A Guide to IMF Stress Testing
Chapter

Chapter 32. Banking Stability Measures

Author(s):
Li Ong
Published Date:
December 2014
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Author(s)
Miguel A. Segoviano and Charles A. E. Goodhart This chapter is an abridged version of IMF Working Paper No. 09/04 (Segoviano and Goodhart, 2009). The authors would like to thank S. Neftci, R. Rigobon, H. Shin, D. Tsomocos, M. Sydow, J. Fell, T. Bayoumi, K. Habermeier, and A. Tieman for helpful comments and discussions; and to acknowledge inputs from participants in the seminar series/conferences at the IMF, European Central Bank, Morgan Stanley, Bank of England, Banca d’ltalia, Columbia University, Deutsche Bundesbank, Banque de France, and Riksbank.

This chapter defines a set of banking stability measures that take account of distress dependence among the banks in a system, thereby providing a set of tools to analyze stability from complementary perspectives by allowing the measurement of (1) common distress of the banks in a system; (2) distress between specific banks; and (3) distress in the system associated with a specific bank. Our approach defines the banking system as a portfolio of banks and infers the Banking System’s (Portfolio) Multivariate Density (BSMD) from which the proposed measures are estimated. The BSMD embeds the banks’ default interdependence structure that captures linear and nonlinear distress dependencies among the banks in the system and its changes at different times of the economic cycle. The BSMD is recovered using the Consistent Information Multivariate Density Optimizing approach, a new approach that in the presence of restricted data improves density specification without explicitly imposing parametric forms that, under restricted data sets, are difficult to model. Thus, the proposed measures can be constructed from a very limited set of publicly available data and can be provided for a wide range of both developing and developed economies.

Method Summary

Method Summary
OverviewBanking stability measures embed banks’ linear (correlation) and nonlinear distress dependence and their changes through the economic cycle. The approach defines the banking system as a portfolio of banks and infers its multivariate density from which the proposed measures are estimated. These can be provided for developed and developing economies and can be estimated under data-restricted environments.
ApplicationThe proposed stability measures incorporate changes in distress dependence that are consistent with the economic cycle; this is a key advantage over traditional risk models that typically incorporate only linear dependence (correlation structure) and assume it to be constant throughout the economic cycle. The framework is appropriate in situations where

1. data are limited; and

2. distress dependence among financial institutions in a financial system needs to be quantified.
Nature of approachStatistical nonparametric density estimation.
Data requirementsProbabilities of default (PDs): these can be market based (credit default swap spreads, bond spreads, structural models); they also can be estimated with supervisory information. Stock prices of financial institutions.
StrengthsThe framework allows the estimation of financial stability measures that provide complementary perspectives on risk, with very limited data requirements. These measures can be updated with daily frequency.
WeaknessesIt is necessary to have robust estimations of PDs. If PDs are not available, it is necessary to make assumptions about this variable.
ToolProprietary estimation codes are available from the authors upon request. Contact author: M. Segoviano.

Economics is a quantitative science. Macroeconomics depends on data for national income, expenditure, and output variables. Macromonetary policy requires measures of inflation. Microeconomics is based on data for prices and quantities of inputs and outputs. Even when the variables of concern are difficult to measure, such as the output gap, expectations, and “happiness,” economists use various techniques, for example, survey data, to provide quantitative proxy variables—often according these data more weight than is consistent with their inherent measurement errors. Without such quantification, comparisons over time, and on a cross-sectional basis, cannot be made; nor would it be easy to provide a quantified analysis of the determinants of such variables.

However, there is currently no such widely accepted measure, quantification, or time series for measuring either financial or banking stability. What is most often used instead is an on/off (1/0) assessment of whether a “crisis” has occurred.1 This then has been used to review whether there have been common factors preceding, possibly even causing, such crises and to assess what official responses have best mitigated such crises (see Hoelscher, 2006). Although much useful research employing a crisis on/off dichotomy has been done, this approach has several inherent deficiencies.2 In particular, the lack of a continuous scale makes it impossible to measure with sufficient accuracy either (1) the relative riskiness of the system in noncrisis mode; or (2) the intensity of a crisis once it has started. If the former could be measured, it may be easier to take early remedial action as the danger of a systemic crisis increases, while measurement of the latter would facilitate decision making on the most appropriate measures to address the crisis.

In our view, a precondition for improving the analysis and management of financial (banking) stability is to be able to construct a metric for it.3 The purpose of this study therefore is to present a method for estimating a set of stability measures for the banking system, or banking stability measures (BSMs). There is no unique, best way to estimate BSMs, any more than there is a unique best way to measure such concepts as “output” or “inflation,” but we hope to demonstrate that our approach is reasonable. Moreover, there are very few alternative available measures that might be used.

Briefly, and as described in far more detail in the subsequent discussion, we conceptualize the banking system as a portfolio of banks comprising the core, systemically important banks in any country. Thus, we infer the Banking System’s (Portfolio) Multivariate Density (BSMD), from which we construct a set of BSMs. These measures embed the banks’ distress interdependence structure, which captures not only linear (correlation) but also nonlinear distress dependencies among the banks in the system. Moreover, the structure of linear and nonlinear distress dependencies changes as banks’ probabilities of distress (PoDs) change; hence, the proposed stability measures incorporate changes in distress dependence that are consistent with the economic cycle. This is a key advantage over traditional risk models that most of the time incorporate only linear dependence (correlation structure) and assume it constant throughout the economic cycle.4 Consequently, the proposed BSMs represent a set of tools to analyze (define) stability from four different yet complementary perspectives, by allowing the quantification of (1) “common” distress in the banks of the system, (2) distress between specific banks, and (3) distress in the system associated with a specific bank.

In this chapter, we estimate the proposed BSMs using publicly available information from 2005 up to the beginning of October 2008. These estimations are to illustrate the methodology rather than to make an assessment of the conjunctural financial stability of any particular system. We examine relative changes in stability over time and among different banks’ business lines in the U.S. banking system. We also analyze cross-region effects between American and European banking groups. Last, we show how our technique can be extended to incorporate the effect of foreign banks on sovereigns with banking systems with cross-border institutions. For this purpose, we estimate the BSMs for major foreign banks and sovereigns in Latin America, Eastern Europe, and Asia. This implementation flexibility is of relevance for banking stability surveillance, because cross-border financial linkages are growing and becoming significant, as has been highlighted by the financial market turmoil of recent years. Thus, surveillance of banking stability cannot stop at national borders.

