Interplay of insurance and financial risks in a discrete-time model with strongly regular variation

Abstract

Consider an insurance company exposed to a stochastic economic environment that contains two kinds of risk. The first kind is the insurance risk caused by traditional insurance claims, and the second kind is the financial risk resulting from investments. Its wealth process is described in a standard discrete-time model in which, during each period, the insurance risk is quantified as a real-valued random variable X equal to the total amount of claims less premiums, and the financial risk as a positive random variable Y equal to the reciprocal of the stochastic accumulation factor. This risk model builds an efficient platform for investigating the interplay of the two kinds of risk. We focus on the ruin probability and the tail probability of the aggregate risk amount. Assuming that every convex combination of the distributions of X and Y is of strongly regular variation, we derive some precise asymptotic formulas for these probabilities with both finite and infinite time horizons, all in the form of linear combinations of the tail probabilities of X and Y . Our treatment is unified in the sense that no dominating relationship between X and Y is required.

A necessary and sufficient condition for the subexponentiality of the product convolution

“…The concept of the product convolution plays an important role in applied probability and has attracted increasing interest in recent years; see, for example, Cline and Samorodnitsky (1994), Tang (2006), Li and Tang (2015), and Samorodnitsky and Sun (2016). Note that the present value of a random future claim, which is one of the most fundamental quantities in finance and insurance, is expressed as the product of the random claim amount and the corresponding stochastic present value factor.…”

“…,Zhang et al (2009), Chen (2011),Yang and Wang (2013),Huang et al (2014), andLi and Tang (2015). By using Theorem 1.3 of this paper, we can obtain a different version of the finite-time ruin probability which is stated later.…”

Iterated random functions and regularly varying tails

“…To see a few examples, how (3.8) and (3.9) come into play, we state the following Corollary, which treats the case when the tail of A ∧ B is negligible. This covers the possibility that A and B are independent, as treated in [34], and the possibility that the tail of B is negligible, as treated by Kevei [30]. For simplicity we will assume that B ≥ 0 so that f − (y) = 0 for any y ∈ L(X) ⊆ [0, +∞).…”

“…Up to our best knowledge, this case was not studied in the literature apart form two specific cases: independent A and B treated in [34] and so-called exponential functional of Lévy processes studied in [37]. Our aim is to present a robust approach to treat the scenario (1.5) and its counterpart for the iterated random functions {R n } n≥0 and R.…”

Asymptotics of convolution with the semi-regular-variation tail and its application to risk

“…Most of the existing works assume that the insurance risk and the financial risk respectively follow the same distribution F and G, and that the financial risk is dominated by the insurance risk with a subexponeantial distribution, or more general convolution equivalent distribution, namely G(x) = o F (x) and F ∈ S(α) for some α ∈ R + ∪ {0}. In Li and Tang (2015), the dominating relationship between financial risk and insurance risk is not required, however, the distributions of every convex combination of F and G are required to belong to the class R * (α) ⊂ R 2 (α). In order to remove the restrictions on the dominated relationship and enlarge the range of the corresponding distribution class, Hashorva and Li (2014) give the following result.…”