ASYMPTOTIC ANALYSIS AND BOUNDS OF MULTIVARIATE COHERENT RISK

Li Zhu

A central topic in modern financial and insurance mathematics is the search for new methods to estimate extreme risk (or tail risk) for multivariate financial assets. This research targets this fundamental question about tail risk, and analyzes tail risk for multivariate financial portfolios, using tail conditional expectation (TCE) and tail distortion risk. Extreme dependence has been observed in diverse fields, such as data network, financial risk management, environmental impact assessment, etc. The tail risk fueled by extreme dependence and its contagious adverse effects have been best illustrated from the recent financial crisis. TCE used in continuous risk analysis describes the expected amount of risk that could be experienced given that a potential risk exceeds a threshold and is especially effective for analyzing tail risks. In this research, the well-known Karamata theorem of regular variation is extended and explicitly tractable tail approximations of TCE in terms of extreme dependence for the portfolios that are multivariate regularly varying are derived. The vector-valued TCE, as a multivariate coherent risk measure, corresponds to a set of deterministic vectors which represent portfolios of extra capitals needed so that the resulting positions are acceptable to regulator/supervisor. We present tractable sharp lower and upper bounds for vector-valued TCEs. Several simulation results for various multivariate distributions are also provided to illustrate our bounds and their monotonicity properties. Our results can be applied for accurate estimates and analysis of extremal risks in quantitative risk management. A distortion risk measure is defined as the expected value of potential loss under a scenario with some transformed probability measure and has been used widely in finance and insurance. The tail distortion risk measure is introduced to assess tail risks of exccess losses modelled by the right tails of loss distributions. The asymptotic linear relation between tail distortion and Value-at-Risk is derived for heavy tailed losses with the linear proportionality constant depending only on the distortion function and heavy tail index. Various examples involving tail distortions for location, scale and shape invariant loss distribution families are also presented to illustrate the results.

ASYMPTOTIC ANALYSIS AND BOUNDS OF MULTIVARIATE COHERENT RISK

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