Contributions To The Theory Of Empirical Linear Bayes Estimation And Its Application To Claims Reserving In Non Life Insurance

This book first provides a review of various aspects of Bayesian statistics. It then investigates three types of claims reserving models in the Bayesian framework: chain ladder models, basis expansion models involving a tail factor, and multivariate copula models. For the Bayesian inferential methods, this book largely relies on Stan, a specialized software environment which applies Hamiltonian Monte Carlo method and variational Bayes.

Claims reserving is central to the insurance industry. Insurance liabilities depend on a number of different risk factors which need to be predicted accurately. This prediction of risk factors and outstanding loss liabilities is the core for pricing insurance products, determining the profitability of an insurance company and for considering the financial strength (solvency) of the company. Following several high-profile company insolvencies, regulatory requirements have moved towards a risk-adjusted basis which has lead to the Solvency II developments. The key focus in the new regime is that financial companies need to analyze adverse developments in their portfolios. Reserving actuaries now have to not only estimate reserves for the outstanding loss liabilities but also to quantify possible shortfalls in these reserves that may lead to potential losses. Such an analysis requires stochastic modeling of loss liability cash flows and it can only be done within a stochastic framework. Therefore stochastic loss liability modeling and quantifying prediction uncertainties has become standard under the new legal framework for the financial industry. This book covers all the mathematical theory and practical guidance needed in order to adhere to these stochastic techniques. Starting with the basic mathematical methods, working right through to the latest developments relevant for practical applications; readers will find out how to estimate total claims reserves while at the same time predicting errors and uncertainty are quantified. Accompanying datasets demonstrate all the techniques, which are easily implemented in a spreadsheet. A practical and essential guide, this book is a must-read in the light of the new solvency requirements for the whole insurance industry.

"Offers a mathematical introduction to non-life insurance and, at the same time, to a multitude of applied stochastic processes. It gives detailed discussions of the fundamental models for claim sizes, claim arrivals, the total claim amount, and their probabilistic properties....The reader gets to know how the underlying probabilistic structures allow one to determine premiums in a portfolio or in an individual policy." --Zentralblatt für Didaktik der Mathematik

We revisit the gamma-gamma Bayesian chain-ladder (BCL) model for claims reserving in non-life insurance. This claims reserving model is usually used in an empirical Bayesian way using plug-in estimates for variance parameters, because this empirical Bayesian framework allows us for closed form solutions. The main purpose of this paper is to develop the full Bayesian case also considering prior distributions for variance parameters, and to study the resulting sensitivities.

This is a single comprehensive reference source covering the key material on this subject, and describing both theoretical and practical aspects.

In this monograph, authors Greg Taylor and Gráinne McGuire discuss generalized linear models (GLM) for loss reserving, beginning with strong emphasis on the chain ladder. The chain ladder is formulated in a GLM context, as is the statistical distribution of the loss reserve. This structure is then used to test the need for departure from the chain ladder model and to consider natural extensions of the chain ladder model that lend themselves to the GLM framework.

Our article considers the class of recently developed stochastic models that combine claims payments and incurred losses information into a coherent reserving methodology. In particular, we develop a family of hierarchical Bayesian paid-incurred-claims models, combining the claims reserving models of Hertig and Gogol. In the process we extend the independent log-normal model of Merz and Wuethrich by incorporating different dependence structures using a Data-Augmented mixture Copula paid-incurred claims model.The usefulness of incorporating both payment and incurred losses into estimating of the full predictive distribution of the outstanding loss liabilities and the resulting reserves is demonstrated in the following cases: (i) an independent payment data model; (ii) the independent payment and incurred claims data model of Merz and Wuethrich; (iii) a novel dependent lag-year telescoping block diagonal Gaussian copula payment and incurred claim data model incorporating conjugacy via transformation; (iv) a novel data-augmented mixture Archimedean copula dependent payment and incurred claim data model.Inference in such models is developed by adaptive Markov chain Monte Carlo sampling algorithms. These incorporate a data-augmentation framework utilised to efficiently evaluate the likelihood for the copula based payment and incurred claim model in the loss reserving triangles. The adaptation strategy of the Markov chain Monte Carlo is based on two components. The first component uses an adaptive strategy for learning the posterior structures for the parameters defined over a Euclidean space and the second component deals with an adaptive learning of the posterior for the covariance matrices restricted to the Riemann manifold corresponding to the space of positive definite matrices for the linear dependence structure specified for the payment and incurred claim model.