In contrast to a portfolio generalisation of risk, as is the case of many market price based SRIs, the spectral method determines the stability of the networked system using the power iteration fixed point algorithm for the matrix representing capital adjusted bilateral exposures, to characterize the impact of individual node failure on counterparties of counterparties, to a very high order, in the chain of indebtedness.
The main focus here is on the Core Global Banking System Network (CGBSN) (18 BIS reporting countries). Here we will show how the Spectral SRI applies to the instability dynamics that can arise from the cross border balance sheet interconnectedness, which involves only a subset of the balance sheets. We start with a stylized balance sheet of a bank headquartered in a country and has foreign assets and liabilities vis-á-vis banks headquartered in other countries. It is also shown how this relates to the BIS CBS which aggregates cross border exposures of banks headquartered in a country to give the foreign claims at the level of national banking systems.
Stylized balance sheet for a cross border bank
The balance sheet equation for a bank headquartered in country i denoted as \(b_i\) with \(A_{b_i 0}\) representing total assets, \(L_{b_i 0}\), total liabilities and, \(C_{b_i 0}\), equity capital of the bank in country i at some initial date, 0:
$$\begin{aligned} A_{b_i 0} - L_{b_i 0} = C_{b_i 0}. \end{aligned}$$
(1)
We distinguish between the assets and liabilities in the balance sheet of bank \(b_i\) that are part of the cross-border interbank system and those that are not. This is given in equation (2) with \(X_{b_j b_i}\) denoting the liabilities of bank \(b_j\) in country j to bank \(b_i\) in country i, while \(X_{b_i b_j}\) denotes liabilities of bank \(b_i\) in country i to bank \(b_j\) in country j. The terms \(A_{b_i 0}^{\#}\) and \(L_{b_i 0}^{\#}\), respectively, denote \(b_i\)’s non-interbank domestic and foreign assets and liabilities.
$$\begin{aligned} \sum _{b_j}\left( X_{b_j b_i 0} - X_{b_i b_j 0}\right) + A_{b_i 0}^{\#} - L_{b_i 0}^\# = C_{b_i 0}. \end{aligned}$$
(2)
In order to develop a cross border banking contagion model and the rates of failure of banks, which we will denote by \(u_{b_i t}\ge 0\), we track the depletion of equity capital at time \(t+1\) that arises only from cross border interbank defaults from net foreign debtors of \(b_i\). To focus on this, we assume no losses or revaluations from non-interbank assets and liabilities and also no recapitalizations for each bank \(b_i\). The second term in (3), \(\sum _{b_j}\left( X_{b_{j 0} b_{i 0}} - X_{b_{i 0} b_{j 0}} \right) ^+ u_{b_j t}^L\) yields the total potential losses given as the weighted sum of net exposures of bank \(b_i\) vis-á-vis each of its foreign counterparties headquartered in all j countries \(j \ne i\) which are pairwise net debtors of \(b_i\). Note, as losses from net debtor banks can result in negative equity in (3), it is not possible to restrict failure rates of banks, \(u_{b_i t}\), to be less than 1.
$$\begin{aligned} C_{b_{i t+1}} = C_{b_{i0}} - \sum _{b_j}\left( X_{b_{j 0} b_{i 0}} - X_{b_{i 0} b_{j 0}} \right) ^+ u_{b_j t}^L - \left( 1-\rho _i \right) \left( C_{b_{i 0}}- C_{b_{i t}} \right) . \end{aligned}$$
(3)
The third term in (3) reflects the role played by the regulatory capital requirement on solvency contagion dynamics for banks. Regulatory capital requirement in country i specifies that cumulative loss of capital at each t, should not exceed a fixed threshold, \(\rho _i\), of initial capital viz \(\left( C_{b_{i 0}}- C_{b_{i t}} \right) \le \rho _i C_{b_{i 0}}\). Likewise, incremental capital losses between t and \(t+1\) should not exceed the same percentage \(\rho _i\) of capital \(C_{b_{i t}}\), viz. \(\left( C_{b_{i t}} - C_{b_{i t+1}} \right) \le \rho _i C_{b_{i t}}\).Footnote 4 Losses exceeding this threshold signal financial distress, while losses up to the threshold can be offset by capital buffers. The term \(\left( 1-\rho _i \right) \left( C_{b_{i 0}}- C_{b_{i t}} \right) \) in (3) gives the residual cumulative losses till t that are not offset by the permitted initial capital buffer.
