unrecoverable loads cause the bank with bigger size distressed and the one with the smaller size completely default in Phase 2. Subsequently, the complete default of the smaller bank causes the remaining bank’s loan to it unrecoverable so that the remaining bank can also default at this time as shown in Phase 3. Although the real interbank system always contain more banks than our simple example, the mechanism of risk contagion is similar. Note that when part default appears, Eqs (4) and (5) are useful to determine how much repaid money can be received within the mechanism of risk contagion displayed in Fig 2.

2.3 Risk distance, risk degree and m-order risk degree

The considered interbank system consists of n banks indexed by N = {1, 2, ⋯, n}, and the lending- borrowing matrix, denoted as X, is expressed as X = [x_ij]_i,j∈N, where x_ij (j ≠ i) denote the amount of money borrowed by bank j from bank i (we pose x_ii = 0). Based on the defined matrix X, the deduced system matrix G = [g_ij]_i,j∈N is defined as below.

(6)

Here, g_ij (i ≠ j) represents the ratio of the amount of money lent from bank i to bank j to bank i’s total the total available liquidities. Then, it is not difficult to find that g_ij ≥ 0 (j ∈ N) and ∑_j∈Ng_ij = 1 so that the deduced matrix G can be understood as a Markov transfer matrix. Note that the above designed matrix does not contain y_ij expressed in Eq (4), but is dependent on x_ij, which implies that the designed matrix G does not change once the balance sheet is given. In fact, the unchanged G facilitates the calculations because we do not need to change G frequently with the change of y_ij (y_ij is endogenous variable of our model). Besides, in order to check the designed G is good or not, we test the designed G whether to satisfy much more desirable properties. In other words, if the designed G can meet much more good properties in measuring the system risk of risk contagion, it is a good design; otherwise, we should try the other forms of G. To be honest, the reported G in Eq (6) is selected from numerous possible forms and is found to meet much more desirable properties compared to the other tried forms, which will be presented in details in the latter part of this paper.

Let S denote the set of initial default banks; for example, if the initial default set only contains bank j, then S = {j}, and if the initial default set contains banks i and j, then S = {i, j}. The risk distance from a non-default bank to the initial default set S is given in Definition 1.

[Definition 1] (risk distance). The risk distance from each non-default bank to the initial default set S is expressed in the vector R_−s: (7) Where 0 < β < 1 guaranteeing that the inverse matrix exists, I is an #(N − S) × #(N − S) identity matrix, 1 is an #(N − S) vector consisting of 1, and G_−s is G deleting the rows and the columns of the banks contained in the set S. In addition, the ith element of R_−s is denoted as r_i,S that represents the risk distance from ith bank to the default bank set S.

By recalling the defined Eq (6), the normalization is a useful procedure, whose theoretical basis is Markov chain, by noting that the normalized matrix can be understood or regarded as a Markov transfer matrix. If the status of default is regarded as the absorbing state, the defined risk distance in Eq (7) can be similarly understood as a likelihood that each remaining bank does not reach the absorbing state, according to the principle of finite-state Markov chain. Thus, a longer risk distance means a lower likelihood of default, which accords with our common sense. Interestingly, this form is quite similar to the famous Katz-Bonacich centrality in the field of network science [11].

Further, taking g_ij (i ≠ j) as an example without loss of generality, the mechanism of risk contagion explained in Section 2.2 means the following statements: when bank i face the risk of complete default, bank j should first returned the money borrowed from bank i, so g_ij = 0 at this time; and then even if bank i has received all the money that was lent out to the other banks, bank i can also completely default so that it appears in the initial default set S, and at this time, g_ij = 0 because complete default means that bank i cannot return the money borrowed from bank j (∀j ≠ i). By considering the above mechanism, we should delete the rows and the columns of the banks in the initial default set S, which interprets why we adopt the G_−s to calculate the risk distance in Eq (7).

As a result, the normalization is important here for three reasons. First, the normalization guarantees that the sum of elements in each row equals 1, which accords with the definition of Markov transfer matrix, and therefore, the principle of Markov transfer matrix can be adopted directly as we have explained. Second, the normalization makes each element in the normalized matrix reflect the ratio of the corresponding liabilities, so that the bank size is potentially considered. In fact, the bank size, defined as the total amount of assets (or liabilities) of one bank, is an important factor influencing the risk contagion. To make it clear, if two banks borrow the same amount of money from another bank but the two banks have different sizes, the normalization will shows different ratios of the borrowed money in the two banks, but without the normalization, the effect of bank size cannot be reflected. Moreover, the normalization can also reflect the effect from each bank’s discount factor η_i (∀i ∈ N), which is also important in the process of risk contagion.

