Allocation of risk capital in a cost cooperative game induced by a modified Expected Shortfall

The standard theory of coherent risk measures fails to consider individual institutions as part of a system which might itself experience instability and spread new sources of risk to the market participants. In compliance with an approach adopted by Shapley and Shubik (1969), this paper proposes a cooperative market game where agents and institutions play the same role can be developed. We take into account a multiple institutions framework where some of them jointly experience distress events in order to evaluate their individual and collective impact on the remaining institutions in the market. To carry out this analysis, we define a new risk measure (SCoES), generalising the Expected Shortfall of Acerbi (2002) and we characterise the riskiness profile as the outcome of a cost cooperative game played by institutions in distress (a similar approach was adopted by Denault 2001). Each institution's marginal contribution to the spread of riskiness towards the safe institutions in then evaluated by calculating suitable solution concepts of the game such as the Banzhaf--Coleman and the Shapley--Shubik values.


Introduction
The assessment of financial risk in a multi-institution framework when some institutions are subject to systemic or non-systemic distress is one of the main topics of the latest years which received large attention from scholars in Mathematical Finance, Statistics, Management, see, e.g. Adrian and Brunnermeier (2016), Billio, Getmansky, Lo, and Pellizon (2012), Acharya, Engle, and Richardson (2012), Girardi and Erg€ un (2013), Hautsch, Schaumburg, and Schienle (2015), Engle, Jondeau, and Rockinger (2015), Lucas, Schwaab, and Zhang (2014), Bernardi and Catania (2019), Sordo, Su arez-Llorens, and Bello (2015), just to quote a few of the most relevant approaches. For an extensive and up to date survey on systemic risk measures, see Bisias, Flood, Lo, and Valavanis (2012), while the recent literature on systemic risk is reviewed by Benoit, Colliard, Hurlin, and P erignon (2017). Especially since 2008, Lehmann Brothers' collapse and the subsequent debt crisis contagion across Europe raised crucial issues and questions to be addressed by new macroeconomic and financial models. In particular, such events and the related consequences provoked a wide increase in investigation of instruments possibly suitable for risk evaluation, thereby targeting the occurrence of contagion among institutions in distress (see, among others, Drehmann & Tarashev, 2013 on the theoretical side and Zhu, 2012a andZhu, 2012b on the empirical side).
In this paper we are going to investigate circumstances where one or more market institutions undergo financial distress and may spread the contagion to the remaining safe institutions in the market. Financial distress has a straightforward meaning: a condition of dramatic financial instability of an institution, or lack of ability to fully repay its creditors, or even the preliminary to bankruptcy or liquidation. Although "financial distress" and "default" are sometimes used interchangeably, there may be some differences, often depending either on each country's specific bankruptcy law or on the institution's rating and so on (see Altman & Hotchkiss, 2010 for an extensive discussion).
When a financial distress takes place in the market, each distressed institution affects the safe ones and individually contributes to the systemic risk. A crucial aspect of the contagion frameworks is the assessment of each agent's (i.e. each institution's) marginal effect on systemic risk. In order to evaluate such contributions, a natural and intuitive tool is the Shapley value (see Shapley, 1988), as well as the Banzhaf value (see Banzhaf III, 1965;Owen, 1995). It is interesting to note that, although his world famous theories did not concern financial markets specifically, L.S. Shapley, Nobel Laureate in Economic Sciences in 2012, proposed a relevant application to market analysis in a paper with M. Shubik in 1969 (see Shapley & Shubik, 1969). In that paper, Shapley and Shubik defined and investigated the direct markets (see Shapley and Shubik (1969), pp. 15-18), i.e. markets where, since each trader possessed one initial unit of a personal commodity, traders might be identified with commodities. As the Authors put it literally, "A canonical market formthe direct marketis introduced, in which the commodities are in effect the traders themselves, made infinitely divisible … " (p. 11), hence that was the first attempt to model a cooperative game where agents had a twice nature: traders and assets at the same time.
Hence, classical solution concepts widely used in Cooperative Game Theory such as the Shapley value and the Banzhaf value fit our evaluation purposes. In more recent times, cooperative game theory has already been employed for risk capital allocation by Denault (2001) and in the latest years by Cs oka, Herings, and K oczy (2009) and Abbasi and Hosseinifard (2013). 1 However, in none of the above papers a characteristic value expressing the difference between an expected value in a non-distressed case and an expected value in a distressed-case has been taken into account.