A network epidemic model for solvency contagion across national banking systems
We will now show how the Eq. (3) can be used to track capital depletion and so called solvency contagion at the level of national banking systems which are denoted pairwise simply by \(\left( i, j\right) \). For this note that \(C_{i0} = \sum _{b_i} C_{b_i 0}\) gives the total initial capital of country i’s cross border banks and \(X_{ji}\) is the cross border liabilities of country j banks to the banking system of country i. Likewise, \(X_{ij}\) denotes the cross border liabilities of country i banking system vis-á-vis banking system j. Thus the Core Global Banking System Network (CGBSN) (18 BIS reporting countries) is defined by a \(N\times N\) directed weighted network represent by the matrix \({\mathbf {X}}\) such that the sum of an i-th row \(\sum _{j=1}^N X_{ij}\) represents the total foreign liabilities of the banking system i to all j banking systems represented by the BIS reporting banks of each country j. The sum of the j-th column \(\sum _{i=1}^N X_{ij}\) represents the total receivable amount in the form of foreign claims of BIS reporting banks in country j. Note that \(X_{ii}=0\).
$$\begin{aligned} \ {\mathbf {X}} = \left[ \begin{array}{c c c c c c} 0 &{} X_{12} &{} \cdots &{} X_{1j} &{} \cdots &{} X_{1N} \\ X_{21} &{} 0 &{} \cdots &{} X_{2j} &{} \cdots &{} X_{2N} \\ \vdots &{} \vdots &{} 0 &{} \cdots &{} \cdots &{} \cdots \\ X_{i1} &{} \vdots &{} \cdots &{} 0 &{} \cdots &{} X_{iN} \\ \vdots &{} \vdots &{} \cdots &{} \cdots &{} 0 &{} \vdots \\ X_{N1} &{} X_{N2} &{} \cdots &{} X_{Nj} &{} \cdots &{} 0 \\ \end{array} \right] . \end{aligned}$$
(4)
To model cross border financial contagion between national banking systems we have
$$\begin{aligned} C_{i t+1} = C_{i 0} - \sum _j \left( X_{ji} - X_{ij} \right) ^+ u_{jt}^L - \left( 1 - \rho _i \right) \left( C_{i 0} - C_{i t} \right) . \end{aligned}$$
(5)
The cumulative rate of capital depletion that is used to assess the rate of failure of the ith national banking system at \(t+1\), denoted by \(u^{L}_{it+1}\), is given by subtracting the equivalent \(C_{it+1}\) from \(C_{i0}\) and dividing through by \(C_{i0}\) in (5). Hence,
$$\begin{aligned} u^{L}_{i t+1} = \frac{C_{i0} - C_{i t+1}}{C_{i0}} = \frac{\sum _j\left( X_{ji} - X_{ij}\right) ^{+}}{C_{i0}} u^{L}_{jt} + \left( 1 - \rho _i \right) u^{L}_{it}. \end{aligned}$$
(6)
This yields the form of the dynamics for the failure rates of national banking systems from a loss of capital due to exposures to counterparty national banking systems in a cross border interbank setting. Here, \(u^{L}_{i t+1}\) gives the potential failure rate of banking system i at \(t+1\) as arising from two factors given in (6).
-
(i)
The first factor in (6) for the failure of i arises from the balance sheet interconnectedness term \(\sum _j \frac{\left( X_{ji}-X_{ij}\right) ^{+}}{C_{i0}}u^{L}_{jt}\), which gives the weighted sum of heterogeneous rates of being “infected” at t from the bilateral net exposures of i to those js which are net debtors of i, adjusted for i’s capital. The weights \(u^{L}_{jt}\) are the individual rates of failure of counterparties j, discussed below.
-
(ii)
Banking systems can be vulnerable due to factors relating to themselves, viz. primarily, the size of initial capital. The term \(u^{L}_{it}\) is the percentage cumulative loss of capital suffered at t by i and is defined as \(u^{L}_{it} = \left( 1- \frac{C_{it}}{C_{i0}} \right) ,~t>0\), where \(C_{it}\) is the remaining capital at t and \(C_{i0}\) represents initial capital. In the epidemiology studies, \(\rho _i\) is called the “cure rate” and hence \(\left( 1- \rho _i\right) \) is the rate of not surviving due to the insufficiency of the cure (see Chakrabarti et al. 2008). The factor \(\left( 1 - \rho _i \right) u^{L}_{it}\) defines the extent to which i cannot mitigate its failure using a percentage \(\rho _i\) of its capital as buffer against losses.
-
(iii)
As already noted, \(u_{it}^L\) cannot be constrained to be less than 1 as bank losses can exceed 100% of initial capital and \(C_{it}\) can be negative at \(t >0\). Hence, \(u_{it}\) cannot be modeled as probability of failure. Further, it is known (Bardoscia et al. 2017) that \(u_{it}^L\) in the first order dynamic model in (6) apply proportionately rather than follow a \(\left( 0,1 \right) \) indicator function as is the case in Furfine (2003) type contagion stress tests when counterparty losses are propagated only when the counterparty has been deemed to have failed and \(u_{it}^L = 1\).