We show how these BSMs can be constructed from a very limited set of data, for example, empirical measurements of distress of individual banks. Such measurements can be estimated using alternative approaches, depending on data availability; thus, the data set that is necessary to estimate the BSMs is available in most countries. Consequently, such measures can be provided for a wide group of developing as well as developed economies. Establishing such a set of measures with a minimum of basic components makes it feasible to undertake a wider range of comparative analysis, both time series and cross section. It is important to note that when measurements of distress of nonbanking financial institutions (NBFIs), that is, insurance companies, hedge funds, and others, are available, our methodology can be extended easily to incorporate the effects of such institutions in the measurement of stability, hence allowing us to estimate a set of stability measures for the financial system. This could be of relevance for countries where NBFIs have systemic importance in the financial sector.

Economically, our approach is based on the microfounded, general equilibrium theoretical framework of Goodhart, Sunirand, and Tsomocos (2006), which indicates that financial instability can arise through systemic shocks, contagion after idiosyncratic shocks, or a combination of both. In this respect, our proposed banking system stability measures represent a clear improvement over purely statistical or mathematical models that are economically atheoretical and therefore difficult to interpret.

Over the past nearly two decades, we have completed a research agenda that has allowed us to gain important insights into the analysis of financial stability. For example, in Segoviano (1998), Segoviano and Lowe (2002), and Goodhart and Segoviano (2004), we present simple approaches that vary in the degree of sophistication to quantify portfolio credit risk (all of these approaches were based on parametric models and accounted only for correlations that were fixed through the cycle) and analyze the procyclicality of banking regulation and its implications for financial stability. In Segoviano and Padilla (2006), we present a framework for macroeconomic stress testing combined with a model for portfolio credit risk evaluation, which accounts for linear and nonlinear dependencies among the assets in banks’ portfolios and their changes across the economic cycle; however, in all these cases, we focus on individual banks’ portfolios or on the overall aggregate banking system. In Goodhart, Hofmann, and Segoviano (2004, 2006) and Aspachs and others (2006), we focus on systemic risk, making use of average measurements of distress of the system, which do not incorporate banks’ distress dependencies or their changes across the economic cycle.

Thus, we hope that the proposed BSMs will allow us to complement our previous research and gain further insights into our understanding of financial stability. We therefore are extending our research as follows in order to achieve specific aims:

  • We examine relative changes in the BSMs over time and between countries, in order to identify the occasion and determinants of changes in the riskiness of the banking system.

  • We try to predict future movements of the BSMs for use as an early-warning mechanism.

  • We explore the significant macroeconomic and financial factors and shocks influencing the BSMs, in order to identify macro-financial linkages.

  • We explore the factors that can limit and reverse tendencies toward instability, so as to discover what instruments may be available (and under what conditions) to control such instability.

In Section 1, we explain the importance of incorporating banks’ distress dependence in the estimation of the stability of the banking system and describe the modeling steps followed in our framework. In Section 2, we present the Consistent Information Multivariate Density Optimizing (CIMDO) methodology to infer the BSMD from which the proposed BSMs are estimated. These are defined in Section 3. In Section 4, we present empirical estimates of the proposed BSMs as described. Finally, conclusions are presented in Section 5.5

1. Distress Dependence Among Banks and Stability of the Banking System

The proper estimation of distress dependence among the banks in a system is of key importance for the surveillance of stability of the banking system. Financial supervisors recognize the importance of assessing not only the risk of distress—that is, large losses and possible default of a specific bank—but also the impact that such an event would have on other banks in the system. Clearly, the event of simultaneous large losses in various banks would affect a banking system’s stability and thus represents a major concern for supervisors. Banks’ distress dependence is based on the fact that banks are usually linked—either directly, through the interbank deposit market and participations in syndicated loans, or indirectly, through lending to common sectors and proprietary trades. Banks’ distress dependence varies across the economic cycle and tends to rise in times of distress because the fortunes of banks decline concurrently through either contagion after idiosyncratic shocks, affecting interbank deposit markets and participations in syndicated loans (direct links), or through negative systemic shocks, affecting lending to common sectors and proprietary trades (indirect links). Therefore, in such periods, the banking system’s joint probability of distress (JPoD)—that is, the probability that all the banks in the system experience large losses simultaneously, which embeds banks’ distress dependence—may experience larger and nonlinear increases than those experienced by the probabilities of distress (PoDs) of individual banks.

Consequently, it becomes essential for the proper estimation of the banking system’s stability to incorporate banks’ distress dependence and its changes across the economic cycle. Quantitative estimation of distress dependence, however, is a difficult task. Information restrictions and difficulties in modeling distress dependence arise due to the fact that distress is an extreme event and can be viewed as a tail event that is defined in the “distress region” of the probability distribution that describes the implied asset price movements of a bank (Figure 32.1).6 The fact that distress is a tail event makes the often used correlation coefficient inadequate to capture bank distress dependence and the standard approach to model parametric copula functions difficult to implement. Our methodology embeds a reduced-form or nonparametric approach to model copulas that seems to adequately capture default dependence and its changes at different points of the economic cycle. This methodology is easily implementable under the data constraints affecting bank default dependence modeling and produces robust estimates under the Probability Integral Transformation (PIT) criterion.7

Figure 32.1The Probability of Distress

In our modeling of banking systems’ stability and distress dependence, we follow four steps (Figure 32.2):

Figure 32.2The Banking System’s Multivariate Density

Source: Authors.

Note: FI = financial institution.

  • Step 1: We conceptualize the banking system as a portfolio of banks.

  • Step 2: For each of the banks included in the portfolio, we obtain empirical measurements of PoDs.

  • Step 3: Making use of the CIMDO methodology, presented in Segoviano (2006b) and summarized in subsequent discussion, and taking as input variables the individual banks’ PoDs estimated in the previous step, we recover the BSMD.

  • Step 4: On the basis of the BSMD, we estimate the proposed BSMs.

Section 2 describes the procedure to recover the BSMD.

2. Banking System’s Multivariate Density

The BSMD characterizes both the individual and joint asset value movements of the portfolio of banks representing the banking system. The BSMD is recovered using the CIMDO methodology (Segoviano, 2004, 2006a, 2006b). The BSMD embeds the banks’ distress dependence structure—characterized by the CIMDO copula function (Segoviano and Goodhart, 2009)—that captures linear and nonlinear distress dependencies among the banks in the system and allows for these to change throughout the economic cycle, reflecting the fact that dependence increases in periods of distress. These are key technical improvements over traditional risk models, which usually account only for linear dependence (correlations) that are assumed to remain constant over the cycle or a fixed period of time. In order to show such improvements in the modeling of distress dependence—thus, in our proposed measures of stability—in what follows, we (1) model the BSMD using the CIMDO methodology; and (2) illustrate the advantages embedded in the CIMDO copula to characterize distress dependence among the banks in the banking system.

