We introduce a continuous-time framework for the prediction of outstanding liabilities, in which chain-ladder development factors arise as a histogram estimator of a cost-weighted hazard function running in reversed development time. We use this formulation to show that under our assumptions on the individual data chain-ladder is consistent. Consistency is understood in the sense that both the number of observed claims grows to infinity and the level of aggregation tends to zero. We propose alternatives to chain-ladder development factors by replacing the histogram estimator with kernel smoothers and by estimating a cost-weighted density instead of a cost-weighted hazard. Finally, we provide a real-data example and a simulation study confirming the strengths of the proposed alternatives.We start by putting the unique sampling scheme of chain-ladder into a micro-structure framework. We observe counting processes (N i (t)) t∈[0,T ] , T > 0, for claims i = 1, . . . n and call t development time. Each counting process starts with value zero at the underwriting date underlying its claim. It jumps, with jump-size one, whenever a payment is made. Additionally to every jump, we observe a mark indicating the size of the payment made. The number of counting processes, n, varies over calendar-time: We follow retrospectively only those claims for which at least one payment has been observed, i.e., we do not follow every claim in the policy book. In this paper, we make the following assumptions.[M1] All claims are independent.[M2] Every claim consists of only one payment.Assumptions [M1] and [M2] are rather strong but are made to simplify the mathematical derivations yielding a first and clean step towards a better understanding of chain-ladder on a
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.
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