-
(iv)
Finally, as will be shown, it is important that when only losses arising from a subset of the balance sheet is being modelled, a pro rata adjustment to capital should be made to represent this subset of the balance sheet. Or else, the contagion effects will be severely underestimated.
The capital adjustment of the net liabilities of banking system j’s counterparties is expressed as \(\theta _{ij} = \frac{(X_{ij}-X_{ji})^{+}}{C_{j0}}\), where the numerator takes only positive net liabilities from i to j and this is equal to zero if \((X_{ij}-X_{ji})<0\). The denominator \(C_{j0}\) is the initial capital of country j’s BIS reporting banks adjusted on a pro rata basis for foreign exposures.Footnote 5 This matrix \(\Theta \) will also be referred to as the stability matrix:
$$\begin{aligned} \ \Theta = \left[ \begin{array}{c c c c c c} 0 &{} \frac{(X_{12}-X_{21})^{+}}{C_{20}} &{} \cdots &{} 0 &{} \cdots &{} \frac{(X_{1N}-X_{N1})^{+}}{C_{N0}} \\ 0 &{} 0 &{} \cdots &{} \frac{(X_{2j}-X_{j2})^{+}}{C_{j0}} &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} 0 &{} \cdots &{} \cdots &{} \cdots \\ \frac{(X_{i1}-X_{1i})^{+}}{C_{10}} &{} \vdots &{} \cdots &{} 0 &{} \cdots &{} \frac{(X_{iN}-X_{Ni})^{+}}{C_{N0}} \\ \vdots &{} \vdots &{} \cdots &{} \cdots &{} 0 &{} \vdots \\ 0 &{} \frac{(X_{N2}-X_{2N})^{+}}{C_{20}} &{} \cdots &{} \frac{(X_{Nj}-X_{jN})^{+}}{C_{j0}} &{} \cdots &{} 0 \\ \end{array} \right] . \end{aligned}$$
(7)
Eigen-pair method for stability of cross border exposures of national banking systems
The capital loss thresholds \(\rho _i\) are important in the determination of the tipping point for the cross border banking system as a whole. We will consider two cases for this. In the homogeneous case, this threshold is assumed to be the same for both banks and the national banking systems, denoted by \(\rho \) (without a subscript). A loss of capital in excess of 30% is a good benchmark for impending crisis. Alternatively, we will apply the equivalent in terms of absolute capital losses of the Basel II or Basel III criteria of capital adequacy for individual banks of a national banking system. The determination of the \(\rho _i\) in the heterogeneous case, the absolute capital loss thresholds for the national banking systems that are equivalent to the regulatory capital requirements for individual banks, is based on the empirical data as will be shown in Sect. 5.2.
In the matrix notation, the above dynamics in (6) in the general case of heterogeneous capital loss thresholds is given by:
$$\begin{aligned} {\mathbf {U}}^{L}_{t+1} = \left[ \varvec{\Theta }' + \varvec{\Gamma } \right] {\mathbf {U}}^{L}_{t} = {\mathbf {Q}}' {\mathbf {U}}^{L}_{t}. \end{aligned}$$
(8)
Here \({\mathbf {U}}^{L}_{t}\) is the \(1 \times N\) column vector of elements \(\left( u^{L}_{1t}, u^{L}_{2t}, \ldots , u^{L}_{Nt} \right) \) defined in (6) for the potential failure rates for each of the banking systems; \(\varvec{\Theta }'\) is the transpose of the \(N\times N\) non-negative stability matrix \(\varvec{\Theta }\) given in (7) with each element \(\Theta _{ji}=\frac{\left( X_{ji} - X_{ij}\right) ^{+}}{C_{i0}}\) and \(\varvec{\Gamma }\) is the \(N\times N\) identity matrix with the diagonal elements being 1 minus the country specific absolute capital thresholds, \(\rho _i\). Note, matrix \({\mathbf {Q}}\) or \(\mathbf {Q'}\) will be referred to as the generalized stability matrix.
