Banking Stability Measures
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Authors’ E-Mail Addresses: msegoviano@imf.org; caegoodhart@aol.com

This paper defines a set of banking stability measures which 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 (i) common distress of the banks in a system, (ii) distress between specific banks, and (iii) distress in the system associated with a specific bank. Our approach defines the banking system as a portfolio of banks and infers the system's multivariate density (BSMD) from which the proposed measures are estimated. The BSMD embeds the banks' default inter-dependence structure that captures linear and non-linear distress dependencies among the banks in the system, and its changes at different times of the economic cycle. The BSMD is recovered using the CIMDO-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 countries.

Abstract

This paper defines a set of banking stability measures which 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 (i) common distress of the banks in a system, (ii) distress between specific banks, and (iii) distress in the system associated with a specific bank. Our approach defines the banking system as a portfolio of banks and infers the system's multivariate density (BSMD) from which the proposed measures are estimated. The BSMD embeds the banks' default inter-dependence structure that captures linear and non-linear distress dependencies among the banks in the system, and its changes at different times of the economic cycle. The BSMD is recovered using the CIMDO-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 countries.

I. Introduction

Economics is a quantitative science. Macroeconomics depends on data for national income, expenditure and output variables. Macro-monetary 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’, we use various techniques, e.g. 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-section 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.2 This has then 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 Bank Restructuring and Resolution, ed. Hoelscher, (2006). While much useful research has been done employing a crisis on/off dichotomy, it has several inherent deficiencies. In particular, the lack of a continuous scale makes it impossible to measure with sufficient accuracy either (i) the relative riskiness of the system in non-crisis mode, or (ii) 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. The purpose of this paper is therefore to present a method for estimating a set of stability measures of the banking system (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 which might be used.

Briefly, and as described in far more detail below, 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 inter-dependence structure, which captures not only linear (correlation) but also non-linear 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. 3 Consequently, the proposed BSMs represent a set of tools to analyze (define) stability from four different, yet, complementary perspectives, by allowing the quantification of (a) “common” distress in the banks of the system, (b) distress between specific banks, and (c) distress in the system associated with a specific bank.

In this paper 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 US banking system. We also analyze cross-region effects between American and European banking groups. Lastly, 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, since cross-border financial linkages are growing and becoming significant; as has been highlighted by the financial market turmoil of recent months. 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, e.g., 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, countries. Being able to establish 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 non-banking financial institutions (NBFIs) i.e., insurance companies, hedge funds etc. are available, our methodology can be easily extended 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 micro-founded, general equilibrium theoretical framework of Goodhart, Sunirand and Tsomocos (2006), which indicates that financial instability can arise either 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 a-theoretical and therefore difficult to interpret.

Over the past decade, 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 non-linear 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 in the overall aggregate banking system. In Goodhart, Hofmann and Segoviano (2004, 2006), and Aspachs et al. (2006) we focus on systemic risk, making use of average measurements of distress of the system, which do not incorporate banks’ distress dependencies, nor 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 are therefore extending our research as follows in order to achieve specific aims.

  1. 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;

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

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

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

In Section II, 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 III, we present the consistent information multivariate density (CIMDO) methodology to infer the banking system’s multivariate density (BSMD) from which the proposed BSMs are estimated. These are defined in Section IV. In Section V, we present empirical estimates of the proposed BSMs as described above. Finally, conclusions are presented in Section VI.

II. Distress Dependence among Banks and Stability of the Banking System

The proper estimation of distress dependence amongst 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 i.e., 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. Bank’s distress dependence is based on the fact that banks are usually linked— either directly, through the inter-bank 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 since the fortunes of banks decline concurrently through either contagion after idiosyncratic shocks, affecting inter-bank 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); i.e., 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 1). 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.

Figure 1.
Figure 1.

The Probability of Distress

Citation: IMF Working Papers 2009, 004; 10.5089/9781451871517.001.A001

Source: Authors’ calculations.

Our methodology embeds a reduced-form or non-parametric approach to model copulas that seems to capture adequately 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 PIT criterion. 4

In our modeling of banking systems’ stability and distress dependence, we follow four steps (Figure 2).

Figure 2.
Figure 2.

The Banking System’s Multivariate Density

Citation: IMF Working Papers 2009, 004; 10.5089/9781451871517.001.A001

Source: Authors’ estimations.