The financial setup we are going to investigate involves a set of possibly distressed institutions (for example a set of banks) and a set of safe institutions. Our purpose is an assessment of the contagion risk, i.e. the extent to which the safe institutions can be affected and damaged by the financial distress of the unsafe ones. For this purpose, risk measures have been extensively theorized and analyzed, giving rise to a rich strand of literature. Some of them might provide helpful techniques for risk allocation among a number of distinct systemic agents. In this respect, the coherent risk measures have been and are a crucial tool for the assignment of risk capital, both for the richness of properties due to their axiomatic structure and for their adaptability to fit several financial frameworks. The basic studies on coherent risk measures have been published by Artzner, Delbaen, Eber, and Heath (1999) in 1999, who first provided an axiomatic characterization. Subsequently, in 2001Denault (2001 developed the analysis by taking into account a cooperative game structure, where players are the firms involved in risky financial activities, and the risk induced by the net worth of firms is the value of the characteristic function of the game, and the subadditivity axiom naturally induces a cost cooperative game. The coherence of the Expected Shortfall (ES, hereafter) was taken into account by Acerbi and Tasche (2002) in 2002, whereas an axiomatic approach was adopted in 2005 by Kalkbrener (2005), to characterize the related capital allocation procedures in portfolio management or performance measurement. In recent years, further mathematical results were achieved by Stoica (2006) (equivalence between properties of some coherent measures with no arbitrage conditions), K oczy (2007), Cs oka et al. (2009) (refinement of measures in a general equilibrium setup, stability of risk allocations), Kountzakis and Polyrakis (2013) (applications to general equilibrium models). Other relevant results in risk allocation theory were provided in Acciaio (2007), Filipovic andSvindland (2008), Drehmann and Tarashev (2013), Ravanelli and Svindland (2014) and Cs oka and Pint er (2016). Furthermore, as a slightly different tool, distortion risk measures have been applied to a financial contagion framework by Cherubini and Mulinacci (2014).
The new approach we are going to propose builds on an alternative risk measure which is an extension of ES and which is supposed to incorporate the effect of the distressed institutions on a safe one. Such a measure, denoted by SCoES s 1 js 2 ijD , represents the expected value of a non-distressed institution i conditional upon a set D of institutions under distress, given two different thresholds s 1 , s 2 2 ð0, 1Þ, specifically related to the institutions' returns. Subsequently, in order to evaluate the marginal contribution of each distressed institution to overall systemic risk, we will construct a cost function. Such a function aims to measure the cost of risk by evaluating the mean of the differences between the standard Expected Shortfalls and the values of SCoES of all non-distressed institutions. This formula will be the characteristic value of a cooperative game played by distressed institutions. Adopting a classical approach, we will examine the properties of the game and calculate the marginal impacts of each distressed institution, pointing out the differences between the Shapley value and the Banzhaf value of the game.
The introduced risk measurement framework is then adopted to empirically analyze the evolution of the SCoES risk measure and the output of the associated cooperative game of risk for eight Eurozone sovereign Credit Default Swaps (CDS) over the period 2008-2014. Our example assumes the Germany as the "safe" country, and explores how the evolution of the remaining countries' debt conditions affect the health and financial stability of Germany considered as a "proxy" for the overall European system. As a byproduct of the proposed risk framework, the evolution of the total impact of the failure of the European system is monitored. Our results show that the risk of failure of the European system displays a transitory high level during period in between the Greece and Portugal bailouts (November 2010 -May 2011), but effectively remains at high levels when the ECB president Mario Draghi announces the implementation of the Outright Monetary Transactions (OMT) and the European Stability Mechanism (ESM) in the months thereafter. It is only after the implementation of the OMT, and the ESM, that the systemic risk of Germany settles more permanently at a level that is roughly 60% lower than during the crisis.
The remainder of the paper is structured as follows. In Section 2 the basic concepts about risk measures will be recalled, together with some axiomatic details of coherent measures. Section 3 introduces the indicators that will be crucial to our analysis, whereas in Section 4 the cooperative game of risk and its solution concepts are exposed and discussed. Section 5 is devoted to the empirical application and Section 6, conclusions and possible future developments are laid out.

Risk measures in portfolio management
Here we will outline the standard notation for risk measurement in recent literature, largely borrowed from the seminal paper by Artzner et al. (1999) and Kalkbrener (2005).