Result 1: System stability with heterogeneous loss threshold \(\rho _i\)
The system stability for Eq. (8) is evaluated on the basis of the power iteration of the matrix \(\mathbf {Q'} = \left[ \varvec{\Theta }' + \varvec{\Gamma } \right] \), which yields:
$$\begin{aligned} {\mathbf {U}}^{L}_{t+1} = \left[ \varvec{\Theta }' + \varvec{\Gamma } \right] ^t {\mathbf {U}}^{L}_{1} = \mathbf {Q'}^t {\mathbf {U}}^{L}_{1} \cong g_1 \lambda _{max}^t\left( \mathbf {Q'} \right) {\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) . \end{aligned}$$
(9)
Following the Perron–Frobenius theorem, a non-negative real square matrix \({\mathbf {Q}}\) yields non-negative real values for eigenvalues and corresponding eigenvectors with the largest eigenvalue denoted as \(\lambda _{max}\left( {\mathbf {Q}} \right) \) being unique and greater than the next largest eigenvalue. Expressing the eigenvalue equation in matrix notation, the matrix \(\mathbf {Q'}\) to the power t in (9) takes the form: \(\mathbf {Q' V} = \mathbf {D V} \rightarrow \mathbf {Q'}^t {\mathbf {V}}^L = {\mathbf {D}}^t {\mathbf {V}}^L\). Here \({\mathbf {D}}\) is the \(N\times N\) identity matrix with eigenvalues of \({\mathbf {Q}}\) arranged from the highest to the lowest along the diagonal and \({\mathbf {V}}^L = \left[ {\mathbf {v}}^L_1 {\mathbf {v}}^L_2 \dots {\mathbf {v}}^L_N\right] \) is the \(N\times N\) matrix of the corresponding left eigenvectors of \({\mathbf {Q}}\). Note the latter are the right eigenvectors of \(\mathbf {Q'}\). As the set of eigenvectors of \({\mathbf {V}}^L\) can span the N dimensional vector space of real numbers \({\mathbf {R}}^N\), any non-zero, non-negative vector \({\mathbf {U}}_1\) can be expressed as a linear combination of the vectors in \({\mathbf {V}}^L\). Hence, we have \(\mathbf {Q'}^t {\mathbf {U}}^{L}_{1} = \lambda ^{t}_{1} \left[ g_1 {\mathbf {v}}^{L}_{1}+ g_2 \left( \frac{\lambda _2}{\lambda _1} \right) ^t {\mathbf {v}}^{L}_2 + \ldots + g_N \left( \frac{\lambda _N}{\lambda _1} \right) ^t {\mathbf {v}}^{L}_N \right] \). Note, \(g_1, g_2, \dots , g_n\) are scalars. For some \(t>0\), a speedy convergence to the term with the maximum eigenvalue in (9) is assured as \(\lambda _1> \lambda _2\). Finally, note that \({\mathbf {v}}^{L\#}\) in (9) is the dominant left eigenvector associated with the maximum eigenvalue \(\lambda _{max}\left( {\mathbf {Q}}\right) \) which is normalized by the 1-norm. This is discussed in Corollary 2b below.
Corollary 1a: Tipping point condition and R number
The stability of the network system in (9) involving the non-negative matrix \({\mathbf {Q}}\) requires that the following tipping point condition is fulfilled:
$$\begin{aligned} \lambda _{max}\left( {\mathbf {Q}}\right) < 1. \end{aligned}$$
(10)
This is analogous to the R number in epidemiology. The key is to see that only \(\lambda _{max}\left( {\mathbf {Q}}\right) \) will determine the tipping point for the system in (9) and drives the long term dynamics of the epidemic in terms of containment (\(\lambda _{max}\left( {\mathbf {Q}}\right) <1\)) or explosive growth (\(\lambda _{max}\left( {\mathbf {Q}}\right) >1\)).Footnote 6 The latter case implies that in Eq. (9) as time progresses, without external interventionsFootnote 7, the system cannot be made stable and there will be full system failure from contagion from any arbitrary size external initial shock with any non-zero \({\mathbf {U}}_1\ge 0\). A further property of \(\lambda _{max}\left( {\mathbf {Q}}\right) \) is that its upper bound is given by the infinity norm \(|| {\mathbf {Q}} ||_{\infty }\)Footnote 8. This implies that the instability of the system can be exacerbated by a single participant who is typically highly leveraged with its largest net debtor status given by the sum of its positions vis-á-vis counterparties in matrix \({\mathbf {Q}}\).
