Chain Ladder Research Articles

A machine learning approach for individual claims reserving in insurance

AbstractAccurate loss reserves are an important item in the financial statement of an insurance company and are mostly evaluated by macrolevel models with aggregate data in run‐off triangles. In recent years, a new set of literature has considered individual claims data and proposed parametric reserving models based on claim history profiles. In this paper, we present a nonparametric and flexible approach for estimating outstanding liabilities using all the covariates associated to the policy, its policyholder, and all the information received by the insurance company on the individual claims since its reporting date. We develop a machine learning–based method and explain how to build specific subsets of data for the machine learning algorithms to be trained and assessed on. The choice for a nonparametric model leads to new issues since the target variables (claim occurrence and claim severity) are right‐censored most of the time. The performance of our approach is evaluated by comparing the predictive values of the reserve estimates with their true values on simulated data. We compare our individual approach with the most used aggregate data method, namely, chain ladder, with respect to the bias and the variance of the estimates. We also provide a short real case study based on a Dutch loan insurance portfolio.

The collective reserving model

This paper sets out a model for analysing claims development data, which we call the collective reserving model (CRM). The model is defined on the individual claim level and it produces separate IBNR and RBNS reserve estimators at the collective level without using any approximations. The CRM is based on ideas from a paper by Verrall, Nielsen and Jessen (VNJ) from 2010 in which a model is proposed that relies on a claim giving rise to a single payment. This is generalised by the CRM to the case of multiple payments per claim. All predictors of outstanding claims payments for the VNJ model are shown to hold for this new model. Moreover, the quasi-Poisson GLM estimation framework will be applicable as well, but without using an approximation. Furthermore, analytical expressions for the variance of the total outstanding claims payments are given, with a subdivision on IBNR and RBNS claims. To quantify the effect of allowing only one payment per claim, the model is related and compared to the VNJ model, in particular by looking at variance inequalities. The double chain ladder (DCL) method is discussed as an estimation method for this new model and it is shown that both the GLM- and DCL-based estimators are consistent in terms of an exposure measure. Lastly, both of these methods are shown to asymptotically reproduce the regular chain ladder reserve estimator when restricting predictions to the lower right triangle without the tail, motivating the chain ladder technique as a large-exposure approximation of this model.

A comparison of in-sample forecasting methods

In-sample forecasting is a recent continuous modification of well-known forecasting methods based on aggregated data. These aggregated methods are known as age-cohort methods in demography, economics, epidemiology and sociology and as chain ladder in non-life insurance. Data is organized in a two-way table with age and cohort as indices, but without measures of exposure. It has recently been established that such structured forecasting methods based on aggregated data can be interpreted as structured histogram estimators. Continuous in-sample forecasting transfers these classical forecasting models into a modern statistical world including smoothing methodology that is more efficient than smoothing via histograms. All in-sample forecasting estimators are collected and their performance is compared via a finite sample simulation study. All methods are extended via multiplicative bias correction. Asymptotic theory is being developed for the histogram-type method of sieves and for the multiplicatively corrected estimators. The multiplicative bias corrected estimators improve all other known in-sample forecasters in the simulation study. The density projection approach seems to have the best performance with forecasting based on survival densities being the runner-up.

A Cape Cod model for the exponential dispersion family

The defining feature of the Cape Cod algorithm in current literature is its assumption of a constant loss ratio over accident periods. This is a highly simplifying assumption relative to the chain ladder model which, in effect, allows loss ratio to vary freely over accident period.Much of the literature on Cape Cod reserving treats it as essentially just an algorithm. It does not posit a parametric model supporting the algorithm. There are one or two exceptions to this. The present paper extends them by introducing a couple of more general stochastic models under which maximum likelihood estimation yields parameters estimates closely resembling those of the classical Cape Cod algorithm.For one of these models, these estimators are shown to be minimum variance unbiased, and so are superior to the conventional estimators, which rely on the chain ladder.A Bayesian Cape Cod model is also introduced, and a MAP estimator calculated.A numerical example is included.

Advantages and disadvantages of loss reserving methods in non-life insurance

We analyse characteristics of the three most commonly used methods for estimating loss reserves in non life insurance: the chain ladder method, the loss ratio method, and the Bornhuetter-Ferguson method. Our aim is to give a comparative analysis of the results obtained from applying these methods to a practical case, and to put emphasis on their advantages and disadvantages.

Estimation of Conditional Mean Squared Error of Prediction for Claims Reserving

This paper studies estimation of conditional mean squared error of prediction, conditional on what is known at the time of prediction. The particular problem considered is the assessment of actuarial reserving methods given data in the form of runoff triangles (trapezoids), where the use of prediction assessment based on out-of-sample performance is not an option. The prediction assessment principle advocated here can be viewed as a generalization of Akaike's final prediction error. A direct application of this simple principle in the setting of a data generating process given in terms of a sequence of general linear models yields an estimator of conditional mean square error of prediction that can be computed explicitly for a wide range of models within this model class. Mack's distribution-free chain ladder model and the corresponding estimator of the prediction error for the ultimate claim amount is shown to be a special case. It is demonstrated that the prediction assessment principle easily applies to quite different data generating processes and results in estimators that have been studied in the literature.