Consider a finite set of states of nature X, whose cardinality is jXj ¼ m: Call Xðx j Þ a random variable indicating the final net worth of a position in state x j 2 X after a certain time interval, i.e. the possible profit and loss realization of a portfolio in state x j . We can identify the set of all real-valued functions (which can also be viewed as the set of all risks) on X as R m , whose elements are of the kind X ¼ ðXðx 1 Þ, :::, Xðx m ÞÞ: In the rigorous construction proposed in Artzner et al. (1999), a measure of risk is a mapping q : R m ! R such that qðXÞ corresponds to the minimum amount of extra cash an agent has to add to her risky portfolio, in order to ensure that this investment is still acceptable to the regulator 2 when using a suitable reference instrument. The basic requirement concerns the price of the asset: 1 is the initial price of the asset and r > 0 is the total return on the reference instrument at a final date T, in all possible states of nature. In Artzner et al. (1999) the Definition of a coherent measure is then provided based on an axiomatic structure.
Definition 2.1. A function q : R S ! R is called a coherent measure of risk if it satisfies the following axioms: Yðx j Þ ! Xðx j Þ for all x j 2 X), qðYÞ qðXÞ: Subadditivity (S): for all X, Y 2 R m , qðX þ YÞ qðXÞ þ qðYÞ: Positive Homogeneity (PH): for all X 2 R m and k 2 R þ , qðkXÞ ¼ kqðXÞ: Translation Invariance (TI) 3 : for all X 2 R m and h 2 R, qðX þ hrÞ ¼ qðXÞÀh: We have to point out that the Value-at-Risk (VaR) satisfies all of them except Subadditivity, as is precisely exposed in Artzner et al. (1999) as well as the standard Definition of VaR, which is recalled here: Definition 2.2. Given s 2 ð0, 1Þ and the return on a reference instrument r > 0, the VaR at level s of the final net worth X with probability PfÁg is the negative of the quantile at level s of X/r, i.e.
VaR s ðXÞ ¼ Àinffx j PfX r Á xg ! sg: Without loss of generality, in the remainder of the paper, r will be normalized to 1.
Back to coherent measures, when the outcomes are equiprobable, i.e. when the state of nature x j occurs with probability p j and p 1 ¼ Á Á Á ¼ p m ¼ 1=m, a special and relevant case can be treated, as was investigated by Acerbi (2002). In particular, we take into account an ordered statistics given by the ordered values of the m-tuple Xðx 1 Þ, :::, Xðx m Þ, i.e. fX 1 , :::,X m g, rearranged in increasing order:X 1 Á Á Á X m : The definition of spectral measure of risk we are going to present is due to Cs oka et al. (2007), who slightly modified Acerbi's original definition by employing a positive discount factor d, not necessarily equal to 1 as in Acerbi (2002).
Definition 2.3. If the outcomes are equiprobable, given a vector U 2 R m and a discount factor d>0, the risk measure M U : R m ! R defined as follows: is called a spectral measure of risk if U satisfies the following axioms: A well-known spectral measure of risk is the one indicating the discounted average of the worst s outcomes, i.e. the s-Expected Shortfall of X. For all s 2 f1, :::, mg, it is given by When X is a continuous random variable, its formula reads as follows: where f ðÁÞ is the law ofX and s is the related confidence level. The proof of the coherence of ES 1Àk ðÁÞ in the continuous random variable setup can be found in Acerbi and Tasche (2002).

The SCoES as a risk measure
Here we propose a generalisation of the Adrian and Brunnermeier (2016)'s CoVaR and CoES, namely System-CoVaR (SCoVaR) and System-CoES (SCoES). The proposed risk measures aim to capture interconnections among multiple connecting market participants which is particularly relevant during periods of financial market crisis, when several institutions may contemporaneously experience distress instances. Let P ¼ f1, :::, pg be a set of p institutions, and assume that the conditioning event is the distress of a subset of P: Call D ¼ fj 1 , :::, j d g & P the set of d institutions potentially under distress, whose cardinality is such that 0<d<p, hence meaning that at least one institution is not under distress and that at least one is under distress. If we consider a set S D of distressed institutions, such set represents a group of institutions picked among the ones that may be under distress. As is usual in typical Cooperative Game Theory literature, we will denominate any group S as a coalition. In Figure 1 we show the set structure of the model: P is the set of all institutions, which contains D, subset including the distressed institutions, whereas S is a generic subset (coalition) of institutions in distress. Thus, S is a subset of D: Given the confidence levels s 1 , s 2 , we are going to define the SCoVaR s 1 js 2 ijS of institution i 2 P for all institutions not belonging to D as follows. We are going to assume that at least one distressed institution has a negative return, i.e. that there exists at least one X j <0, to ensure possible positivity of the Expected Shortfall as has been defined by (2), when d, s>0: Let X ¼ ðX 1 , :::, X p Þ be a vector of p institution returns with probability PfÁg: Given any group S D of distressed institutions, and a related confidence level s 2 , we implicitly define VaR s 2 S : Hence, the conditioning event is the one described in (4) and whose probability equals the confidence level s 2 .