Corollary 1b: Speed of system failure
Temporally, the time taken for N banking systems to lose all of their capital, \(u_{it} =1\) implies \(|| {\mathbf {U}}_t ||_1 = \sum _i u_{it}\) scales with N and \(\lambda _{max}\left( {\mathbf {Q}}\right) >1\) . The exogenous arbitrary size shock at time 1 depletes the initial capital of one or more participants and renders the vector \({\mathbf {U}}_1\ge 0\) such that at least one element of \({\mathbf {U}}_1\) is strictly positive. The constant \(g_1\) in (9), \(g_1 = \frac{|| \mathbf {Q'} {\mathbf {U}}_1||_1}{\lambda _{max}\left( {\mathbf {Q}}\right) } \), is a function of the initial vector of losses as a percentage of capitalFootnote 9. Taking logs, the time taken till all N banking systems have lost 100% of capital denoted as \(t^F\):
$$\begin{aligned} ln \left( N \right) = ln \left( g_1\right) + t~ ln \left( \lambda _{max}\left( {\mathbf {Q}}\right) \right) \rightarrow t^F = \frac{ln \left( N\right) - ln \left( g_1\right) }{ln \left( \lambda _{max}\left( {\mathbf {Q}}\right) \right) }. \end{aligned}$$
(11)
Clearly, smaller \(g_1\) and \(0< g_1 < 1\) , \(ln \left( g_1\right) <0\) and hence time to full system failure, \(t^F\), increases, while higher \(\lambda _{max}\left( {\mathbf {Q}}\right) >0\) and higher \(g_1\), the sooner is the point of full system failure. We hold the view that tipping condition in (10) gives a good ex ante indication whether or not system failure can occur. For \(\lambda _{max}\left( {\mathbf {Q}}\right) >1\), systemic failure is not a matter of if but a case of when, on the basis of bilateral data of extant contractual obligations and information in matrix \({\mathbf {Q}}\). Importantly, it is invariant to the size of the initial shocks, which may be difficult to quantify. Hence, like the epidemic R number, we recommend that regulators as a matter of routine monitor the status of \(\lambda _{max}\left( {\mathbf {Q}}\right) \) for purposes of systemic risk management.
Result 2: The eigen-pair indexes for systemic risk, systemic importance and vulnerability of banking systems
The eigen-pair result, defined as \(\left( \lambda _{max} \left( {\mathbf {Q}}\right) ;{\mathbf {v}}^{L\#} \left( {\mathbf {Q}}\right) , {\mathbf {v}}^{R\#} \left( {\mathbf {Q}}\right) \right) \), yields the systemic risk index in terms of the maximum eigenvalue \(\lambda _{max} \left( {\mathbf {Q}}\right) \) of the matrix \({\mathbf {Q}}\) with \({\mathbf {v}}^{L\#} \left( {\mathbf {Q}}\right) \) being the 1-norm dominant left eigenvector associated with the vulnerability indexes for the banking systems and \({\mathbf {v}}^{R\#} \left( {\mathbf {Q}}\right) \) is the 1-norm dominant right eigenvector for matrix \({\mathbf {Q}}\), and gives the network centrality of the systemically important banking systems.
Corollary 2a: Unique steady state fixed point result incorporated in temporal dynamics for state variables \({\mathbf {U}}_{t+1}^L\)
Inherent to the dynamics of the financial contagion in (9) is the near term time independent fixed point solution:
$$\begin{aligned} {\mathbf {U}}^{L\#} = \lambda _{max} \left( {\mathbf {Q}} \right) {\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) = \mathbf {Q'} {\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) . \end{aligned}$$
(12)
Key to the eigen-pair result is the von Mises and Pollaczek-Geiringer (1929) power iteration algorithm where for any real initial non-zero vector \({\mathbf {U}}^{L}_{1}\ge 0\) as in (9), the \(t^{th}\) power of the matrix, denoted as \(\mathbf {Q'}^{t}\), can be solved iteratively using an appropriate norm. As we will see, for the economic interpretation of the \(\lambda _{max}\) as a systemic risk index, it is important to use the 1-norm \(||~.~||_1\)Footnote 10 to normalize the vector as in the following power iteration algorithm:
$$\begin{aligned} {\mathbf {U}}^{L}_{t+1} = \frac{\mathbf {Q'} {\mathbf {U}}^{L}_{t}}{|| \mathbf {Q'} {\mathbf {U}}^{L}_{t} ||_1} = \frac{\mathbf {Q'}^{t} {\mathbf {U}}^{L}_{1}}{|| \mathbf {Q'}^{t} {\mathbf {U}}^{L}_{1} ||_1}. \end{aligned}$$
(13)
The iteration is said to have converged at some t when \({\mathbf {U}}^{L}_{t+1} = {\mathbf {U}}^{L}_{t} = {\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) \) with an epsilon margin of error. The vector \({\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) \) is the left eigenvector of the matrix \({\mathbf {Q}}\) and \(|| {\mathbf {Q}}' {\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) ||_1 = \lambda _{max} \left( {\mathbf {Q}} \right) \).