A. The CIMDO approach: modeling the banking system’s multivariate density

We recover the BSMD by employing the CIMDO methodology and empirical measures of PoDs of individual banks. There are alternative approaches to estimate individual banks’ PoDs. For example, we analyzed (1) the structural approach; (2) credit default swaps (CDS); and (3) out-of-the-money (OOM) option prices. These are discussed further in Section 4.8 It is important to emphasize that individual banks’ PoDs are exogenous variables in the CIMDO framework; thus, the framework can be implemented with any alternative approach to estimate PoDs. Consequently, this provides great flexibility in the estimation of the BSMD.

The CIMDO methodology is based on the minimum cross-entropy approach (Kullback, 1959). Under this approach, a posterior multivariate distribution p—the CIMDO density—is recovered using an optimization procedure by which a prior density q is updated with empirical information via a set of constraints. Thus, the posterior density satisfies the constraints imposed on the prior density. In this case, the banks’ empirically estimated PoDs represent the information used to formulate the constraint set. Accordingly, the CIMDO density—the BSMD—is the posterior density that is closest to the prior distribution and that is consistent with the empirically estimated PoDs of the banks making up the system.9

In order to formalize these ideas, we proceed by defining a banking system—portfolio of banks—comprising two banks, Bank X and Bank Y, whose logarithmic returns are characterized by the random variables x and y. Hence, we define the CIMDO objective function as follows:10

It is important to point out that the prior distribution follows a parametric form q that is consistent with economic intuition (e.g., default is triggered by a drop in the firm’s asset value below a threshold value) and with theoretical models (i.e., the structural approach to modeling risk). However, the parametric density q is usually inconsistent with the empirically observed measures of distress. Hence, the information provided by the empirical measures of distress of each bank in the system is of prime importance for the recovery of the posterior distribution. In order to incorporate this information into the posterior density, we formulate consistency-constraint equations that have to be fulfilled when optimizing the CIMDO objective function. These constraints are imposed on the marginal densities of the multivariate posterior density and are of the form

where p(x, y) is the posterior multivariate distribution that represents the unknown to be solved. PoDtx and PoDtx are the empirically estimated PoDs of each of the banks in the system, and χ[xdx,),χ[xdy,) are indicating functions defined with the distress thresholds xdx,xdy, estimated for each bank in the portfolio. In order to ensure that the solution for p(x, y) represents a valid density, the conditions that p(x, y) ≥ 0 and the probability additivity constraint ∬ p(x, y) dxdy = 1, also need to be satisfied. Once the set of constraints is defined, the CIMDO density is recovered by minimizing the functional:

where λ1, λ2 represent the Lagrange multipliers of the consistency constraints, and μ represents the Lagrange multiplier of the probability additivity constraint. By using the calculus of variations, the optimization procedure can be performed. Hence, the optimal solution is represented by a posterior multivariate density that takes the following form:

Intuitively, imposing the constraint set on the objective function guarantees that the posterior multivariate distribution—the BSMD—contains marginal densities that satisfy the PoDs observed empirically for each bank in the banking portfolio. CIMDO-recovered distributions outperform the most commonly used parametric multivariate densities in the modeling of portfolio risk under the PIT criterion.11 This is because when recovering multivariate distributions through the CIMDO approach, the available information embedded in the constraint set is used to adjust the “shape” of the multivariate density via the optimization procedure described. This appears to be a more efficient manner of using the empirically observed information than under parametric approaches, which adjust the “shape” of parametric distributions via fixed sets of parameters. A detailed development of the PIT criterion and Monte Carlo studies used to evaluate specifications of the CIMDO density are presented in Segoviano (2006b).

B. The CIMDO copula: distress dependence among banks in the system

The BSMD embeds the structure of linear and nonlinear default dependence among the banks included in the portfolio that is used to represent the banking system. Such dependence structure is characterized by the copula function of the BSMD, that is, the CIMDO copula, which changes at each period of time consistently with changes in the empirically observed PoDs. In order to illustrate this point, we heuristically introduce the copula approach to characterize dependence structures of random variables and explain the particular advantages of the CIMDO copula.

The Copula Approach

The copula approach is based on the fact that any multivariate density, which characterizes the stochastic behavior of a group of random variables, can be broken into two subsets of information: (1) information of each random variable, that is, the marginal distribution of each variable; and (2) information about the dependence structure among the random variables. Thus, in order to recover the latter, the copula approach sterilizes the marginal information of each variable, consequently isolating the dependence structure embedded in the multivariate density. Sterilization of marginal information is done by transforming the marginal distributions into uniform distributions: U(0,1), which are uninformative distributions.12 For example, let x and y be two random variables with individual distributions x~ F, y ~ H and a joint distribution (x, y) ~ G. To transform x and y into two random variables with uniform distributions U(0,1), we define two new variables as u = F(x), v = H(y) both distributed as U(0,1) with joint density c[u, v]. Under the distribution of transformation of random variables, the copula function c[u, v] is defined as

where g, f, and h are defined densities. From equation (32.4), we see that copula functions are multivariate distributions, whose marginal distributions are uniform on the interval [0,1]. Therefore, because each of the variables is individually (marginally) uniform—that is, their information content has been sterilized—their joint distribution will contain only dependence information. Rewriting equation (32.4) in terms of x and y, we get

From equation (32.5), we see that the joint density of u and v is the ratio of the joint density of x and y to the product of the marginal densities. Thus, if the variables are independent, equation (32.5) is equal to one.

The copula approach to model dependence possesses many positive features when compared with correlations (see Box 32.1). In comparison to correlation, the dependence structure as characterized by copula functions describes linear and nonlinear dependencies of any type of multivariate densities and along their entire domain. In addition, copula functions are invariant under increasing and continuous transformations of the marginal distributions. Under the standard procedure, first, a given parametric copula is chosen and calibrated to describe the dependence structure among the random variables characterized by a multivariate density. Then, marginal distributions that characterize the individual behavior of the random variables are modeled separately. Last, the marginal distributions are “coupled” with the chosen copula function to “construct” a multivariate distribution. Therefore, the modeling of dependence with standard parametric copulas embeds two important shortcomings:

  • 1. It requires modelers to deal with the choice, proper specification, and calibration of parametric copula functions—that is, the Copula Choice Problem (CCP). The CCP is in general a challenging task, because results are very sensitive to the functional form and parameter values of the chosen copula functions (Frey and McNeil, 2001). In order to specify the correct functional form and parameters, it is necessary to have information on the joint distribution of the variables of interest, in this case, joint distributions of distress, which are not available.

