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.

Determining accurate capital requirements is a central activity across the life insurance industry. This is computationally challenging and often involves the acceptance of proxy errors that directly impact capital requirements. Within simulation-based capital models, where proxies are being used, capital estimates are approximations that contain both statistical and proxy errors. Here, we show how basic error analysis combined with targeted exact computation can entirely eliminate proxy errors from the capital estimate. Consideration of the possible ordering of losses, combined with knowledge of their error bounds, identifies an important subset of scenarios. When these scenarios are calculated exactly, the resulting capital estimate can be made devoid of proxy errors. Advances in the handling of proxy errors improve the accuracy of capital requirements.

Climate risks are systemic risks and may be clustered according to so-called volatilities, uncertainties, complexities, and ambiguities (VUCA) criteria. We analyze climate risk in the VUCA concept and provide a framework that allows to interpret systemic risks as model risk. As climate risks are characterized by deep uncertainties (unknown unknowns), we argue that precautionary and resilient principles should be applied instead of capital-based risk measures (reasonable for known unknows). A prominent example of the proposed principles is the precommitment approach (PCA). Within the PCA, subjective probabilities allow to discriminate between tolerable risks and acceptable ones. The amount of determined solvency capital for acceptable risks and estimations of model risk may be aggregated by means of a multiplier approach. This framework is in line with the three-pillar approach of Solvency II, especially with the recovery and resolution plan. Furthermore, it fits smoothly to a hybrid approach of micro- and macroprudential supervision.

Given the assumption that weather risks affect crop yields, we designed a weather index insurance product for soybean producers in the US state of Illinois. By separating the entire vegetation cycle into four growth stages, we investigate whether the phase-division procedure contributes to weather–yield loss relation estimation and, hence, to basis risk mitigation. Concretely, supposing stage-variant interaction patterns between temperature-based weather index growing degree days and rainfall-based weather index cumulative rainfall, a nonparametric weather–yield loss relation is estimated by a generalized additive model. The model includes penalized B-spline (P-spline) approach based on nonlinear optimal indemnity solutions under the expected utility framework. The P-spline analysis of variance (PS-ANOVA) method is used for efficient estimation through mixed model re-parameterization. The results indicate that the phase-division models significantly outperform the benchmark whole-cycle ones either under quadratic utility or exponential utility, given different levels of risk aversions. Finally, regarding hedging effectiveness, the expected utility ratio between the phase-division contract and the whole-cycle contract, and the percentage changes of mean root square loss and variance of revenues support the proposed phase-division contract.

One complication in mortality modelling is capturing the impact of risk factors that contribute to mortality differentials between different populations. Evidence has suggested that mortality differentials tend to diminish over age. Classical methods such as the Gompertz law attempt to capture mortality patterns over age using intercept and slope parameters, possibly causing an unjustified mortality crossover at advanced ages when applied independently to different populations. In recent research, Richards (Scandinavian Actuarial Journal 2020(2), 110–127) proposed a Hermite spline (HS) model that describes the age pattern of mortality differentials using one parameter and circumvents an unreasonable crossover by default. The original HS model was applied to pension data at individual level in the age dimension only. This paper extends the method to model population mortality in both age and period dimensions. Our results indicate that in addition to possessing desirable fitting properties, the HS approach can produce accurate mortality forecasts, compared with the Gompertz and P-splines models.

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.

Climate change is expected to increase the frequency and intensity of extreme weather events. To properly assess the increased economical risk of these events, actuaries can gain in relying on expert models/opinions from multiple different sources, which requires the use of model combination techniques. From non-parametric to Bayesian approaches, different methods rely on varying assumptions potentially leading to very different results. In this paper, we apply multiple model combination methods to an ensemble of 24 experts in a pooling approach and use the differences in outputs from the different combinations to illustrate how one can gain additional insight from using multiple methods. The densities obtained from pooling in Montreal and Quebec City highlight the significant changes in higher quantiles obtained through different combination approaches. Areal reduction factor and quantile projected changes are used to show that consistency, or lack thereof, across approaches reflects the uncertainty of combination methods. This shows how an actuary using multiple expert models should consider more than one combination method to properly assess the impact of climate change on loss distributions, seeing as a single method can lead to overconfidence in projections.

A detailed analysis of management and performance fees for asset managers and investment funds is undertaken. While fund fees are considered as a cost of capital for investors, the structuring of such fee mechanisms in a fund can also influence a fund manager’s decisions and investment strategy, thereby also influencing the investment performance of the investors funds. The study undertaken will allow for an assessment of the effect of fee structures and the potential for asymmetric incentives to arise that may promote adverse risk-taking behaviours by the fund manager, to the detriment of the investor or retiree who places a portion of their retirement savings into such a managed fund with such fee structures. As such, understanding the mechanism of fee charging as well as pricing the fees correctly is vital. An exploration of the application of actuarial distortion pricing methods for complete and incomplete market valuation is performed on a variety of path-dependent option-like performance fee structures for various funds in the European and American markets. Furthermore, several scenario analysis and sensitivity studies are undertaken. The class of Net Asset Value models adopted are Lévy processes, and the pricing is performed via Monte Carlo techniques.