At the point of convergence, multiplying through by \(\lambda _{max} \left( {\mathbf {Q}} \right) \) in the power iteration Eq. (13), we have the eigenvalue Eq. (14) which encapsulates the important fixed point solution of the dynamical system in (12):
$$\begin{aligned} \mathbf {Q'} {\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) = \lambda _{max} \left( {\mathbf {Q}} \right) {\mathbf {v}}^{L\#}\left( {\mathbf {Q}} \right) \equiv {\mathbf {U}}^{L\#}. \end{aligned}$$
(14)
Corollary 2b: 1-Norm left eigenvector \({\mathbf {v}}^{L\#}\) as vulnerability index for banking systems
From (14), noting that \(|| {\mathbf {v}}^{L\#} ||_1 = \sum _{i=1}^N v_i^{L\#} = 1\) , \({\mathbf {v}}^{L\#}\) is the 1-norm dominant left eigenvector of \({\mathbf {Q}} \) such that elements \(v_i^{L\#}\) satisfy \(0\le v_i^{L\#} \le 1\). Thus \(v_i^{L\#}\) can be interpreted to be the probability of failure of the ith banking system or its share of expected systemic losses quantified by \(\lambda _{max}\) as a percentage of total capital. Hence, \(v_i^{L\#}\) are characterized as the vulnerability index for the ith banking system.
Corollary 2c: The systemic index \(\lambda _{max}\) measure of maximum expected total (%) loss of capital
Equation (14) and Corollary 2b directly yield the result that \(\lambda _{max} \left( {\mathbf {Q}} \right) \) is the maximum total expected losses as a percentage of capital for the system as whole at each t adjusted for the permissible capital buffers:
$$\begin{aligned} \lambda _{max} \left( {\mathbf {Q}} \right) = || \mathbf {Q'} {\mathbf {v}}^{L\#} \left( {\mathbf {Q}} \right) ||_1 = \sum _{i=1}^N \left[ \sum _{j\ne i} \frac{\left( X_{ji}-X_{ij}\right) ^+}{C_{i0}} v_j^{L\#} + \left( 1 - \rho _i \right) v_i^{L\#} \right] . \end{aligned}$$
(15)
Noting \( \mathbf {Q'} = \varvec{\Theta }' + \varvec{\Gamma }\) from (8), the first term in the square bracket is the maximum expected losses from net exposures to counterparties as a percentage of capital for the ith banking system,while the second term adjusts for the expected percentage of losses not offset by the capital buffer threshold \(\rho _i\). Here we assume that there is a 100% loss of net exposures to counterparties given their probability of failure, \(v_j^{L\#}\). Summing over all i, equation (15) justifies the moniker of \(\lambda _{max} \left( {\mathbf {Q}} \right) \) as the Systemic Risk Index being a system wide measure common to all participants.
There are two further points to mention. From (15), \(\lambda _{max} \left( {\mathbf {Q}} \right) = || \varvec{\Theta }' {\mathbf {v}}^{L\#}||_1 + || {\mathbf {I}} {\mathbf {v}}^{L\#}||_1 - ||\varvec{\Phi } {\mathbf {v}}^{L\#}||_1 = \lambda _{max} \left( \varvec{\Theta } \right) + 1 - ||\varvec{\Phi } {\mathbf {v}}^{L\#}||_1\). Here \(\varvec{\Phi }\) is a diagonal matrix with values \(\rho _i\). Hence, the 1-norm of \(|| \varvec{\Gamma } {\mathbf {v}}^{L\#} ||_1 = || {\mathbf {I}} {\mathbf {v}}^{L\#}||_1 - ||\varvec{\Phi } {\mathbf {v}}^{L\#}||_1 = 1 - \sum _i \rho _i v_i^{L\#}\) where the latter term is the expected value of the (%) capital buffer for the system as a whole. This implies that, in absence of capital buffer, any expected losses from counterparties given by \(|| \varvec{\Theta '} {\mathbf {v}}^{L\#} ||_1\) will lead to \(\lambda _{max}>1\). Therefore the stability condition hinges on whether or not the expected total counterparty losses \(|| \varvec{\Theta '} {\mathbf {v}}^{L\#} ||_1\) are greater or less than the expected total buffers, both as % of total capital. This leads to the interpretation that if \(\lambda _{max} - 1 > 0\), the system will lose capital at this rate at each t. If \(\lambda _{max} - 1 < 0\), then as buffers exceed losses, the latter will be reduced at that rate, yielding negative growth of losses as percentage of capital. This is analogous to the interpretation of the R number being greater or less than 1 in terms of the positive/negative growth of the epidemic in the population. To give an example, if \(\lambda _{max} \left( {\mathbf {Q}} \right) = 1.09\), then \((1.09 - 1) = 0.09\) is the growth rate of expected total losses as percentage of capital at each time t. For any t, the (%) expected losses is \(\left( 1.09^t -1 \right) \). If \(\lambda _{max} \left( {\mathbf {Q}} \right) = 0.83\) then the total expected losses will be offset/reduced by the buffers at the rate of \(( 0.83 - 1)=-0.17\) at each t.
