Abstract
Systemic risk, in a complex system with several interrelated variables, such as a financial market, is quantifiable from the multivariate probability distribution describing the reciprocal influence between the system’s variables. The effect of stress on the system is reflected by the change in such a multivariate probability distribution, conditioned to some of the variables being at a given stress’ amplitude. Therefore, the knowledge of the conditional probability distribution function can provide a full quantification of risk and stress propagation in the system. However, multivariate probabilities are hard to estimate from observations. In this paper, I investigate the vast family of multivariate elliptical distributions, discussing their estimation from data and proposing novel measures for stress impact and systemic risk in systems with many interrelated variables. Specific examples are described for the multivariate Student-t and the multivariate normal distributions applied to financial stress testing. An example of the US equity market illustrates the practical potentials of this approach.
Highlights
In financial systems, stress testing consists in quantifying the ability of the system to cope with a crisis
As simple measure for stress test, I here propose to use the average extra losses on the subset Y provoked by extreme losses on the subset X. This is in analogy with the reasoning beyond the Conditional Value at Risk (CoVaR) approach (Adrian and Brunnermeier 2011), the idea is to stress the set of variables X to some extent and measure the average losses caused in the Y set
In this work I have shown that for the vast class of multivariate models described by elliptical distributions, the conditional probability density function is strictly linked to the original unconditional probability
Summary
Stress testing consists in quantifying the ability of the system to cope with a crisis. This paper adds on to the existing literature by introducing measures for stress propagation and systemic risk that are truly multidimensional quantifying directly effects between sets of variables and not just couples of them These measures are general for the vast family of elliptical distributions and can be extended further, for instance including the multivariate generalized hyperbolic class (Prause 1999). For the normal case the probability distribution shifts its location by the factor in Equation (7) and changes the scale by a factor 1 − ρ2X,Y This implies that any increase in the value at risk of the conditioned variable is only driven by the linear shift because the variance of Y is reduced by the conditioning. Some traditional risk measures and CoVaR applications for systemic risk can be found in Kurosaki and Kim (2013)
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