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This paper proposes a multistate model with a Semi-Markov dependence structure describing the different stages in the settlement process of individual claims in general insurance. Every trajectory, from reporting to closure is combined with a modeling of individual link ratios to obtain the ultimate cost of each claim. Analytical expressions are derived for the moments of ultimate amounts whereas quantile risk measures can be obtained by simulation. In the 1-year view, the proposed matrix calculations avoid the simulation-within-simulation issue and offer a tractable evaluation method. A case study illustrates the relevance of the proposed approach.

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... In this paper, we adopt the multi-state approach to loss reserving proposed by [16] and further considered by [17], [18], and [19]. In particular, we ...

... This section uses the multi-state approach of [19] represented in Figure 2. In this approach, an RBNS claim occured in state S oc and is reported in state S 0 . Once reported, either a first payment can occur, implying a transition from state S 0 to state S 1 , or the claim can go to one of two absorbing states, S tn or S tp . ...

... We use the methodology from discrete time survival analysis, and model the time until an event or transition, from one state to the other. We define an event as being the occurrence of a payment or as the transition to a terminal state without payment as in [17] and [19]. Furthermore, we say that a claim is censored or open when it is not in one of the two absorbing states at the moment of evaluation. ...

This paper presents a multinomial multi-state micro-level reserving model, denoted mCube. We propose a unified framework for modelling the time and the payment process for IBNR and RBNS claims and for modeling IBNR claim counts. We use multinomial distributions for the time process and spliced mixture models for the payment process. We illustrate the excellent performance of the proposed model on a real data set of a major insurance company consisting of bodily injury claims. It is shown that the proposed model produces a best estimate distribution that is centered around the true reserve.

... That is, starting from individual claims, it is natural that you need to keep track of claims staying open as well as those that have been closed, but could become re-opened. Examples of individual claim models with claim closings are, for example, Antonio and Plat (2014), Crevecoeur and Antonio (2019), Bettonville et al. (2020), and Delong et al. (2020), where the latter two also allow for claim reopenings. Another type of computer intensive approach is to use self-exciting processes as in Maciak et al. (2021). ...

... That is, for example, ν ij := exp{α i + β j } and similarly, which in practice will make the number of unique parameters in θ considerably less than O(m 3 ). A concrete example of this is given in Section 6. Bettonville et al. (2020) and Crevecoeur and Antonio (2019). (d) As with the models discussed in Verrall et al. (2010), Martinez-Miranda et al. (2011), andWahl et al. (2019), the current model is defined based on individual claim dynamics. ...

... Another approach is to instead consider individual claims modelling directly, which naturally allows for closing of claims. One such model that has been evaluated on publicly available data generated using the procedure from Gabrielli and Wüthrich (2018) is the semi-Markov model introduced in Bettonville et al. (2020), which allows for claim closings, but no re-openings. The data used in Bettonville et al. (2020) differ from the previously used six LoBs, here a single LoB with approximately 50,000 claims is used, but we refer to Bettonville et al. (2020) for more details. ...

The present paper introduces a simple aggregated reserving model based on claim count and payment dynamics, which allows for claim closings and re-openings. The modelling starts off from individual Poisson process claim dynamics in discrete time, keeping track of accident year, reporting year and payment delay. This modelling approach is closely related to the one underpinning the so-called double chain-ladder model, and it allows for producing separate reported but not settled and incurred but not reported reserves. Even though the introduction of claim closings and re-openings will produce new types of dependencies, it is possible to use flexible parametrisations in terms of, for example, generalised linear models (GLM) whose parameters can be estimated based on aggregated data using quasi-likelihood theory. Moreover, it is possible to obtain interpretable and explicit moment calculations, as well as having consistency of normalised reserves when the number of contracts tend to infinity. Further, by having access to simple analytic expressions for moments, it is computationally cheap to bootstrap the mean squared error of prediction for reserves. The performance of the model is illustrated using a flexible GLM parametrisation evaluated on non-trivial simulated claims data. This numerical illustration indicates a clear improvement compared with models not taking claim closings and re-openings into account. The results are also seen to be of comparable quality with machine learning models for aggregated data not taking claim openness into account.