Subsequently, let rv(S) denote the risk degree of the whole interbank system given the initial default set S. Based on the defined risk distance (see Definition 1), rv(S) is defined as follows.

[Definition 2] (risk degree). Given the initial default set S, the risk degree rv_s is defined as (8) where 1 is an #(N − S) vector whose elements are all 1. In other words,1 ⋅ R_−s equals the sum of all of the elements in R_−s

Recalling the explanations of Definition 1, the longer risk distance means a lower likelihood of default. Here, we first sum all the distances from the remaining banks to the given initial default set, and accordingly the mathematical expression is 1 ⋅ R_−s. Note that a larger value of 1 ⋅ R_−s means a lower system risk degree, which does not accord with our common sense. Thus, we use (1 ⋅ R_−s)⁻¹ to measure the system risk degree to avoid this problem.

Furthermore, from the perspective of the system, we can compare the risk degrees relative to all possible initial default sets and determine the maximal risk degree and the corresponding default set. Considering the amount of default banks that will affect the risk degree, we keep this amount identical for fair comparison. As a result, let rv_m(G) denote the m-order risk degree of the given system G, where m is the number of banks contained in the given initial default set S. For example, when m = 2, rv₂(G) means the maximum sum of risk distance from each non-default bank to any possible default bank sets which contain two banks. Then, the rv_m(G) is defined as below, where m = 1, 2, ⋯, n.

[Definition 3] (m-order risk degree). For all initial default sets containing m banks, the maximal risk degree among all initial default sets is defined as the m-order risk degree, whose mathematical expression is (9) Where i₁, i₁, ⋯, i_m are the different banks contained in the initial default set and, therefore, the total number of their combination is n!/(m!(n − m)!).

Keeping the number of banks in initial default set identical, the defined m-order risk degree identifies not only the maximal risk degree for all possible initial default sets containing m banks but also the corresponding default set that leads to the highest system risk.

4.1 Relationship between financial contagion and risk distance

First, we explore whether the defined risk distance can reflect the risk level of banks by controlling the effect of the discount factor. To this end, this subsection sets η_i = 1 (i ∈ N) to focus on the relationship between financial contagion and risk distance. Specifically, given an initial default set S, if bank i defaults with the financial contagion, then do all the banks that are shorter to the set S than bank i? The following Property E1 gives the answer.

[Property E1] For any given initial default set S and two banks i and j (i, j ∉ S) in a given system G with η_i = 1 for any i ∈ N, if bank j completely defaults with the financial contagion and r_i,S < r_j,S for any β ∈ (0, 1), then bank i must completely default. In addition, there exists Δr, such that bank w defaults if r_w,S < Δr.

Proof. For any given initial default set S and η_i = 1 for any i ∈ N, Eq (5) and Assumption 1 guarantee that the necessary and sufficient condition of bank j completely defaulting is (15) Recalling Eqs (3) and (5), Inequality (15) equals (16)

On the other hand, r_i,S < r_j,S can be further expressed as (17) and the Taylor expansion guarantees that (18) which holds for any β ∈ (0, 1). Then, let β → 0, and we immediately obtain that (19) which equals 1_i(I − G_−S)1^T > 1_j(I − G_−S)1^T > 1 − γ. Thus, the first part of Property E2 holds.

As a deduction, let Δr = max(r_k,S | bank k defaults), and then the second part of Property E2 is obtained based on the result of the first part.

Property E1 provides the answer of the question is YES. Again, given an initial default set S, if bank i defaults with the financial contagion, all the banks shorter to the set S than bank i will default. Thus, the defined risk distance is demonstrated to be reasonable because it can fully reflect the insolvency contagion of banks when banks can liquate their assets freely. Note that the effect of the discount factor will be discussed in Subsection 4.5 to enrich our illustrations.

4.2 Risk measurements as a function of liability size

In this subsection, we first check whether the increase in all pairwise liabilities raises the system risk, and the following Lemma 1 gives the answer.

