Quantitative Financial Risk Management. Galariotis Emilios

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by

1.3

      the distress barrier (in the simplest case) is

      1.4

      and the distance to default can be written as

1.5

      The random factors are the values

of financial assets, and (in an extended model) the credits from outside the system, payable back at time.

      Again, one could stop at this point and analyze the distances to default

, respectively the sum of all individual distances to default in the framework of classical default models. Systemic models in the strict sense, however, go farther.

      Consider now all entities in distress (defaulted banks), that is,

. Each of these entities is closed down and the related debt has to be adjusted, because entity i cannot fully pay back its debts. In a simple setup, this can be done by reducing all debts to other entities as follows:

      1.6

      1.7

      Here, the factor

1.8

      is an estimate for the loss given default of entity i.

It is now possible to calculate new asset values, new distress barriers, and new distances to default, after the default of all entities in

. For this purpose, we replace in (1.3) to (1.5) all occurrences of by and all occurrences of by . This first default triggers further ones and starts a loss cascade: It may happen that after the first adjustment step new defaults can be observed, which results in a new set of bankrupt entities after the second round. In 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

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