Corollary 2d: Right eigenvector centrality and systemic importance of banking systems
Likewise, applying the power iteration algorithm (13) on \({\mathbf {Q}}\), from some initial non-zero real valued vector \({\mathbf {U}}^{R}_{1}\), yields the following power iteration equation \({\mathbf {U}}^{R}_{t+1} = \frac{{\mathbf {Q}} {\mathbf {U}}^{R}_{t}}{|| {\mathbf {Q}} {\mathbf {U}}^{R}_{t} ||_1} = \frac{{\mathbf {Q}}^{t} {\mathbf {U}}^{R}_{1}}{|| {\mathbf {Q}}^{t} {\mathbf {U}}^{R}_{1} ||_1}\), which converges to the right eigenvector \({\mathbf {v}}^{R\#}\left( {\mathbf {Q}} \right) \). The elements of latter represents the systemic importance of each banking system in terms of the percentage loss of capital that it can inflict on the rest of the system in the near term given as a product of \(\lambda _{max} \left( {\mathbf {Q}} \right) \) and the right eigen vector centrality:
$$\begin{aligned} {\mathbf {U}}^{R\#} = \lambda _{max} \left( {\mathbf {Q}} \right) {\mathbf {v}}^{R\#}\left( {\mathbf {Q}} \right) . \end{aligned}$$
(16)
From the eigenvalue equation, it can be seen in (17) that the centrality of a node (banking system) is proportional by \(\frac{1}{\lambda _{max} \left( {\mathbf {Q}} \right) }\) to the weighted sum of the centrality measures of all its neighbours, viz. as a fixed point result, a node is central not only because of the number and size of its weighted links, but also because it is connected to other highly central nodes:
$$\begin{aligned} v^{R\#}_i\left( {\mathbf {Q}} \right) = \frac{1}{\lambda _{max} \left( {\mathbf {Q}} \right) } \sum _{j=1}^{N} \Theta _{ij}v^{R\#}_{j}\left( {\mathbf {Q}} \right) . \end{aligned}$$
(17)
These eigenvector centrality measures provide valuable analytics on systemic importance and systemic vulnerability of the banking systems in that they provide a robust fixed point ex ante quantitative percentage capital loss estimation at the level of each participant banking system and also for the core global banking network as a whole using the SRI \(\lambda _{max} \left( {\mathbf {Q}} \right) \). Other studies such as Cont et al. (2012), Degryse et al. (2010) and Castrén and Rancan (2014) use ex post simulated Furfine (2003) type of stress test calculations for the loss of capital conditional on the failure of a trigger bank.
Result 3: A conservative tipping point condition with heterogeneous capital threshold
As the upper bound of \(\lambda _{max}\left( {\mathbf {Q}}\right) \) is the infinity norm \(||{\mathbf {Q}}||_{\infty }\), the more conservative tipping point condition is given by
$$\begin{aligned} \lambda _{max}\left( \varvec{\Theta }\right) < \rho ^{min}_{i}. \end{aligned}$$
(18)
The infinity norm of \(||{\mathbf {Q}}||_{\infty } = ||\varvec{\Theta }||_{\infty } + || \varvec{\Gamma } ||_{\infty } = ||\varvec{\Theta }||_{\infty } + \left( 1 - \rho ^{min}_{i} \right) \). As \(\lambda _{max}\left( \varvec{\Theta }\right) \) has upperbound of \(||\varvec{\Theta }||_{\infty }\), the stability condition in (9) can now be given as \(\lambda _{max}\left( \varvec{\Theta }\right) + \left( 1 - \rho ^{min}_{i} \right) < 1\). This gives the result in (18). The condition in (18) highlights the role of the so called weakest link identified by the banking system with the lowest absolute capital loss threshold, \(\rho ^{min}_{i}\), at which that constituent unit falls into distress.Footnote 11 Thus, in general, in (9) when many banking systems have low absolute capital loss thresholds, the instability of the system is increased.
