In this paper, we first introduce a simulator of cases estimates of incurred losses called SPLICE (Synthetic Paid Loss and Incurred Cost Experience). In three modules, case estimates are simulated in continuous time, and a record is output for each individual claim. Revisions for the case estimates are also simulated as a sequence over the lifetime of the claim in a number of different situations. Furthermore, some dependencies in relation to case estimates of incurred losses are incorporated, particularly recognising certain properties of case estimates that are found in practice. For example, the magnitude of revisions depends on ultimate claim size, as does the distribution of the revisions over time. Some of these revisions occur in response to occurrence of claim payments, and so SPLICE requires input of simulated per-claim payment histories. The claim data can be summarised by accident and payment “periods” whose duration is an arbitrary choice (e.g. month, quarter, etc.) available to the user. SPLICE is built on an existing simulator of individual claim experience called SynthETIC (introduced in Avanzi et al. 2021b, Insurance: Mathematics and Economics, 100, 296–308), which offers flexible modelling of occurrence, notification, as well as the timing and magnitude of individual partial payments. This is in contrast with the incurred losses, which constitute the additional contribution of SPLICE. The inclusion of incurred loss estimates provides a facility that almost no other simulators do. SPLICE is is a fully documented R package that is publicly available and open source (on CRAN). SPLICE, combined with SynthETIC, provides 11 modules (occurrence, notification, etc.), any one or more of which may be re-designed according to the user’s requirements. It comes with a default version that is loosely calibrated to resemble a specific (but anonymous) Auto Bodily Injury portfolio, as well as data generation functionality that outputs alternative data sets under a range of hypothetical scenarios differing in complexity. The general structure is suitable for most lines of business, with some reparameterisation.

In recent papers, Bonus-Malus Scales (BMS) estimated using data have been considered as an alternative to longitudinal data and hierarchical data approaches to model the dependence between different contracts for the same insured. Those papers, however, did not discuss in detail how to construct and understand BMS models, and many of the BMS’s basic properties were not discussed. The first objective of this paper is to correct this situation by explaining the logic behind BMS models and by describing those properties. More particularly, we will explain how BMS models are linked with simple count regression models that have covariates associated with the past claims experience. This study could help actuaries to understand how and why they should use BMS models for experience rating. The second objective of this paper is to create artificial past claims history for each insured. This is done by combining recent panel data theory with BMS models. We show that this addition significantly improves the prediction capacity of the BMS and provides a temporary solution for insurers who do not have enough historical data. We apply the BMS model to real data from a major Canadian insurance company. Results are analysed deeply to identify specific aspects of the BMS model.

We derive formulae for the moments of the time of ruin in both ordinary and modified Sparre Andersen risk models without specifying either the inter-claim time distribution or the individual claim amount distribution. We illustrate the application of our results in the special case of exponentially distributed claims, as well as for the following ordinary models: the classical risk model, phase-type(2) risk models, and the Erlang( $\mathscr{n}$ ) risk model. We also show how the key quantities for modified models can be found.

The financing of long-term care and the planning of care capacity are of increasing interest due to demographic changes and the ageing population in many countries. Since many care-intensive conditions begin to manifest at higher ages, a better understanding and assessment of the expected costs, required infrastructure, and number of qualified personnel are essential. To evaluate the overall burden of institutional care, we derive a model based on the duration of stay in dependence and the intensity of help provided to elderly individuals. This article aims to model both aspects using novel longitudinal data from nursing homes in the canton of Geneva in Switzerland. Our data contain comprehensive health and care information, including medical diagnoses, levels of dependence, and physical and psychological impairments on 21,758 individuals. We build an accelerated failure time model to study the influence of selected factors on the duration of care and a beta regression model to describe the intensity of care. We show that apart from age and gender, the duration of stay before death is mainly affected by the underlying diseases and the number of different diagnoses. Simultaneously, care intensity is driven by the individual level of dependence and specific limitations. Using both evaluations, we approximate the overall care severity for individual profiles. Our study sheds light on the relevant medical, physical, and psychological health indicators that need to be accounted for, not only by care providers but also by policy-makers and insurers.

We propose a new estimator for the ultimate prediction uncertainty within the famous Mack’s distribution-free chain-ladder model, which can be proved to be unbiased (conditionally given the first triangle column) under some additional technical assumptions. A peculiar behaviour of the unbiased estimator is given by its possible negativity. This is a drawback which might be worth trading off for the unbiasedness property, since there is empirical evidence that the likelihood of a negative realisation is extremely low. This offers an alternative to the well-known Mack and BBMW formulas since the latters can be proved to be biased. However, we also show that this novel estimator, as well as the Mack and BBMW formulas, can (with non-negligible probability) materially fail to estimate the true uncertainty.

The loss count distributions whose probabilities ultimately satisfy Panjer’s recursion were classified between 1981 and 2002; they split into six types, which look quite diverse. Yet, the distributions are closely related – we show that their probabilities emerge out of one formula: the binomial series. We propose a parameter change that leads to a unified, practical and intuitive, representation of the Panjer distributions and their parameter space. We determine the subsets of the parameter space where the probabilities are continuous functions of the parameters. Finally, we give an inventory of parameterisations used for Panjer distributions.

This paper proposes a shift in the valuation and production of long-term annuities, away from the classical risk-neutral methodology towards a methodology using the real-world probability measure. The proposed production method is applied to three examples of annuity products, one having annual payments linked to a mortality index and the savings account and the others having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of the proposed less-expensive production method. In contrast to classical risk-neutral production, which revolves around the savings account as reference unit, the long-term best-performing portfolio, the numéraire portfolio of the equity market, is employed as the fundamental reference unit in the production of the annuity. The numéraire portfolio is the strictly positive, tradable portfolio that when used as denominator or benchmark makes all benchmarked non-negative portfolios supermartingales. Under real-world valuation, the initial benchmarked value of a benchmarked contingent claim equals its real-world conditional expectation. The proposed real-world valuation and production can lead to significantly lower values of long-term annuities and their less-expensive production than suggested by the risk-neutral approach.

This article comments on the paper “Less-expensive long-term annuities linked to mortality, cash and equity” by Kevin Fergusson and Eckard Platen, appearing in this issue of the Annals of Actuarial Science. It adds two perspectives to their thought-provoking contribution. The first is a similarity to some recent work in quantitative finance on “deep hedging” that leverages machine learning models to find the cheapest replication strategy for a derivative payoff in a largely model-free setting. The second perspective engages with some of the interesting implications of their approach and draws parallels to literature in asset pricing and macro-finance. These perspectives point to the potential need for more fundamental shifts than the authors of the paper are advertising.