  • 2. The commonly employed parametric copula functions in portfolio risk measurement require the specification of correlation parameters, which usually are specified to remain fixed through time.13 Thus, the dependence structure that is characterized with parametric copula functions, although improving the modeling of dependence versus correlations, still embeds the problem of characterizing dependence that remains fixed through time.14

Box 32.1Drawbacks to the Characterization of Distress Dependence of Financial Returns with Correlations

Interdependencies of financial returns traditionally have been modeled based on correlation analysis (de Bandt and Hartmann, 2001). However, the characterization of financial returns with correlations presents important drawbacks, the most relevant of which are the following.

Financial Returns and Gaussian Distributions

The popularity of linear correlation stems from the fact that it can be easily calculated, easily manipulated under linear operations, and is a natural scalar measure of dependence in the world of multivariate normal distributions. However, empirical research in finance shows that distributions of financial assets are seldom in this class.1 Thus, using multivariate normal distributions and, consequently, linear correlations, might prove very misleading for describing bank distress dependence (Embrechts, McNeil, and Straumann, 1999). Moreover, when working with heavy-tailed distributions—which usually characterize financial asset returns—their variances might not be finite; hence, correlation becomes undefined.2

Linear and Nonlinear Dependence

Another problem associated with correlation is that the data may be highly dependent, while the correlation coefficient is zero. Equivalently, the independence of two random variables implies they are uncorrelated, but zero correlation does not imply independence. A simple example where the covariance disappears despite strong dependence between random variables is obtained by taking X ~ N(0,1), Y=X2 given that the third moment of the standard normal distribution is zero.

Nonlinear Transformations

In addition, linear correlation is not invariant under nonlinear strictly increasing transformations. For two real-valued random variables, we have in general p(T(X), T(Y)) ±p(X, Y). This is relevant when modeling dependence among financial assets. For example, suppose that we have a copula function describing the dependence structure among bank percentage returns. If we decide to model the dependence among logarithm returns, the copula will not change; only the marginal distributions will (Embrechts, McNeil, and Straumann, 1999).

Dependence of Extreme Events

Furthermore, correlation is a measure of dependence in the center of the distribution, which gives little weight to tail events, that is, extreme events, when evaluated empirically. Hence, because distress is characterized as a tail event, correlation is not an appropriate measure of distress dependence when marginal distributions of financial assets are nonnormal (de Vries, 2005).

1 Empirical support for modeling financial returns with t-distributions can be found in Danielsson and de Vries (1997); Bonti, Hoskin, and Siegel (2000); and Glasserman, Heidelberger, and Shahabuddin (2002).2 Even for jointly elliptical distributed random variables, there are situations where using linear correlation does not make sense. If we modeled asset values using heavy-tailed distributions, for example, t2-distributions, the linear correlation is not even defined because of infinite second moments.

The CIMDO Copula

Our approach to model multivariate densities is the inverse of the standard copula approach. We first infer the CIMDO density as explained in Section 2.A. The CIMDO density embeds the dependence structure among the random variables that it characterizes; therefore, once we have inferred the CIMDO density, we can extract the copula function describing such dependence structure, that is, the CIMDO copula. This is done by estimating the marginal densities from the multivariate density and using Sklar’s theorem (Sklar, 1959).

The CIMDO copula maintains all the benefits of the copula approach:

  • 1. It describes linear and nonlinear dependencies among the variables described by the CIMDO density. Such dependence structure is invariant under increasing and continuous transformations of the marginal distributions.

  • 2. It characterizes the dependence structure along the entire domain of the CIMDO density. Nevertheless, the dependence structure characterized by the CIMDO copula appears to be more robust in the tail of the density, where our main interest lies, that is, to characterize distress dependence.

However, the CIMDO copula avoids the drawbacks implied by the use of standard parametric copulas:

  • 1. It circumvents the CCP. The explicit choice and calibration of parametric copula functions is avoided because the CIMDO copula is extracted from the CIMDO density; therefore, in contrast with most copula models, the CIMDO copula is recovered without explicitly imposing parametric forms that, under restricted data sets, are difficult to model empirically and frequently wrongly specified. It is important to note that under such information constraints, that is, when only information of marginal PoDs exists, the CIMDO copula is not only easily implementable, it outperforms the most common parametric copulas used in portfolio risk modeling under the PIT criterion. This is especially on the tail of the copula function, where distress dependence is characterized.15

  • 2. The CIMDO copula avoids the imposition of constant correlation parameter assumptions. It updates “automatically” when the PoDs are employed to infer the CIMDO density change. Therefore, the CIMDO copula incorporates banks’ distress dependencies that change, according to the dissimilar effects of shocks on individual banks’ PoDs, and that are consistent with the economic cycle.

In order to formalize these ideas, note that if the CIMDO density is of the form presented in equation (32.3), the CIMDO copula, cc(u, v), is represented by

where u=Fc(x)x=Fc1(u), and v=Hc(y)y=Hc1(v).16

Equation (32.6) shows that the CIMDO copula is a nonlinear function of λ1λ2, and μ, the Lagrange multipliers of the CIMDO functional presented in equation (32.2). Like all optimization problems, the Lagrange multipliers reflect the change in the objective function’s value as a result of a marginal change in the constraint set. Therefore, as the empirical PoDs of individual banks change at each period of time, the Lagrange multipliers change, the values of the constraint set change, and the CIMDO copula changes; consequently, the default dependence among the banks in the system changes.

Thus, as already mentioned, the default dependence gets updated “automatically” with changes in empirical PoDs at each period of time. This is a relevant improvement over most risk models, which usually account only for linear dependence (correlation) that is also assumed to remain constant over the cycle or a fixed period of time.

3. Banking Stability Measures

The BSMD characterizes the PoD of the individual banks included in the portfolio, their distress dependence, and changes across the economic cycle. This is a rich set of information that allows us to analyze (define) banking stability from three different, yet complementary, perspectives. For this purpose, we define a set of BSMs to quantify

  • 1. common distress in the banks of the system;

  • 2. distress between specific banks; and

  • 3. distress in the system associated with a specific bank.

