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Author's personal copy Journal of International Money and Finance 32 (2013) 990–1007

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Journal of International Money and Finance journal homepage: www.elsevier.com/locate/jimf

Marking-to-market government guarantees to ﬁnancial systems – Theory and evidence for Europe Angelo Baglioni a, *, Umberto Cherubini b a b

Catholic University of Milan, Largo Gemelli, n. 1, 20123 Milano, Italy University of Bologna, Italy

a b s t r a c t JEL codes: G21 H63 Keywords: Bank bail out Government budget Systemic risk Financial crisis

We propose a new index for measuring the systemic risk of default of the banking sector, which is based on a homogeneous version of multivariate intensity based models (Cuadras–Augé distribution). We compute the index for 10 European countries, exploiting the information incorporated in the CDS premia of 44 large banks over the period January 2007–September 2010. In this way, we provide a market based measure of the liability incurred by the Governments, due to the implicit bail-out guarantees they provide to the ﬁnancial sector. We ﬁnd that during the ﬁnancial crisis the systemic component of the default risk in the banking sector has signiﬁcantly increased in all countries, with the exception of Germany and the Netherlands. As a consequence, the Governments’ liability implicit in the bail out guarantee amounts to a quite relevant share of GDP in several countries: it is huge for Ireland, lower but still important for the other PIIGS (Italy is the least affected within this group) and for the UK. Finally, our estimate is very close to the overall amount of money already committed in the rescue plans adopted in Europe between October 2008 and March 2010, despite strong cross-country differences: in particular, Germany and Ireland seem to have committed an amount of resources much larger than needed; to the contrary, the Italian Government has committed much less than it should. 2012 Elsevier Ltd. All rights reserved.

* Corresponding author. Tel.: þ39 02 72344024; fax: þ39 02 72342781. E-mail addresses: [email protected] (A. Baglioni), [email protected] (U. Cherubini). URL: https://sites.google.com/site/angelobaglionihomepage/Home 0261-5606/$ – see front matter 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jimonﬁn.2012.08.004

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1. Introduction In the current phase of the ﬁnancial crisis, the question attracting most of attention and debate is the mutation and transmission of the credit virus from the banking system, where the epidemic started, to the Governments’ balance sheets. According to some interpretations of the crisis, the diffusion of the virus across ﬁnancial institutions and markets was magniﬁed by the accounting rules that were imposing to mark-to-market assets and liabilities of all the players in the ﬁnancial system. Someone could argue, with somewhat of a joke, that the most acute phase of the crisis was over when the problem exposures ended up in the only budget that is not evaluated at fair-value yet, namely the Government budget. This argument calls for a question: if the implicit guarantee provided by each Government to its ﬁnancial system were marked-to-market, what would be the impact on public debt ﬁgures? This paper addresses this question, focussing on the situation of Europe. Addressing this question amounts to asking how much each Government should pay in order to buy insurance against default of one ﬁnancial institution or even of the whole ﬁnancial system. This can be done, as it is done for budgets of companies, by extracting this cost from market quotes, in this case the values of Credit Default Swaps (CDS) written on “names” of ﬁnancial institutions. Notice that this question has nothing to do with arguments concerning whether the CDS markets provide faithful representations of credit risk, or whether their forecasting power is hampered by the risk premium embedded in them. No matter if one ﬁnds this risk premium excessive or not, this is the market price a Government should pay to buy protection against default of the ﬁnancial system. This issue is relevant not only for academics and ﬁnancial practitioners. It is becoming increasingly important in the policy debate on the reform of the Stability Pact in Europe, where some policy-makers argue that private debt should be included in the evaluation of the ﬁnancial soundness of member countries. Our paper contributes to this debate by introducing a methodology to estimate – using market data – the burden to be added to the debt/GDP ratio in each European country, due to the implicit bail-out guarantee that Governments provide to the ﬁnancial sector. Following the dramatic consequences of the Lehman Brothers default, many Governments have announced and implemented rescue plans for the ﬁnancial sector, creating the expectation of a wide bail-out policy. This expectation clearly emerges from CDS quotes: the default risk priced by the market shows a remarkable comovement between the ﬁnancial sector and the Government sector. The plan of the paper is the following. After reviewing the literature most closely related to our work (Section 2), we lay out a model of credit risk for the banking sector (Section 3), which provides the framework for our empirical methodology to measure the fragility of the ﬁnancial sector of a country: we introduce here a new ﬁnancial stability index (named “Cuadras–Augé index”). In Section 4 we present our empirical evidence, using a sample of 44 large banks located in 10 European countries: among other things, we provide here an estimate of the liability incurred by the European Governments, due to the implicit bail-out guarantee. Section 5 summarizes our main results. The detailed derivation of the ﬁnancial stability index is postponed to Appendix 1. Finally, Appendix 2 provides detailed information on our sample. 2. Related literature The idea that the implicit guarantee of bail out, provided by the Government to the ﬁnancial sector, should be taken into account in the balance sheet of the public sector has been developed by Gray et al. (2006), and by Gapen et al. (2005). They apply the contingent claim analysis, where the bail out guarantee – particularly relevant for the “too-big-to-fail” intermediaries – is modelled as a put option enabling a bank to sell its own assets to the Government, which pays a strike price equal to the value of the bank liabilities backed by the guarantee. They provide a methodology for quantifying the value of the put option, which should be included in the asset side of the balance sheet of ﬁnancial intermediaries and in the liability side of the public sector balance sheet. Although different from our work on technical grounds – they take a structural approach while we use an intensity based model – they share with us the idea that an accurate measurement of sovereign risk cannot ignore the interlinks between the different sectors of the economy, in particular between the ﬁnancial and the public sectors. The recent ﬁnancial crisis has dramatically increased the need to assess the impact of the bail out measures taken by several Governments in order to avoid the collapse of the ﬁnancial system. Through