[Lemma 1] For any given financial system G and an initial default set S, let for all i ≠ j and α > 1 in the new financial system denoted as . If bank i (i ∉ S) completely defaults with the risk contagion in G, then it must completely default in .

Proof. Here, we first investigate the relationship of the available assets of bank i (i ∉ S) between G and , and its results are as follows: (20) because of Eq (5) and the fact that . Then, because α > 1 and ∑_jx_ij > ∑_jy_ij, it is not difficult to obtain that (21)

Then, according to Inequality (14), for the given default set S and bank i (i ∉ S), it holds that (22) and Inequalities (20) and (21) guarantee that (23)

Overall, Lemma 1 holds.

The proven Lemma 1 gives us a hint that increasing all pairwise liabilities in the network will raise the systemic risk, because the banks that completely default in G also completely default in . Note that, here, we use the number of banks that completely default to depict the level of system risk. Then, we further check whether the defined risk measurements can correctly reflect the change in the system risk when all pairwise liabilities rise, as discussed above. As a result, by keeping all liquidities c_i, securities s_i and other assets p_i unchanged, the following Property E2 gives the answer.

[Property E2] For any given financial system G, if for all i ≠ j and α > 1 in the new financial system denoted as , for any given initial default set S, the defined risk distance is shorter, and the defined risk degree and the m-order risk degree are larger in the new financial system.

Proof. Based on the definition of G = [g_ij]_i,j∈N in Eq (5), the relationship between and G is (24) and (25) whose elements are all more than 1. Then, (26)

Now, is considered the function of α, and we further have (27)

On one hand, all of the elements in are non-negative, which has been proven in Property B1. On the other hand, (28) and therefore, Eq (27) is equivalent to (29)

Note that and because all of the elements in are less than 0. Thus, all of the elements in are decreasing functions of α, meaning that the defined risk distance obtains its maximal value when α → 1 and obtains its minimum value when α → +∞. Furthermore, because the inequality of risk degrees holds for any initial default set S, the inequality is naturally inherited by the defined m-order risk degree. Thus, Property E2 holds, and we also obtain the boundary of the defined measurements when the liability size changes.

Overall, Property E2 reflects that the rise in pairwise liabilities will increase the systemic risk, regardless of the type of network structure. Hence, controlling the level of total credits in the system is a method of managing systemic risk. From another perspective, the proposed measurements can properly reflect the increase in systemic risk, which indicates that the definition of the proposed measurement is rational.

4.3 Risk measurements as a function of liability distribution

Intuitively, a bank with positive net interbank lending is more likely to be harmed by the risk contagion because it will get loss from its counterparties’ default. Accordingly, this subsection focuses on the effect of liability distributions and explores how to arrange the liability distribution among banks to obtain a smaller system risk degree. To this end, consider any given system G, and suppose that each bank’s liability, on one hand, is kept unchanged, (that is, keeping ∑_{i ≠ j}x_ij unchanged for any i ∈ N to avoid the effect from the amount of liability that has been proven influential), and, on the other hand, each bank’s total lending amount is kept equal to its borrowing amount (that is, ∑_ix_ij = ∑_jx_ij to ignore the effect of the difference between the loan and borrowing amounts). This subsection focuses on the effect of liability distributions and explores how to arrange the liability distribution among banks to obtain a smaller system risk degree. The following Property E3 gives the answer.

[Property E3] Given any system G with the precondition that ∑_ix_ij = ∑_jx_ij for any i, j ∈ N, let for any i, j ∈ N in the newly generated system so the operation does not change each bank’s liability. Then, if the same default set S is given for the two systems, we have and the equation holds if and only if x_ij = x_ji holds in the given system G.

Proof. Here, system changes the liability distribution of the given system G and does not change each bank’s liability, noting that the precondition guarantees the following equation: (30)

Thus, the above operation, which changes the liability distribution, does not change each bank’s liability. In addition, the above operation does not change c_i + η_i ⋅ (p_i + s_i) for any i ∈ N, meaning that . Moreover, we have (31)

Accordingly, it holds that .