Step1: 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 probabilities of distress (PoDs).

Step 3: Making use of the Consistent Information Multivariate Density Optimizing (CIMDO) methodology, presented in Segoviano (2006) and summarized below, and taking as input variables the individual banks’ PoDs estimated in the previous step, we recover the banking system’s (portfolio) multivariate density (BSMD).

Step 4: Based on the BSMD, we estimate the proposed banking stability measures (BSMs).

Section III describes the procedure to recover the BSMD.

III. Banking System Multivariate Density

The Banking System Multivariate Density (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 Consistent Information Multivariate Density Optimizing (CIMDO) methodology (Segoviano, 2006b). The BSMD embeds the banks’ distress dependence structure—characterized by the CIMDO-copula function (Segoviano, 2009)—that captures linear and non-linear 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 (i) model the BSMD using the CIMDO methodology, and (ii) illustrate the advantages embeded in the CIMDO-Copula to characterize distress dependence among the banks in the banking system.

A. The CIMDO Approach: Modeling the Banking System Multivariate Density

We recover the BSMD employing the CIMDO methodology and empirical measures of probabilities of distress (PoDs) of individual banks. There are alternative approaches to estimate individual banks’ probabilities of distress. For example, we analyzed (i) the Structural Approach (SA), (ii) Credit Default Swaps (CDS), and (iii) Out of the Money Option Prices (OOM). These are discussed further in Section V. It is important to emphasize the fact that individual banks’ PoDs are exogenous variables in the CIMDO framework; thus, it 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.

In order to formalize these ideas, we proceed by defining a banking system—portfolio of banks—comprising two banks; i.e., 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:5

C[p,q]=p(x,y)ln [p(x,y)q(x,y)]dxdy, where q(x,y) and p(x,y) 2.

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 model 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:

p(x,y)χ(xdx,)dxdy=PoDtx,p(x,y)χ(xdy,)dydx=PoDty(1)

where p (x, y) is the posterior multivariate distribution that represents the unknown to be solved. PoDtx and PoDty are the empirically estimated probabilities of distress (PoDs) of each of the banks in the system, and χ[xdx,), χ[xdy,) are indicating functions defined with thedistress 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:

L[p,q]=p(x,y) lnp(x,y)dxdyp(x,y)lnq(x,y)dxdy+(2)λ1[p(x,y)χ[xdy,)dxdyPoDtx]+λ2[p(x,y)χ[xdy,)dydxPoDty]+μ[p(x,y)dxdy1]

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 form

p(x,y)=q(x,y)exp {[1+μ+(λ1χ[xdx,))+(λ2χ[xdy,))]}(3)

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 Probability Integral Transformation (PIT) criterion.6 This is because when recovering multivariate distributions through the CIMDO approach, the available information embeded in the constraint set is used to adjust the “shape” of the multivariate density via the optimization procedure described above. 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). In Appendix 3, we provide a summary of the evaluation criterion and its results.

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; i.e., 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. For further details see Segoviano (2008).

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: (i) information of each random variable; i.e., the marginal distribution of each variable; and (ii) 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.7 For example, let x and y be two random variables with individual distributions xF, yH 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:

c[u,v]=g [F(1)(u),H(1)(v)]f [F(1)(u)]h[H(1)(v)],(4)

where g, f, and h are defined densities. From equation (4), we see that copula functions are multivariate distributions, whose marginal distributions are uniform on the interval [0,1]. Therefore, since each of the variables is individually (marginally) uniform; i.e. their information content has been sterilized, their joint distribution will only contain dependence information. Rewriting equation (4) in terms of x and y we get:

c[F(x),H(y)]=g[x,y]f[x]h[y],(5)

From equation (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 (5) is equal to one.

The copula approach to model dependence possesses many positive features when compared to correlations (See Box 1). In comparison to correlation, the dependence structure as characterized by copula functions, describes linear and non-linear dependencies of any type of multivariate densities, and along their entire domain. Additionally, 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. Lastly, 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; i.e., the copula choice problem (CCP). The CCP is in general a challenging task, since 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 are usually specified to remain fixed through time (see Appendix 1). Thus, the dependence structure that is characterized with parametric copula functions, although improving the modeling of dependence vs. correlations, still embeds the problem of characterizing dependence that remains fixed through time.8

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

Interdependencies of financial returns have been traditionally 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—that 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 XN (0, 1),Y = X2, since the third moment of the standard normal distribution is zero.

