Optimal periodic dividend strategies for spectrally positive L\'evy risk processes with fixed transaction costs

We consider the general class of spectrally positive L\'evy risk processes, which are appropriate for businesses with continuous expenses and lump sum gains whose timing and sizes are stochastic. Motivated by the fact that dividends cannot be paid at any time in real life, we study $\textit{periodic}$ dividend strategies whereby dividend decisions are made according to a separate arrival process. In this paper, we investigate the impact of fixed transaction costs on the optimal periodic dividend strategy, and show that a periodic $(b_u,b_l)$ strategy is optimal when decision times arrive according to an independent Poisson process. Such a strategy leads to lump sum dividends that bring the surplus back to $b_l$ as long as it is no less than $b_u$ at a dividend decision time. The expected present value of dividends (net of transaction costs) is provided explicitly with the help of scale functions. Results are illustrated.Comment: Accepted for publication in Insurance: Mathematics and Economic

Optimal Dividends in the Dual Model with Diffusion

In the dual model, the surplus of a company is a Lévy process with sample paths that are skip-free downwards. In this paper, the aggregate gains process is the sum of a shifted compound Poisson process and an independent Wiener process. By means of Laplace transforms, it is shown how the expectation of the discounted dividends until ruin can be calculated, if a barrier strategy is applied, and how the optimal dividend barrier can be determined. Conditions for optimality are discussed and several numerical illustrations are given. Furthermore, a family of models is analysed where the individual gain amount distribution is rescaled and compensated by a change of the Poisson paramete

On the optimality of joint periodic and extraordinary dividend strategies

In this paper, we model the cash surplus (or equity) of a risky business with a Brownian motion. Owners can take cash out of the surplus in the form of "dividends", subject to transaction costs. However, if the surplus hits 0 then ruin occurs and the business cannot operate any more. We consider two types of dividend distributions: (i) periodic, regular ones (that is, dividends can be paid only at countable many points in time, according to a specific arrival process); and (ii) extraordinary dividend payments that can be made immediately at any time (that is, the dividend decision time space is continuous and matches that of the surplus process). Both types of dividends attract proportional transaction costs, and extraordinary distributions also attracts fixed transaction costs, a realistic feature. A dividend strategy that involves both types of distributions (periodic and extraordinary) is qualified as "hybrid". We determine which strategies (either periodic, immediate, or hybrid) are optimal, that is, we show which are the strategies that maximise the expected present value of dividends paid until ruin, net of transaction costs. Sometimes, a liquidation strategy (which pays out all monies and stops the process) is optimal. Which strategy is optimal depends on the profitability of the business, and the level of (proportional and fixed) transaction costs. Results are illustrated

Stable Dividends under Linear-Quadratic Optimization

The optimization criterion for dividends from a risky business is most often formalized in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for stability of dividends. In particular, within actuarial risk theory, maximization of future dividends have been intensively studied as the so-called de Finetti problem. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable and suboptimal affine dividend strategies have therefore received attention recently. In contrast, in the class of linear-quadratic problems a demand for stability if explicitly stressed. These have most often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these patterns of thinking by deriving optimal affine dividend strategies under a linear-quadratic criterion for a general L\'evy process. We characterize the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximizing expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, as e.g. in classical risk theory, is explicitly demanded and explicitly obtained

Modelling and understanding count processes through a Markov-modulated non-homogeneous Poisson process framework

The Markov-modulated Poisson process is utilised for count modelling in a variety of areas such as queueing, reliability, network and insurance claims analysis. In this paper, we extend the Markov-modulated Poisson process framework through the introduction of a flexible frequency perturbation measure. This contribution enables known information of observed event arrivals to be naturally incorporated in a tractable manner, while the hidden Markov chain captures the effect of unobservable drivers of the data. In addition to increases in accuracy and interpretability, this method supplements analysis of the latent factors. Further, this procedure naturally incorporates data features such as over-dispersion and autocorrelation. Additional insights can be generated to assist analysis, including a procedure for iterative model improvement. Implementation difficulties are also addressed with a focus on dealing with large data sets, where latent models are especially advantageous due the large number of observations facilitating identification of hidden factors. Namely, computational issues such as numerical underflow and high processing cost arise in this context and in this paper, we produce procedures to overcome these problems. This modelling framework is demonstrated using a large insurance data set to illustrate theoretical, practical and computational contributions and an empirical comparison to other count models highlight the advantages of the proposed approach.Comment: For simulated data sets and code, please go to https://github.com/agi-lab/reserving-MMNP

SynthETIC: an individual insurance claim simulator with feature control

Recent years have seen rapid increase in the application of machine learning to insurance loss reserving. They yield most value when applied to large data sets, such as individual claims, or large claim triangles. In short, they are likely to be useful in the analysis of any data set whose volume is sufficient to obscure a naked-eye view of its features. Unfortunately, such large data sets are in short supply in the actuarial literature. Accordingly, one needs to turn to synthetic data. Although the ultimate objective of these methods is application to real data, the use of synthetic data containing features commonly observed in real data is also to be encouraged. While there are a number of claims simulators in existence, each valuable within its own context, the inclusion of a number of desirable (but complicated) data features requires further development. Accordingly, in this paper we review those desirable features, and propose a new simulator of individual claim experience called SynthETIC. Our simulator is publicly available, open source, and fills a gap in the non-life actuarial toolkit. The simulator specifically allows for desirable (but optionally complicated) data features typically occurring in practice, such as variations in rates of settlements and development patterns; as with superimposed inflation, and various discontinuities, and also enables various dependencies between variables. The user has full control of the mechanics of the evolution of an individual claim. As a result, the complexity of the data set generated (meaning the level of difficulty of analysis) may be dialled anywhere from extremely simple to extremely complex

Leprosy in wild chimpanzees

Humans are considered as the main host for Mycobacterium leprae1, the aetiological agent of leprosy, but spillover has occurred to other mammals that are now maintenance hosts, such as nine-banded armadillos and red squirrels2,3. Although naturally acquired leprosy has also been described in captive nonhuman primates4-7, the exact origins of infection remain unclear. Here we describe leprosy-like lesions in two wild populations of western chimpanzees (Pan troglodytes verus) in Cantanhez National Park, Guinea-Bissau and Taï National Park, Côte d'Ivoire, West Africa. Longitudinal monitoring of both populations revealed the progression of disease symptoms compatible with advanced leprosy. Screening of faecal and necropsy samples confirmed the presence of M. leprae as the causative agent at each site and phylogenomic comparisons with other strains from humans and other animals show that the chimpanzee strains belong to different and rare genotypes (4N/O and 2F). These findings suggest that M. leprae may be circulating in more wild animals than suspected, either as a result of exposure to humans or other unknown environmental sources