economic aspects. Such an approach could be seen as “systemic risk in the narrow sense” and we state it (slightly modified) as follows:
Systemic risk is the risk of breakdowns in an entire system, as opposed to breakdowns in individual parts or components.
Three issues have to be substantiated, if one wants to apply such a definition in concrete situations: system, breakdowns, and risk.
System
In financial applications, the focus lies on parts of the financial system (like the banking system, insurance, hedge funds) or the financial system as a whole. Any analysis has to start with describing the agents (e.g., banks in the banking system) within the analyzed system. This involves their assets and liabilities and the main risk factors related to profit and loss.
For a systemic view, it is important that the agents are not isolated entities at all. Systematic risk can be modeled by joint risk factors, influencing all profit and losses. Systemic risk in financial systems usually comes by mutual debt between the entities and the related leverage.
Breakdowns
In single-period models, breakdown is related to bankruptcy in a technical sense – that is, that the asset value of an agent at the end of the period does not reach a certain level (e.g., is not sufficient to pay back the agents debt). A lower boundary than debt can be used to reflect the fact that confidence into a bank might fade away even before bankruptcy, which severely reduces confidence between banks. In a systemic view, it is not sufficient to look at breakdowns of individual agents: Relevant are events that lead to the breakdown of more than one agent.
Risk
Risk is the danger that unwanted events (here, breakdowns) may happen or that developments go in an unintended direction. Quantifiable risk is described by distributions arising from risk. For financial systems this may involve the probability of breakdowns or the distribution of payments necessary to bring back asset values to an acceptable level. Risk measures summarize favorable or unfavorable properties of such distributions.
It should be mentioned that such an approach assumes that a good distributional model for the relevant risk factors can be formulated and estimated. During this chapter, we will stick to exactly this assumption. However, it is clear that in practice it is often difficult to come up with good models, and data availability might be severely restricted. Additional risk (model risk) is related to the quality of the used models and estimations; see Hansen (2012) for a deeper discussion of this point.
From Structural Models to Systemic Risk
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
1.1
The relation between asset value and distress barrier is usually closely related to leverage, the ratio between debt and equity.
Finally, let
denote the distance to default of the individual entities. Note that alternatively the distance to default can also be defined in terms of
In a one period setup – as used throughout this chapter – one is interested at 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