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these measures – capital injections, debt guarantees, asset purchases and asset guarantees – the public sector has committed a huge amount of resources in several countries (we report some data below in Section 4.3). The overall impact of this kind of interventions has been positive as far as the credit risk of ﬁnancial institutions is concerned. BIS (2009) shows that the announcement and the implementation of rescue packages have been followed by a fall in bank CDS spreads, so they have indeed been able to reduce the default probability of banks as perceived by the market. However, they have at the same time increased the market price of sovereign risk. Ejsing and Lemke (2009) show that, following the announcement of rescue packages in the fall of 2008, a marked increase of sovereign CDS spreads has come along with the reduction of bank CDS spreads. The analysis by Berndt and Obreja (2010) is closely related to ours. They study the correlation structure of weekly CDS returns for a large sample of European ﬁrms over the period 2003–2008, showing that the co-movement among them increases dramatically after the onset of the ﬁnancial crisis in August 2007. They trace back this result to the larger weight of a factor mimicking the economic catastrophe risk. These ﬁndings are in line with ours, which point to a signiﬁcant increase of the systemic component in the default risk of European banks during the ﬁnancial crisis, as we shall see below. Finally, Huang et al., 2009 propose a framework to extract the systemic risk of ﬁnancial institutions from CDS market spreads. Their work share our same goal. Nevertheless, our model innovates on that approach on theoretical grounds, dropping the assumption that the public guarantee covers all the losses incurred by the banking system in a systemic crisis. We base our analysis on a model in which the moral hazard arising in this situation is addressed, showing that only partial insurance is incentive compatible. On technical grounds, we propose a fully intensity-based technique to extract the systemic risk from the data, while keeping the structural model leading to Gaussian dependence as a benchmark. 3. The model In this section we present a model of credit risk for the banking sector, with the aim of assessing the expected liability incurred by the government providing a bail out guarantee to the banks of the country. The task is to represent the banking system of each country as a homogeneous set of obligors. We remind that a credit portfolio, or a set of obligors, is called homogeneous, if each obligor has the same default probability and the same dependence with all the other obligors. For this reason, we present the model in two steps. We ﬁrst present a simple univariate model (subsection 3.1) of the representative bank of each country, where the whole banking system is supposed to share the same risk proﬁle. Then we turn to a multivariate version of the model addressing the question of the joint probability of default of the banks in the system, comparing a structural approach and an intensity based approach (3.2). This will provide the basis for the analysis of the probability of systemic shock and the burden that it would put on the Government budget.

3.1. A univariate model of bank bail out The task of this model is to derive a threshold for the insurance coverage provided by the Government to the banking system, consistently with the requirement of ruling out moral hazard behaviour from the bank. It is in fact straightforward to observe that if the Government were to provide full insurance to banks, the banks’ managers would have the incentive to take more risk. This model then tries to capture in the simplest way the idea that the moral hazard, created by a bail out policy of banks, can be avoided by the government by committing only to a partial insurance coverage of the possible losses incurred by banks. In particular, it is essential that the government guarantee does not rule out the losses accruing to shareholders. Consider a representative bank and a Government. The bank is run by a (risk neutral) shareholder, who owns the whole equity and tries to maximize the expected value of his claim on the bank.1 He can x2 . The ﬁrst one takes value x1 > 0 with probability p1 choose between two investment strategies: ~ x1 , ~

1

We are of course abstracting from any issue arising from the separation between ownership and control.

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and zero otherwise.2 The second one has a similar return structure, with the following assumptions: x2 > x1, p2 < p1, and p2x2 < p1x1. So strategy one is clearly more efﬁcient, since it has a larger expected value; a self-ﬁnanced ﬁrm would trivially choose strategy one. However, strategy two gives a higher return in the positive state of nature, and this might give a shareholder the incentive to take that strategy if he is indebted. By deﬁnition, a banking ﬁrm has some debt, say D. The bank shareholder’s claim can be written as: max[xi D,0], for i ¼ 1,2. The expected value of this claim is: pi(xi D). It can be easily veriﬁed that the shareholder will choose strategy one iff:

p x p2 x2 D < D h 1 1 p1 p2

(1)

In the following, we make the assumption that it is D < D*, so the bank shareholder takes the efﬁcient strategy: so far, no moral hazard issue has emerged. Moral hazard comes into the picture if the Government provides a bail out guarantee, that we design as an insurance on the value of bank assets: in the low state of nature, the Government takes care of a share g˛[0,1] of the losses hitting bank assets. As a consequence, the insured value of bank assets in the low state becomes: gxi, for i ¼ 1,2. It is immediate to see that a complete insurance (g ¼ 1) induces the bank to take the inefﬁcient strategy (2), since the shareholder’s payoff is (xi D) for sure. In other words, a complete bail out guarantee would generate moral hazard. Note that this choice hurts the Government, since its expected liability is (1 pi)gxi, and this is larger if i ¼ 2 (for any given g). What is the maximum amount of guarantee compatible with the bank taking the more efﬁcient investment strategy? The answer comes from the following incentive compatibility (IC) constraint:

p1 ðx1 DÞ þ ð1 p1 Þ½gx1 D p2 ðx2 DÞ þ ð1 p2 Þ½gx2 D

(2)

which gives the condition g g*, where:

p1 x1 p2 x2 g h ð1 p2 Þx2 ð1 p1 Þx1

(3)

and g*˛[0,1] under our assumptions. The above IC constraint has been written under the implicit assumption that gx1 D (implying that gx2 D holds as well). Therefore lenders are fully insured. However, what is crucial in this framework is that shareholders are given only a partial insurance, so that their payoffs internalize the consequences of risky choices. In particular, if g*x1 D, our framework shows that it is possible (by setting g g*) to provide full insurance to lenders and avoid moral hazard at the same time. If the assumption that g*x1 D does not hold, we have two cases. First, if g*x2 < D, for any g g* the IC constraint becomes p1(x1 D) p2(x2 D), which is satisﬁed by assumption. In this case the guarantee provides a partial insurance to lenders, and no insurance at all to shareholders. Hence the IC constraint is trivially met, thanks to the assumption that D < D*. In the intermediate case where g*x2 D g*x1, for values of g close to g* the IC constraint becomes:

p1 ðx1 DÞ p2 ðx2 DÞ þ ð1 p2 Þ½gx2 D

(4)

and the new threshold value for g is:

p x p2 x2 þ Dð1 p1 Þ b gh 1 1 ð1 p2 Þx2

(5)

So it is still possible to avoid moral hazard by setting g b g : since in this case the shareholder chooses strategy one, it turns out that in equilibrium lenders are only partially insured and the shareholder is not actually given any insurance.