Nonlinear Transformations

In addition, linear correlation is not invariant under nonlinear strictly increasing transformations T:□ → □. For two real-valued random variables we have in general ρ (T (X),T (Y)) ≠ ρ (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 (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; i.e., extreme events, when evaluated empirically. Hence, since distress is characterized as a tail event, correlation is not an appropriate measure of distress dependence when marginal distributions of financial assets are non-normal. (De Vries, 2005).

1/ Empirical support for modeling financial returns with t-distributions can be found in Danielsson and de Vries (1997), Hoskin, Bonti, and Siegel (2000), and Glasserman, Heidelberger, and Shahabuddin (2000).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; e.g., 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 III.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, i.e., 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 non-linear 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 (see discussion below), where our main interest lies i.e., to characterize distress dependence.

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

  1. It circumvents the Copula Choice Problem. The explicit choice and calibration of parametric copula functions is avoided because the CIMDO-copula is extracted from the CIMDO-density (as explained above); 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, i.e., when only information of marginal probabilities of distress exist; 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 specially on the tail of the copula function, where distress dependence is characterized. See Appendix 3 for a summary of this evaluation criterion and its results.

  2. The CIMDO-copula avoids the imposition of constant correlation parameter assumptions. It updates “automatically” when the probabilities of distress 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’ probabilities of distress, 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 (3), Appendix 2 shows that the CIMDO-copula, cc (u,v), is represented by

cc(u,v)=q[Fc1(u),Hc1(v)]exp{[1+μ]}+q[Fc1(u),y]exp{λ2χxdy(y)}dy+q[x,Hc1(v)]exp{λ1χxdx(x)}dx(6)

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

Equation (6) shows that the CIMDO-copula is a nonlinear function of λ1 λ2 and μ, the Lagrange multipliers of the CIMDO functional presented in equation (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.

IV. Banking Stability Measures

The BSMD characterizes the probability of distress 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.

  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 superviors to identify how risks are evolving and where contagion might most easily develop.

For illustration purposes, and to make it easier to present definitions below, 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 III.A, we infer the CIMDO-density function, which takes the form:

p(x,y,r)=q(x,y,r)exp{[1+μ+(λ1χ[xdx,))+(λ2χ[xdx,))+(λ3χ[xdr,))]}.(7)

where q(x,y,r) and p(x,y,r) ∈ ℝ3.

A. Common Distress in the Banks of the System

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

The Joint Probability of Distress

The Joint Probability of Distress (JPoD) represents the probability of all the banks in the system (portfolio) becoming distressed, i.e., 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 (7) the JPoD is defined as P(XYR) and it is estimated by integrating the density (BSMD) as follows:

xdrxdyxdxp(x,y,r)dxdydr=JPoD(8)

The Banking Stability Index

The Banking Stability Index (BSI) is based on the conditional expectation of default probability measure developed by Huang (1992).9 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:

BSI=P(Xxdx)+P(Yxdy)1P(X<xdx,Y<xdy).(9)

The BSI represents a probability measure that conditions on any bank becoming distressed, without indicating the specific bank.10

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 probabilities of distress, which are presented in the Distress Dependence Matrix (DiDe). This matrix contains the probability of distress 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 (7), at a given date, the DiDe is represented in Table 1.

Table 1.

Distress Dependence Matrix

article image
Source: Authors’ calculations.

Where for example, the probability of distress of bank X conditional on bank Y becoming distressed is estimated by

P(Xxdx|Yxdy)=P(Xxdx,Yxdy)P(Yxdy).(10)

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 probabability 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

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 3). In this example, the PAO can be defined as follows:

Figure 3.
Figure 3.

