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

The quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. In this paper, we establish second-order asymptotics of the risk concentration based on several risk measures for a portfolio of n identically distributed but dependent deflated risks Xj=RjS, j=1,2,…,n under the assumptions of second-order regular variation on the survival functions of the risks Rj and the deflator S, where R1,R2,…,Rn are n independent and identically distributed random variables with a common survival function and S is a random variable being independent of R1,R2,…,Rn. Examples are also given to illustrate our main results.

THE SECOND-ORDER REGULAR VARIATION OF ORDER STATISTICS

Let X1, …, Xn be non-negative, independent and identically distributed random variables with a common distribution function F, and denote by X1:n ≤ ··· ≤ Xn:n the corresponding order statistics. In this paper, we investigate the second-order regular variation (2RV) of the tail probabilities of Xk:n and Xj:n − Xi:n under the assumption that $\bar {F}$ is of the 2RV, where 1 ≤ k ≤ n and 1 ≤ i < j ≤ n.

PROPERTIES OF SECOND-ORDER REGULAR VARIATION AND EXPANSIONS FOR RISK CONCENTRATION

The purpose of this study is two-fold. First, we investigate further properties of the second-order regular variation (2RV). These properties include the preservation properties of 2RV under the composition operation and the generalized inverse transform, among others. Second, we derive second-order expansions of the tail probabilities of convolutions of non-independent and identically distributed (i.i.d.) heavy-tail random variables, and establish second-order expansions of risk concentration under mild assumptions. The main results extend some ones in the literature from the i.i.d. case to non-i.i.d. case.