Asymptotic Analysis for Spectral Risk Measures Parameterized by Confidence Level

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.