We hope that the complementary perspectives of financial stability brought by the proposed BSMs represent a useful tool set to help financial supervisors to identify how risks are evolving and where contagion might most easily develop.

For illustration purposes, and to make it easier to present definitions, we proceed by defining a banking system—portfolio of banks—comprising three banks, whose asset values are characterized by the random variables x and y and r. Hence, following the procedure described in Section 2.A, we infer the CIMDO density function, which takes the following form:

where q(x, y, r) and p(x, y, r) εR3.

A. Common distress in the banks of the system

In order to analyze common distress in the banks comprising the system, we propose using the JPoD and the Banking Stability Index (BSI).

Joint Probability of Distress

The JPoD represents the probability of all the banks in the system (portfolio) becoming distressed, that is, the tail risk of the system. The JPoD embeds not only changes in the individual banks’ PoDs; it also captures changes in the distress dependence among the banks, which increases in times of financial distress; therefore, in such periods, the banking system’s JPoD may experience larger and nonlinear increases than those experienced by the (average) PoDs of individual banks. For the hypothetical banking system defined in equation (32.7), the JPoD is defined as P(XI YI R), and it is estimated by integrating the density (BSMD) as follows:

Banking Stability Index

The BSI is based on the conditional expectation of the default probability measure developed by Huang (1992).17 The BSI reflects the expected number of banks becoming distressed given that at least one bank has become distressed. A higher number signifies increased instability.

For example, for a system of two banks, the BSI is defined as follows:

The BSI represents a probability measure that conditions on any bank becoming distressed, without indicating the specific bank.18

B. Distress between specific banks

Distress Dependence Matrix

For each period under analysis, for each pair of banks in the portfolio, we estimate the set of pairwise conditional PoDs, which are presented in the Distress Dependence Matrix (DiDe). This matrix contains the PoD of the bank specified in the row, given that the bank specified in the column becomes distressed.

Although conditional probabilities do not imply causation, this set of pairwise conditional probabilities can provide important insights into interlinkages and the likelihood of contagion between the banks in the system. For the hypothetical banking system defined in equation (32.7), at a given date, the DiDe is represented in Table 32.1.

Table 32.1Distress Dependence Matrix
BankBank XBank YBank R
Bank X1P(X/Y)P(X/R)
Bank YP(Y/X)1P(Y/R)
Bank RP(R/X)P(R/Y)1
Source: Authors.
Source: Authors.

For example, the PoD of Bank X conditional on Bank Y becoming distressed is estimated by

Distress in Specific Banks/Groups of Banks Associated with Distress in Other Banks/Groups of Banks

Note that the BSMD allows us to estimate any conditional probability of distress, including conditional probabilities of groups or specific banks. This feature provides great flexibility to analyze linkages among diverse groups of banks. For example, we can estimate conditional probabilities between groups or individual banks in different business lines or geographical zones.

C. Distress in the system associated with a specific bank

The probability that at least one bank becomes distressed (PAO) given that a specific bank becomes distressed characterizes the likelihood that one, two, or more institutions, up to the total number of banks in the system, become distressed. Therefore, this measure quantifies the potential “cascade” effects in the system given distress in a specific bank. Consequently, we propose this measure as an indicator to quantify the systemic importance of a specific bank if it becomes distressed. Again, it is worth noting that conditional probabilities do not imply causation; however, we consider that the PAO can provide important insights into systemic interlinkages among the banks comprising a system.

For example, in a banking system with four banks, X, Y, Z, and R, the PAO given that Bank X becomes distressed corresponds to the probability set marked in the Venn diagram (Figure 32.3). In this example, the PAO can be defined as follows:

Figure 32.3Probability That at Least One Bank Becomes Distressed

Source: Authors.

4. Banking Stability Measures: Empirical Results

To illustrate the methodology, in this section we estimate the proposed BSMs to

  • 1. examine relative changes in stability over time and among different banks’ business;

  • 2. analyze cross-regional effects between different banking groups; and

  • 3. analyze the effect of foreign banks on sovereigns with banking systems with cross-border institutions.

Our estimations are performed from 2005 up to October 2008 using only publicly available data and include major American and European banks and sovereigns in Latin America, Eastern Europe, and Asia. Implementation flexibility in our approach is of relevance for banking stability surveillance, because cross-border financial linkages are growing and becoming increasingly significant, as has been highlighted by the financial market turmoil of recent years. Thus, surveillance of banking stability cannot stop at national borders. An important feature of this methodology is that it can be implemented with alternative measures of PoDs of individual banks, which we will describe. We continue by presenting the estimated BSMs and analyzing them.

A. Estimation of probabilities of distress of individual banks

There are alternative approaches by which PoDs of individual banks can be empirically estimated. The most well known include the structural approach, PoDs derived from CDS spreads (CDS-PoDs), or PoDs derived from OOM option prices. These alternative approaches present diverse advantages and disadvantages, in terms of availability of data necessary for their implementation, parameterization of quantitative techniques, and consistency of empirical estimations. We performed an extensive empirical analysis of these approaches. The structural approach presented significant difficulties for the proper parameterization of its quantitative framework. It also produced estimates that appeared inconsistent. The OOM approach suffered from the latter problem, in addition to data restrictions for its implementation across time. Nor were CDS-PoDs free of problems. There are arguments against the trustworthiness of the CDS spreads as a reliable barometer of firms’ financial health. In particular, CDS spreads may exaggerate a firm’s “fundamental” risk when there is (1) lack of liquidity in the particular CDS market and (2) generalized risk aversion in the financial system. Although such arguments might be correct to some degree, these factors can become self-fulfilling if they affect the market’s perception and therefore have a real impact on the market’s willingness to fund a particular firm. Consequently, this can cause a real effect on the firm’s financial health, as has been seen in the recent financial turmoil. Moreover, although CDS spreads may overshoot at times, they do not generally stay wrong for long. Rating agencies have mentioned that CDS spreads frequently anticipate rating changes. Though the magnitude of the moves may at times be unrealistic, the direction is usually a good distress signal. For these reasons, and because of the problems encountered with the other approaches (which we consider more serious), we decided to use CDS-PoDs to estimate the proposed BSMs. Although we consider that CDS-PoDs represent reasonable input variables to estimate the proposed BSMs, we keep in mind their potential shortcomings when drawing conclusions in our analysis. Furthermore, because none of these estimators represents a “first best” choice, we continue performing empirical research to improve the estimation of individual banks’ PoDs and to investigate which of the alternative approaches (already investigated or to be investigated) is the most appropriate for specific countries and types of banks. Thus, if we found a better approach, it would be straightforward to replace the chosen PoD approach in the estimation of the BSMs, because PoDs are exogenous variables in the CIMDO framework.