2 The choice of assuming zero recovery rate is without loss of generality, and it helps to highlight the link between Government guarantee and moral hazard. Thanks to this assumption, in fact, we are able to deﬁne a base model with no moral hazard at all, and to study how it could arise from a bail out policy taken by the Government.

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Summing up, a government guarantee providing full insurance of bank assets generates moral hazard, by inducing shareholders to take an inefﬁcient strategy, leading to a lower expected value of bank assets and to a higher default probability. This moral hazard can be avoided by limiting the guarantee to a fraction of the value of bank assets. Despite its simplicity, the model is able to identify (in principle) the maximum amount of incentive compatible insurance. If the leverage of the bank is low enough, it is still incentive compatible to provide full insurance to lenders (depositors and bond holders), provided that shareholders are only partially insured. To the contrary, for higher levels of leverage the incentive compatible guarantee must provide a partial insurance to lenders, and no coverage at all for shareholders. It is quite clear that in the case of ﬁnancial intermediaries and banks in particular, for which the degree of leverage is structurally high, the latter case is the most likely, and so in practical applications the need arise for the identiﬁcation of a plausible level for the bail-out threshold b g. 3.2. A multivariate intensity based model Building on the univariate model described above, we now build a model of the banking sector of the economy. To keep the model simple, and amenable to calibration from market data, we assume that the banks in the system are a homogeneous set of obligors. This means that each bank faces that same investment problem described in Section 3.1 and as a result it has the same default probability. Furthermore, this implies that there is a single factor, that in the model is the state of the economy, that brings about multiple default of the obligors in the system. Apart from this factor, it is assumed that default of each obligor may be determined by idiosyncratic factors, that is factors that are speciﬁc and unique to each obligor, and that are independent across different obligors. In more technical words, this approach is also called conditionally independent default model, meaning that conditional on a common factor, the credit drivers that cause default of the obligors are independent from each other. We now lay out formally the speciﬁcation of our model, and the choices that will be made in its calibration. The ﬁrst choice, that refers to the speciﬁcation of the default probability of each obligor, is between the class of structural models and reduced form (or intensity based) models. In fact, it is well known that credit risk models can be built either on the basis of economic information on the business activity and the balance sheet of the obligors, or on statistical information based on the probability distribution of the default event (default probability, DP) and the loss incurred in case of default (loss given default, LGD). The former class of models is referred to as the structural approach to credit risk, while the second is known as the class of reduced form or intensity based models. In cases in which the default of the obligor may occur for reasons that do not depend, or not depend only, on the balance sheet structure, the reduced form approach is clearly preferable. This is actually the case when ﬁnancial institutions are the obligors. As recent experience proves (and the ancient conﬁrms), the default of a ﬁnancial institution may occur for many reasons that are not strictly dependent on the balance sheet, such as a panic or a bank-run, or contagion and liquidity crises. For the reasons above, we then settle for an intensity-based representation of the default probability of the bank. We then assume that the default event may occur at any time and it is described by a Poisson l. This means that the time at which default occurs is a variable endowed with an process with intensity b exponential distribution, so that the probability of obligor i to survive beyond time T is given by

l ðT tÞ Pðq > TÞ ¼ exp b

(6)

where q denotes the default time of the obligor. The default probability is of course the complement to one of the survival probability. Notice that the assumption of a homogeneous banking system is embedded in the fact that the intensity is the same for all the banks in the system. Also notice that this representation also gives a time reference to the model in section 3.1, which only considered a binomial setting for simplicity. In a sense, this can be considered a slight further speciﬁcation of that model in which we state that the realization of the state of the economy, and default of the obligor, may arrive at a random time in the future. Before extending the approach to the multivariate setting, a comment is needed concerning the modelling choices for the intensity parameter, that is the only parameter in the univariate model. The

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choice that is typically made in the market, at that we also make, is to consider the intensity parameter as a constant. We stick to this simpliﬁcation because our main task is to investigate the risk and the dependence structure of the “average” bank in the system, and making the assumptions on the marginal default probability more complex would only make the multivariate model much more involved while adding little to the description of the implementation strategy and the empirical results of the cross-country comparison. Nevertheless, it may be important to remind the reader of the choices available for an “industrial” implementation of the model. The assumption of a constant intensity may be dropped in two different ways. First, we may assume that intensity is a variable that reacts to changes in the economic setting: in the simplest example, it may be assumed to co-move with some indicator of the business cycle. This could be implemented by simply specifying a regression model for the intensity of each obligor, or for the “average” obligor of each country. Second, we may assume that the intensity of each obligor is a process, rather than a variable, and specify a particular dynamics for it. This would change the model from a Poisson process to a Cox process (see Lando, 2004 for a review). To implement this, one would need, for each date, the price of insurance against default of each obligor for different maturities. From these prices one could recover, by a standard algorithm called bootstrapping, different intensity values for different maturities. Notice that the two extensions have different meanings: the ﬁrst allows the level of intensity to move in response to other variables across time, while preserving the same term structure shape of the average intensity across different maturities (a ﬂat shape); the second allows for shapes of the intensity term structure different from the ﬂat one. In both cases, implementing this in a multivariate, cross-country model would be very costly. For the reasons discussed above, and for the fact that our target in this paper is to describe the credit risk of the “average” obligor of each country, implying a substantial degree of approximation, in our model we decided to stick to the standard constant intensity representation. The next step is to model the dependence among the banks in the system. The standard tool used in the literature, and in the industry, to model credit risk dependence is represented by copula functions (see Cherubini et al., 2012 for a review of the basic concepts and applications to ﬁnance). A copula function enables to specify the joint probability of survival, or default, of two, or several obligors in the system, simply taking the marginal survival or default probabilities as input. In our model, the homogeneity assumption implies that the dependence structure must be the same for any subset of obligors. So, without loss of generality, it is sufﬁcient to represent the bivariate case, so that for every couple of obligors, i and j, we have