Definition 3.1. Given a set S D of institutions in distress and s 1 , s 2 2 ð0, 1Þ, for all i 2 P n D, the SCoVaR s 1 js 2 ijS is implicitly defined as follows: An alternative Definition of SCoVaR s 1 js 2 ijS is the following one.
Definition 3.2. Let X ¼ ðX 1 , :::, X p Þ be a vector of p institution returns with probability PfÁg: Given a set S D of institutions in distress and s 1 , s 2 2 ð0, 1Þ, for all i 2 P n D, the SCoVaR Basically, SCoVaR s 1 js 2 ijS is the Value-at-Risk of an institution subject to the condition that the sum of the realizations of the institutions under distress do not exceed the Value-at-Risk of their sum, when two different confidence levels are in general taken into account. The two following Remarks aim to point out two circumstances where SCoVaR coincides with the marginal VaR.
Remark 1. In this context it is quite natural to consider the sum as aggregated measure of risk since we are considering profits and losses. Of course alternative definitions are possible, as for example, the maximum loss of the distressed institutions, see, e.g. Bernardi, Durante, Jaworski, Lea, and Salvadori (2018) and discussion therein.
Remark 2. If all the returns of the institutions in W are independent of all the returns of institutions in D, then we have: Remark 3. When no institution is under distress, S ¼ ;, i.e. X j k ¼ 0 for all j k 2 S: In this case, (6) is well-defined too and in particular it collapses to the standard VaR. Namely, The SCoVaR is particularly useful to formulate the risk measure we are going to investigate.
Definition 3.3. Given s 1 , s 2 2 ð0, 1Þ and a set S of institutions in distress, the SCoES s 1 js 2 ijS is the expected value of institution i 2 P n D, provided that it does not exceed SCoVaR s 1 js 2 ijS and conditional upon the set of institutions S being at the level of their joint ES s 2 -level: As in Remark 3, Equation (7) can be evaluated when no distress occurs too, collapsing to the standard Expected Shortfall: A short explanation may be helpful to clarify the formulation of Equation (7): the condition P j k 2S X j k ÀES s 2 ð P j k 2S X j k Þ is always verified when all X j k <0: On the other hand, just one negative institution return is enough to determine an open interval for d such that the condition holds. More precisely, if X j 1 , :::, X j l are negative, where j k 2 S for k ¼ 1, :::, l, the condition boils down to: d !
: Such an estimate is trivially true whenever all institution returns in a coalition S are negative. 4 Getting to analyze possible relations between SCoVaR and SCoES, we can prove some results.
Proposition 3.4. For any pair of confidence levels s 1 , s 2 2 ð0, 1Þ, for any set S D, and for any i 2 P n D, we have that: Proof. Given any random position X, we first recollect the Definition of ES(X) in our framework: where s 2 ð0, 1Þ: It is already well-known that, since for every c 2 R we have that: then X ÀVaR s ðXÞ ) E X ½ ÀVaR s ðXÞ, which implies ES s ðXÞ ¼ ÀE½XjX ÀVaR s ðXÞ ! VaR s ðXÞ: Given any S D, we know that X for all s 2 2 ð0, 1Þ, hence the condition for the Definition of SCoVaR is satisfied, i.e.
P ( X i ÀSCoVaR Consequently, the same argument which is applied to ES and VaR can be employed to establish an inequality involving SCoES s 1 js 2 ijS and SCoVaR Previous result is obtained by applying the property of the expectation operation to the conditional distribution X i jX i ÀSCoVaR s 1 js 2 ijS : w Proposition 3.5. Given two coalitions S, S 0 2 2 D , if the following hypotheses are verified: Proof. If S, S 0 2 2 D , the first hypothesis ensures that whereas the second and the third hypotheses guarantee that SCoES In the remainder of the paper, whenever there is no misunderstanding, we are going to simply denote the above quantities with ES, SCoES, VaR and SCoVaR, to lighten the notation.

A cooperative game for risk allocation induced by SCoES
The risk allocation problems were introduced by Denault (2001), where the problem of allocating the risk of a given firm, as measured by a coherent measure of risk, among its N constituents, was taken into account, closely resembling the typical Cooperative Game Theory approach. In Denault (2001) cooperative games of cost were employed to model risk allocation problems, and the chosen solution concepts were the Shapley value and the Aumann-Shapley value. Such approach was subsequently adopted and improved in Cs oka et al.