Customer churn, which insurance companies use to describe the non-renewal of existing customers, is a widespread and expensive problem in general insurance, particularly because contracts are usually short-term and are renewed periodically. Traditionally, customer churn analyses have employed models which utilise only a binary outcome (churn or not churn) in one period. However, real business relationships are multi-period, and policyholders may reside and transition between a wider range of states beyond that of the simply churn/not churn throughout this relationship. To better encapsulate the richness of policyholder behaviours through time, we propose multi-state customer churn analysis, which aims to model behaviour over a larger number of states (defined by different combinations of insurance coverage taken) and across multiple periods (thereby making use of readily available longitudinal data). Using multinomial logistic regression (MLR) with a second-order Markov assumption, we demonstrate how multi-state customer churn analysis offers deeper insights into how a policyholder’s transition history is associated with their decision making, whether that be to retain the current set of policies, churn, or add/drop a coverage. Applying this model to commercial insurance data from the Wisconsin Local Government Property Insurance Fund, we illustrate how transition probabilities between states are affected by differing sets of explanatory variables and that a multi-state analysis can potentially offer stronger predictive performance and more accurate calculations of customer lifetime value (say), compared to the traditional customer churn analysis techniques.

Random effects are particularly useful in insurance studies, to capture residual heterogeneity or to induce cross‐sectional and/or serial dependence, opening hence the door to many applications including experience rating and microreserving. However, their nonobservability often makes existing models computationally cumbersome in a multivariate context. In this paper, it is shown that the multivariate extension to the Gamma distribution based on Wishart distributions for random symmetric positive‐definite matrices (considering diagonal terms) is particularly tractable and convenient to model correlated random effects in multivariate frequency, severity and duration models. Three applications are discussed to demonstrate the versatility of the approach: (a) frequency‐based experience rating with several policies or guarantees per policyholder, (b) experience rating accounting for the correlation between claim frequency and severity components, and (c) joint modeling and forecasting of the time‐to‐payment and amount of payment in microlevel reserving, when both are subject to censoring.

The aim of this project is to develop a stochastic simulation machine that generates individual claims histories of non-life insurance claims. This simulation machine is based on neural networks to incorporate individual claims feature information. We provide a fully calibrated stochastic scenario generator that is based on real non-life insurance data. This stochastic simulation machine allows everyone to simulate their own synthetic insurance portfolio of individual claims histories and back-test thier preferred claims reserving method.

It is probably fair to date loss reserving by means of claim modelling from the late 1960s [...]

Traditionally, actuaries have used run-off triangles to estimate reserve ("macro" models, on agregated data). But it is possible to model payments related to individual claims. If those models provide similar estimations, we investigate uncertainty related to reserves, with "macro" and "micro" models. We study theoretical properties of econometric models (Gaussian, Poisson and quasi-Poisson) on individual data, and clustered data. Finally, application on claims reserving are considered.

We construct a simple parametric multi-state gamma distributed aggregate claims reserving model. It is based on the multi-state claims number reserving model by Orr, and adds the simplest possible modelling of the claims size process. Predictive power and advantages of the new model are discussed and illustrated.

This paper discusses a statistical modeling strategy based on extreme value theory to describe the behavior of an insurance portfolio, with particular emphasis on large claims. The strategy is illustrated using the 1991-92 group medical claims database maintained by the Society of Actuaries. Using extreme value theory, the modeling strategy focuses on the ”excesses over threshold” approach to fit generalized Pareto distributions. The proposed strategy is compared to standard parametric modeling based on gamma, lognormal, and log-gamma distributions. Extreme value theory outperforms classical parametric fits and allows the actuary to easily estimate high quantiles and the probable maximum loss from the data.

The claims generating process for a non-life insurance portfolio is modelled as a marked Poisson process, where the mark associated with an incurred claim describes the development of that claim until final settlement. An unsettled claim is at any point in time assigned to a state in some state-space, and the transitions between different states are assumed to be governed by a Markovian law. All claims payments are assumed to occur at the time of transition between states. We develop separate expressions for the IBNR and RBNS reserves, and the corresponding prediction errors.