P qi > T; qj > T ¼ C Pðqi > TÞ; P qj > T

(7)

where C(.,.) is the bivariate copula function specifying the dependence structure in the default of the two obligors. Notice that homogeneity allows C(.,.) to be the same for all the pairs of obligors in the system. Allowing for heterogeneous dependence would lead to a complex (and unsolved) problem of multivariate analysis, called compatibility (see Joe, 1997). Furthermore, notice that since we assume to work in a system of “average” obligors, we are interested in the function C(u;v) only on the subset of values u ¼ v, which is called the trace of the copula function. In this paper, we use the copula function stemming from the simplest multivariate extension of an intensity based approach: this is called the Marshall and Olkin (1967) model. This model was ﬁrst applied to credit risk by Esposito (2002), and was recently extended to a hierarchical structure by Durante et al. (2009). The idea of the model is very simple: the intensity of the average obligor is the sum of two parts:

b l ¼ lþl

(8)

where l is a component that is common to all the obligors, and l is the component speciﬁc to each one of them. Given this structure, it is possible to prove (see Nelsen, 2006) that the copula function in equation (7) takes the shape

Cðu; vÞ ¼ uvmin ua ; va

(9)

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where the parameter a is given by

a¼

l

b l

(10)

The copula is called exchangeable3 Marshall–Olkin copula, or Cuadras and Augé (1981) copula. The parameter a may be considered a measure of the systemic content of the credit risk of the average obligor in the system, that is how much of the instantaneous probability of default of the average bank is due to a systemic event: the closer the parameter to 1, the higher the relevance of a systemic shock for the default of the average obligor. For comparison, it may be useful to evaluate this homogeneous market model against the standard approach used in the market. The standard approach is due to Vasicek (1991), and is based on the speciﬁcation of Gaussian copula, namely:

Cðu; vÞ ¼ F2 F1 ðuÞ; F1 ðvÞ; r

(11)

with F2(x,y;r) the bivariate normal distribution with standardized marginals and correlation equal to r, and F(x) is the univariate standard normal distribution. Based on this assumption, and the limit central theorem, Vasicek (1991) derives an asymptotic formula for the percentage loss on a portfolio. 3.3. The Cuadras–Augé ﬁnancial stability index Having laid out the basics of the multivariate homogeneous intensity based model, we propose here the methodology of our empirical analysis (see Appendix 1 for analytical details). Our task is to apply a methodology to approximate a banking system by a homogeneous one with same average credit standing and same average dependence. For this, we use a synthetic ﬁnancial stability index representing both the average level of risk of each ﬁnancial institution in the system and the correlation among them. The index should be able to represent the average degree of systemic and idiosyncratic risk in the ﬁnancial system of a given country. As a warning, remember that this task is entirely different from that of specifying and estimating the joint probability distribution of the system. When we build a ﬁnancial stability index what we are actually doing is to adapt a model, typically a homogeneous credit exposure model, to provide a close enough, albeit synthetic representation of risk. As the most famous example, one of the ﬁrst indexes that was proposed in the industry, from the Moody’s rating agency, was called the diversity score. This index provided a representation of the credit risk of a portfolio in terms of a basket of homogeneous independent exposures. In this case, homogeneous only means that each exposure has the same probability of default, that is actually meant to represent the average credit risk in the basket. Other ﬁnancial stability indexes are computed and published by the IMF on a regular basis. Here we propose a new ﬁnancial stability index, which is based on the homogeneous version of the multivariate intensity based models. For this reason we call it “Cuadras–Augé index”. The index is l which is the same for all the exposures in the synthetically represented in terms of a marginal intensityb set, and a dependence structure that is the same across all pairs of exposures (measured either in terms of concordance index, rank correlation or the a parameter). A natural choice to calibrate the index is to set: i) the marginal intensity equal to the average marginal intensity of the basket, and ii) the correlation ﬁgure equal to some average of the ﬁgures in the correlation matrix. Collapsing the correlation matrix in a single ﬁgure is common usage in the basket credit derivatives market, where for some products the implied correlation is also used as a quoting device. The main difference with respect to our model is that while the Gaussian copula is typically used in the market, here we apply the Cuadras–Augé one. It is also worth noting that there is an interesting symmetry in the fact that while the Gaussian copula correlation stems directly from structural models of credit risk, the Cuadras–Augé is the direct offspring of intensity based models.

3

We remind that the concept of exchangeability means the property P(u,v) ¼ P(v,u), and was introduced by De Finetti in the 30s.

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4. Empirical analysis 4.1. The data set We report here a general description of the data set, which is described in greater detail in Appendix 2 (Table A1). We collected data for the banking systems of 10 European countries, with a total of 44 banks. The selected banks were those used for the stress test exercise performed by the Committee of European Banking Supervisors (CEBS) in July 2010. From this sample, we dropped those banks for which Credit Default Swap data were not available. Our sample includes the banks with the highest systemic relevance in Europe, since all the major institutions are included, accounting for a large market share in each country. For all the banks in the sample, as well as for Governments, we collected a data set of CDS spreads for the 5 year maturity; the sample period goes from January 2007 to September 2010. The choice to use CDS was made to make the sample homogeneous, meaning that the quotes for banks of different countries are taken from the same market. The alternative of using bond spreads would have required to estimate, from different markets, an issuer-speciﬁc curve for each and every bank in the sample and for each day. Moreover, we remind that the technical structure of the CDS, which is a derivative contract having the whole debt of an obligor as the underlying asset, is considered by market participants the best possible representation of his credit standing. Finally, it is well known that the CDS and the bond markets are tightly linked by an arbitrage relationship, which is technically known as “basis” (see Choudhry, 2006). The marginal intensities were computed using the standard approximation rule

b l

it

¼

CDSit LGD

(12)

where CDSit denotes the CDS quote of the i-th name at time t. By market convention, the reference LGD ﬁgure used to compute the price is set equal to 60%. Once the ﬁve-year intensity ﬁgure has been extracted, we computed survival probabilities of all the names in the sample. Then, for each country we reckoned the rank correlation ﬁgure using a rolling window of one year of data. This way, for most of the countries we have a time series of rank correlation matrices starting from January 2008. In cases in which data quality was particularly poor, the time series started later. In all cases, we have all the data for 2009 and 2010 (with the slight exception of Greece for which the ﬁrst quarter is missing). Having done this, we applied the Cuadras–Augé ﬁlter described in Section 3.3 to represent an index of the systemic risk component. Concerning the computation of the ﬁlter, we faced the problem of some negative correlation ﬁgures, while the computation of harmonic mean can only be computed for positive variables. In this case, we could have either excluded these negative ﬁgures from the computation of the mean or computed the arithmetic mean instead of the harmonic one. We decided to follow the latter route. This leads to an overvaluation of the ﬁlter, because the harmonic mean is lower than the arithmetic one: however, this effect is somewhat mitigated by the fact that negative correlations bring about a decrease in the value of the mean.

