A COLLECTIVE RESERVING MODEL WITH CLAIM OPENNESS

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.

Calendar Effect and In-Sample Forecasting Applied to Mesothelioma Mortality Data

In this paper, we apply and further illustrate a recently developed extended continuous chain ladder model to forecast mesothelioma deaths. Making such a forecast has always been a challenge for insurance companies as exposure is difficult or impossible to measure, and the latency of the disease usually lasts several decades. While we compare three approaches to this problem, we show that the extended continuous chain ladder model is a promising benchmark candidate for asbestosis mortality forecasting due to its flexible and simple forecasting strategy. Furthermore, we demonstrate how the model can be used to provide an update for the forecast of the number of deaths due to mesothelioma in Great Britain using in recent Health and Safety Executive (HSE) data.

One-Year and Ultimate Reserve Risk in Mack Chain Ladder Model

We investigate the relation between one-year reserve risk and ultimate reserve risk in Mack Chain Ladder model in a simulation study. The first goal is to validate the so-called linear emergence pattern formula, which maps the ultimate loss to the one-year loss, in case when we measure the risks with Value-at-Risk. The second goal is to estimate the true emergence pattern of the ultimate loss, i.e., the conditional distribution of the one-year loss given the ultimate loss, from which we can properly derive a risk measure for the one-year horizon from the simulations of ultimate losses. Finally, our third goal is to test if classical actuarial distributions can be used for modelling of the outstanding loss from the ultimate and the one-year perspective. In our simulation study, we investigate several synthetic loss triangles with various duration of the claims development process, volatility, skewness, and distributional assumptions of the individual development factors. We quantify the reserve risks without and with the estimation error of the claims development factors.

A robust random coefficient regression representation of the chain-ladder method

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.

PARAMETRIC METHOD OF SOLVING PROBLEMS OF MATHEMATICAL SAFE ON GRAPHS

We consider one method of solving the problem of mathematical safe on certain graphs called parametric. Its gist consist in denoting some variables, corresponding to graph vertices, by certain parameters. Other unknown variables are expressed through these parameters. Then unknown variables chosen in special way are compared and the mentioned parameters are found by solving additional system of equations for these parameters. Dimension of this system is equal to the number of parameters. Solution to the problem i.e. all unknown variables of the original system, are found by solving additional system of equations. In the paper this method is described on specially chosen examples. The method is demonstrated by solving the mathematical safe problem on the graphs of «chain», «ladder» and «window» types that showed its efficiency. Besides special attention is paid to special cases when solution does not exist. This occurs in the cases when the weighed sum of system equations is not divisable without remainder to its modulo. In such cases, to find solution the initial state of the vector b is corrected in such a way that the weighted sum of equations satisfies the above mentioned condition. Then solution of the problem is performed according to the general method scheme.

Implementation of Gaussian Process Regression in Estimating Motor Vehicle Insurance Claims Reserves

This study aims to calculate the allowance for losses by applying Gaussian Process regression to estimate future claims. Modeling is done on motor vehicle insurance data. The data used in this study are historical data on PT XYZ's motor vehicle insurance business line during 2017 and 2019 (January 2017 to December 2019). Data analysis will be carried out on the 2017 - 2019 data to obtain an estimate of the claim reserves in the following year, namely 2018 - 2020. This study uses the Chain Ladder method which is the most popular loss reserving method in theory and practice. The estimation results show that the Gaussian Process Regression method is very flexible and can be applied without much adjustment. These results were also compared with the Chain Ladder method. Estimated claim reserves for PT XYZ's motor vehicle business line using the chain-ladder method, the company must provide funds for 2017 of 8,997,979,222 IDR in 2018 16,194,503,605 IDR in 2019 amounting to Rp. 1,719,764,520 for backup. Meanwhile, by using the Bayessian Gaussian Process method, the company must provide funds for 2017 of 9,060,965,077 IDR in 2018 amounting to 16,307,865,130 IDR, and in 2019 1,731,802,871 IDR for backup. The more conservative Bayessian Gaussian Process method. Motor vehicle insurance data has a short development time (claims occur) so that it is included in the short-tail type of business.