Second-order tail asymptotics of deflated risks

Abstract

Random deflated risk models have been considered in recent literatures. In this paper, we investigate second-order tail behavior of the deflated risk X = RS under the assumptions of second-order regular variation on the survival functions of the risk R and the deflator S. Our findings are applied to approximation of Value at Risk, estimation of small tail probability under random deflation and tail asymptotics of aggregated deflated risk.The main contributions of this paper concern the second-order expansions of the tail probability of the deflated risk X = RS which are then illustrated by several examples. Our main findings are utilized for the formulations of three applications, namely approximation of Value-at-Risk, estimation of small tail probability of the deflated risk, and the derivation of the tail asymptotics of aggregated risk under deflation.The rest of this paper is organized as follows. Section 2 gives our main results under second-order regular variation conditions. Section 3 shows the efficiency of our second-order asymptotics through some illustrating examples. Section 4 is dedicated to three applications. The proofs of all results are relegated to Section 5. We conclude the paper with a short Appendix. Main resultsWe start with the definitions and some properties of regular variation followed by our principal findings. A measurable function f : [0, ∞) → IR with constant sign near infinity is said to be of second-order regular variation with parameters α ∈ IR and ρ ≤ 0, denoted by f ∈ 2RV α,ρ , if there exists some function A with constant sign near infinity satisfying lim t→∞ A(t) = 0 such that for all x > 0 (cf. Bingham et al. (1987) and Resnick (2007))Here, A is referred to as the auxiliary function of f . Noting that (2.1) implies lim t→∞ f (tx)/f (t) = x α , i.e., f is regularly varying at infinity with index α ∈ IR, denoted by f ∈ RV α ; RV 0 is the class of slowly varying functions. When f is eventually positive, it is of second-order Π-variation with the second-order parameter ρ ≤ 0, denoted by f ∈ 2ERV 0,ρ , if there exist some functions a and A with constant sign near infinity and lim t→∞ A(t) = 0 such that for all x positive lim t→∞ f (tx)−f (t) a(t)

Second-Order Asymptotics of the Risk Concentration of a Portfolio with Deflated Risks

“…Several classes of parametric survival functions are shown to possess 2RV properties; see Hashorva et al [7]. For more details on RV and 2RV, see Hua and Joe [29] and Lv et al [5].…”

“…Before proving the above results, we introduce some lemmas. The first one gives a second-order form of Breiman's theorem (see Breiman [31]), which is from Hashorva et al [7].…”

“…Measuring a risk and quantifying its diversification benefits have become an important task. Especially when the underlying risk factors show a heavy-tailed pattern, many papers discussed diversification benefits; see, for instance, Degen et al [1] (2010), Ibragimov and Walden [2], Ibragimov et al [3], Mao et al [4], Lv et al [5,6], Hashorva et al [7], and references therein.…”

“…Degen et al [10] derived second-order approximations of VaR ( ) for independent and identically distributed (i.i.d) loss variables with a common survival function possessing the property of second-order regular variation (2RV). Secondorder approximations of the risk concentrations CTE ( ) and ( ) as ↑ 1 for i.i.d loss random variables were derived by Mao et al [4], Mao and Hu [12], Lv et al [6], and Hashorva et al [7]. For a portfolio of i.i.d.…”

Tail asymptotic expansions for L-statistics

“…Obviously, equation (2.1) implies F ∈ RV −α and the second-order parameter ρ controls the convergence rate of F (tx)/F (t) − x −α . Several classes of parametric survival functions are shown to possess 2RV properties, see e.g., Hashorva et al [15].…”