Finally, we would like to explain our definition of “distress” risk. Assessing at what point “liquidity risk” becomes solvency risk, that is, credit risk, is difficult, and disentangling these risks is a complex issue. In addition, CDS often covers not only the event of default of an underlying security but a wider set of “credit events,” that is, downgrades. We consider the combined effects of these factors, which are embedded in CDS spreads, to be “distress” risk: large losses and the possible default of a specific bank. Thus, our definition of “distress” risk is broader than “default,” “credit,” or “liquidity” risks.

B. Examination of relative changes of stability over time

The analysis of risks among banks in specific countries and among different business lines is illustrated by estimating our proposed measures of stability for a set of large U.S. banks as it was up to October 2008 using only publicly available data.19 For this purpose, we focus on the largest U.S. banking groups. The bank holding companies (BHCs) that are included are Citigroup, Bank of America, JP Morgan, and Wachovia. The investment banks included are Goldman Sachs, Lehman Brothers, Merrill Lynch, and Morgan Stanley.20 In addition to the major U.S. banks, we included Washington Mutual (WaMu) and American International Group (AIG; a thrift and an insurance company both under intense market pressure in September 2008). The results can be summarized as follows:

Perspective 1. Common Distress in the Banks of the System: BSI and JPoD

  • U.S. banks are highly interconnected, with distress in one bank associated with high PoD elsewhere. This is clearly indicated by the JPoD. Moreover, movements in the JPoD and BSI coincide with events that were considered relevant by the markets on specific dates (Figures 32.4 and 32.5).

  • Distress dependence across banks rises during times of crisis, indicating that systemic risks, as implied by the JPoD and the BSI, rise faster than idiosyncratic risks. The JPoD and the BSI not only take account of individual banks’ PoDs, but these measures also embed banks’ distress dependence. Therefore, these measures may experience larger and nonlinear increases than those experienced by the PoDs of individual banks. Figure 32.6 shows that daily percentage changes of the JPoD are larger than daily percentage changes of the individual (average) PoDs. This empirical fact provides evidence that in times of distress, not only do individual PoDs increase, but so does distress dependence.

  • Risks vary by the business line of the banks. Figures 32.4 and 32.5 show that IBs’ JPoD and BSI are larger than those for BHCs. This chart also shows that for IBs, risks were higher at the time of Lehman’s collapse.

Figure 32.4Joint Probability of Distress: January 2007-October 2008

Source: Authors.

Note: AIG = American International Group; BHC = bank holding company; FNM/FRE = Fannie Mae/Freddie Mac; IB = investment bank; TAF = Term Auction Facility; TARP = Troubled Asset Relief Program; WaMu = Washington Mutual.

Figure 32.5Banking Stability Index: January 2007-October 2008

Source: Authors.

Note: AIG = American International Group; BHC = bank holding companies; FNM/FRE = Fannie Mae/Freddie Mac; IB = investment bank; TAF = Term Auction Facility; TARP = Troubled Asset Relief Program; WaMu = Washington Mutual.

Figure 32.6Daily Percentage Increase: Joint and Average Probability of Distress

Source: Authors.

Note: BHC = bank holding company; IB = investment bank; JPoD = joint probability of distress; PoD = probability of distress.

Perspective 2. Distress between Specific Banks: Distress Dependence Matrix

The DiDe presented in Table 32.2 shows the (pairwise) conditional PoDs of the bank in the column, given that the bank in the row falls into distress. The DiDe is estimated daily. For purposes of analysis, we have chosen July 1, 2007, and September 12, 2008; thus, we can show how conditional PoDs have changed from a precrisis date to the day before Lehman Brothers filed for bankruptcy. We have also broken these matrices into four quadrants, top left (quadrant 1), top right (quadrant 2), bottom left (quadrant 3), and bottom right (quadrant 4), to make explanations clearer. From these matrices we can observe the following:

Table 32.2Distress Dependence Matrix: American and European Banks, July 2007 and September 2008
BankCitiBACJPMWachoWaMuGSLEHMERMSAIGRow