Neural networks applied to chain–ladder reserving

Classical claims reserving methods act on so-called claims reserving triangles which are aggregated insurance portfolios. A crucial assumption in classical claims reserving is that these aggregated portfolios are sufficiently homogeneous so that a coarse reserving algorithm can be applied. We start from such a coarse reserving method, which in our case is Mack's chain-ladder method, and show how this approach can be refined for heterogeneity and individual claims feature information using neural networks.

A Robust General Multivariate Chain Ladder Method

The chain ladder method is a popular technique to estimate the future reserves needed to handle claims that are not fully settled. Since the predictions of the aggregate portfolio (consisting of different subportfolios) do not need to be equal to the sum of the predictions of the subportfolios, a general multivariate chain ladder (GMCL) method has already been proposed. However, the GMCL method is based on the seemingly unrelated regression (SUR) technique which makes it very sensitive to outliers. To address this issue, we propose a robust alternative that estimates the SUR parameters in a more outlier resistant way. With the robust methodology it is possible to automatically flag the claims with a significantly large influence on the reserve estimates. We introduce a simulation design to generate artificial multivariate run-off triangles based on the GMCL model and illustrate the importance of taking into account contemporaneous correlations and structural connections between the run-off triangles. By adding contamination to these artificial datasets, the sensitivity of the traditional GMCL method and the good performance of the robust GMCL method is shown. From the analysis of a portfolio from practice it is clear that the robust GMCL method can provide better insight in the structure of the data.

An Iterated Bornhuetter-Ferguson Model

The iterated Bornhuetter-Ferguson loss reserving method generates an infinite sequence of reserve formulas, with the chain ladder and Bornhuetter-Ferguson formulas at opposite extremes. The sequence also contains the Benktander-Hovinen formula. Although the literature contains parametric stochastic models for the first two of these three, it does not contain models for the remainder of the sequence. The present paper fills this gap with a sequence of Bayesian chain ladder models, one for each iterate. Each is an ODP chain ladder model, with each row parameter drawn from a gamma prior. The parameters vary from iterate to iterate. Equality between each Bayesian chain ladder model and its corresponding iterate is achieved by suitable choice of the coefficients of variation of the row parameters. Thus, each iterate is a special case of the general Gamma-ODP model. Applications of this sequence of models are illustrated, first the application of the sequence of Bayesian chain ladder models directly to loss reserving, and second to a hierarchy of proportional reinsurance arrangements. In the first of these applications, the parameter selection for the Gamma-ODP special cases will not always be natural or intuitive. The use of the general Gamma-ODP model with more natural parameter values may be preferable.

Claim Reserving Estimation by Using the Chain Ladder Method

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Estimating loss reserves using hierarchical Bayesian Gaussian process regression with input warping

In this paper, we visualize the loss reserve runoff triangle as a spatially-organized data set. We apply Gaussian Process (GP) regression with input warping and several covariance functions to estimate future claims. We then compare our results over a range of product lines, including workers’ comp, medical malpractice, and personal auto. Even though the claims development of the lines are very different, the GP method is very flexible and can be applied to each without much customization. We find that our model generally outperforms the classical chain ladder model as well as the recently proposed hierarchical growth curve models of Guszcza (2008) in terms of point-wise predictive accuracy and produces dramatically better estimates of outstanding claims liabilities.

Photochemical Structural Transformation of a Linear 1D Coordination Polymer Impacts the Electrical Conductivity.

A pair of 4-(1-naphthylvinyl)pyridine (4-nvp) ligands has been successfully aligned in head-to-tail fashion in a one-dimensional (1D) double chain ladder polymer [Cd(adc)(4-nvp)2(H2O)] n (1; H2adc = acetylenedicarboxylic acid) that undergoes a photochemical [2 + 2] cycloaddition reaction accompanied by single-crystal to single-crystal (SCSC) structural transformation from a 1D chain to a 2D layer structure. These structural changes have a significant impact on the conductivity and Schottky nature of the compound.

A Cape Cod Model for the Exponential Dispersion Family

The defining feature of the Cape Cod algorithm in current literature is its assumption of a constant loss ratio over accident periods. This is a highly simplifying assumption relative to the chain ladder model which, in effect, allows loss ratio to vary freely over accident period. Much of the literature on Cape Cod reserving treats it as essentially just an algorithm. It does not posit a parametric model supporting the algorithm. There are one or two exceptions to this. The present paper extends them by introducing a more general stochastic model under which maximum likelihood estimation yields parameters estimates closely resembling those of the classical Cape Cod algorithm. These estimators are shown to be minimum variance unbiased, and so are superior to the classical estimators, which rely on the chain ladder.

