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Systemic Risk Contributions

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Part of the Studies in Systems, Decision and Control book series (SSDC,volume 409)


Several critical issues should be considered when we attempt to measure sovereign default risk in a systemic context. First, we need to derive multivariate joint probabilities of default. As we saw in the previous chapter, multivariate joint PoDs provide more comprehensive information about the risk in the euro area than the individual PoDs of each sovereign or the bivariate PoDs of sovereign couples: In addition to the individual and bilateral country risks, they capture the complex dependence patterns and interactions among euro area countries. Second, we should be able to analyze how various scenarios involving defaults of one or several governments affect systemic risk.

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  1. 1.

    The overall joint default risk of the sovereign system is the probability of all sovereigns to default simultaneously. Following Lehar (2005), Adrian and Brunnermeier (2016) and Tarashev et al. (2010), we assume that the financial system is a portfolio of a finite number of European sovereigns.

  2. 2.

    We constrain the euro area system to include only sovereigns.

  3. 3.

    See Table 5.2 for an example of the correlation matrix for the counterfactual that we use to derive the individual contribution of Greek government default to sovereign systemic risk.

  4. 4.

    The literature on credit default swap pricing uses the terms “CDS spreads”, “CDS prices” and “CDS premia” interchangeably. For the remainder of the chapter, we will use these three terms as synonyms.

  5. 5.

    If the average correlation is high, we have a smaller correction, but the bias still exists.

  6. 6.

    For a full definition of the Shapley value, see Tarashev et al. (2010).

  7. 7.

    The reason why we focus on the most simple case first is that systemic risk measures need to be understandable to policymakers that make many parallel decisions in a short period of time.

  8. 8.

    The default thresholds are defined in the sense of the classical structural model by Merton (1974): an entity defaults if its assets drop below a specific value. Note that as in Segoviano (2006) and Segoviano and Goodhart (2009), the default region is in the right tail of the distribution. This does not affect our results, due to the assumption of a symmetrical prior distribution, but simplifies our estimation procedure.

  9. 9.

    As in Segoviano and Goodhart (2009), we define the default threshold to be:

    $$\begin{aligned} \overline{\mathbf{x }}_{k} = \Phi ^{-1}(1-\overline{PoD^k}), \end{aligned}$$

    where \(\Phi ^{-1}(\cdot )\) is the inverse of the cumulative distribution function of a standard normal distribution and \(\overline{PoD^k}\) is the through-the-period average of the CDS-derived individual probabilities of default of sovereign k.

  10. 10.

    The individual probability of default of sovereign k is defined as

    $$\begin{aligned} \int \limits _{-\infty }^{+\infty } \int \limits _{-\infty }^{+\infty } \cdots \int \limits _{-\infty }^{+\infty } p(x_1,x_2,\ldots ,x_n) \mathbf{I} _{[\overline{\mathbf{x }}_{1}, \infty )} dx_1 \cdots dx_{n-1} dx_n = PoD^{k}, \end{aligned}$$

    where \(\mathbf{I} _{[\overline{\mathbf{x }}_{1}, \infty )}\) is an indicator variable that takes the value of 1 if the latent asset process of sovereign k crosses the sovereign-specific default threshold and 0 otherwise. In practice, it is estimated empirically from CDS spreads using a bootstrapping procedure. The estimation procedure is outlined in Sect. 3.1.

  11. 11.

    Find Tables 4.5 and 5.2 in Sect. 5.3.3. In the particular example in Table 5.2, we use Greece as the defaulting sovereign.

  12. 12.

    The observed difference in average outcomes can be summarized as (Angrist and Pischke 2009):

    $$\begin{aligned} E\left[ Y_i|D_i=1\right] - E\left[ Y_i|D_i=0\right] = E\left[ Y_{1i}|D_i=1\right] - E\left[ Y_{0i}|D_i=1\right] + E\left[ Y_{1i}|D_i=1\right] - E\left[ Y_{0i}|D_i=0\right] , \end{aligned}$$

    where \(D_i\) is a dummy equal to 1, in case the object (the sovereign system) is treated (a sovereign k defaults) and 0 otherwise; \(Y_i\) is the observed outcome, which is further equal to \(Y_{1i}\) if the sovereign system experiences a default of sovereign k and \(Y_{0i}\) in case that it does not.

    Examining Eq. (5.6) more carefully and realizing that the CDS-derived probabilities of default are market expectations about the solvency of sovereign, we can translate the \(\Delta CoJPoD\) measure to the term

    $$\begin{aligned} E\left[ Y_{1i}|D_i=1\right] - E\left[ Y_{0i}|D_i=1\right] , \end{aligned}$$

    which is the Average Treatment Effect on the Treated, with \(CoJPoD'\) being the counterfactual of the system, when it is not affected by the default of sovereign k, and hence equal to \(E\left[ Y_{0i}|D_i=1\right] \). The second difference on the right-hand side of Eq. (5.7) is the selection bias.

  13. 13.

    The spikes in certain periods are due to minor rounding errors.

  14. 14.

    CDS data for some of the big banks in the European Union was not available from CMA at the time of download. Therefore, we complement the CDS data for banks with data from Bloomberg to achieve a higher coverage of the EU banking system.

  15. 15.

    Of course, this measure is only meaningful when there are no bank runs. Since bank runs will affect not only the deposits, but also the general funding availability, the information content of this liquidity measure is reduced during such periods.

  16. 16.

    We “fix” the composition of the subportfolios in favour of a dynamic composition, in order to make the multivariate probability results comparable across periods.

  17. 17.

    We present and interpret the results for several financial characteristics only. The rest of the results are available upon request.

  18. 18.

    We should note that our definition of the counterfactual state of the world differs from the median state used for the final version of the \(\Delta CoVaR\). In fact, or measure is related to an earlier version of the latter, where an institution in distress is excluded from the counterfactual. In our probability setting, this earlier concept is more appropriate in identifying the information content from a default.

  19. 19.

    Furthermore, since regulators can either observe a default or a non-default, and never both counterfactuals, the calculation of the ATE suffers from a selection bias (see, Angrist and Pischke 2009, p. 22), since a sovereign can self-select to default or not and is therefore of limited interest for causal analysis. This, however, is more of a statistical issue, rather than probabilistic, and both types of systemic risk contributions can be readily calculated, once we recover the overall sovereign asset distribution.


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Correspondence to Deyan Radev .

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Radev, D. (2022). Systemic Risk Contributions. In: Measuring Systemic Risk. Studies in Systems, Decision and Control, vol 409. Springer, Cham.

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