Probability That At Least One Bank Becomes Distressed

Citation: IMF Working Papers 2009, 004; 10.5089/9781451871517.001.A001

Source: Authors’ estimations.
PAO=P(Y/X)+P(Z/X)+P(R/X)[P(YR/X)+P(YZ/X)+P(ZR/X)]+P(YRZ/X)(11)

V. 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.

  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, since cross-border financial linkages are growing and becoming increasingly significant, as has been highlighted by the financial market turmoil of recent months. 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 probabilities of distress of individual banks, which we describe below. 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 probabilities of distress (PoDs) of individual banks can be empirically estimated. The most well known include the structural approach (SA), PoDs derived from Credit Default Swap (CDS) spreads (CDS-PoDs), or from out-of-the-money (OOM) option prices. These alternative approaches present diverse advantages and disadvantages, in terms of availability of data necessary for their implementation, parametrization of quantitative techniques, and consistency of empirical estimations. We performed an extensive empirical analysis of these approaches.11 The SA presented significant difficulties for the proper parametrization 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 (i) lack of liquidity in the particular CDS market, and (ii) 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 due to 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, since 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, since 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, i.e., credit risk, is difficult, and disentangling these risks is a complex issue. Additionally, note that many times, CDS cover not only the event of default of an underlying security but a wider set of “credit events”, i.e., downgrades. We consider the combined effects of these factors, which are embedded in CDS spreads, to be “distress” risk, i.e., 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 Time12

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 US banks as it was up to October 2008 using only publicly available data. For this purpose, we focus on the largest U.S. banking groups. The bank holding companies (BHC) that are included are Citigroup, Bank of America, JPMorgan, and Wachovia. The investment banks (IB) included are Goldman Sachs, Lehman Brothers, Merrill Lynch, and Morgan Stanley.13 In addition to the major U.S. banks, we included Washington Mutual (WaMu) and AIG (a thrift and an insurance company both under intense market pressure in September 2008). The results can be summarized as follows:

Common Distress in the Banks of the System: BSI and JPoD

Distress Between Specific Banks: Distress Dependence Matrix

The DiDes presented in Table 2 show the (pairwise) conditional probabilities of distress 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 probabilities of distress have changed from a pre-crisis date to the day before Lehman Brothers filed for bankruptcy. We have also broken these matrices into four quadrants; i.e., 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 2.

Distress Dependence Matrix: American and European Banks

Distress Dependence Matrices, July 2007 and September 2008 1/

article image
Source: Authors’ calculations.

Probability of distress of the bank in the row, conditional on the bank in the column becoming distressed.

  • Links across major U.S. banks have increased greatly. This is clearly shown by the conditional PoDs presented in quadrant 1 of the DiDe’s presented in Table 2. On average, if any of the US 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.

  • On 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, Washington Mutual, 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.

Distress in the System Associated with a Specific Bank: Probability That At Least One Bank Becomes Distressed

On September 12, using equation (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 7). This analysis (perspective 3) is in line with the insights brought by the DiDe (perspective 2), which indicated Lehman’s distress would be associated with distress in several institutions.

Figure 7.
Figure 7.

PAO: Lehman

Citation: IMF Working Papers 2009, 004; 10.5089/9781451871517.001.A001

Source: Authors’ calculations.

C. Analysis of Cross-Region Effects Between 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 and HSBC from the UK, UBS and Credit-Suisse (CSFB) from Switzerland, and Deutsche from Germany).

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 probability of distress elsewhere in Europe. The European JPoD and BSI move in tandem with movements in the US indicators, coinciding also with relevant market events. (Figures 3 and 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 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 US IBs’ and very similar to those for US BHCs’ across time. Figures 3 and 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.

Distress Between Specific Banks: Distress Dependence Matrix

  • Links across major European banks have increased significantly (Table 2). This is clearly shown by the conditional PoDs presented in quadrant 4 of the DiDe matrices. 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 probability of distress conditional on UBS becoming distressed was estimated to reach 31 percent on September 12, 2008. This was a signifiant increase from 18 percent estimated on July 17, 2007.

  • Among the European banks under analysis, distress at Credit Suisse (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. The (average) PoD of American banks conditional on Deutsche falling into distress reached 35 percent. (quadrant 2, column-average Deutsche). This might be related to the high integration of Deutsche in some markets at the global level.

  • On September 12, 2008, while 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) probability of distress 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 Washington Mutual, 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, as indicated by the ALPoD.

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 country 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 probabilities of distress of banks and sovereigns. Therefore, such PoDs represent markets’ views of risks of distress for these banks and countries. While 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 country banks with larger regional presences in these regions. The countries and banks analyzed are:

  • Latin America. Countries: Mexico, Colombia, Brazil and Chile. Banks: BBVA, Santander, Citigroup, Scotia Bank and HSBC.