The actuarial and insurance industries frequently use the lognormal and the Pareto distributions to model their payments data. These types of payment data are typically very highly positively skewed. Pareto model with a longer and thicker upper tail is used to model the larger loss data, while the larger data with lower frequencies as well as smaller data with higher frequencies are usually modeled by the lognormal distribution. Even though the lognormal model covers larger data with lower frequencies, it fades away to zero more quickly than the Pareto model. Furthermore, the Pareto model does not provide a reasonable parametric fit for smaller data due to its monotonic decreasing shape of the density. Therefore, taking into account the tail behavior of both small and large losses, we were motivated to look for a new avenue to remedy the situation. Here we introduce a two-parameter smooth continuous composite lognormal-Pareto model that is a two-parameter lognormal density up to an unknown threshold value and a two-parameter Pareto density for the remainder. The resulting two-parameter smooth density is similar in shape to the lognormal density, yet its upper tail is larger than the lognormal density and the tail behavior is quite similar to the Pareto density. Parameter estimation techniques and properties of this new composite lognormal-Pareto model are discussed and we compare its performance with the other commonly used models. A simulated example and a well-known fire insurance data set are analyzed to show the importance and applicability of this newly proposed composite lognormal-Pareto model.

This paper adopts the new loss reserving approach proposed by Denuit and Trufin (2016), inspired from the collective model of risk theory. But instead of considering the whole set of claims as a collective, two types of claims are distinguished, those claims with relatively short development patterns and claims requiring longer developments. In each case, the total payment per cell is modelled by means of a Compound Poisson distribution with appropriate assumptions about the severities. A case study based on a motor third party liability insurance portfolio observed over 2004–2014 is used to illustrate the approach proposed in this paper. Comparisons with Chain-Ladder are performed and reveal significant differences in best estimates as well as in Value-at-Risk at high probability levels.

This article proposes a new loss reserving approach, inspired from the collective model of risk theory. According to the collective paradigm, we do not relate payments to specific claims or policies, but we work within a frequency-severity setting, with a number of payments in every cell of the run-off triangle, together with the corresponding paid amounts. Compared to the Tweedie reserving model, which can be seen as a compound sum with Poisson-distributed number of terms and Gamma-distributed summands, we allow here for more general severity distributions, typically mixture models combining a light-tailed component with a heavier-tailed one, including inflation effects. The severity model is fitted to individual observations and not to aggregated data displayed in run-off triangles with a single value in every cell. In that respect, the modeling approach appears to be a powerful alternative to both the crude traditional aggregated approach based on triangles and the extremely detailed individual reserving approach developing each and every claim separately. A case study based on a motor third-party liability insurance portfolio observed over 2004–2014 is used to illustrate the relevance of the proposed approach.

These notes are strongly motivated by practitioners who have been seeking for advise in stochastic claims reserving modeling under Solvency 2 and under the Swiss Solvency Test. There have been tremendous developments since the publication of our first book Stochastic Claims Reserving Methods in Insurance in 2008. Particularly the new solvency guidelines have added a dynamic component to claims reserving which has not been present before. This new viewpoint has motivated numerous new developments, for instance, the claims development result and the risk margin were introduced. The present text considers these new aspects, not treated in our previous book, and it should be viewed as completion to our first book.

Insurance companies hold reserves to be able to fulll future liabilities with respect to the policies they write. Micro-level reserving methods focus on the development of individual claims over time, providing an alternative to the classical techniques that aggregate the development of claims into run-o triangles. This paper presents a discrete-time multi-state framework that reconstructs the claim development process as a series of transitions between a given set of states. The states in our setting represent the events that may happen over the lifetime of a claim, i.e. reporting, intermediate payments and closure. For each intermediate payment we model the payment distribution separately. To this end, we use a body-tail approach where the body of the distribution is modeled separately from the tail. Generalized Additive Models for Location, Scale and Shape introduced by Stasinopoulos and Rigby (2007) allow for exible modeling of the body distribution while incorporating covariate information. We use the toolbox from Extreme Value Theory to determine the threshold separating the body from the tail and to model the tail of the payment distributions. We do not correct payments for in ation beforehand, but include relevant covariate information in the model. Using these building blocks, we outline a simulation procedure to evaluate the RBNS reserve. The method is applied to a real life data set, and we benchmark our results by means of a back test.

