We investigate bottom-up risk aggregation applied by insurance companies facing reserve risk from multiple lines of business. Since risk capitals should be calculated in different time horizons and calendar years, depending on the regulatory or reporting regime (Solvency II vs IFRS 17), we study correlations of ultimate losses and correlations of one-year losses in future calendar years in lines of business. We consider a multivariate version of a Hertig’s lognormal model and we derive analytical formulas for the ultimate correlation and the one-year correlations in future calendar years. Our main conclusion is that the correlation coefficients that should be used in a bottom-up aggregation formula depend on the time horizon and the future calendar year where the risk emerges. We investigate analytically and numerically properties of the ultimate and the one-year correlations, their possible values observed in practice, and the impact of misspecified correlations on the diversified risk capital.

The distribution-free chain ladder of Mack justified the use of the chain ladder predictor and enabled Mack to derive an estimator of conditional mean squared error of prediction for the chain ladder predictor. Classical insurance loss models, that is of compound Poisson type, are not consistent with Mack’s distribution-free chain ladder. However, for a sequence of compound Poisson loss models indexed by exposure (e.g., number of contracts), we show that the chain ladder predictor and Mack’s estimator of conditional mean squared error of prediction can be derived by considering large exposure asymptotics. Hence, quantifying chain ladder prediction uncertainty can be done with Mack’s estimator without relying on the validity of the model assumptions of the distribution-free chain ladder.

This paper introduces gemact, a Python package for actuarial modeling based on the collective risk model. The library supports applications to risk costing and risk transfer, loss aggregation, and loss reserving. We add new probability distributions to those available in scipy, including the (a, b, 0) and (a, b, 1) discrete distributions, copulas of the Archimedean family, the Gaussian, the Student t and the Fundamental copulas. We provide an implementation of the AEP algorithm for calculating the cumulative distribution function of the sum of dependent, nonnegative random variables, given their dependency structure specified with a copula. The theoretical framework is introduced at the beginning of each section to give the reader with a sufficient understanding of the underlying actuarial models.

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.

One of the most fundamental tasks in non-life insurance, done on regular basis, is risk reserving assessment analysis, which amounts to predict stochastically the overall loss reserves to cover possible claims. The most common reserving methods are based on different parametric approaches using aggregated data structured in the run-off triangles. In this paper, we propose a rather non-parametric approach, which handles the underlying loss development triangles as functional profiles and predicts the claim reserve distribution through permutation bootstrap. Three competitive functional-based reserving techniques, each with slightly different scope, are presented; their theoretical and practical advantages – in particular, effortless implementation, robustness against outliers, and wide-range applicability – are discussed. Theoretical justifications of the methods are derived as well. An evaluation of the empirical performance of the designed methods and a full-scale comparison with standard (parametric) reserving techniques are carried on several hundreds of real run-off triangles against the known real loss outcomes. An important objective of the paper is also to promote the idea of natural usefulness of the functional reserving methods among the reserving practitioners.

We propose a stochastic model for claims reserving that captures dependence along development years within a single triangle. This dependence is based on a gamma process with a moving average form of order $p \ge 0$ which is achieved through the use of poisson latent variables. We carry out Bayesian inference on model parameters and borrow strength across several triangles, coming from different lines of businesses or companies, through the use of hierarchical priors. We carry out a simulation study as well as a real data analysis. Results show that reserve estimates, for the real data set studied, are more accurate with our gamma dependence model as compared to the benchmark over-dispersed poisson that assumes independence.

Claims reserving models are usually based on data recorded in run-off tables, according to the origin and the development years of the payments. The amounts on the same diagonal are paid in the same calendar year and are influenced by some common effects, for example, claims inflation, that can induce dependence among payments. We introduce hierarchical generalized linear models (HGLM) with risk parameters related to the origin and the calendar years, in order to model the dependence among payments of both the same origin year and the same calendar year. Besides the random effects, the linear predictor also includes fixed effects. All the parameters are estimated within the model by the h-likelihood approach. The prediction for the outstanding claims and an approximate formula to evaluate the mean square error of prediction are obtained. Moreover, a parametric bootstrap procedure is delineated to get an estimate of the predictive distribution of the outstanding claims. A Poisson-gamma HGLM with origin and calendar year effects is studied extensively and a numerical example is provided. We find that the estimates of the correlations can be significant for payments in the same calendar year and that the inclusion of calendar effects can determine a remarkable impact on the prediction uncertainty.

We revisit the “full picture” of the claims development uncertainty in Mack’s (1993) distribution-free stochastic chain ladder model. We derive the uncertainty estimators in a new and easily understandable way, which is much simpler than the derivation found so far in the literature, and compare them with the well known estimators of Mack and of Merz–Wüthrich.

Our uncertainty estimators of the one-year run-off risks are new and different to the Merz–Wüthrich formulas. But if we approximate our estimators by a first order Taylor expansion, we obtain equivalent but simpler formulas. As regards the ultimate run-off risk, we obtain the same formulas as Mack for single accident years and an equivalent but better interpretable formula for the total over all accident years.

The chain-ladder method is one of the most popular claims reserving techniques. The aim of this study is to back-test the chain-ladder method. For this purpose, we use a stochastic scenario generator that allows us to simulate arbitrarily many upper claims reserving triangles of similar characteristics for which we also know the corresponding lower triangles. Based on these simulated triangles, we analyse the performance of the chain-ladder claims reserving method.