4.2. The probability of a banking crisis In Fig. 1 we report the intensities of a systemic event for our sample. It is easy to note an upsurge of this risk in February 2009 when the effects of the banking crisis of 2008 displayed their effects in full. Following the Greek crisis the increase in the probability of a banking crisis reached heights that had never been seen before, and not only for Greece. The countries that are exposed the most are, beyond Greece, Ireland, Portugal and Spain. The other countries are clearly part of a different cluster, with Italy, Austria and UK with the highest risk among those countries. Table 1 reports the values of the probability of a systemic shock priced in the CDS for each country. Fig. 2 enables to appreciate the relevance of the systemic risk out of the overall average risk. The dynamics of the ﬁlter represents the ratio of the systemic to the average intensity, which is actually the a parameter of the Cuadras–Augé formulas. This can be actually seen as a dependence parameter of the

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Table 1 Risk adjusted probability of a systemic shock to the banking system. Date

17/03/2008

15/09/2008

16/03/2009

15/09/2009

15/03/2010

15/09/2010

Portugal Ireland Italy Greece Spain Germany France UK Netherlands Austria

6.18% 16.06% 9.22%

7.05% 21.46% 6.03%

6.47% 40.61% 11.86%

5.26% 15.97% 4.43%

6.27% 11.27% 12.96% 10.79% 14.01%

4.43% 8.97% 10.07% 7.65% 12.14%

19.03% 8.12% 8.82% 13.91% 10.45% 31.96%

9.20% 3.46% 4.83% 5.10% 4.46% 12.58%

10.15% 15.70% 6.17% 29.02% 9.49% 5.85% 4.56% 8.05% 5.84% 11.42%

26.06% 30.05% 12.42% 45.45% 21.06% 4.57% 6.56% 9.85% 8.15% 13.02%

ﬁnancial sector of each country. The main feature emerging from the graph is the recent increase of the systemic content of default intensity for all countries, with the noticeable exceptions of Germany and the Netherlands (see also Table 2 below). The counterparty of this evidence is that for all the countries, where we have found an increase of average rank correlation in the latest period of the sample, we also observe an increase of concentration of the correlation values between pairs. Table 2 reports, for the same dates as those in Table 1, the values of the ﬁlter a. The economic meaning is the percentage relevance of a systemic shock out of the average probability of a credit event. Note that in the ﬁnal observations of the sample the ﬁlter is very close to 1 for all countries, except Germany and the Netherlands. The case of Germany is peculiar, since the ﬁlter is around 55%, much lower than anywhere else. In Table 3 we ﬁnally report, for the ﬁnal date of our sample, that is 15/09/2010, a comparison of our model and the standard Gaussian one. The Gaussian copula was calibrated on the same average marginal distributions as that used for the exchangeable Marshall–Olkin system and the same average correlation. We then reported the dependence structure between the systemic shock and the default of

Fig. 1. Systemic intensities. Financial sectors of selected countries.

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999

Fig. 2. Cuadras–Augé ﬁlter. Maximum, minimum and average values. 10 European countries.

a representative obligor of the system. Notice that the Gaussian model produces a joint probability slightly lower than that of the Marshall–Olkin system. The difference is motivated by the fact that the Gaussian model does not allow for perfect dependence between the systemic shock and default, that is instead enforced by construction in the Marshall–Olkin system. However, given the high level of correlation that is common to the two models, the distance between them is not huge, with the exception of the Netherlands and Germany. 4.3. The bail-out cost of a banking crisis European Governments have committed a huge amount of money to restore public conﬁdence in the ﬁnancial sector, starting from the most acute phase of the ﬁnancial crisis, namely in the aftermath of the Lehman Brothers collapse. In the period running from October 2008 through March 2010, the member States of EU-27 have committed an overall amount of resources equal to 4131 bn euro, equivalent to 32.6% of their GDP.4 Table 4 reports a breakdown of the intervention measures taken by the Governments of the ten countries covered by our analysis. The bulk of the involved resources have been devoted to guarantee schemes. Governments have relied extensively on this tool, particularly until mid-2009, as it is the most cost-effective way for restoring the conﬁdence of investors. Banks’ liabilities are backed by the guarantee provided by the State; at the same time the Government budget is not hit by an immediate outlay. The “take-up rate” – the actual use of funds relative to the allocated amounts – is on average 32%, but it is much higher in some individual countries (like Portugal: 51%). Many banks turned out to be under-capitalized during the ﬁnancial crisis. Governments have reacted by approving both recapitalization schemes for the bank sector as a whole and ad hoc measures for individual troubled institutions. The take-up rate differs remarkably between the two kinds of intervention: 27% for schemes and 90% for ad hoc cases. The reason is that individual measures have

4

See EU (2010).

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Table 2 Systemic content of default intensity. Date