Multistate analysis of life histories with R is an introduction to multistate event history analysis. It is an extension of survival analysis, in which a single terminal event (endpoint) is considered and the time-to-event is studied. Multistate models focus on life histories or trajectories, conceptualized as sequences of states and sequences of transitions between states. Life histories are modeled as realizations of continuous- time Markov processes. The model parameters, transition rates, are estimated from data on event counts and populations at risk, using the statistical theory of counting processes.
The Comprehensive R Network Archive (CRAN) includes several packages for multistate modeling. This book is about Biograph. The package is designed to (a) enhance exploratory analysis of life histories and (b) make multistate modeling accessible. The package incorporates utilities that connect to several packages for multistate modeling, including survival, eha, Epi, mvna, etm, mstate, msm, and TraMineR for sequence analysis. The book is a ‘hands-on’ presentation of Biograph and the packages listed. It is written from the perspective of the user. To help the user master the techniques and the software, a single data set is used to illustrate the methods and software. It is the subsample of the German Life History Survey, which was also used by Blossfeld and Rohwer in their popular textbook on event history modeling. Another data set, the Netherlands Family and Fertility Survey, is used to illustrate how Biograph can assist in answering questions on life paths of cohorts and individuals.
The book is suitable as a textbook for graduate courses on event history analysis and introductory courses on competing risks and multistate models. It may also be used as a self-study book.

A fully time-continuous approach is taken to the problem of predicting the total liability of a non-hfe insurance company Claims are assumed to be generated by a non-homogeneous marked Polsson process, the marks representing the developments of the individual claims. A first basic result is that the total claim amount follows a generalized Poisson distribution. Fixing the time of consideration, the claims are categorized into settled, reported but not settled, incurred but not reported, and covered but not incurred. It is proved that these four categories of claims can be viewed as arising from independent marked Polsson processes By use of this decomposition result predictors are constructed for all categories of outstanding claims. The claims process may depend on observable as well as unobservable risk characteristics, which may change in the course of time, possibly in a random manner Special attention ~s gwen to the case where the claim Intensity per risk unit IS a stationary stochastic process. A theory of continuous linear pre&ctlon ~s mstrumental.

Generalized additive models for location, scale and, shape define a flexible, semi-parametric class of regression models for analyzing insurance data in which the exponential family assumption for the response is relaxed. This approach allows the actuary to include risk factors not only in the mean but also in other key parameters governing the claiming behavior, like the degree of residual heterogeneity or the no-claim probability. In this broader setting, the Negative Binomial regression with cell-specific heterogeneity and the zero-inflated Poisson regression with cell-specific additional probability mass at zero are applied to model claim frequencies. New models for claim severities that can be applied either per claim or aggregated per year are also presented. Bayesian inference is based on efficient Markov chain Monte Carlo simulation techniques and allows for the simultaneous estimation of linear effects as well as of possible nonlinear effects, spatial variations and interactions between risk factors within the data set. To illustrate the relevance of this approach, a detailed case study is proposed based on the Belgian motor insurance portfolio studied in Denuit and Lang (2004).

This is a follow-up of a previous paper by the author, where claims reserving in non-life insurance is treated in the framework of a marked Poisson claims process. A key result on decomposition of the process is generalized, and a number of related results are added. Their usefulness is demonstrated by examples and, in particular, the connection to the analogous discrete time model is clarified. The problem of predicting the outstanding part of reported but not settled claims is revisited and, by way of example, solved in a model where the partial payments are governed by a Dirichlet process. The process of reported claims is examined, and its dual relationship to the process of occurred claims is pointed out.

We present some relatively simple structural ideas about how probabilistic modeling, and in particular, the modern theory of point processes and martingales, can be used in the estimation of claims reserves.

Recently, Cooray & Ananda (2005) proposed a composite lognormal-Pareto model for use with loss payments data of the sort arising in the actuarial and insurance industries. Their model is based on a lognormal density up to an unknown threshold value and a two-parameter Pareto density thereafter. Here we identify and discuss limitations of this composite lognormal-Pareto model which are likely to severely curtail its potential for practical application to real world data sets. In addition, we present two different composite models based on lognormal and Pareto models in order to address these concerns. The performance of all three composite models is discussed and compared in the context of an example based upon a well-known fire insurance data set.

Semi-Markov multistate individual loss reserving model in general insurance

- C Bettonville
- M Denuit
- J Trufin
- R Van Oirbeek

A stochastic model for loss reserving

- C A Hachemeister

The long road to enlightenment: loss reserving models from the past

- G Taylor