Claustro de San Agustín.
Classical Reserving – Double Chain Ladder and its Extensions
Carolin Margraf*, Cass Business School, London; Jens Nielsen, Cass Business School, London; Maria Martinez Miranda, University of Granada, Spain; Munir Hiabu, Cass Business School, London
In this paper, we propose different methods based on the Double Chain Ladder (DCL) framework introduced by Martinez-Miranda, Nielsen, and Verrall (2012a). The aim of reserving in non-life insurance is to forecast the amount of claims which have been underwritten in the past, but are not settled yet. For as long as anyone remembers, non-life insurance companies have used the so called chain ladder method to reserve for outstanding liabilities. In the DCL framework, we build on this classical reserving method and also add the information of reported counts data to the classical reserving data. By using more data, it is expected that the method will have less volatility than the celebrated chain ladder method.
Not only do we derive a surprisingly simple method for forecasting the outstanding liabilities but we are also able to estimate RBNS (reported but not settled) and IBNR (incurred but not reported yet) claims separately. Furthermore, we benefit from the advantages of having a full stochastic cash flow model of outstanding liabilities for the model developed in Verrall, Nielsen and Jessen (2010).This way, we can take advantage of the simple relationship between development factors which allows us to involve and then estimate the reporting and payment delay.
In this work, we investigate the Double Chain Ladder model further and consider the case when other knowledge is available, namely prior knowledge or expert knowledge. Initially we focus on prior knowledge on the number of zero-claims for each underwriting year and prior knowledge about the relationship between the development of the claim and its mean severity. Both types of prior knowledge readily lend themselves to be included in the DCL framework. Furthermore, we want to include a mixture of paid data and expert knowledge, incurred data, in different ways. More precisely, we use incurred data to rectify one weak point for DCL and CLM where the underwriting year inflation might be estimated with significant uncertainty. A key feature of the new method is that the underwriting year inflation can be estimated from the less volatile incurred data and then transferred into the DCL model. In addition, we take advantage of this expert knowledge and use it as an estimate for the RBNS claims. We include an empirical illustration that illustrates the differences between the estimates of the IBNR and RBNS cash flows from DCL and the new method.
This paper constructs for the first time a full statistical cash flow model of the incurred chain ladder method useful for asset-liability hedging, capital allocation and other management decision tools.
We also apply bootstrap estimation to approximate the predictive distributions. This is an alternative to the widely used England and Verrall (1999) bootstrap which might have to restrictive assumptions.
Risk Measure Preserving Piecewise Linear Approximation of Empirical Distributions
William Guevara Alarcón*, Université de Lausanne; Philipp Arbenz, SCOR.
Stochastic models used for pricing, reserving, or capital modelling in insurance companies are often very complex, which is why resulting distributions are typically approximated by Monte Carlo simulations. Both the market and regulators exert increasing pressure not to discard the resulting sample distributions, but rather to store them for future review, audit, or validation, as well as to transfer them between actuarial systems. The present work introduces a compression algorithm which approximates an empirical univariate distribution function through a piecewise linear distribution. In contrast to keeping the full sample, such an approximation facilitates the storage and data transfer of the results by drastically reducing memory requirements. The approximation algorithm preserves the mean and imposes a uniformly bounded relative error over a space of coherent risk measures (TVaR). An efficient, open source implementation is provided.
In-sample forecasting with local linear survival densities – A continuous chain ladder approach
Munir Hiabu*, Cass Business School; Maria Martinez Miranda, University of Granada, Spain; Enno Mammen, ; Jens Nielsen, Cass Business School, London
Non-life insurance companies mostly use the so called chain ladder method for reserving outstanding liabilities.
Chain ladder and all its extensions are based on aggregated run-off triangles. In this talk we will show how to translate the chain ladder method into a continuous framework using granular data. In contrast to the few granular methods which already exist, we will keep the basic structure of having observations on a triangle. As it turns out, chain ladder, and thus also our continuous analogue, is an in-sample technique where no extrapolation is needed to forecast the reserve. The in-sample area is defined as one triangle (the “upper triangle”) and the forecasting area as the second triangle (the “lower triangle”) that added to the first triangle produces a square. We call our approach in-sample forecasting. The in-sample forecasting will be performed with non-parametric methods in a survival analysis framework. In the first part of the talk we will focus on the multiplicative density structure which also is the underlying assumption of chain ladder. Calendar and seasonal effects do not follow this multiplicativity assumption. Therefore, in the second part of the talk, we will show how to go beyond this multiplicativity assumption in order to get more accurate forecasts. A real data example and a simulation study will be provided to support the theory.