17/03/2008

15/09/2008

16/03/2009

15/09/2009

15/03/2010

15/09/2010

Portugal Ireland Italy Greece Spain Germany France UK Netherland Austria

62.97% 99.62% 89.70%

62.44% 97.41% 81.60%

46.90% 98.68% 86.79%

68.22% 98.24% 85.42%

64.90% 99.05% 98.98% 92.34% 99.76%

51.59% 97.96% 93.98% 87.93% 98.07%

82.13% 65.25% 88.69% 90.10% 55.03% 96.68%

65.68% 41.74% 84.20% 70.85% 54.01% 96.95%

89.12% 99.43% 93.38% 96.49% 62.09% 68.65% 93.03% 92.99% 68.60% 99.62%

98.38% 98.69% 95.87% 98.77% 95.24% 54.45% 98.18% 94.68% 79.15% 96.68%

been designed to meet well deﬁned and urgent needs to restore a sufﬁcient capital base of some institutions. Two countries, namely UK and Ireland, account for the bulk (80%) of the third kind of intervention: impaired assets relief, where the Government either provides an insurance against assets devaluation or it directly buys some bank troubled assets. While Ireland has approved a major asset relief scheme, the intervention in UK has been on an individual basis. Table 5 shows – for each of the countries included in our sample – the overall cost of the different kinds of intervention outlined above, both as a proportion of the banking sector size and as a ratio to GDP. The numbers reported in the table highlight the existence of great differences across countries. Ireland has committed an incredibly large amount of resources, both in terms of bank assets and as a ratio to GDP. It is followed by UK and the Netherlands: they have implemented interventions for amounts equivalent to about a half of their GDP. All the other countries have committed important amounts of resources, putting at stake a signiﬁcant share of their GDP, although lower than in such extreme cases. The only exception is Italy: the Italian Government distinguishes for having spent a negligible amount of money. The burden of the state interventions as a ratio to GDP is the outcome of two factors: the amounts of public resources committed as a share of the size of the banking sector, and in turn the size of this sector relative to GDP. An interesting question is whether any relationship exists between such two factors. Fig. 3 provides a tentative answer, and it is positive. It is easy to see that in those countries where the banking sector is larger in proportion to the economy, Governments have been induced to intervene more heavily in support of the ﬁnancial system. In particular, the huge amount of funds pledged by Ireland and the UK can be partially explained by the very large size of their ﬁnancial sectors.

Table 3 Model comparison: Marshall–Olkin vs. Gaussian. Country

Portugal Ireland Italy Greece Spain Germany France UK Netherland Austria a b c

Marginal intensity

Copulac

Dependence

Systemic

Mean

M–Oa

Gaussianb

M–O

Gaussian

0.060385314 0.071468375 0.026522934 0.121200797 0.047301457 0.009365111 0.013570467 0.020741324 0.017006822 0.027896716

0.061376667 0.072418333 0.027665833 0.122708333 0.0496675 0.01719963 0.013822708 0.021907917 0.021485833 0.028854167

0.983848058 0.986882355 0.958689131 0.987714476 0.952362353 0.544494911 0.981751645 0.946750194 0.791536521 0.966817591

0.980511214 0.984165851 0.950327449 0.985168641 0.942771281 0.489991668 0.977987986 0.936080367 0.755786338 0.960055793

0.260607645 0.300466697 0.124197499 0.454473549 0.210619869 0.045746151 0.065601557 0.098510264 0.081519044 0.130192694

0.23669905 0.27730207 0.10052261 0.42919108 0.17555606 0.01564391 0.05543721 0.0758931 0.04598697 0.10809208

Alpha parameter of the exchangeable Marshall–Olkin copula with homogeneous marginals. Correlation parameter of the Gaussian copula of the homogeneous set of obligors. Bivariate copula function giving the joint probability of a systemic shock and the default of an obligor.

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Table 4 Government commitments supporting the ﬁnancial sector (Oct. 2008–March 2010) (billions euro).

Austria France Germany Greece Ireland Italy Netherlands Portugal Spain UK Total a

Guarantee schemes

Recapitalization schemes

75 265 400 15 376

15 23.95 80 5

Othera

8 54

Individual cases

Total

0.5 62.2 107.6

90.5 351.15 587.6 28 455.6 20 256.2 20.5 329 850.26 2988.81

25.6

20 200 16 200 381.87 1928.87

56.2 0.5

4 99 62.79 309.74

30 92

405.6 658.2

Liquidity and asset relief schemes.Source: EU (2010).

The well known “too-big-to-fail” doctrine states that a Government cannot let a very large ﬁnancial institution go bust, since the implied cost for the whole economic system would be too high. The preliminary evidence reported here points to a sort of “too-big-to-fail” doctrine at the macro level: the incentive for a Government to bail out the ﬁnancial system altogether is stronger, the larger the size of that sector relative to the economy of the country. Of course, this is an issue deserving further analysis and explanation. 4.4. Marking-to-market the bail-out guarantees to the ﬁnancial system We ﬁnally gather (see Table 7 below) the risk-neutral ﬁgures concerning the probability of a systemic shock to the ﬁnancial system and those concerning the bail out programs, in order to address the issue of the adequacy of such programs with respect to the actuarial cost of systemic events. In particular, we provide a measure of the liability incurred by the public sector, due to a systemic shock to the ﬁnancial sector. On technical grounds, the actuarial value of the insurance provided to the banking system is computed multiplying the probability of a systemic shock by the percentage of losses covered in case of default. While the probability of a systemic shock was recovered in Section 4.2, a discussion is needed to assess the percentage of losses that can be covered by the insurance. As we have shown in the model in Section 3.1, insurance of the full amount of losses could not be conceived, because it would cause moral hazard behaviour from the bank insured. For this reason, going back to the analysis of Section 3.1, we try to assume a scenario in which, given a range of leverage typical of a bank, a bank faces an investment opportunity that may induce moral hazard. We ask which is the threshold of insurance coverage above which this moral hazard behaviour is actually triggered (see Table 6). To be clear, in the ﬁrst example we assume a bank with a leverage ratio of 27.5 (the average value of the German system) that faces the alternative between a fair game investment, giving 100 with probability 50% and zero Table 5 Government commitments over bank assets and GDP.

Austria France Germany Greece Ireland Italy Netherlands Portugal Spain UK

Government commitments (% of bank sector total assets) (A)

Bank sector total assets over GDP (B)

Government commitments (% of GDP) (A*B)

8.78 4.59 7.90 5.69 27.88 0.53 11.56 3.94 9.55 8.99

3.72 4.01 3.09 2.07 9.99 2.46 3.89 3.17 3.28 6.04

32.68 18.41 24.41 11.79 278.58 1.32 44.93 12.51 31.30 54.27

Sources: Bank sector total assets (end-2009): ECB (BoE for UK). GDP (2009): Eurostat. Government commitments: EU (2010).