, where risk allocation games and totally balanced games are compared to ensure the existence of a stable allocation of risk. In particular, they define a risk environment characterized by a set of portfolios, a set of states of nature, a discrete probability density of realization of states, a matrix of realization vectors and a coherent measure of risk, from which they construct and analyze a risk allocation game. In both approaches, the portfolios of a firm are looked upon as the players of a subadditive cooperative game. In our setting, we are ready to apply the measures defined in Section 3 to institutional circumstances where distress occurs, and in particular we are going to rely on some typical tools borrowed from Cooperative Game Theory. In more details, we will look upon any possible set of distressed institutions as a coalition of a cooperative game (or TU-game, see Owen, 1995). The effect of distress on the remaining institutions, corresponding to risk of contagion, will be evaluated by means of a cost function.
Call W the set of institutions not belonging to D, i.e. W ¼ P n D: We can evaluate the cost of risk induced by any coalition S D by taking a weighted arithmetic mean over all differences between the unconditioned ES s 1 and the SCoES for all institutions in W, i.e.
where a i ! 0 for all i ¼ 1, :::, jWj, P jWj i¼1 a i ¼ jWj, for all S D: In the frame of risk allocation, we introduce a cooperative game C ¼ ðc W , DÞ, where D represents the the set of involved portfolios and c : 2 D ! R is as in (9), and assigns a cost to each coalition S D: The cooperative game approach appears very suitable, in that in an uncertain financial framework it allows to take into account all possible combinations of institutions undergoing distress.
Moreover, the couple ðc W , DÞ actually defines a cooperative game. In fact, when no institution in D is under distress, then c W ð;Þ ¼ 0 because all the differences in the numerator of Equation (9) vanish. Essentially, this hypothesis, which is necessary to define a cooperative game on D, may have a clearcut financial interpretation: all the safe institutions are collected in W, meaning that all of them are secured beyond a reasonable doubt. They can be viewed as states or companies issuing either government bonds or securities guaranteed by top-quality collateral, namely all the kinds of agents which do not involve any risk factors. Also note the positive sign in Equation (9): in standard payoff cooperative games such sign is reversed. But because we are assuming that below the confidence level s 1 some realizations X J k are negative, positivity of ES s 1 ðÁÞ is ensured, hence the level of risk induced by distress can be positive. However, it may occur that for some coalition S, c W ðSÞ is non-positive, but this would mean that the contagion is even less likely to spread from such a group of institutions to the nondistressed ones.
Formula (9) needs some further explanation, in terms of what the differences between ES and SCoES actually measure. Each difference provides the spread between the standard risk and the risk which is correlated to the distress of a coalition S, composed of one institution at least. In order to completely assess the risk effect caused by any coalition S, the sum of those differences is taken over the whole set of safe institutions W. Perhaps some structural differences may occur among the safe institutions, including insurance contracts, implementation of hedging strategies, and so on. Such heterogeneity can be captured by weights a i in Equation (9), which can also be interpreted as directly dependent of each single institution, in compliance with its size or its systemic relevance. In order to lighten the notation, we are going to hypothesize a simplified scenario where all institutions' weights are equal, then we are going to posit a 1 ¼ Á Á Á ¼ a jWj ¼ 1: The issue concerning the properties of the game (9) is somewhat complex, due to the fact that the SCoES of an institution subject to an external distress is a kind of measure of correlation, or also a measure of how distress spreads its contagion towards non-distressed institutions. Consequently, the standard axioms associated to coherent risk measures can hardly be demonstrated. Instead of an axiomatization, we are going to outline some characteristics of c W ðÁÞ, which are listed in the next Propositions. Some of these properties resemble the axioms stated by Denault (2001).
