Chapter 1
Measuring Systemic Risk: Structural Approaches

Raimund M. Kovacevic

ISOR, Department of Statistics and OR, University of Vienna, Austria

Georg Ch. Pflug

ISOR and IIASA, Laxenburg

-Ben Bernanke, ex-chairman of the US Federal Reserve Bank.

The global financial crisis of 2007-2008, often considered as the worst financial crisis since the Great Depression of the 1930s, resulted in a change of paradigms in the financial and banking sector. These crisis years saw collapses of large financial institutions, bailouts of banks by governments, and declines of stock markets. Triggered by the U.S. housing bubble, which itself was caused by giving easy access to loans for subprime borrowers, financial distress spread over the banking sector and led to failure of key businesses and to the 2008-2012 global recession. Finally, this also contributed to the European sovereign-debt crisis, with lots of aftereffects in our present times.

Uncertainties about bank solvency, declines in credit availability, and reduced investor confidence had an impact on global stock markets. Governments responded with fiscal measures and institutional bailouts, which in the long term resulted in extreme public debts and necessary tax increases.

This negative experience demonstrates that the economy as a whole, but especially the financial sector is subject to risks, which are grounded in the interdependencies between the different economic actors and not in the performance of individual actors. This type of risk is generally called systemic risk. While aspects of systemic risk (e.g., bank run and contagion) were always an issue in discussions about the financial system, the recent crises have increased the interest in the topic, not only in academic circles, but also among regulators and central banks.

Systemic Risk: Definitions

If one aims at measuring-and in a further step managing and mitigating-systemic risk, it is important to start with a definition. However, despite the consent that systemic risk is an important topic, which is reflected by an increasing number of related papers and technical reports, there is still not a single generally accepted definition.

As a first step, one should distinguish between systemic and systematic risk. Systematic risks are aggregate (macroeconomic) risks that cannot be reduced by hedging and diversification. Systemic risk, on the other hand, is a different notion. It refers to the risk of breakdown or at least major dysfunction of financial markets. The Group of Ten (2001) gave the following, often cited definition:

Systemic financial risk is the risk that an event will trigger a loss of economic value or confidence in, and attendant increases in uncertainly about, a substantial portion of the financial system that is serious enough to quite probably have significant adverse effects on the real economy. Systemic risk events can be sudden and unexpected, or the likelihood of their occurrence can build up through time in the absence of appropriate policy responses. The adverse real economic effects from systemic problems are generally seen as arising from disruptions to the payment system, to credit flows, and from the destruction of asset values.

This formulation describes many aspects related to systemic risk but can hardly be called a definition in the technical sense, as it is very broad and hard to quantify. In addition, it seems to confuse cause (confidence) and consequence (breakdown).

As an alternative, Kaufmann and Scott (2003) introduced the following definition:

Systemic risk refers to the risk or probability of breakdowns in an entire system, as opposed to breakdowns in individual parts or components, and is evidenced by co-movements among most or all the parts.

In similar manner, but naming the cause and again considering larger consequences, the European Central Bank (2004) defines systemic risk as follows:

The risk that the inability of one institution to meet its obligations when due will cause other institutions to be unable to meet their obligations when due. Such a failure may cause significant liquidity or credit problems and, as a result, could threaten the stability of or confidence in markets.

All discussed definitions focus on the banking or financial system as a whole, and relate systemic risk to the interconnectedness within the system. Often, they stress the risk of spillovers from the financial sector to the real economy and the associated related costs. This effect is emphasized even more after the financial crisis as described in the following definition of systemic risk from Adrian and Brunnermeier (2009):

The risk that institutional distress spreads widely and distorts the supply of credit and capital to the real economy.

A similar definition can be found in Acharya et al. 2009.

Given the described diversity of definitions, which are similar but also different with respect to their focus, it is hard to develop universally accepted measures for systemic risk. Different definitions refer to different important nuances of systemic risk, which means that on the operational level a robust framework for monitoring and managing systemic risk should involve a variety of risk measures related to these different aspects. See Hansen (2012) for a deeper discussion of the basic difficulties in defining and identifying systemic risk.

We will focus on the first part of the definition by Kaufmann and Scott (2003), which summarizes the most important aspect of systematic risk in financial systems, without addressing more general 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 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 Di, the...