Eigen-Pair Method for systemic risk analysis

The eigenvector centrality measure (Newman, 2010) of the matrix Θ is found by Markose (2012) and Markose et al. (2012) to be the best correlated with the contagion losses created by the Furfine (2003) type of stress test47. It was found that the higher the

47 _{Furfine (2003) is a popular methodology of assessing contagion (used for instance in seminal papers by}

Upper and Worms (2004), Degryse et al. (2010)). It is based on assuming that a country (originally a financial institution) defaults and is unable to pay its cross-border liabilities imposing the loss on a

centrality measure of a node (an agent – for example a financial institution), the larger the losses caused by the triggering economic agent on others during the contagion. This finding is in line with the nature of eigenvector centrality, which is based on the principle that the centrality score of a node is higher if the neighbouring nodes are high-scoring nodes themselves, which translates, in case of financial contagion, into the following: the higher the systemic threat posed by the neighbours of a node, the higher is its centrality score (and its systemic importance).

The matrix Θ has two sets of eigenvectors corresponding to the highest eigenvalue: right (𝑣𝑅) and left (𝑣𝐿). Since the ith node’s centrality is proportional to the centrality measure

of all its neighbours, then denoting the right eigenvector centrality of ith node as the ith

element of the right eigenvector (𝑣_𝑖𝑅), we have:

𝑣_𝑖𝑅 = 𝜆𝑚𝑎𝑥−1∑𝑁𝑗=1𝜃𝑖𝑗𝑣𝑗𝑅, (4.3) where 𝜆_𝑚𝑎𝑥 is the largest eigenvalue and 𝑣𝑅, the corresponding eigenvector as the eigenvector centrality is proportional to the leading eigenvector of the adjacency matrix (see section 7.2 of Newman, 2010). Translating (4.3) into matrix form we obtain:

Θ𝑣𝑅 = 𝜆_𝑚𝑎𝑥𝑣𝑅_{. (4.4)}

Similarly for the 𝜆𝑚𝑎𝑥 there exists a corresponding eigenvector 𝑣𝐿, which by the definition of eigenvector is a transpose of the right eigenvector of the transposed matrix

Θ𝑇, viz.:

𝑣𝐿Θ = Θ𝑇𝑣𝐿 = 𝜆𝑚𝑎𝑥𝑣𝐿. (4.5)

counterparty, if the capital cushion of the counterparty is not enough to cover the losses, the counterparty becomes insolvent and its default triggers the second wave of contagion.

The matrix Θ is non-negative and has real entries, hence 𝜆_𝑚𝑎𝑥 is a real positive number. The Perron-Frobenious theorem guarantees a positive eigenvectors 𝑣𝑅 and 𝑣𝐿 under the assumption that Θ represents the irreducible network48_.

Taking the systemic risk perspective again, the right eigenvector centrality measure of a matrix Θ will be taken as a measure of the systemic importance of a respective country and will be referred to as the Systemic Importance Index. Moreover, the left eigenvector centrality is expected to be correlated with the vulnerability of a respective banking system to the contagion losses across all the triggering scenario of a Furfine (2003) algorithm and will be referred to as the Systemic Vulnerability Index.

Failure of a national banking system to its cross-border exposures is usually determined by the criteria that losses exceed a predetermined buffer ratio, ρ, of equity capital. In the epidemiology literature, Chakrabarti et al. (2008), ρ is the common cure rate and (1 - ρ) is the rate of not surviving in the worst case scenario.

The dynamics of the contagion and the rates of failure of country i’s banking system

from default of debtor countries can be given as:

𝑢_𝑖𝑞+1 = (1 − 𝜌)𝑢_𝑖𝑞+ ∑ (𝑥𝑗𝑖−𝑥𝑖𝑗)+

𝐶𝑖0

𝑗 𝑢𝑗𝑞1 , (4.6)

where 𝑢𝑖𝑞, gives probability of banking system i being “infected” at the q-th iteration and 𝑢_𝑖𝑞1 represent the banking system that fail at the q-th iteration and infect all non-failed counterparties with probability 1. The initial probability of failure is assumed to be

ui0=1/ci0, therefore the probability is determined by the rate at which the banking system of country i is depleted by losses from failed countries. In the matrix form the above dynamics is given by:

𝑈_𝑞+1= [Θ′_{+ (1 − 𝜌)𝐼]𝑈}

𝑞, (4.7)

48 _{For any randomly selected pairs of nodes (i,j) in an irreducible network, there is a path between them,}

where Θ’ is the transpose of matrix Θ with each element θ’ij= θ’ji and I is identity

matrix. The system stability of equation (4.7) is evaluated on the basis of the power iteration of the initial matrix:

Q=[ Θ’+(1- ρ)I]. (4.8)

The Uq takes form:

𝑈_𝑞 = [Θ′_{+ (1 − 𝜌)𝐼]}𝑞_𝑈

0 = 𝑄𝑞𝑈0, (4.9)

In the framework ρ is the permissible capital loss threshold49 typically determined by the regulatory requirements for a bank and hence also for the national banking system. We equate failure of the net creditor country with failure of its banking system. In sequence the latter triggers failure of its counterparties.

Using the power iteration algorithm for equation (4.7), it can be shown that the steady state potential percentage capital loss for country i can be estimated as the product of

𝜆_𝑚𝑎𝑥(Θ′) and i’s vulnerability index whilst using the infinity norm, denoted as vi∞50.

𝑢_𝑖# = 𝜆_𝑚𝑎𝑥(Θ′)𝑣_𝑖∞. (4.10)

It was shown in Markose (2012) that the stability of the network system involving matrix Θ requires that the following stability condition is fulfilled:

𝜆_𝑚𝑎𝑥(Θ) < 𝜌, (4.11)

If this condition is violated, any negative shock, in the absence of outside interventions, can propagate through the networked system as a whole and cause system failure.

49 _{It has been found by Markose (Reserve Bank of India, 2011), that the Basel III capital ratio of 6% for risk}

weighted assets typically implies capital ratio of 25% for total assets. Thus the ρ = 0.25 can be considered a proxy for capital adequacy ratios of banking systems.q

The formula for calculating the threshold is as follows 𝑇𝐶 = 1 − (𝑇𝑅𝑊𝐴∗ 𝑅𝑊𝐴/𝑇𝑖𝑒𝑟1𝐶𝑎𝑝𝑖𝑡𝑎𝑙), where

RWA are Risk Weighted Assets, TRWA is the Basel II criteria for capital adequacy, i.e. Tier1 Capital has to be

higher than 6% of RWA. TC is the permissible loss in terms of Tier1Capital.

50 _{For this analysis, it is important to make sure that the right and left eigenvectors associated with the}

largest eigenvalue are given using the infinity norm. The infinity norm of a vector x denoted as ||x||∞ is the largest number in the vector. Hence, the highest ranked country will have an index of 1 in terms of its eigenvector centrality. There is a simple conversion from the eigenvector produced using the Euclidean norm to one using the infinity norm.