Result 4: Global banking system stability with homogeneous loss threshold \(\rho \)
In this case, as matrix \({\mathbf {Q}}\) differs from matrix \(\varvec{\Theta }\) only by a constant multiple of the identity matrix, the so called shift eigenvector theorem gives the result that
$$\begin{aligned} \lambda _{max}\left( {\mathbf {Q}}\right) = \lambda _{max}\left( \varvec{\Theta }\right) + \left( 1 - \rho \right) . \end{aligned}$$
(19)
The eigenvector centrality of \({\mathbf {Q}}\) (\(\mathbf {Q'}\)) in this case is given by the right (left) dominant eigenvector of the matrix \(\varvec{\Theta }\) (see also Newman 2010, p.663). As discussed in Markose (2012), the stability condition that \(\lambda _{max}\left( {\mathbf {Q}}\right) < 1\) in (19), for the case of a homogeneous loss threshold the following tipping point with \(\lambda _{max}\left( \varvec{\Theta }\right) \) as the Spectral SRI is given by
$$\begin{aligned} \lambda _{max}\left( \varvec{\Theta }\right) < \rho . \end{aligned}$$
(20)
Note the condition for system stability in Eq. (20) implies that the expected percentage loss of capital of the system as a whole cannot exceed the homogeneous regulatory capital threshold \(\rho \).
Operationalizing spectral SRI for BIS consolidated banking statistics (CBS)
We will now use this framework to fit the cross border banking system data given in BIS Consolidated Banking Statistics (CBS). This is done in terms of the new BIS CBS based on the banking sector breakdown.Footnote 12 As the CBS are reported to the BIS at an aggregate (banking system) level rather than individual bank level, the netting is done at the level of the banking systems.
The reporting banks in country i with foreign claims on the banking sector of each country j in the BIS CBS are denote by \(b_i^R\). Those with no foreign claims in a time period are the non-reporting banks, \(b_i^{NR}\). Note that some of these can have liabilities to reporting banks in another country. Thus, banking system of country i in BIS CBS can be defined as \(B_{i} = \sum b_i^R + \sum b_i^{NR}\). In order to show how there is an analogous derivation as in (6), consider the stylized bank balance sheet of a BIS reporting bank, \(b_i^R\), in country i:
$$\begin{aligned} \sum _j X_{B_j b_i^R}^F + A_{b_i^R}^{\#DF} - \sum _j X_{b_i^R B_{j}}^F - L_{b_i^R}^{\#DF} = C_{b_i^R}. \end{aligned}$$
(21)
\(X_{B_j b_i^R}^F\): Assets/claims of reporting bank \(b_i^R\) of country i on the banking system \(B_{j}\) of country j, with F denoting the foreign status.
\(X_{b_i^R B_{j}}^F\): Liabilities of bank \(b_i^R\) of country i owed to the banking system of country j. The latter, by definition, can only be the BIS reporting banks of country j.
\(A_{b_i}^{\#DF}\): Domestic interbank and non-interbank domestic and foreign assets of bank \(b_i^R\) of country i.
\(L_{b_i^R}^{\#DF}\): Domestic interbank and non-interbank domestic and foreign liabilities of bank \(b_i^R\) of country i.
\(C_{b_i^R}\): Capital of the cross border reporting bank \(b_i^R\) in country i.
As in (2), aggregating over all reporting banks in a BIS country i and netting bilaterally with their liabilities to country j’s reporting banks, we obtain \(\sum _{b^R} C_{b_i^R}\) as in (22), which is the total capital of the BIS reporting banks in country i:
$$\begin{aligned} \sum _{b^R}\sum _{j\ne i} \left( X_{B_j b_i^R}^F - X_{b_i^R B_{j}}^F \right) + \sum _{b^R} A_{b_i^R}^{\#DF} - \sum _{b^R} L_{b_i^R}^{\#DF} = \sum _{b^R} C_{b_i^R}. \end{aligned}$$
(22)
Here, in the first term in brackets we have \(\sum _{b^R} X_{B_j b_i^R}^F\), which corresponds to the BIS CBS definition of foreign claims of the reporting banks of country \(b_i^R\) on the banking sector \(B_j\) of country j. This corresponds to \(X_{ji}\) in (6). However, the cross-border liability side of reporting banks in country i, \(\sum _{b^R} X_{b_i^R B_j^R}^F\), is not available. We proxy this by the sum of the foreign claims of banking system j represented by the BIS reporting banks in that country on the banking system of country i, i.e. \(\sum _{b^R} X_{B_{_i} b_j^R}^F \equiv X^F_{B_i B_j}\) noting that \(B_j\) by definition includes only the BIS reporting banks in country j.Footnote 13 Thus, the capital of the reporting banks of country i given in (22), denoted simply as \(C_i\), will be an overestimation of what is needed in terms of their net exposure to the another country j’s banking system, which is a net debtor vis-á-vis i. With this proviso, this yields a failure model analogous to (6) for the BIS CBS reporting country banking systems. Finally, the above discussion implies a number of adjustments given in the Online Appendix A.