  • Eastern Europe. Countries: Bulgaria, Croatia, Hungary and Slovakia. Banks: Intesa, Unicredito, Erste, Societe Generale, and Citigroup.

  • Asia. Countries: China, Korea, Thailand, Malaysia, the Philippines, and Indonesia. Banks: Citigroup, JP Morgan Chase, HSBC, Standard and Chartered, BNP, Deutsche, and DBS.

The key observation from the analysis is that concerns about bank solvency and emerging market instability appear to be highly interlinked. The markets’ evolving views of risks of distress for these banks and countries, based on market credit default swaps, are presented in Figure 8.

Figure 8.
Figure 8.

Foreign-Bank and Sovereign Risks

Citation: IMF Working Papers 2009, 004; 10.5089/9781451871517.001.A001

Source: Authors’ calculations.

The level of risk in Latin America and Asia was falling in the run-up to the crisis, and was about to converge with the low level of concern about advanced country banks and Eastern Europe. However, since August 2008, both sovereign risk and bank risk have risen sharply and moved increasingly in tandem. This points to close interlinkages between concerns about bank solvency and emerging market instability.

Distress between specific banks: distress dependence matrix

To gain insight into these interlinkages and how they have evolved, in Tables 3, 4, and 5, we present the DiDe matrices estimated for Latin America, eastern Europe and Asia respectively. We have chosen July 1, 2007, and September 16, 2008; thus, we can show how conditional probabilities of distress have changed from a pre-crisis period to the post-Lehman episode. The DiDe matrices report probabilities that a bank/country in a row will become distressed if the bank/country in the column fails. Most importantly, as well as links across countries (bottom right, quadrant 4) and across banks (top left, quadrant 1) they report cross dependencies between these groups. The bottom left (quadrant 3) reports how bank problems can presage sovereign distress, while the top right (quadrant 2) indicates the opposite link. As noted above, while dependence does not imply causation, such correlations provide important insights. In particular:

Table 3.

Distress Dependence Matrix: Latin America. Sovereigns And Banks

article image
Source: Authors’ calculationsThese matrices present the probability of distress of the bank/sovereign in the row, conditional on the bank/sovereign in the column.
Table 4.

Distress Dependence Matrix: Eastern Europe. Sovereigns And Banks

article image
Source: Authors’ calculationsThese matrices present the probability of distress of the bank/sovereign in the row, conditional on the bank/sovereign in the column.
Table 5.

Distress Dependence Matrix: Asia. Sovereigns And Banks

article image
Source: Authors’ calculationsThese matrices present the probability of distress of the bank/sovereign in the row, conditional on the bank/sovereign in the column.
  • Cross dependencies have risen sharply over the crisis, implying that systemic risks have leapt. This is clearly seen when comparing the probability of distress of the sovereigns conditional on distress on the banks between July 2007, when sovereigns appeared to have low risk of contamination, and September 2008. For Latin America, the average conditional PoD increased from 39 to 51 percent, for eastern Europe from 16 to 46 percent and for Asia, from 12 to 34 percent (quadrant 3). The increase in the probability of distress of the banks, conditional on distress of the sovereigns, is even more significant, as eroding capital has increased exposure. For Latin America, the average conditional PoD of the relevant banks increased from 13 to 54 percent, for eastern Europe from 11 to 45 percent and for Asia, from 4 to 30 percent (quadrant 2).

  • Of particular interest is to see how bank problems can presage sovereign distress. We can see that before the crisis, the conditional probabilities of distress for Chile and Mexico were the lowest in Latin America (quadrant 3, row-average); however, on September 16, 2008, there was a significant increase for Mexico, making it the country with the second highest conditional PoD in its region. Chile also recorded an increase, although it still remains the country with the lowest conditional PoD in its region. In eastern Europe, on September 16, our model was already signaling that Bulgaria and Hungary were under significant stress. In Asia, Indonesia, Korea, and the Phillipines were at highest risk. However Slovakia and China remained the least stressed in those regions (quadrant 3, row-average).

  • Country distress conditional on bank distress appears high (quadrant 3). Distress of Spanish banks would be associated with the highest distress in Latin America and Italian banks in eastern Europe. Distress of Standard Chartered would be associated with significant stress in Asia (quadrant 3, column-average). These results suggest that geographic roles matter, since these banks have a substantial presence in the respective regions under analysis.

