  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.