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Commitments as % of bank sector assets

IRELAND 25,00

20,00

15,00 NETHERLANDS SPAIN

10,00

AUSTRIA

UK

GERMANY

GREECE 5,00

PORTUGAL FRANCE 0,00 0,00

ITALY 2,00

4,00

6,00

8,00

10,00

12,00

Bank sector total assets over GDP

Fig. 3. Government commitments and bank sector size.

otherwise, and another investment that in the good state (taking place with probability 5%) gives an extra return of 10%. If we rule out public insurance, there would be no moral hazard (condition (1) in Section 3.1 holds). If we address the question of the threshold insurance coverage beyond which there is moral hazard (g*), we ﬁnd that it is almost 89%. This decreases to 81% if the excess gain from the alternative investment doubles to 20%. Most importantly, if we assume the lowest leverage level of the banking systems of the countries of our sample (Austria: 8.6), we ﬁnd that the coverage threshold falls just below 78%. So, assuming a conservative level of coverage of 60% we are quite sure that moral hazard behaviour from the banks can be safely ruled out. In fact, we checked that 60% would be enough to prevent moral hazard, in our simple model, even if the leverage ratio were one to one, a level that is much smaller than any ﬁgure that can be found in a banking institution. In Table 7 we ﬁnalize our empirical analysis computing the actuarial value of the liability of each Government in the sample. More speciﬁcally, we report the intensity of a systemic event and its probability (DP) over a horizon of ﬁve years. We then report an estimate of the cost that the insurance would bring about to the public balance sheet: following the discussion above, this coverage is assumed to be 60% of the total amount of bank assets in each country. Next, the actuarial value of the Government liability is computed by multiplying the probability of a systemic shock times the euro value of the Government coverage. Finally, the Government liability is compared with the commitment actually made available by the same Government of each country. We can see that the estimated market values of the bail-out guarantees needed to face a systemic crisis in the banking sector are associated to the commitments actually devoted by European Governments to such purpose. On one side, the total amounts are very similar, namely 2245 billion euros against 2330. On the other side, the degree of association can be clearly appreciated Table 6 Analysis of moral hazard. Assets

Liabilities

P1

P2

Excess Gaina

Leverageb

Threshold

100 100 100 100

96.5 96.5 89.6 89.6

50% 50% 50% 50%

5% 5% 5% 5%

10% 20% 10% 20%

27.5 27.5 8.6 8.6

88.76% 80.92% 85.45% 77.89%

a b

Gain to be exploited from moral hazard over the current investment. Leverage is deﬁned as assets over equity.

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Table 7 Mark-to-market of the implicit guarantee to a systemic shock (bn euro).

Portugal Ireland Italy Greece Spain Germany France UK Netherland Austria Total

Intensity

DP

Coverage 60%

Government liability

Commitments

Liability Commitments

6.04% 7.15% 2.65% 12.12% 4.73% 0.94% 1.36% 2.07% 1.70% 2.79%

26.06% 30.05% 12.42% 45.45% 21.06% 4.57% 6.56% 9.85% 8.15% 13.02%

312.12 980.4 2248.62 295.14 2068.08 4461.66 4594.02 5677.2 1330.2 618.12

73.68 266.85 252.98 121.51 394.57 184.89 273.00 506.61 98.23 72.90 2245.23

20 430 20 28 329 480 288.95 444.66 200 90 2330.61

53.68 163.15 232.98 93.51 65.57 295.11 15.95 61.95 101.77 17.10 85.38

Fig. 4. Government liabilities due to systemic shocks and actual commitments.

in Fig. 4. Note that if this association is quite clear across all countries, the difference between guarantees and commitments shows quite a large degree of variation from one country to the others. The two extreme cases are: Germany, for which the commitments are largely higher than the actual value of the guarantee, and this can be associated to the low systemic content of default intensity; and Italy, which reports the lowest level of commitments along with Portugal, in spite of much greater dimension of the banking system. A surprise is that Ireland is the country that, after Germany, has the highest positive difference between the value of commitments and that of the guarantees.5 Finally, in Table 8 we report the impact of the bail-out guarantee on the public debt/GDP ﬁgures. We can see that the problem is worst for Ireland, and the reason is the huge dimension of the banking system relative to GDP (see Table 5, column B). The impact is lower but still quite relevant for the other

5

We have run the exercise shown in Fig. 4 also by subtracting the exposure to the Government of the own country from the total assets of each bank, since in several countries this exposure is quite relevant. In case of distress, a bank can transfer its portfolio of Government bonds to its creditors, thus reducing the liability of the Government involved in a bail-out. However the results obtained are very similar to those reported here, so we decided not to show them to save space (of course, they are available upon request).

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Table 8 Bail-out Government liability and Debt/GDP.

Portugal Ireland Italy Greece Spain Germany France UK Netherlands Austria

Debt/GDP

Liability/GDP

Total

76.80% 64.00% 115.80% 115.10% 53.20% 73.20% 77.60% 68.10% 60.90% 66.50%

44.96% 163.17% 16.63% 51.16% 37.54% 7.68% 14.31% 32.34% 17.23% 26.33%

121.76% 227.17% 132.43% 166.26% 90.74% 80.88% 91.91% 100.44% 78.13% 92.83%

PIIGS; among these countries, Italy is the one which suffers the lowest burden, thanks to the relative strength of its banking system. Among the other countries, UK is the one most hit by the bail-out liability: just like it happens for Ireland, the size of the banking sector plays a crucial role. 5. Summary and conclusions We have introduced a new methodology for measuring the systemic risk of default of the banking sector and we have applied it to the European countries. Our methodology is built within the framework of multivariate intensity based models, and it is based on a homogeneous version of such models, called Cuadras–Augé distribution. The ﬁlter we derive measures the relevance of systemic risk (due to the likelihood of a shock spreading to the whole banking system) relative to the average default probability of individual banks. We have computed the index for 10 European countries, exploiting the information incorporated in the CDS premia of 44 banks over the period January 2007–September 2010. In this way, we can provide a market based measure of the liability incurred by the Governments, due to the implicit bail-out guarantees they provide to the ﬁnancial sector of the economy. We then compare this estimate with the actual resources committed so far by the European Governments through the rescue plans adopted between October 2008 and March 2010. Our main results may be summarized as follows. During the ﬁnancial crisis, the systemic component of the default risk in the banking sector has signiﬁcantly increased in all countries, with the exception of Germany and the Netherlands. The most recent data (September 2010) show that almost all the risk of default of individual banks is accounted for by the systemic component. As a consequence, the Governments’ liability implicit in the bail out guarantee amounts to a quite relevant share of GDP in several countries: it is huge for Ireland, lower but still important for the other PIIGS (Italy is the least affected within this group) and for the UK. The two crucial factors determining these results are: (i) the correlation among the default probabilities of banks within a country; (ii) the size of the banking system relative to the economy of each country. Finally, our estimate is very close to the overall amount of money committed in rescue plans at the European level. However, strong cross-country differences emerge under this regard: in particular, Germany and Ireland seem to have committed an amount of resources much larger than needed; to the contrary, the Italian Government has committed much less than it should. This evidence supports the view that a common European rescue fund should be established, able to overcome these cross-country differences. Appendix 1. Cuadras–Augé Filter In this Appendix we report the main idea of the Cuadras–Augé ﬁlter. We start from the Marshall– Olkin non-homogeneous probability model:

l i þ bl j l ðT tÞ P qi > T; qj > T ¼ exp b

(A1)