We consider a Tweedie's compound Poisson regression model with fixed and random effects, to describe the payment numbers and the incremental payments, jointly, in claims reserving. The parameter estimates are obtained within the framework of hierarchical generalized linear models, by applying the h-likelihood approach. Regression structures are allowed for the means and also for the dispersions. Predictions and prediction errors of the claims reserves are evaluated. Through the parameters of the distributions of the random effects, some external information (e.g. a development pattern of industry wide-data) can be incorporated into the model. A numerical example shows the impact of external data on the reserve and prediction error evaluations.

This paper shows how the well-known Bornhuetter–Ferguson claims-reserving method can be extended by applying fuzzy methods. The a priori information for the ultimate claims derives from market statistics, organisational data, etc. and might contain vagueness. Likewise, the parameters of the claims development pattern can be vague or are adapted, retrospectively, due to subjective judgement. With the help of fuzzy numbers we develop new predictors for the ultimate claims. Furthermore, we quantify the uncertainty of the ultimate claims for single and aggregated accident years.

The Cape Cod (CC) method was designed by Bühlmann and Straub in order to overcome some shortcomings of the chain ladder (CL) method. Owing to its simplicity and because of the advantages over the CL method, the CC method has become a well-established method in practice. In this paper we consider a distribution-free stochastic model for the CC method. Within this model we give the parameter estimates and we derive estimates for the conditional mean square error of prediction for the CC method. In addition, we derive an estimate for the uncertainty in the claims development result.

The claims development result (CDR) is the difference between the best estimate predictions of the ultimate claim in 2 successive years. With best estimate reserves it is often argued that CDR’s in consecutive years should fluctuate randomly around zero. However, in practice one frequently observes that CDR’s in a given line of business have the same sign over several consecutive years. We show that this is a phenomenon which is not unusual and to be expected in situations of change. Moreover, we show how situations of change can adequately be described by a model, taking into account the evolving external information.

The paid-incurred chain (PIC) reserving method is a claims reserving method that allows to combine claims payments and incurred losses information in a mathematical consistent way. The main criticism on the original Bayesian log-normal PIC model presented in Merz–Wüthrich [5] is that it does not respect dependence properties within the observed data. In the present paper, we extend the original Bayesian log-normal PIC model so that dependence is modeled in an appropriate way.

A central issue in claims reserving is the modelling of appropriate dependence structures. Most classical models cannot cope with this task. We define a multivariate log-normal model that allows to model both, dependence between different sub-portfolios and dependence within sub-portfolios such as claims inflation. In this model we derive closed form solutions for claims reserves and the corresponding prediction uncertainty.

We consider the Bayesian over-dispersed Poisson (ODP) model for claims reserving in general insurance. We choose two different types of prior distributions for the parameters and then study the different Bayesian predictors. This study leads, on the one hand, to the classical chain ladder predictor and, on the other hand, to Bornhuetter & Ferguson predictors. We highlight (either analytically or numerically) how these predictors are obtained and how their prediction uncertainty can be determined.

We investigate the question how the development pattern in the Bornhuetter-Ferguson method should be estimated and derive the corresponding conditional mean square error of prediction (MSEP) of the ultimate claim prediction. An estimator of this conditional MSEP in a distribution-free model was given by Mack [9], whereas in Alai et al. [2] this conditional MSEP was studied in an over-dispersed Poisson model using the chain ladder development pattern. First we consider distributional models and derive estimators (maximum likelihood) for the development pattern taking all relevant information into account. Moreover, we suggest new estimators of the correlation matrix of these estimators and new estimators of the conditional MSEP. Our findings supplement some of Mack's results. The methodology is illustrated at two numerical examples.

This paper presents a statistical model underlying the chain-ladder technique. This is related to other statistical approaches to the chain-ladder technique which have been presented previously. The statistical model is cast in the form of a generalised linear model, and a quasi-likelihood approach is used. It is shown that this enables the method to process negative incremental claims. It is suggested that the chain-ladder technique represents a very narrow view of the possible range of models.

This paper examines the chain-ladder technique using the recently developed theory for age-period-cohort models. The theory was set out by Kuang et al. (2008a), and we believe that it has some significant implications for claims reserving and the chain-ladder technique. This paper applies the age-period-cohort model using the over-dispersed Poisson framework, and examines a number of experiments in order to understand better how the chain-ladder technique deals with calendar year effects. The conclusions from these investigations are that the basic chain-ladder technique may have some fundamental difficulties in many circumstances. We would therefore recommend that it should be used with caution, and that the data are examined in detail before any projections are made. This has particular importance in the context of solvency calculations since the chain-ladder technique can impose some specific patterns into the projections.

The prediction of adequate claims reserves is a major subject in actuarial practice and science. Due to their simplicity, the chain ladder (CL) and Bornhuetter–Ferguson (BF) methods are the most commonly used claims reserving methods in practice. However, in contrast to the CL method, no estimator for the conditional mean square error of prediction (MSEP) of the ultimate claim has been derived in the BF method until now, and as such, this paper aims to fill that gap. This will be done in the framework of generalized linear models (GLM) using the (overdispersed) Poisson model motivation for the use of CL factor estimates in the estimation of the claims development pattern.