  • Moreover, these banks seem less dependent than others on country risks (quadrant 2, row-average). This seemingly paradoxical pattern is likely because the Spanish/Italian banks are seen as relatively safe, and hence their distress would signal larger systemic financial problems that would feed through to emerging markets in which they have a large presence.

  • Vulnerabilities between countries (quadrant 4) were highest in Brazil, Mexico, Bulgaria Hungary, Indonesia, Phillipines, and Korea (quadrant 4, row-average). Vulnerabilities between banks (quadrant 1) were highest for Citi, Scotia, Erste (quadrant 1, row-average).

  • Direct links between banks and countries matter. Distress in countries with a particularly large foreign bank presence—such as Mexico and Croatia—is highly linked with potential banking distress (quadrant 2). At a more specific level, direct links from individual banks to countries also matter—for example, distress at Citigroup, Intesa, and DBS are relatively more important for Mexico, Hungary, and Indonesia, respectively, than for other countries (quadrant 3).

  • These results also illustrate how the leap in systemic risk, and hence indirect links, has affected Asia, over and above direct regional and bilateral ties. This is because direct ownership and lending by foreign banks is generally lower in Asia than in eastern Europe or Latin America. Since the banking systems are more insulated from these direct links, the results are more likely to reflect indirect ones through overall risks linked to bank/sovereign distress. In addition, links between banks may be somewhat less important for emerging Asia, as borrowing through equity markets tends to play a larger role in local financial markets. Thus, powerful indirect effects appear evident in all cases, particularly for Korea and Indonesia.

  • An important strength of our approach is that market prices reflect perceptions of direct links and indirect links. For the former, market presence might be an important element, as in Latin America and eastern Europe; however, for the latter, liquidity pressures and systemic banking distress/macroeconomic spillovers might play an important role. This feature of our approach appears to be particularly relevant in Asia.

The results confirm that systemic bank risks and emerging market vulnerabilities are highly dependent. This likely largely reflects the fact that distress in individual banks is a bellweather for the state of the overall financial system, via direct or indirect links. In financial terms, the world is increasingly a global village. The bottom line is that policies to limit systemic risks in advanced country financial systems would also sharply reduce risks to emerging markets.

VI. Conclusions

The purpose of this paper is 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, i.e., 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, both developed and developing, as long as there is reasonable data to reflect individual banks’ PoDs.

  • It embeds the banks’ default interdependence structure (copula function), which captures linear and non-linear 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 paper applied this methodology to a number of country and regional examples using publicly available information up to October 2008. This implementation flexibility is of relevance for banking stability surveillance, since cross-border financial linkages are growing and becoming significant, as has been highlighted by the financial market turmoil of recent months. 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. For example, by tryiun to predict future movements of the BSMs for use as an early-warning mechanism; exploring the significant macroeconomic and financial factors and shocks influencing the BSMs, in order to identify macro-financial linkages; and exploring the factors that can limit and reverse tendencies towards instability, so as to discover what instruments may be available (and under what conditions) to control such instability.

Appendix I. Copula Functions

Let x and y be two random variables with individual distributions xF, yH and a joint distribution (x, y) □ G. The joint distribution contains three types of information. Individual (marginal) information on the variable x, individual (marginal) information on the variable y and information on the dependence between x and y. In order to model the dependence structure between the two random variables, the copula approach sterilizes the marginal information on x and y from their joint distribution; consequently, isolating the dependence structure. Sterilization of marginal information is done by transforming the distribution of x and y into a uniform distribution; U(0,1), which is uninformative. Under this distribution the random variables have an equal probability of taking a value between 0 and 1 and a zero probability of taking a value outside [0,1]. Therefore, this distribution is typically thought of as being uninformative. In order to transform x and y into U(0,1) we use the Probability Integral Transformation (PIT), presented in Appendix 3.

Under the PIT, two new variables are defined 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 (Cassella and Berger, 1990), the copula function c[u,v] is defined as:

c[u,v]=g[F(1)(u),H(1)(v)]f[F(1)(u)]h[H(1)(v)],(12)

where g, f, and h are defined densities.