It is well known that in this model the correlation between the default times of obligors i and j is given by the Pearson correlation formula

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rij ¼

1005

l

(A2)

b l i þ bl j l

for each pair and we want to come up with a representation of the average dependence of the system based on

l

rij ¼ r ¼

(A3)

ll 2b

for all the pairs. The straightforward choice for marginal intensities is to set them equal to the average, namely

b l ¼ bl m ¼

Pn

i¼1

n

b l

i

Pn ¼

i ¼ 1 li

n

þl

(A4)

As for the solution of this problem, in Cherubini et al. (2012) it is shown that if the harmonic mean is l m in the used to represent average correlation, this yields a ﬁlter that amounts to the ﬁgure ahl=b Cuadras–Augé system, where

a¼

2 1þ

(A5)

1 H

and H denotes the harmonic mean of the Pearson coefﬁcients. The same procedure could be applied to rank correlation, remembering that in Marshall–Olkin structures we have

3l

rs;ij ¼

(A6)

l i þ bl j l 2 b

where rs,ij denotes rank correlation. Finally, it can be shown that

1

0

4B

a¼ B 3@1 3

1 þ

C C 1A

(A7)

Hs

where Hs denotes the harmonic mean of the rank-correlation coefﬁcients. Appendix 2. The data set Table A.1 below provides a description of our data set, which includes 44 banks representing 10 European countries. Table A1 Our sample. Assets (bn euro) (A)

Allied Irish Banks Bank of Ireland Ireland Piraeus Bank Alpha Bank EFG Eurobank National Bank Greece

179.02 181.82 360.84 48.95 67.83 84.62 113.99

Sample banks’ total assets over bank sector assets (%) (B)

22.1

Risk-weighted assets (bn euro) (C)

Exposure to own country Government (% of assets) (D)

Leverage (assets/equity) (E)

Tier 1 ratio (F)

121.6 104.6 226.2 37.4 51.1 47.6 67.4

2.31 0.65 1.48 16.97 7.47 8.81 17.33

14.6 14.0 14.3 13.6 10.1 14.3 15.0

7.0 9.2 8.1 9.1 11.6 11.2 11.3

(continued on next page)

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Table A1 (continued )

Greece Banco Pastor Bankinter Caixa Cataluna Sabadel Banco Popular Caja Madrid BBVA Santander Spain Banco BPI Espirito Santo BCP Caixa Geral Portugal UBI Banca Montepaschi Intesasanpaolo Unicredit Italy SNS Bank ABN AMRO Rabobank ING Netherland Lloyds Barclays HSBC RBS UK Bayern LB LBBW West LB HSH Nordbank Postbank DZ Bank HYPO Commerz Deutsche Bank Germany Societe Generale Credit Agricole BNP France RZB Erste BANK Austria

Assets (bn euro) (A)

Sample banks’ total assets over bank sector assets (%) (B)

Risk-weighted assets (bn euro) (C)

Exposure to own country Government (% of assets) (D)

Leverage (assets/equity) (E)

Tier 1 ratio (F)

315.38 32.17 54.55 273.43 83.22 129.37 193.01 394.41 423.08 1583.22 47.55 85.31 13.99 121.68 268.53

64.1

203.5 18.7 30.7 52.9 58.0 92.6 223.1 290.1 562.6 1328.5 26.1 67.9 65.6 71.0 230.6 85.7 120.9 361.8 452.4 1020.7 25.9 118.7 236.3 332.4 713.3 580.4 450.2 871.7 515.7 2418.0 135.7 142.5 35.7 71.4 68.7 95.0 81.0 280.1 273.5 1183.5 324.1 538.9 620.7 1483.7 94.5 125.5 219.9

12.65 8.20 3.18 1.49 5.85 5.85 12.55 13.22 11.97 7.79 8.88 5.49 6.81 5.56 6.69

13.3 14.0 17.0 50.6 13.6 14.5 7.6 10.0 5.2 16.6 17.0 11.7 1.9 13.7 11.1

12.29 15.00 10.34 12.54 6.00 4.83 2.21 0.47 3.38 0.93 0.69 0.00 1.79 0.85

15.7 9.9 6.9 10.8 22.6 9.8 12.2 19.9 16.1 9.0 20.9 7.1 15.6 13.2

6.75

14.9 34.0 19.3 41.6 14.7 40.7 27.5 21.2 8.6 14.1 14.7 12.1 5.0 8.6

10.8 10.5 7.5 6.6 9.0 9.1 8.6 9.4 10.0 8.8 8.5 7.7 9.3 8.4 8.5 8.0 7.5 8.3 8.6 8.1 10.7 13.0 14.1 10.2 12.0 9.6 13.0 10.8 14.4 12.0 10.9 9.8 14.4 10.5 7.1 9.9 9.4 10.5 12.7 10.0 10.7 9.7 10.1 10.2 9.3 9.2 9.3

225.87 424.48 375.52 1025.87 58.74 203.50 425.87 888.11 1576.22 648.25 1561.54 851.05 1311.89 4372.73

171.33 227.97 219.58 362.24 609.79 1542.66 3133.57 891.61 522.38 1246.15 2660.14 148.95 79.72 228.67

45.9

51.6

27.4

71.1

46.2

42.1

34.7

22.2

7.02 6.88 1.69 4.86 1.45 2.67 2.78 7.27 5.03

Country lines show total values for columns (A)–(B)–(C) and mean values for columns (D–(E)–(F). Sources. Sample banks’ assets: Bankscope. Bank sector total assets: ECB (BoE for UK). Other items: Unicredit (2010) and our own computations.

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