The paper is concerned with common shock models of claim triangles. These are usually constructed as linear combinations of shock components and idiosyncratic components. Previous literature has discussed the unbalanced property of such models, whereby the shocks may over- or under-contribute to some observations. The literature has also introduced corrections for this. The present paper discusses “auto-balanced” models, in which all shock and idiosyncratic components contribute to observations such that their proportionate contributions are constant from one observation to another. The conditions for auto-balance are found to be simple and applicable to a wide range of model structures. Numerical illustrations are given.

Modelling loss reserve data in run-off triangles is challenging due to the complex but unknown dynamics in the claim/loss process. Popular loss reserve models describe the mean process through development year, accident year, and calendar year effects using the analysis of variance and covariance (ANCOVA) models. We propose to include in the mean function the persistence terms in the conditional autoregressive range model for modelling the persistence of claim across development years. In the ANCOVA model, we adopt linear trends for the accident and calendar year effects and a quadratic trend for the development year effect. We investigate linear or log-transformed mean functions and four distributions, namely generalised beta type 2, generalised gamma, Weibull, and exponential extension, with positive support to enhance the model flexibility. The proposed models are implemented using the Bayesian user-friendly package Stan running in the R environment. Results show that the models with log-transformed mean function and persistence terms provide better model fits. Lastly, the best model is applied to forecast partial loss reserve and calendar year reserve for three years.

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.

It is well known that the presence of outliers can mis-estimate (underestimate or overestimate) the overall reserve in the chain-ladder method, when we consider a linear regression model, based on the assumption that the coefficients are fixed and identical from one observation to another. By relaxing the usual regression assumptions and applying a regression with randomly varying coefficients, we have a similar phenomenon, i.e., mis-estimation of the overall reserves. The lack of robustness of loss reserving regression with random coefficients on incremental payment estimators leads to the development of this paper, aiming to apply robust statistical procedures to the loss reserving estimation when regression coefficients are random. Numerical results of the proposed method are illustrated and compared with the results that were obtained by linear regression with fixed coefficients.

The projection of outstanding liabilities caused by incurred losses or claims has played a fundamental role in general insurance operations. Loss reserving methods based on individual losses generally perform better than those based on aggregate losses. This study uses a parametric individual information model taking not only individual losses but also individual information such as age, gender, and so on from policies themselves into account. Based on this model, this study proposes a computation procedure for the projection of the outstanding liabilities, discusses the estimation and statistical properties of the unknown parameters, and explores the asymptotic behaviors of the resulting loss reserving as the portfolio size approaching infinity. Most importantly, this study demonstrates the benefits of individual information on loss reserving. Remarkably, the accuracy gained from individual information is much greater than that from considering individual losses. Therefore, it is highly recommended to use individual information in loss reserving in general insurance.

Incurred but not reported (IBNR) loss reserving is of great importance for Property & Casualty (P&C) insurers. However, the temporal dependence exhibited in the claim arrival process is not reflected in many current loss reserving models, which might affect the accuracy of the IBNR reserve predictions. To overcome this shortcoming, we proposed a marked Cox process and showed its many desirable properties in Badescu et al. (2016).

In this paper, we consider the model estimation and applications. We first present an expectation–maximization (EM) algorithm which guarantees the efficiency of the estimators unlike the moment estimation methods widely used in estimating Cox processes. In addition, the proposed fitting algorithm can be implemented at a reasonable computational cost. We examine the performance of the proposed algorithm through simulation studies. The applicability of the proposed model is tested by fitting it to a real insurance claim data set. Through out-of-sample tests, we find that the proposed model can provide realistic predictive distributions.

The paper is concerned with multiple claim arrays. In recognition of the extensive use by practitioners of large correlation matrices for the estimation of diversification benefits in capital modelling, we develop a methodology for the construction of such correlation structures (to any dimension). Indeed, the literature does not document any methodology by which practitioners, who often parameterise those correlations by means of informed guesswork, may do so in a disciplined and parsimonious manner.

We construct a broad and flexible family of models, where dependency is induced by common shock components. Models incorporate dependencies between observations both within arrays and between arrays. Arrays are of general shape (possibly with holes), but include the usual cases of claim triangles and trapezia that appear in the literature. General forms of dependency are considered with cell-, row-, column-, diagonal-wise, and other forms of dependency as special cases. Substantial effort is applied to practical interpretation of such matrices generated by the models constructed here.

Reasonably realistic examples are examined, in which an expression is obtained for the general entry in the correlation matrix in terms of a limited set of parameters, each of which has a straightforward intuitive meaning to the practitioner. This will maximise chance of obtaining a reliable matrix. This construction is illustrated by a numerical example.

The cross-classified chain ladder has a number of versions, depending on the distribution to which observations are subject. The simplest case is that of Poisson distributed observations, and then maximum likelihood estimates of parameters are explicit. Most other cases, however, including Bayesian chain ladder models, lead to implicit MAP (Bayesian) or MLE (non-Bayesian) solutions for these parameter estimates, raising questions as to their existence and uniqueness. The present paper investigates these questions in the case where observations are distributed according to some member of the exponential dispersion family.

Suppose that N is a ℤ+-valued random variable and that X,X1,X2,… is a sequence of independent and identically distributed ℤ+ random variables independent of N. In this paper we are interested in properties of the conditional variable In particular, we want to know the asymptotic behavior of the conditional mean ENk or the conditional variance varNk as k→∞. We consider the cases when X is Poisson and when X is mixed Poisson. The problem is motivated by modeling loss reserves in nonlife insurance.

This paper examines various forms of individual claim model for the purpose of loss reserving, with emphasis on the prediction error associated with the reserve. Each form of model is calibrated against a single extensive data set, and then used to generate a forecast of loss reserve and an estimate of its prediction error.

The basis of this is a model of the “paids” type, in which the sizes of strictly positive individual finalised claims are expressed in terms of a small number of covariates, most of which are in some way functions of time. Such models can be found in the literature.

The purpose of the current paper is to extend these to individual claim “incurreds” models. These are constructed by the inclusion of case estimates in the model's conditioning information. This form of model is found to involve rather more complexity in its structure.

For the particular data set considered here, this did not yield any direct improvement in prediction error. However, a blending of the paids and incurreds models did so.

We investigate the usual method of discretizing loss reserving data by calendar year and show how this procedure may introduce fluctuations in the delay probabilities. These fluctuations, when treated as random fluctuations, possess a special correlation structure and we present a simple credibility method accounting for these fluctuations. The results are illustrated by a numerical example.