average
BARCHSBCUBSCSFBDBRow

average
July 17, 2007
Citigroup1.000.140.110.110.080.090.080.090.090.080.190.070.070.080.060.070.07
Bank of America0.121.000.270.270.110.110.100.120.120.150.240.080.070.090.060.100.08
JP Morgan0.150.421.000.310.130.190.160.190.180.170.290.100.080.120.090.140.10
Wachovia0.120.330.241.000.110.120.100.120.120.140.240.070.050.070.050.080.07
Washington Mutual0.160.280.210.231.000.120.120.160.130.150.260.090.080.090.060.090.08
Goldman Sachs0.170.250.280.210.111.000.310.280.310.170.310.130.110.150.120.180.14
Lehman0.220.320.320.260.150.431.000.350.330.200.360.140.120.150.140.220.15
Merrill Lynch0.190.320.330.250.170.330.311.000.310.200.340.150.150.190.150.210.17
Morgan Stanley0.190.310.280.240.140.350.280.301.000.160.330.140.120.140.120.180.14
AIG0.070.140.100.100.050.070.060.070.061.000.170.050.060.070.040.060.06
Column average0.240.350.310.300.210.280.250.270.260.240.270.100.090.110.090.130.11
Barclays0.040.050.040.040.020.040.030.040.040.040.041.000.180.180.120.120.32
HSBC0.040.040.030.020.020.030.020.030.030.040.030.161.000.130.090.110.30
UBS0.040.050.040.030.020.040.030.040.030.040.040.170.131.000.210.150.33
CSFB0.050.060.050.040.030.050.050.060.050.050.050.190.150.361.000.210.38
Deutsche Bank0.050.090.080.060.030.070.060.070.060.060.060.170.160.220.191.000.35
Column average0.050.060.050.040.020.050.040.050.040.050.040.340.320.380.320.320.34
September 12, 2008
Citigroup1.000.200.190.140.070.170.130.140.160.110.230.150.170.170.150.160.16
Bank of America0.141.000.310.180.050.160.100.130.150.110.230.120.130.130.110.150.13
JP Morgan0.130.291.000.160.050.190.110.140.160.090.230.110.100.120.110.150.12
Wachovia0.340.600.551.000.170.360.270.310.340.290.420.270.230.270.250.310.27
Washington Mutual0.930.970.950.941.000.910.880.920.910.890.930.870.860.860.830.860.86
Goldman Sachs0.150.190.240.130.061.000.180.200.270.110.250.140.130.150.150.190.15
Lehman0.470.530.580.430.250.751.000.590.620.370.560.390.370.400.420.520.42
Merrill Lynch0.320.410.470.300.160.530.371.000.480.260.430.310.330.350.350.390.35
Morgan Stanley0.210.280.290.190.090.400.220.271.000.140.310.180.180.180.180.230.19
AIG0.500.660.590.530.290.540.430.490.471.000.550.490.530.530.490.530.52
Column average0.420.510.520.400.220.500.370.420.460.340.410.300.300.320.310.350.32
Barclays0.100.110.100.080.040.100.070.090.090.070.081.000.360.310.300.280.45
HSBC0.060.060.050.030.020.050.040.050.050.040.050.201.000.160.160.170.34
UBS0.110.110.110.070.040.110.070.100.090.080.090.320.301.000.470.340.48
CSFB0.070.070.070.050.030.070.050.070.060.050.060.200.200.311.000.260.40
Deutsche Bank0.060.080.090.050.030.090.060.070.070.050.060.180.200.210.241.000.36
Column average0.080.090.080.060.030.090.060.070.070.060.070.380.410.400.430.410.41
Source: Authors.Note: The table shows the probability of distress of the bank in the row conditional on the bank in the column becoming distressed. AIG = American International Group; BAC = Bank of America; BARC = Barclays; Citi = Citigroup; CSFB = Credit Suisse; DB = Deutsche Bank; GS = Goldman Sachs; HSBC = Hongkong and Shanghai Banking Corporation; JPM = JP Morgan; LEH = Lehman; MER = Merrill Lynch; MS = Morgan Stanley; Wacho = Wachovia; WaMu = Washington Mutual.
Source: Authors.Note: The table shows the probability of distress of the bank in the row conditional on the bank in the column becoming distressed. AIG = American International Group; BAC = Bank of America; BARC = Barclays; Citi = Citigroup; CSFB = Credit Suisse; DB = Deutsche Bank; GS = Goldman Sachs; HSBC = Hongkong and Shanghai Banking Corporation; JPM = JP Morgan; LEH = Lehman; MER = Merrill Lynch; MS = Morgan Stanley; Wacho = Wachovia; WaMu = Washington Mutual.
  • Links across major U.S. banks have increased greatly. This is clearly shown by the conditional PoDs presented in Quadrant 1 of the DiDe presented in Table 32.2. On average, if any of the U.S. banks fell into distress, the average probability of the other banks being distressed increased from 27 percent on July 17, 2007, to 41 percent on September 12, 2008.

  • In September, Lehman was the bank under highest stress. This is revealed by Lehman’s large PoD conditional on any other bank falling into distress, which on September 12 reached on average 56 percent (row-average Lehman). Moreover, a Lehman default was estimated on September 12 to raise the chances of a default elsewhere by 46 percent. In other words, the PoD of any other bank conditional on Lehman falling into distress went from 25 percent on July 17, 2007, to 37 percent on September 12, 2008 (column-average Lehman).

  • AIG’s connections to the other major U.S. banks were similar to Lehman’s. This can be seen by comparing the chances of each one of the U.S. banks being affected by distress in AIG and Lehman (column AIG vs. column Lehman) on September 12. Links were particularly close between Lehman, AIG, WaMu, and Wachovia, all of which were particularly exposed to housing. On September 12, a Lehman bankruptcy implied chances of 88, 43, and 27 percent that WaMu, AIG, and Wachovia, respectively, would fall into distress.

Perspective 3. Distress in the System Associated with a Specific Bank: Probability That at Least One Bank Becomes Distressed

On September 12, using equation (32.11), we estimated the probability that one or more banks in the system would become distressed, given that Lehman became distressed. This reached 97 percent. Thus, the possible “domino” effect observed in the days after its collapse were signaled by the PAO (Figure 32.7). This analysis (Perspective 3) is in line with the insights brought by the DiDe (Perspective 2), which indicated that Lehman’s distress would be associated with distress in several institutions.

Figure 32.7Probability That at Least One Bank Becomes Distressed: Lehman

Source: Authors.

Note: AIG = American International Group.

C. Analysis of cross-region effects among different banking groups

In order to gain insight into the cross-region effects between American and European banking groups, we included five major European banks: Barclays (BARC) and Hongkong and Shanghai Banking Corporation (HSBC) from the United Kingdom, UBS and Credit Suisse (CSFB) from Switzerland, and Deutsche Bank from Germany.

Perspective 1. Common Distress in the Banks of the System: BSI and JPoD

  • The JPoD indicates that risks among European banks are highly interconnected, with distress in one bank associated with a high PoD elsewhere in Europe. The European JPoD and BSI move in tandem with movements in the U.S. indicators, coinciding also with relevant market events (Figures 32.3 and 32.4).

  • Distress dependence among banks in Europe also rises during times of crisis, indicating that systemic risks, as implied by the JPoD and the BSI, rise faster than idiosyncratic risks. In Figure 32.5, it is clear that also for European banks, the daily percentage changes of the JPoD are larger than those of the individual (average) PoDs.

  • Risks for European banks as measured by JPoD and the BSI appear lower than those for U.S. IBs and very similar to those for U.S. BHCs across time. Figures 32.3 and 32.4 also show that risks among European banks were similar at the time of the Bear Stearns debacle (March 17) and Lehman’s collapse (September 15). This is in contrast to American banks, for which risks appear larger at the time of Lehman’s collapse. However, the situation in Europe appeared to be deteriorating fast in mid-September.

Perspective 2. Distress between Specific Banks: Distress Dependence Matrix

  • Links across major European banks have increased significantly (Table 32.2). This is clearly shown by the conditional PoDs presented in Quadrant 4 of the DiDe. On average, if any of the European banks appeared in distress, the probability of the other banks being distressed increased from 34 percent on July 17, 2007, to 41 percent on September 12, 2008.

  • Among the European banks under analysis, UBS appeared to be the bank under highest stress on September 12, 2008. It showed the largest PoD conditional on any other bank falling into distress, reaching on average 48 percent (row-average UBS). UBS’s distress would also be associated with high stress on Barclays, whose PoD conditional on UBS becoming distressed was estimated to reach 31 percent on September 12, 2008. This was a significant increase from 18 percent estimated on July 17, 2007.