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.

Wind derivatives are financial instruments designed to mitigate losses caused by adverse wind conditions. With the rapid growth of wind power capacity due to efforts to reduce carbon emissions, the demand for wind derivatives to manage uncertainty in wind power production is expected to increase. However, existing wind derivative literature often assumes normally distributed wind speed, despite the presence of skewness and leptokurtosis in historical wind speed data. This paper investigates how the misspecification of wind speed models affects wind derivative prices and proposes the use of the generalized hyperbolic distribution to account for non-normality. The study develops risk-neutral approaches for pricing wind derivatives using the conditional Esscher transform, which can accommodate stochastic processes with any distribution, provided the moment-generating function exists. The analysis demonstrates that model risk varies depending on the choice of the underlying index and the derivative’s payoff structure. Therefore, caution should be exercised when choosing wind speed models. Essentially, model risk cannot be ignored in pricing wind speed derivatives.

This article examines the impact of the largest claims reinsurance treaties on loss reserve of the ceding company. The largest claims reinsurance, known as LCR, and ECOMOR reinsurance treaties are considered to be the two most appropriate reinsurance treaties for large or catastrophe claims. Then, it studies the impact of such treaties on loss reserves. Through a simulation study, it shown that, under a more general situation, the LCR treaty can be a more efficient (in some sense, see below) treaty than the ECOMOR treaty for the ceding company.

This paper considers the pricing of long-term options on assets such as housing, where either government intervention or the economic nature of the asset limits large falls in prices. The observed asset price is modelled by a geometric Brownian motion (“the notional price”) reflected at a lower barrier. The resulting observed price has standard dynamics but with localised intervention at the barrier, which allows arbitrage with interim losses; this is funded by the government’s unlimited powers of intervention, and its exploitation is subject to credit constraints. Despite the lack of an equivalent martingale measure for the observed price, options on this price can be expressed as compound options on the arbitrage-free notional price, to which standard risk-neutral arguments can be applied. Because option deltas tend to zero when the observed price approaches the barrier, hedging with the observed price gives the same results as hedging with the notional price and so exactly replicates option payoffs. Hedging schemes are not unique, with the cheapest scheme for any derivative being the one which best exploits the interventions at the barrier. The price of a put is clear: direct replication has a lower initial cost than synthetic replication, and the replication portfolio always has positive value. The price of a call is ambiguous: synthetic replication has a lower initial cost than direct replication, but the replication portfolio may give interim losses. So the preferred replication strategy (and hence price) of a call depends on what margin payments need to be made on these losses.

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.

This paper proposes a two-phase deep reinforcement learning approach, for hedging variable annuity contracts with both GMMB and GMDB riders, which can address model miscalibration in Black-Scholes financial and constant force of mortality actuarial market environments. In the training phase, an infant reinforcement learning agent interacts with a pre-designed training environment, collects sequential anchor-hedging reward signals, and gradually learns how to hedge the contracts. As expected, after a sufficient number of training steps, the trained reinforcement learning agent hedges, in the training environment, equally well as the correct Delta while outperforms misspecified Deltas. In the online learning phase, the trained reinforcement learning agent interacts with the market environment in real time, collects single terminal reward signals, and self-revises its hedging strategy. The hedging performance of the further trained reinforcement learning agent is demonstrated via an illustrative example on a rolling basis to reveal the self-revision capability on the hedging strategy by online learning.

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.

A quantification of the financial implications of the design of a funded, collective defined contribution (CDC) pension scheme is presented and illustrated. It is done through an attribution analysis, which allows the importance of various elements of CDC scheme design to be determined. The model of a CDC scheme analysed is based lightly on the first CDC scheme set to be approved in the UK. In the CDC scheme analysed, contributions are fixed and the initial benefit accrued by each contribution is fixed. Once accrued, benefits are subsequently adjusted annually in response to changes in assumptions and returns. An attribution of the benefit payments shows that this design gives higher benefits to the first generations and lower benefits to the last generations, for a scheme which starts with no members. The contributions paid also affect the balance of benefits paid between generations. Too high a contribution is to the advantage of the first generations. Too low a contribution is in the interests of the later generations. The conclusion, within the simple model considered, is that a constant benefit accrual is an important design choice. Its financial consequences across all generations should be carefully analysed, if it is intended to be implemented. Additionally, contributions should be reviewed regularly in such a CDC scheme, to ensure that cross-subsidies are not borne excessively by particular generations.