A Guide to IMF Stress Testing

Chapter 19. Introduction to the Extreme Value Theory Approach to Stress Testing

Li Ong
Published Date:
December 2014
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Srobona Mitra

As risks build up, systemic events could materialize very rapidly. The speed at which a financial crisis spreads depends on the degree of interconnectedness of financial institutions, within and across countries. This interconnectedness may be attributable to specific interinstitutional exposures, such as interbank loans and deposits and repo transactions, or to exposures to aggregate risk, such as investor confidence and real economic and financial cycles. Thus, it is important to try and gauge the extent of potential spillovers well in advance of an actual stress event.

There are two major obstacles to modeling spillovers: the lack of published data of actual exposures between institutions and the (in)frequency of large systemic events, often referred to as “tail-risk events.” Instead, data on market prices, such as equity prices, bond prices, and credit default swap (CDS) spreads, may be used to derive the dependence of a financial institution or a country on an event in another institution or a country.

There are various ways of modeling spillovers or distress dependence. Some are based on econometric techniques, which assume that the distribution of the shocks is known, or “parametric” methods. The “nonparametric” methods allow the data to speak for themselves without imposing restrictions on the distribution of the shocks:

  • The CoRisk model provides a sense of the contribution of a financial institution to systemwide financial risk (e.g., Chan-Lau in Chapters 17 and 18). The Diebold-Yilmaz spillover matrix (Diebold and Yilmaz, 2009) is another parametric method that uses vector autoregressions and the associated variance decompositions to construct directional spillovers. The estimates from the latter model are based on central moments and do not provide a sense of tail-risk movements.

  • Nonparametric “copula-based” measures of spillover risk have become increasingly popular in recent years (e.g., Jobst and Gray in Chapter 26 and Segoviano and Goodhart in Chapter 32). The concept is based on the dissection of the joint multivariate distribution of returns (or other quantities) of several financial systems or institutions into the individual probability distribution and the dependence structure between the institutions. By “letting the data speak,” the spillover estimates from these methods are not tainted by the model. Moreover, these models provide several outputs including probability of default (PD) of an individual institution, the joint PD of the entire system, the PD of an institution conditional on one or more institutions defaulting, and the dependence structure of the institutions over time. However, these models are still in their infancy and are difficult to manipulate.

The extreme value theory (EVT) approach sits somewhere in between. It was introduced by Gropp and Moerman (2004) and Gropp, Lo Duca, and Vesala (2006) and provides a framework for analyzing extreme events (tail risks) that are rare but which could have a severe impact on the soundness of financial institutions or systems. It employs econometric and statistical analyses to estimate the potential for domino effects during a tail-risk event. The CoRisk method would be the closest comparator.

The EVT approach presented in this book first identifies all extreme events in the data by looking at, say, the 1st or the 5th percentile of the joint distribution of returns. These typically comprise weekly or daily observations of equity or CDS prices or the market value of assets. All returns lying in the left tail, that is, the ones below the defined thresholds, are called “exceedances.” Distress dependence then is estimated by using a logit model to account for the fatness of the tails of the distribution of exceedances. The probability of an exceedance is estimated conditional on exceedances in other financial institutions or centers, after controlling for common shocks such as extreme conditions in the world equity markets, the country’s stock markets, and real sector indicators. The results are, in most cases, unchanged if we assume an extreme value distribution instead of logistic distribution.

The EVT approach is more versatile in capturing the extent of spillover risks from a particular financial system or institution. Take, for example, a country in which financial markets have been enjoying a calm period for a number of years. The average correlations in movements (given by the mean of the distribution) in CDS spreads or equity returns across financial institutions might appear to be very low. However, a lack of correlation (a linear concept) may not imply independence. Extreme movements on a day-to-day or week-on-week basis—for instance, negative weekly returns or volatility exceeding the 5th-percentile left tail of the joint distribution of weekly returns—could point to high spillover risks during distress episodes.

The EVT models applied by Duggar and Mitra (Chapter 20) and by Chan-Lau and others (Chapter 21) are easy to maneuver and apply, with any standard econometric software package sufficient for analysis. Specific precrisis examples explored include (1) the risk of spillovers across borders for the Irish banking system; and (2) the direction of spillovers among the 25 largest banks in the world and that of the major banks within the European Union countries. In these examples, the distress-dependence matrices are largely static, and the sample periods are fairly long. The analyses can be made time varying by repeating the exercise over a rolling window, albeit one that is sufficiently long to provide an adequate number of observations of extreme movements.


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