Knowing rv_S(G) = (1 ⋅ R_−S(G))⁻¹, we next focus on the item 1 ⋅ R_−S(G) whose detailed form is 1 ⋅ (I − βG_−S)⁻¹ ⋅ 1^T / n. Accordingly, the problem of proving is changed into the problem of proving , where . The following part is to prove the above inequality. First, we have (32)

Then, because 0.5([I − βG_−S]⁻¹ + [I − βG_−S]^−T) and are both symmetric matrices, the theorem of Rayleigh-Ritz guarantees that (33)

Since |λ ([I − βG_−S]⁻¹[I − βG_−S]^T)| = |λ([I − βG_−S]^−T[I − βG_−S])| = 1 and ([I − βG_−S]⁻¹[I − βG_−S]^T)⁻¹ = ([I − βG_−S]^−T[I − βG_−S], let exp(iw) and exp(−iw) be the eigenvalue of [I − βG_−S]⁻¹[I − βG_−S]^T and [I − βG_−S]^−T[I − βG_−S], respectively, where w ∈ [0, 2π). As a result, it holds that (34) and if and only if w = 0, the equality holds.

To summarize, if (or ), , and if and only if (or ), . Property E3 holds.

As Property E2 has proven that liability size is an influencing factor of systemic risk, we here fix the liability size and keep each bank’s total loan and borrowing amounts unchanged in order to precisely study the influence of liability distribution on system risk. Although various liability distributions could exist in real interbank system, Property E3 uncovers that pairwise liabilities symmetrically distributed among the banks, without changing the liability size, can lead to a smaller system risk degree irrespective of the variety of liability distributions.

4.4 Risk measurements as a function of bank size

Here we define the bank size as the amount of total liquidity a bank holds. Considering the bank size as the main factor and keeping the lending-borrowing matrix X unchanged, when bank i does not belong to the initial default set S and its size becomes larger, this subsection further checks whether the defined risk degree can reflect the change of bank size. The following Property E4 provides the result.

[Property E4] For any given system G, a new system is generated by keeping the lending-borrowing matrix X unchanged and the size of bank i becomes larger, that is, and . Then, for any given default set S that does not contain bank i, it holds that and .

Proof. Based on the operation described here, the relationship between and G_−S is (35)

Then, we have and, further, , where T_i is a #(N − S) × #(N − S) diagonal matrix whose ith element is and the remaining elements are all 1. Then, can be expressed as a function of T_i as follows, where 1_i is an #(N − S) vector whose ith element is 1 and the remaining elements are 0. Then, we further have (36)

Here, (I − βG_−S) + β(T_i − I) ⋅ (I − G_−S) = (I − βG_−S)[I + β(I − βG_−S)⁻¹(T_i − I) ⋅ (I − G_−S)], and note that (37)

Thus, 1_i ⋅ [(I − βG_−S) + β(T_i − I) ⋅ (I − G_−S)]⁻¹ ⋅ 1^T > 1_i ⋅ (I − βG_−S)⁻¹ ⋅ 1^T. That is, . In addition, by replacing with 1^T in Eqs (36) and (37), we can immediately obtain that (38) which equals .

Property E4 demonstrates the intuition that a larger bank size decreases the systemic risk, which means larger size is a good buffer for insolvency contagion, and indicates that the proposed measurements can well reflect the effect of bank size on system risk.

4.5 Risk measurements as a function of the discount factor

This subsection focuses on how each bank’s discount factor η_i affects the defined risk distance and risk degree. Intuitively, an increase in η_i means a stronger ability to liquidate assets; thus, systemic risk should decrease. Here, for any given default set S, we check whether the risk distance from bank i to S increases and the corresponding risk degree rv_S decreases. The following Property E5 provides the answer.

[Property E5] For any given default set S, if there exists a bank set Ξ such that Ξ ∩ S = Φ and for any i ∈ Ξ in the new system , then the risk distance r_i,S is longer in and the risk degree is smaller compared to the original G, where i ∈ Ξ.

Proof. The relationship between and G is , and further, , where Q_i is a #(N − S) × #(N − S)diagonal matrix whose ith element is for i ∈ Ξ and the remaining elements are all 1. Then, this problem has the same structure with Property E4, and therefore, according to the conclusion provided in Property E4, Property E5 holds.

Property E5 validates the intuition that a rise in the discount factor of some banks will increase the risk distance and decrease the level of system risk. More importantly, the proposed measurements make it possible to reflect the effect of the discount factor, which implies that the effective liquidity management of every bank will benefit the safety of a financial system.