Consider a company whose business carries the potential for investment losses and is additionally vulnerable to exogenous shocks. The unpredictability of the shocks makes it challenging for both the company and the regulator to accurately assess their impact, potentially leading to an underestimation of solvency capital when employing traditional approaches. In this paper, we utilize a stylized model to conduct an extreme value analysis of the tail risk of the company under a Fréchet-type and a Gumbel-type shock. Our main results explicitly demonstrate the different roles of investment risk and shock risk in driving large losses. Furthermore, we derive asymptotic estimates for the value at risk and expected shortfall of the total loss. Numerical studies are conducted to examine the accuracy of the obtained estimates.

The model uncertainty issue is pervasive in virtually all applied fields but especially critical in insurance and finance. To hedge against the uncertainty of the underlying probability distribution, which we refer to as ambiguity, the worst case is often considered in quantifying the underlying risk. However, this worst-case treatment often yields results that are overly conservative. We argue that, in most practical situations, a generic risk is realized from multiple scenarios and the risk in some ordinary scenarios may be subject to negligible ambiguity so that it is safe to trust the reference distributions. Hence, we only need to consider the worst case in the other scenarios where ambiguity is significant. We implement this idea in the study of the worst-case moments of a risk in the hope to alleviate the over-conservativeness issue. Note that the ambiguity in our consideration exists in both the scenario indicator and the risk in the corresponding scenario, leading to a two-fold ambiguity issue. We employ the Wasserstein distance to construct an ambiguity ball. Then, we disentangle the ambiguity along the scenario indicator and the risk in the corresponding scenario, so that we convert the two-fold optimization problem into two one-fold problems. Our main result is a closed-form worst-case moment estimate. Our numerical studies illustrate that the consideration of partial ambiguity indeed greatly alleviates the over-conservativeness issue.

The study of desirable structural properties that define a marketable insurance contract has been a recurring theme in insurance economic theory and practice. In this article, we develop probabilistic and structural characterizations for insurance indemnities that are universally marketable in the sense that they appeal to all policyholders whose risk preferences respect the convex order. We begin with the univariate case where a given policyholder faces a single risk, then extend our results to the case where multiple risks possessing a certain dependence structure coexist. The non-decreasing and 1-Lipschitz condition, in various forms, is shown to be intimately related to the notion of universal marketability. As the highlight of this article, we propose a multivariate mixture model which not only accommodates a host of dependence structures commonly encountered in practice but is also flexible enough to house a rich class of marketable indemnity schedules.

In recent decades, there has been significant growth in the capital market for mortality- and longevity-linked bonds. Therefore, modeling and forecasting the mortality indexes underlying these bonds have crucial implications for risk management in life insurance companies. In this paper, we propose a hierarchical reconciliation approach to constructing probabilistic forecasts for mortality bond indexes. We apply this approach to analyzing the Swiss Re Kortis bond, which is the first “longevity trend bond” introduced in the market. We express the longevity divergence index associated with the bond’s principal reduction factor (PRF) in a hierarchical setting. We first adopt time-series models to obtain forecasts on each hierarchical level, and then apply a minimum trace reconciliation approach to ensure coherence of forecasts across all levels. Based on the reconciled probabilistic forecasts of the longevity divergence index, we estimate the probability distribution of the PRF of the Kortis bond, and compare our results with those stated in Standard and Poor’s report on pre-sale information. We also illustrate the strong performance of the approach by comparing the reconciled forecasts with unreconciled forecasts as well as those from the bottom-up approach and the optimal combination approach. Finally, we provide first insights on the interest spread of the Kortis bond throughout its risk period 2010–2016.

Frequent large losses from recent catastrophes have caused great concerns among insurers/reinsurers, who then turn to seek mitigations of such catastrophe risks by issuing catastrophe (CAT) bonds and thereby transferring the risks to the bond market. Whereas, the pricing of CAT bonds remains a challenging task, mainly due to the facts that the CAT bond market is incomplete and that the pricing usually requires knowledge about the tail of the risks. In this paper, we propose a general pricing framework based on a product pricing measure, which combines a distorted probability measure that prices the catastrophe risks underlying the CAT bond with a risk-neutral probability measure that prices interest rate risk. We also demonstrate the use of the peaks over threshold (POT) method to uncover the tail risk. Finally, we conduct case studies using Mexico and California earthquake data to demonstrate the applicability of our pricing framework.

We study the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtain explicit formulae for the Laplace transforms associated with the two-sided exit problem. The expected present value of tax payments until default, the two-sided exit probabilities, and, hence, the nondefault probability with the default threshold equal to the lower bound are solved as immediate corollaries. A sufficient and necessary condition for the tax identity in ruin theory is discovered.

Consider a general bivariate Lévy-driven risk model. The surplus process Y, starting with Y 0=x > 0, evolves according to dY t = Y t- dR t -dP t for t > 0, where P and R are two independent Lévy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x→∞, which confirms Paulsen's conjecture.

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.

In this paper we consider the probabilities of finite- and infinite-time absolute ruins in the renewal risk model with constant premium rate and constant force of interest. In the particular case of the compound Poisson model, explicit asymptotic expressions for the finite- and infinite-time absolute ruin probabilities are given. For the general renewal risk model, we present an asymptotic expression for the infinite-time absolute ruin probability. Conditional distributions of Poisson processes and probabilistic techniques regarding randomly weighted sums are employed in the course of this study.

Let X1 and X2 be two independent and nonnegative random variables with distributions F1 and F2, respectively. This paper proves that if both F1 and F2 are of Weibull type and fulfill certain easily verifiable conditions, then the distribution of the product X1X2, called the product convolution of F1 and F2, belongs to the class 𝒮* and, hence, is subexponential.

Recently, Albrecher and his coauthors have published a series of papers on the ruin probability of the Lévy insurance model under the so-called loss-carry-forward taxation, meaning that taxes are paid at a certain fixed rate immediately when the surplus of the company is at a running maximum. In this paper we assume periodic taxation under which the company pays tax at a fixed rate on its net income during each period. We devote ourselves to deriving explicit asymptotic relations for the ruin probability in the most general Lévy insurance model in which the Lévy measure has a subexponential tail, a convolution-equivalent tail, or an exponential-like tail.

In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.

In this paper we study the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

Consider a discrete-time insurance risk model with risky investments. Under the assumption that the loss distribution belongs to a certain subclass of the subexponential class, Tang and Tsitsiashvili (Stochastic Processes and Their Applications 108(2): 299–325 (2003)) established a precise estimate for the finite time ruin probability. This article extends the result both to the whole subexponential class and to a nonstandard case with associated discount factors.

In this paper, we establish a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and subexponential claims in the case that the initial surplus is large. The formula is consistent with known results for the ultimate ruin probability and, in particular, is uniform for all time horizons when the claim size distribution is regularly varying tailed.

Let {X k , k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {w k , k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1 n w k X k and the maximum of weighted sums max1≤m≤n ∑k=1 m w k X k , subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss X k during the kth year, with a discount or inflation factor w k .

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.

In this paper, we discuss max-sum equivalence and convolution closure of heavy-tailed distributions. We generalize the well-known max-sum equivalence and convolution closure in the class of regular variation to two larger classes of heavy-tailed distributions. As applications of these results, we study asymptotic behaviour of the tails of compound geometric convolutions, the ruin probability in the compound Poisson risk process perturbed by an α-stable Lévy motion, and the equilibrium waiting-time distribution of the M/G/k queue.

Let {X k , k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X k , k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X (τ), S τ and S (τ), where X 0 = 0 by convention and h > 0 is arbitrarily fixed.