From equation (12), we see that copula functions are multivariate distributions, whose marginal distributions are uniform on the interval [0,1]. Therefore, since each of the variables is individually (marginally) uniform (i.e. their information content has been sterilized via the PIT), their joint distribution will only contain dependence information. Rewriting equation (1) in terms of x and y we get

c[F(x),H(y)]=g[x,y]f[x]h[y],(13)

From equation (13), 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. Therefore, if the variables are independent, equation (13) is equal to one.

Sklar’s Theorem

The following theorem is due to Sklar (1959) and is known as Sklar’s Theorem. It is a relevant result regarding copula functions, and is used in all applications of copulas. Let G be a joint distribution function with marginals F and H. Then there exists a copula C such that for all x, y in □ ,

G[x,y]=C[F(x),H(y)].(14)

If F and H are continuous, then C is unique; otherwise, C is uniquely determined on RanF x RanH. Conversely, if C is a copula and F and H are distribution functions, then the multivariate function G defined by equation (14) is a joint distribution function with univariate margins F and H. Then, the dependence structure is completely characterized by the copula C (Nelsen, 1999). Nelsen also provides the following corollary to Sklar’s theorem.

Corollary: Let G be any joint distribution with continuous marginals F and H. Let F(−1) (u), H(−1) (v) denote the (quasi) inverses of the marginal distributions. Then there exists a unique copula C: [0,1] x [0,1]→[0,1] such that, g [F(−1) (u), H(−1) v) ∀ ∈ [0,1] x [0,1]. If the cross partial derivatives of equation (3) are taken, we obtain:

g[x,y]=f[x]h[y]c[F(x),H(y)].(15)

The converse of Sklar’s theorem implies that we can couple together any marginal distributions, of any family, with any copula function and a valid joint density will be defined. The corollary implies that from any joint distribution we can extract the implied copula and marginal distributions (Nelsen, 1999).

Parametric Copula Functions

In the finance literature, it is common to see the Gaussian-copula and the t-copula for modeling dependence among financial assets. These are defined as follows (Embrechts, Lindskog, McNeil, 2001):

Gaussian-copula: The copula of the bivariate normal distribution can be written as:

CRGa(u,v)=Φ1(u)Φ1(v)12π(1ρ2)1/2exp{s22ρst+t22(1ρ2)}dsdt.(16)

Where ρ is the linear correlation coefficient of the corresponding bivariate normal distribution, and Φ−1 denotes the inverse of the distribution function of the univariate standard normal distribution.

t-copula: The copula of the bivariate t-distribution with υ degrees of freedom and correlation ρ is:

Cυρt(u,v)=tυ1(u)tυ1(v)12π(1ρ2)1/2{1+s22ρst+t2υ(1ρ2)}(υ+2)/2dsdt.(17)

Where tυ1(v) denotes the inverse of the distribution function of the standard univariate t-distribution with υ degrees of freedom. As it can be seen, this copula depends only on ρ and υ.

Appendix II. CIMDO-copula

In order to provide a heuristic explanation of the CIMDO-copula, we compare the copula of a bivariate CIMDO-distribution and a bivariate distribution of the form that the prior density in the entropy functional would be set; e.g., a t-distribution. First, we recall from equation (1) that copula functions were defined as

c[u,v]=g[F(1)(u),H(1)(v)]f[F(1)(u)]h[H(1)(v)].

Assume that the prior has a density function q (x, y). Hence, its marginal cumulative distribution functions are of the form, u=F(x)=x¯+q(x,y)dydx, andv=H(y)=x¯+q(x,y)dxdy; where u = F(x) ⇔ x = F−1(u), and v = H (y) ⇔ y = H−1(v). Therefore, its marginal densities are of the form, f(x)=+q(x,y)dy, and h(y)=+q(x,y)dx.

Substituting these into the copula definition we get, the copula of the prior, cq (u, v),

cq(u,v)=q[F1(u),H1(v)]+q[F1(u),y]dy+q[x,H1(v)]dx.(18)

Similarly, assume that the CIMDO distribution with q(x,y) as the prior is of the form, p(x,y)=q(x,y)exp{[1+μ+(λ1χ[xdx,))+(λ2χ[xdy,))]}. We also define u=Fc(x)x=Fc1(u), and v=Hc(y)y=Hc1(v).

Its marginal densities take the form, fc(x)=+q(x,y)exp{[1+μ+(λ1χxdx(x))+(λ2χxdy(y))]}dy, and hc(y)=+q(x,y)exp{[1+μ+(λ1χxax(x))+(λ2χxdy(y))]}dx.

