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addition, bankruptcy of additional entities may reduce even further the insolvent assets of entities that already defaulted in the first round.

      This process can be continued, leading to new values

, and an augmented set of defaulted entities after each iteration k. The loss cascade terminates, when no additional entity is sent to bankruptcy in step k, that is, .

      The sequences

and of debt are nonincreasing in k and, furthermore, are bounded from below by zero values for all components, which implies convergence of debt. At this point we have

1.9

This system describes the relation between the positive and negative parts of the distances to default

for all entities i. It holds with probability 1 for all entities. Note that previous literature, such as Chan-Lau et al. (2009a; 2009b), uses fixed numbers instead of the estimated loss given defaults in (1.9).

      In fact, the system (1.9) is ambiguous, and we search for the smallest solution, the optimization problem

      1.10

      has to be solved in order to obtain the correct estimates for

and .

      This basic setup can be easily extended to deal with different definitions of the distress barrier, involving early warning barriers, or accounting for different types of debt (e.g., short-term and long-term, as indicated earlier).

      Measuring Systemic Risk

      The distances to default, derived from structural models, in particular from systemic models in the strict sense, can be used to measure systemic risk. In principle, the joint distribution of distances to default for all involved entities contains (together with the definition of distress barriers) all the relevant information. We assume that the joint distribution is continuous and let

denote the joint density of the distances to default for all entities.

      Note that the risk measures discussed in the following are often defined in terms of asset value, which is fully appropriate for systemic models in the broader sense. In view of the previous discussion of systemic models in the strict sense, we instead prefer to use the distances to default or loss variables derived from the distance to default.

      The first group of risk measures is based directly on unconditional and conditional default probabilities. See Guerra et al. (2013) for an overview of such measures. The simplest approach considers the individual distress probabilities

      1.11

      The term in squared brackets is the marginal density of

, which means that it is not necessary to estimate the joint density for this measure. In similar manner, one can consider joint distributions for any subset of entities by using the related (joint) marginal density , which can be obtained by integrating the joint density over all other entities, that is, .

       Joint probabilities of distress for a subset I can be achieved by

1.12

      where the set I contains the elements

. Of special interest are the default probabilities of pairs of entities (see, e.g., Guerra, et al., 2013). Joint probabilities of distress describe tail risk within the chosen set I. If I represents the whole system (i.e., it contains all the entities), then the joint probability of distress can be considered as a tail risk measure for systemic risk (see, e.g., Segoviano & Goodhart, 2009).

      Closely related are conditional probabilities of distress, that is, the probability that entity j is in distress, given that entity j is in distress, which can be written as

      1.13

      These conditional probabilities can be presented by a matrix with

as its ijth matrix element, the distress dependency matrix.

      While conditional distress probabilities contain important information, it should be noted that they only reflect the two-dimensional marginal distributions. Conditional probabilities are often used for analyzing the interlinkage of the system and the likelihood of contagion. However, such arguments should not be carried to extremes. Finally, conditional probabilities do not contain any information about causality.

      Another systemic measure related to probabilities is the probability of at least one distressed entity; see Segoviano and Goodhart (2009) for an application to a small system of four entities. It can be calculated as

      1.14

      Guerra et al. (2013) propose an asset-value-weighted average of individual probabilities of distress as an upper bound for the probability of at least one distressed entity. Probabilities of exactly one, two, or another number of distressed entities are hard to calculate for large systems because of the large number of combinatorial possibilities.

      An important measure based on probabilities is the banking stability index, measuring the expected number of entities in distress, given that at least one entity is in distress. This measure can be written as

      1.15

      Other systemic risk measures based directly on the distribution of distances to default. Adrian and Brunnermeier (2009) propose a measure called conditional value at risk,1

. It is closely related to value at risk, which is the main risk measure for banks under the Basel accord.

      

is based on conditional versions of the quantile at level for an entity given that entity i reaches the -quantile. In terms of distances to default, this reads

1.16

      where

1.17

      The contribution of entity i to the risk of entity j then is calculated as

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