Continuous chain-ladder with paid data

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.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-Sample Hazard Forecasting Based on Survival Models with Operational Time

“…As shown in a simulation study in Baudry and Robert (2019), even when enough data are available for monthly aggregation, chain-ladder reserve estimates based on monthly data show very high variance, making them effectively unreliable in practice; however, monthly data are necessary for chain-ladder if one is interested, for instance, in the estimation of monthly cash-flows. This phenomenon has been confirmed in a simulation in Bischofberger et al (2019), in which kernel estimators picked larger bandwidths while still being able to yield monthly cash-flow predictions. Furthermore, chain-ladder is typically used on at least quarterly aggregated data to prevent columns that contain only zeros in the run-off triangle.…”

“…All mentioned continuous chain-ladder publications including this present paper focus on claim numbers instead of payment amounts. Recently, Bischofberger et al (2019) have shown how to extend the models and estimators for payment amounts. This extension is also feasible for our approach; however, adding extensive additional technicalities is beyond the scope of this paper.…”

Analysis of IBNR Liabilities with Interevent Times Depending on Claim Counts

“…( 2019 ), Bischofberger et al. ( 2020 ) extended the chain ladder to individual claims. However, they used hazard rates associated with payment-producing claims to develop a continuous-time version of the chain ladder.…”

Collective reserving using individual claims data

“…Lately, machine learning methods find one's way into individual claims reserving, allowing for more flexible regression structures. Some recent papers are based on regression trees and gradient boosting, see Wüthrich (2018), Lopez et al (2019), Lopez & Milhaud (2021), De Felice & Moriconi (2019 and Duval & Pigeon (2019); others are based on non-parametric and kernel methods, see Rosenlund (2012), Bischofberger et al (2019), Baudry & Robert (2019), or on neural networks, see Gabrielli (2020), Kuo (2020) and Delong & Wüthrich (2020). Surprisingly, many of these approaches pay little attention to the data itself, but they describe the methods used in much detail.…”