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Asymptotic Analysis for Spectral Risk Measures Parameterized by Confidence Level

DOI: 10.4236/jmf.2018.81015, PP. 197-226

Keywords: Spectral Risk Measures, Quantitative Risk Management, Asymptotic Analysis, Extreme Value Theory, Euler Contribution

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Abstract:

We study the asymptotic behavior of the difference \"\" as \"\", where \"\" is a risk measure equipped with a confidence level parameter \"\" , and where X and Y are non-negative random variables whose tail probability functions are regularly varying. The case where is the value-at-risk (VaR) at α, is treated in [1]. This paper investigates the case where \"\" is a spectral risk measure that converges to the worst-case risk measure as \"\" . We give the asymptotic behavior of the difference between the marginal risk contribution \"\" and the Euler contribution \"\" of Y to the portfolio X+Y . Similarly to [1], our results depend primarily on the relative magnitudes of the thicknesses of the tails of X and Y. Especially, we find that \"\" is asymptotically equivalent to the expectation (expected loss) of Y if the tail of Y is sufficiently thinner than that of X. Moreover, we obtain the asymptotic relationship \"\" as \"\", where \"\" is a constant whose value likewise changes according to the relative magnitudes of the thicknesses of the tails of X and Y. We also conducted a numerical experiment, finding that when the tail of X is sufficiently thicker than that of Y, \"\" does not increase monotonically with α and takes a maximum at a confidence level strictly less than 1.

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