Substituting these into the copula definition we get, the CIMDO-copula, cc(u, v),

cc(u,v)=q[Fc1(u),Hc1(v)]exp{[1+μ]}+q[Fc1(u),y]exp{λ2χxdx(y)}dy+q[x,Hc1(v)]exp{λ1χxdx(x)}dx.(19)

Equation (14) shows that the CIMDO-copula is a nonlinear function of μ, λ1, and λ2, which change as the PoDs of the banks under analysis change. Therefore, the CIMDO-copula captures changes in PoDs, as these change at different periods of the economic cycle.

Appendix III. CIMDO-Density and CIMDO-copula Evaluation Framework

We consider extremely important the evaluation framework used to assess whether CIMDO-distributions and CIMDO-copulas provide improvements over the most commonly parametric distributions, copulas used for portfolio credit risk modeling. In order to evaluate these densities, we follow the Probability Integral Transformation approach (PIT) developed by Diebold et al (1999). In what follows in this section, we provide a summary of the detailed theoretical results supporting the chosen evaluation criterion. A complete development and Monte Carlo exercise can be found in (Segoviano 2006b).

The Probability Integral Transformation Approach (PIT)

Density evaluation is not a trivial problem, since there is no way to rank two incorrect density forecasts such that all users will agree with the ranking. Ranking depends on the specific loss functions of the users.14

However, Diebold et al. (1998) assert that if a forecast coincides with a random variable true data generating process (DGP), then it will be preferred by all forecast users, regardless of loss function. Thus this proposition implies that, regardless of the users’ loss function, the correct density is weakly superior to all forecasts. As a result, Diebold et al. (1998) suggest evaluating forecasts by assessing whether the forecast densities are correct; i.e., by assessing whether the forecast coincide with the true DGP.

Nonetheless, the task of determining whether a forecast equals the true DGP appears to be difficult because the true DGP is never observed. Moreover, the true density may exhibit structural changes at every point in time. To overcome this difficulty, they developed a method based on the Rosenblatt (1952) Probability Integral Transformation (PIT).

Diebold et al. (1998) prove that the series of the Probability Integral Transformations of a series of density forecasts are distributed iid U(0,1) if the density forecasts coincide with the true DGP. Thus, to assess whether a series of density forecasts coincides with the corresponding series of true conditional densities, it is only necessary to test whether the sequence of density forecast PITs are iid U(0,1).

Diebold, Hahn, and Tay (1999) extend their results to the Mth-multivariate case. Suppose there is a series of T, Mth-multivariate density forecasts. They factorize each period’s t, joint forecast density into the product of their conditionals:15

pt1(lt1,,ltM)=pt1(ltM|ltM1,..lt1)..pt1(lt2|lt1)pt1(lt1).

This procedure produces a set of (M-1) conditionals and (1) marginal density. The PIT’s of the lm random variable realizations under these M series will be iid U(0,1), individually and also when taken as a whole, if the multivariate density forecasts are correct. The proof of this assertion in a time series framework can be found in Diebold et al. (1999).

CIMDO: Density Evaluation

However, we needed to develop a slightly different test. This is because CIMDO recovers densities with information at each period of time t.16 Thus, the density evaluation needs to be done at specific periods of time t. Alternatively, rather than evaluating a time-series as Diebold et al (1999) do, we would like to evaluate cross-sectionally the multivariate distribution. As a result, we need to prove that at each point in time t, the product of the conditionals and marginal PIT’s on which a multivariate distribution can be decomposed is iid U(0,1). This proof is not explicitly presented in Diebold et al (1999); therefore, we develop the proof in Segoviano (2006b)

In the interest of parsimony, we focus on a portfolio containing two different types of assets (e.g., types of loans or in the case of this paper, two different banks), whose logarithmic returns are characterized by the random variables x and y, where x, y, ∈ li s.t. i = 1,..,M. The portfolio bivariate density can be decomposed into,

p(x,y)=p(x)p(y|x),(20)
p(x,y)=p(y)p(x|y).(21)

We start by analyzing the first case presented in equation (20). Under the PIT, two new variables are defined as

u=P(x)x=P(1)(u),v=P(y|x)y=P(1)(v|x).<