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Structural models for default go back to Merton (2009) and build on the idea that default of a firm happens if the firm's assets are insufficient to cover contractual obligations (liabilities). Simple models such as Merton (2009) start by modeling a single firm in the framework of the Black–Scholes option pricing model, whereas more complex models extend the framework to multivariate formulations, usually based on correlations between the individual asset values. A famous example is Vasicek's asymptotic single factor model (see Vasicek 1987; 1991; and 2002), which is very stylized but leads to a closed-form solution.

In most structural default models, it is not possible to calculate the portfolio loss explicitly; hence, Monte Carlo simulation is an important tool for default calculations. Even then, the models usually make simplifying assumptions.

Consider a system consisting of economic entities (e.g., banks), and let denote the asset processes – that is, the asset values at time for the individual entities. Furthermore, for each entity i a limit D_i, the distress barrier, defines default in the following sense: default occurs if the asset value of entity i falls below the distress barrier:

1.1

The relation between asset value and distress barrier is usually closely related to leverage, the ratio between debt and equity.

Finally, let with

1.2

denote the distance to default of the individual entities. Note that alternatively the distance to default can also be defined in terms of as a percentage of asset value, divided by the asset volatility (see e.g., Crosbie and Bohn 2003).

In a one period setup – as used throughout this chapter – one is interested at values at time T, the end of the planning horizon. Analyzing systemic risk then means analyzing the joint distribution of the distances to default , in particular their negative parts , and the underlying random risk factors are described by the joint distribution of asset values .

Many approaches for modeling the asset values exist in literature. In a classical finance setup, one would use correlated geometric Brownian motions resulting in correlated log-normal distributions for the asset values at the end of the planning horizon. Segoviano Basurto proposes a Bayesian approach (Segoviano Basurto 2006); for applications, see also Jin and Nadal de Simone (2013). In this chapter, we will use copula-based models, as discussed later.

The second component of the approach, the distress barrier, is in the simplest case (Merton 2009), modeled just by the face value of overall debt for each entity. Other approaches distinguish between short-term and long-term debt (longer than the planning horizon). Usually, this is done by adding some reasonable fraction of long-term debt to the full amount of short term debt; see, for example, Servigny and Renault (2007).

Still, such classical credit default models (see, e.g., Guerra et al. 2013), although classified as systemic risk models, neglect an important aspect: Economic entities like banks are mutually indebted, and each amount of debt is shown as a liability for one entity but also as an asset for another entity. Default of one entity (a reduction in liabilities) may trigger subsequent defaults of other entities by reducing their asset values. We call such models systemic models in the strict sense.

Such approaches with mutual debt have been proposed, such as in Chan-Lau et al. (2009a; 2009b). Models neglecting this aspect are systemic models in a broad sense; in fact, they are restricted to the effects of systematic risk related to asset values.

The basic setup of systemic models in the strict sense can be described as follows: Let denote the amount of debt between entities i and j – that is, the amount of money borrowed by entity i from entity j. We also include debt to the nonbank sector, denoted by for each entity i and credit to the nonbanking sector, both repayable (including interest) at the end of the planning horizon, time T. Furthermore, is the value at time T of other financial assets held by entity i. Then the asset value of entity i at the end of the planning horizon is given 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).