Proposition 4.1. Given two coalitions S, S 0 2 2 D , if the following hypotheses are verified: then c W ðSÞ ! c W ðS 0 Þ: Proof. It follows directly from the inequality in Proposition 3.5.
w In the following Proposition, the expression kS means that all the institution returns in S are multiplied by k, i.e. kX j k , for all j k 2 S D: Proof. Applying Definition 3.2 to kS, for all k>0, implies that SCoVaR By the positive homogeneity of VaR (see either Artzner et al., 1999;McNeil, Frey, and Embrechts, 2005, p. 74), that expression coincides with Equation (6), hence SCoVaR Finally, c W ðkSÞ can be written as follows: w Proposition 4.3. For all S D such that all the returns X i , where i 2 W, are independent of all returns X j k , where j k 2 S, c W ðSÞ ¼ c W ð;Þ ¼ 0: Proof. For all institution returns X i which are independent of all X j k (see Remark 2), where i 2 W and j k 2 S D, we have that SCoES s 1 js 2 ijS ¼ SCoES s 1 js 2 ij; ¼ ES s 1 ½X i : If this holds for all i 2 W, the proof is complete.
w Subadditivity is a key feature of cost allocation games. Recall that C is subadditive when its cost function is subadditive, i.e. for all S, T 2 2 D such that S \ T ¼ ;, c W ðS [ TÞ c W ðSÞ þ c W ðTÞ (e.g. Anily & Haviv, 2014). As is shown in the next Proposition, the game ðc W , DÞ is not always subadditive. In particular, some assumptions on the related ES and SCoES are supposed to hold. Proof. By definition of subadditivity, we can note that for all S, T 2 2 D , where S \ T ¼ ; : then, if 8i 2 W, ES s 1 ðX i Þ>0 and SCoES is superadditive, i.e. ijT , then c W ðÁÞ is subadditive. w However, it is difficult and restrictive to impose this condition, because the very definition of ES s 1 allows both its possible positivity and negativity, depending on the level at which the worst outcomes are taken.

Risk allocation
A typical and well-known application of cooperative games is the determination of suitable allocations among players who get a share of a total amount, which is a benefit when they play a payoff game (generally superadditive) and a cost when they play a cost game (generally subadditive). In this case, by Proposition 4.4, subadditivity is not ensured, but the characteristic function of the game represents a contagion risk induced by distress of some institutions, hence its interpretation as a cost game sounds intuitive and natural.
A complete presentation of the several solution concepts and allocation rules in cooperative games can be found in Owen (1995). The two main values we are going to apply to our setup are the Shapley-Shubik (first introduced in 1953, see Shapley, 1988) and the Banzhaf-Coleman (which was formulated in 1965, see Banzhaf III, 1965) values. What follow are the expressions of such allocation principles when employing the characteristic function c W ðÁÞ: The Shapley value of the game ðc W , DÞ is given by the d-dimensional vector Uðc W Þ ¼ ð/ 1 ðc W Þ, :::, / d ðc W ÞÞ such that: for all j k 2 D: On the other hand, the Banzhaf value of ðc W , DÞ is the d-dimensional vector bðc W Þ ¼ ðb 1 ðc W Þ, :::, b d ðc W ÞÞ such that: for all j k 2 D: Their respective axiomatizations 5 point out a crucial difference between (11) and (12) ijSnfj k g ¼ SCoES s 1 js 2 ijS : The economic meaning of a dummy institution is simple: its marginal contribution to overall contagion is always zero.
A special discussion should be devoted to the socalled no undercut property (see Denault (2001), Def. 3), which can be reformulated as follows: given an allocation ðK j 1 , :::, K j d Þ for the game ðc W , DÞ, for all S D, the inequality X must hold. The condition (13) has a twofold meaning. The first one is technical: any allocation ðK j 1 , :::, K j d Þ satisfying it for all S is in the core of the cooperative game, consequently if at least one allocation of this kind exists, the core is non-empty.
The second meaning is strictly connected to a financial aspect (see Denault, 2001): an undercut happens when a portfolio allocation exceeds the risk capital that the whole group of institutions would face. Relying on previous results, we can establish some sufficient conditions for positivity of Uðc W Þ and bðc W Þ, i.e. to ensure that all their coordinates are non-negative, meaning that each distressed institution brings a positive marginal contribution to systemic risk.