A GENERALIZED LOSS RATIO METHOD DEALING WITH UNCERTAIN VOLUME MEASURES

AbstractUnlike chain ladder, the loss ratio method requires volume measures. Typically, these volumes are assumed to be known. In practice, however, accurate volume measures are rarely available. We interpret the available volumes as estimators for the true volume measures and analyze the consequences for the loss ratio method. In particular, we calculate the mean squared error of prediction, including uncertainty of volume measures, and derive approximately optimal weights for the observed incremental loss ratios. We then introduce a generalization of the loss ratio method that is tailored to the situation of uncertain volume measures and calculate the prediction uncertainty of this generalized loss ratio method.

Claims reserving in the presence of excess-of-loss reinsurance using micro models based on aggregate data

This paper addresses a new problem in the literature, which is how to consider reserving issues for a portfolio of general insurance policies when there is excess-of-loss reinsurance. This is very important for pricing considerations and for decision making regarding capital issues. The paper sets out how this is currently often tackled in practice and provides an alternative approach using recent developments in stochastic claims reserving. These alternative approaches are illustrated and compared in an example using real data. The stochastic modelling framework used in this paper is Double Chain Ladder, but other approaches would also be possible. The paper sets out an approach which could be explored further and built on in future research.

Método Bootstrap para el cálculo de provisiones técnicas por siniestros

Proper risk management is one of the main objectives in the insurance companies, the good assessment of uncertainty is one of the keys to success in this business sector, therefore, covers the object of our study. The key to managing uncertainty is to foresee the possible claims and thus estimate the technical provisions to set up by the company for the payment of these. The European directive Solvency II, which came into force in 2016, substantially modified the methodology to be used for the estimation of technical provisions used previously. This regulation establishes a better assessment of the risk to meet payments for claims without having problems of low liquidity or insolvency. Adapting ourselves to this norm, we proceed to estimate technical provisions through the Chain Ladder method implemented by the triangle format, which adapts to the directive and takes into account the behaviour of the payments for claims in different years. To reduce uncertainty in our estimates, we use the Bootstrap method which we will determine the reliability of the data estimating the error. Using the data obtained from an insurance company in its area of cars, we carry out hypothesis to determine the reliability of the Chain Ladder with Bootstrap estimation method.

Separation of small and large claims on the basis of collective models

This paper is inspired by two papers of Riegel who proposed to consider the paid and incurred loss development of the individual claims and to use a filter in order to separate small and large claims and to construct loss development squares for the paid or incurred small or large claims and for the numbers of large claims. We show that such loss development squares can be constructed from collective models for the accident years. Moreover, under certain assumptions on these collective models, we show that a development pattern exists for each of these loss development squares, which implies that various methods of loss reserving can be used for prediction and that the chain ladder method is a natural method for the prediction of future numbers of large claims.

Chain Ladder Method: Bayesian Bootstrap Versus Classical Bootstrap

The intention of this paper is to analyse the mean square error of prediction (MSEP) under the distribution-free chain ladder (DFCL) claims reserving method. We compare the estimation obtained from the classical bootstrap method with the one obtained from a Bayesian bootstrap. To achieve this in the DFCL model we develop a novel approximate Bayesian computation (ABC) sampling algorithm to obtain the empirical posterior distribution. We need an ABC sampling algorithm because we work in a distribution-free setting. The use of this ABC methodology combined with bootstrap allows us to obtain samples from the intractable posterior distribution without the requirement of any distributional assumptions. This then enables us to calculate the MSEP and other risk measures like Value-at-Risk.

State Space Models and the Kalman-Filter in Stochastic Claims Reserving: Forecasting, Filtering and Smoothing

This paper gives a detailed overview of the current state of research in relation to the use of state space models and the Kalman-filter in the field of stochastic claims reserving. Most of these state space representations are matrix-based, which complicates their applications. Therefore, to facilitate the implementation of state space models in practice, we present a scalar state space model for cumulative payments, which is an extension of the well-known chain ladder (CL) method. The presented model is distribution-free, forms a basis for determining the entire unobservable lower and upper run-off triangles and can easily be applied in practice using the Kalman-filter for prediction, filtering and smoothing of cumulative payments. In addition, the model provides an easy way to find outliers in the data and to determine outlier effects. Finally, an empirical comparison of the scalar state space model, promising prior state space models and some popular stochastic claims reserving methods is performed.

Multi-Year Non-Life Insurance Risk for Correlated Loss Portfolios Under Chain Ladder Model Assumptions

In this paper we extend the definition of multi-year claims development results and quantification of multi-year non-life insurance risk to the bivariate chain ladder model as introduced by Braun in 2004. In this model, we assume two correlated loss portfolios each of which is underlying the classical chain ladder model. In accordance with standard literature, multi-year risk is defined through the variation of the multi-year claims development result and quantified in terms of the corresponding mean squared error of prediction. Following previous research on the univariate chain ladder model, for the first time we derive closed analytical expressions for the prediction error of the aggregate multi-year claims development result via first-order Taylor approximation. We reproduce well-known results for the ultimo view from literature. The goodness of our approximation is confirmed by a simulation study. Furthermore, a case study demonstrates the applicability of our analytical formulae.