# Continuous chain-ladder with paid data

## TL;DR

This paper develops a continuous-time, non-parametric framework for predicting outstanding liabilities using hazard functions and kernel smoothing, demonstrating consistency and improved estimation methods.

## Contribution

It introduces a continuous-time chain-ladder model with histogram and kernel estimators, extending traditional methods with new smoothing techniques and theoretical consistency results.

## Key findings

- The proposed methods are consistent under increasing claim data and decreasing aggregation levels.
- Kernel-based estimators outperform traditional development factors in simulations.
- Real-data application confirms the effectiveness of the new estimators.

## Abstract

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.