Proposition 4.5. If for all S 2 2 D n ; and for all j k 2 S the following hypotheses are verified: then / j k ðc W Þ ! 0 and b j k ðc W Þ ! 0 for all j k 2 S: Proof. Given a coalition of distressed institutions S 6 ¼ ; and any element j k 2 S, we can apply Proposition 3.5 to two coalitions S and S n fj k g by reformulating its three hypotheses. Since by Proposition 3.5 we have that SCoES ijS , then all terms in the sums in Equation (11) and Equation (12)

Application
To illustrate how the SCoES risk measure behaves in practice we examine the evolution of European Sovereign Credit Spreads (CDS) over a period that includes the Eurozone sovereign debt crisis of 2012. Specifically, we investigate the evolution over time of the Shapley-Value SCoES induced by the cooperative Game where the Germany acts as the only "safe" country, as described in the previous sections. The potentially distressed countries are: Belgium, France, Greece, Italy, Netherlands, Portugal and Spain. We consider a panel of daily CDS spreads over the period from July 21st, 2008 to December 30th, 2014 except for the Greece for which the data are available only until March 8, 2012. We use US-denominated sovereign CDS for each country using data obtained from Datastream. Our aim is to assess how the events related to the European sovereign debt crisis impact the safety of the most important economy in the Euro area, using the provided risk measure and the associated risk measurement framework based on the cooperative game. A similar empirical investigation has been conducted by Bernardi and Catania (2019) using stock market data of the major European financial indexes, Lucas et al. (2014) using dynamic Generalised Autoregressive Score (GAS) models on CDS, and Engle et al. (2015) again using stock market data of European individual institutions, and Blasques, Koopman, Lucas, and Schaumburg (2016) using spatial GAS models on European sovereign debt CDS. Major financial events affecting the Euro area during the considered period are collected in Table 2. Since EU countries have been affected by the crisis to different degree, sovereign credit spreads in Europe are strongly correlated. Figure 2 shows the evolution of the credit default spreads in log basis points for the period covered by our analysis. Visual inspection of the series reveals clear common patterns particularly between Netherlands and Germany on the one hand and Italy and Spain on the other hand. As expected, the evolution of the Greek CDS strongly differs from those of the other countries in the sample. Summary statistics of log CDS returns multiplied by 100 are reported in Table 1.
In order to calculate the SCoES risk measure a parametric assumption about the joint distribution of the involved CDS log-returns should be made. Several alternative methods have been proposed to estimate the CoVaR. In the seminal paper Adrian and Brunnermeier (2016) approach the estimation issue in a semiparametric framework using conditional quantile regression methods (see, e.g. Koenker, 2005;Koenker & Bassett, 1978). Indeed, they do not make any specific distributional assumption on the response variable. Unfortunately, the semiparametric approach in not suitable for our purposes since it does not allow to evaluate the SCoVaRs of the two groups of institutions (distress and not in distress) as required by the cooperative game here developed. Previous motivations support our decision to empirically investigate the performances of the proposed risk measure and risk measurement framework by opting for a joint Gaussian distribution for the profits&losses of the considered institutions (in the case of our empirical example, sovereign bonds). Of course, the Gaussian assumption is quite restrictive when the assessment of the relevance of distress situations is the major aim, because it implies independence of extreme events (see, for example, McNeil et al., 2015). Nevertheless, the Gaussian assumption remains a valid illustrative example. For the sake of completeness, it should be stressed that the SCoVaR risk measure and the systemic risk measurement based on the solution of a cooperative game remain valid under any assumption for the joint distribution of the random variable involved. For example, one can assume a multivariate Elliptical distribution (Fang, Kotz, & Ng, 1990), or can opt for the copula approach (Demarta & McNeil, 2007), or for a skew distribution (Azzalini, 2013). The only restriction of the proposed methodology is the availability of the conditional quantile/ expectile in closed form.
The Gaussian assumption is not only convenient but it represents also a common choice for practical  Table 2. applications, and it favours the interpretation of the estimation results as the output of a graphical model, see Koller and Friedman (2009). Nevertheless the Gaussian distribution can be easily replaced by either another parametric distribution or by more involved dynamic models that describe the evolution over time of the CDS, see, for example, Bernardi and Catania (2019). Proposition A.1 in Appendix provides the analytical formulas to calculate VaR, ES, SCoVaR and SCoES under the Gaussian assumption. As far as parameter estimation is concerned we apply the Graphical-Lasso algorithm of Friedman, Hastie, and Tibshirani (2008), which allows for sparse covariance estimation. The tuning parameter that regulates the amount of sparsity in the covariance structure has been fixed at k N ¼ 2 ffiffiffiffiffiffiffi log p N q , where N denotes the sample size, as suggested by the theory Ravikumar, Wainwright, Raskutti, and Yu (2011), see also Hastie, Tibshirani, and Wainwright (2015) and references therein.
