Continuous chain-ladder with paid data
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We introduce a model where iid payments generate the traditional paid run-off triangle. Recent literature explains how claim counts data can be embedded into a continuous chain-ladder model. However, when outstanding claim amounts are to be calculated there is a cost associated to every claim. Assuming iid payments, we show under which assumption on the individual data the traditional chain-ladder method is consistent. Consistency is understood under the framework that both the number of observations grows to infinity and the level of aggregation tends to zero. Besides regularity conditions, two major assumptions are needed: Firstly, development delay (column effect) and accident date (row effect) must be independent. Secondly, the expected claims severity conditioned on development delay and accident date must be multiplicative, i.e, the two effects must act independently on the severity of the claim. We introduce a local polynomial estimator of the cost-weighted density. This enables us to forecast the outstanding liabilities. In a real data example, we show how the assumptions can be visually checked. A simulation study shows that our local polynomial kernel estimator outperforms the classical chain-ladder method --- identified as histogram estimator --- significantly.
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