  • Among the European banks under analysis, distress at CSFB would be associated with the highest stress on other European banks on September 12, 2008. The (average) PoD of European banks conditional on CSFB falling into distress reached 43 percent (quadrant 4, column-average CSFB). However, the European bank that would be associated with the highest distress among American banks is Deutsche Bank. The (average) PoD of American banks conditional on Deutsche Bank falling into distress reached 35 percent (Quadrant 2, column-average Deutsche Bank). This might be related to the high integration of Deutsche Bank in some markets at the global level.

  • On September 12, 2008, whereas failure of one of the U.S. banks implied (on average) chances of distress of one European bank of 7 percent (Quadrant 3), the (average) PoD of one American bank, conditional on a European bank becoming distressed is above 30 percent (Quadrant 2). This is possibly because a European default would imply more generalized problems, including in U.S. markets.

Even though distress dependence does not imply causation, these results help explain why the Lehman bankruptcy led to a global crisis. The bankruptcy of Lehman appears to have sealed the fate of AIG and WaMu, while putting greatly increased pressure on Wachovia, as indicated by the DiDe. In market terms, this was equivalent to the failure of a major U.S. institution, with significant reverberations on both sides of the Atlantic.

D. Analysis of foreign banks’ risks to sovereigns with banking systems with cross-border institutions

We extend our methodology to analyze how rising problems in advanced market banking systems are linked with increasing risks to emerging markets. For this purpose, we use CDS spreads written on sovereign and banks’ bonds to derive PoDs of banks and sovereigns. Therefore, such PoDs represent markets’ views of risks of distress for these banks and countries. Although absolute risks are discussed, the focus is largely on cross-distress dependence of risks and what they can say about emerging vulnerabilities (Perspective 2). More precisely, using publicly available data, we estimate cross-vulnerabilities between Latin American, Eastern European, and Asian emerging markets and the advanced market banks with larger regional presences in these regions. The countries and banks analyzed are

  • Latin America. Countries included were Mexico, Colombia, Brazil, and Chile; banks included were Banco Bilbao Vizcaya Argentaria, Santander, Citigroup, Scotia Bank, and HSBC.

  • Eastern Europe. Countries included were Bulgaria, Croatia, Hungary, and Slovakia; banks included were Intesa, Unicredito, Erste, Societe Generale, and Citigroup.

  • Asia. Countries included were China, Korea, Thailand, Malaysia, the Philippines, and Indonesia; banks included were Citigroup, JP Morgan Chase, HSBC, Standard and Chartered, BNP, Deutsche Bank, and Development Bank of Singapore.

The detailed results are presented in Segoviano and Goodhart (2009).

5. Conclusion

The purpose of this study was to seek to provide a set of quantitative measures of the financial stability of the main banks in any country or region, so that this portfolio of banks’ relative stability as a group can be tracked over time and compared in a cross section of comparative groupings. To this end, we have developed a new framework that has several advantages:

  • It provides measures that allow us to analyze (define) stability from three different yet complementary perspectives.

  • It can be constructed from a very limited set of data, that is, the empirical measurements of default probabilities of individual banks. Such measurements can be estimated using alternative approaches, depending on data availability; thus, the data set that is necessary for the estimation is available in many countries, with either developed or developing economies, as long as there are reasonable data to reflect individual banks’

    PoDs.

  • It embeds the banks’ default interdependence structure (copula function), which captures linear and nonlinear default dependencies among the main banks in a system.

  • It allows the quantification of changes in the banks’ default interdependence structure at specific points in time; hence, it can be useful to quantify the empirically observed increases in dependencies in periods of distress and relax the commonly used assumption in risk measurement models of fixed correlations across time.

The empirical part of the chapter applied this methodology to a number of country and regional examples by using publicly available information up to October 2008. This implementation flexibility is of relevance for banking stability surveillance, because cross-border financial linkages are growing and becoming significant, as has been highlighted by the financial market turmoil of recent years. Thus, surveillance of banking stability cannot stop at national borders.

The proposed measures will allow us to complement our previous research and expand our research agenda; thus, we hope to gain further insights in our understanding of financial stability: by trying to predict future movements of the BSMs for use as an early-warning mechanism; by exploring the significant macroeconomic and financial factors and shocks influencing the BSMs, in order to identify macro-financial linkages; and by exploring the factors that can limit and reverse tendencies toward instability, so as to discover what instruments may be available (and under what conditions) to control such instability.

References

For a review of the literature on financial crises, see Bordo and others (2001).

In contrast to correlation, which captures only linear dependence, copula functions characterize the whole dependence structure—that is, linear and nonlinear dependence, embedded in multivariate densities (Nelsen, 1999). Thus, in order to characterize banks’ distress dependence, we employ a novel nonparametric copula approach, that is, the CIMDO copula. In comparison to traditional methodologies to model parametric copula functions, the CIMDO copula avoids the difficulties of explicitly choosing the parametric form of the copula function to be used and calibrating its parameters, because CIMDO copula functions are inferred directly (implicitly) from the joint movements of the individual banks’ PoDs.

The PIT criterion for multivariate density’s evaluation is presented in Diebold, Hahn, and Tay (1999).

See Jaynes (1957, 1984).

A detailed definition and development of the CIMDO objective function and constraint set, as well as the optimization procedure that is followed to solve the CIMDO functional, is presented in Segoviano (2006b).

The standard and conditional normal distributions, the t-distribution, and the mixture of normal distributions.

For further details, proofs, and a comprehensive and didactical exposition of copula theory, see Embrechts, McNeil, and Straumann (1999) and Nelsen (1999), where also properties and different types of copula functions are presented.

Note that even if correlation parameters are dynamically updated using rolling windows, correlations remain fixed within such rolling windows. Moreover, the choice of the length of such rolling windows remains subjective most of the time.

See Appendix III of Segoviano and Goodhart (2009) for a summary of this evaluation criterion and its results.

See Appendix II of Segoviano and Goodhart (2009) for an exposition.

This function is presented in Huang (1992). For empirical applications, see Hartmann, Straetmans, and de Vries (2001).

Huang (1992) shows that this measure also can be interpreted as a relative measure of banking linkage. When the BSI = 1 in the limit, banking linkage is weak (asymptotic independence). As the value of the BSI increases, banking linkage increases (asymptotic dependence).

The authors would like to thank Tamim Bayoumi for insightful discussions and contributions in the analysis of these empirical results.

Although investment banks (IBs) have changed their status recently to BHCs, we keep referring to this group of banks as IBs for the purpose of differentiating their risk profiles in the analysis.

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