To analyse more deeply the impact of the recent European Sovereign debt crisis, we estimate recursively the SCoES over the sample period using a rolling window. Moreover, to obtain results more robust to temporary short-term shocks affecting the considered economies, we consider weekly log-CDS returns. Specifically, at each point in time we estimate the SCoES risk measure using a window of N ¼ 26 more recent weekly observations, and then we run the cooperative game to get the Shapley Value with s 1 ¼ s 2 ¼ 5%: It is worth mentioning that, with p ¼ 8 institutions, we are going to esti- The Graphical-Lasso method of Friedman et al. (2008) delivers consistent estimates of the parameters even when the number of parameters is greater than the dimension of the sample, see Hastie et al. (2015) and Tibshirani (1996Tibshirani ( , 2011 for further details. Results are reported in Figure 3, for the period before the onset of the Greek crisis (3(a)), as well as for the subsequent period (3(b)). For both periods, the bottom panel reports the overall impact of the  Table 2.  Figure 3 and the Banzhaf Value in Figure 4, they can be interpreted as the normalised country risk factors. As expected, the two approaches provide different contributions to risk, in particular during the onset of the European crisis, reflecting their different properties. More precisely, the Shapley solution, suggests that, during the period in between the bailout of Ireland (November 29, 2010) and the bailout of Portugal (May 5, 2011), the most severe source of risk for the Germany's economy is represented by the Greece, while Spain, Italy and France contribute less, see Figure 3. The picture by Banzhaf is a little bit different since, during the same period, the most important source of risk for Germany is Belgium, followed by Greece, a result which is a little bit surprisingly. Afterwards, the overall contributions of European countries converge, for both methods. The failure of the Greek austerity package in February 2012 suddenly increases the riskiness of Greece which is comparable only with that of Portugal at the beginning of 2011. Interestingly, the proposed approach is able to capture the most important events that happened during European sovereign debt crisis of 2012, as reported in Table 2.

Conclusions and further developments
This paper presents a cooperative game among distressed institutions to assess the potential damage done by all possible coalitions in distress. At this aim, a new risk measure which features some properties of the standard Expected Shortfall in a financial framework where some institutions are distressed and contagion threatens the remaining safe institutions is developed. Standard solution concepts like Shapley value and Banzhaf value can be helpful to measure the marginal contributions to systemic risk.
Our study of the European sovereign debt crisis of 2012 provides empirical support for the ability of the proposed cooperative Game approach to systemic risk measurement to effectively capture the dynamic evolution of the overall riskiness of the European countries. Furthermore, the proposed risk measurement framework is able to identify the major sources of risk and the risk contributions.
Further extensions of such a theoretical setup can be conceived, in terms of more complex and precise cost functions to be employed in the cooperative game. Moreover, a detailed analysis of correlations among distressed institutions might give rise to different game structures, such as a priori unions or  Table 2.
bounded forms of cooperation which can be described with the help of graphs. In such cases, a model with some constraints might be necessary to determine the characteristics of risk transmission and the related consequences on the systemic risk. It is possible that a given structure with certain links among institutions can minimize the contagion risk.
Notes 1. A mechanism for allocation of risk capital to subportfolios of pooled liabilities has been proposed by Tsanakas andBarnett in 2003 (see Tsanakas andBarnett, 2003) and by Tsanakas in 2009 (see Tsanakas, 2009), based on the Aumann-Shapley value. 2. This property relies on the acceptance sets, which are axiomatized in Artzner et al. (1999). 3. Property TI is sometimes denoted as Risk Free Condition (RFC). 4. Also note that the estimates on d are as many as the possible coalitions S except the empty set, i.e. 2 jDj À1, hence there are at most 2 jDj À1 levels of d that must be exceeded. Because the choice of d in the definition of (2) is arbitrary, taking the maximum level among such values implies that such condition in (7) is always satisfied, consequently SCoES s 1 js 2 ijS E½X i jX i SCoVaR s1js2 ijS for all i 2 P n D: 5. There exists a large number of contributions on axiomatizations of values in literature, see for example Feltkamp (1995) and van den Brink and van der Laan (1998). 6. Nonetheless, some recent contributions have been published on the Shapley value without the efficiency axiom, see Einy and Haimanko (2011) for simple voting games and Casajus (2014) for different classes of games.

Disclosure statement
No potential conflict of interest was reported by the authors. Funding where P p k¼1, k6 ¼i r i, k denotes the covariance between X i and P p k¼1, k6 ¼i X k , and r 2 ðÁÞ denotes the variance of X i . Under the previous assumptions the VaR, ES, of X i , for i ¼ 1, 2, :::, p are calculated as follows: for i ¼ 1, 2, :::, p, where F X, Y ðÁÞ denotes the joint cdf of the random variables (X, Y).
Equation (5) implicitly defines the SCoVaR as the value of y that solves the conditional cdf of the involved variables equal to s 1 . The solution always exists and is unique because the involved random variables are absolutely continuous.