Qihe Tang

Risk Measures and Comonotonicity: A Review

In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. [11] .

Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments

This paper investigates the finite- and infinite-time ruin probabilities in a discrete-time stochastic economic environment. Under the assumption that the insurance risk - the total net loss within one time period - is extended-regularly-varying or rapidly-varying tailed, various precise estimates for the ruin probabilities are derived. In particular, some estimates obtained are uniform with respect to the time horizon, and so apply in the case of infinite-time ruin.

Randomly weighted sums of subexponential random variables with application to ruin theory

Let {Xk, 1 ≤ k ≤ n} be n independent and real-valued random variables with common subexponential distribution function, and let {θk, 1 ≤ k ≤ n} be other n random variables independent of {Xk, 1 ≤ k ≤ n} and satisfying a ≤ θk ≤ b for some 0 < a ≤ b < ∞ for all 1 ≤ k ≤ n. This paper proves that the asymptotic relations P (max1 ≤ m ≤ n ∑k=1m θkXk > x) ∼ P (sumk=1n θkXk > x) ∼ sumk=1nP (θkXk > x) hold as x → ∞. In doing so, no any assumption is made on the dependence structure of the sequence {θk, 1 ≤ k ≤ n}. An application to ruin theory is proposed.

Large deviations for heavy-tailed random sums in compound renewal model

In the present paper we investigate the precise large deviations for heavy-tailed random sums. First, we obtain a result which improves the relative result in Kluppelberg and Mikosch (J. Appl. Probab. 34 (1997) 293). Then we introduce a more realistic risk model than classical ones, named the compound renewal model, and establish the precise large deviations in this model.

Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails

In this paper we investigate the ruin probability in the classical risk model under a positive constant interest force. We restrict ourselves to the case where the claim size is heavy-tailed, i.e. the equilibrium distribution function (e.d.f.) of the claim size belongs to a wide subclass of the subexponential distributions. Two-sided estimates for the ruin probability are developed by reduction from the classical model without interest force.

Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model

Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and interarrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the interarrival time before the claim. We focus on determining the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim size distribution is extended regularly varying tailed, we show that this asymptotic formula is globally uniform.

The tail probability of discounted sums of Pareto-like losses in insurance

In an insurance context, the discounted sum of losses within a finite or infinite time period can be described as a randomly weighted sum of a sequence of independent random variables. These independent random variables represent the amounts of losses in successive development years, while the weights represent the stochastic discount factors. In this paper, we investigate the problem of approximating the tail probability of this weighted sum in the case when the losses have Pareto-like distributions and the discount factors are mutually dependent. We also give some simulation results.

Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation

Klüppelberg and Stadtmüller (1998, Scand. Actuar. J., no. 1, 49–58) obtained a simple asymptotic formula for the ruin probability of the classical model with constant interest force and regularly varying tailed claims. This short note extends their result to the renewal model. The proof is based on a result of Resnick & Willekens (1991, Comm. Statist. Stochastic Models 7, no. 4, 511–525).

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S[set membership, variant](0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.

Tail Probabilities of Randomly Weighted Sums of Random Variables with Dominated Variation

This paper investigates the asymptotic behavior of tail probabilities of randomly weighted sums of independent heavy-tailed random variables, where the weights form another sequence of nonnegative and arbitrarily dependent random variables. The results obtained are further applied to derive asymptotic estimates for the ruin probabilities in a discrete time risk model with dependent stochastic returns.