A Likelihood Approach to Bornhuetter–Ferguson Analysis

A new Bornhuetter–Ferguson method is suggested herein. This is a variant of the traditional chain ladder method. The actuary can adjust the relative ultimates using externally estimated relative ultimates. These correspond to linear constraints on the Poisson likelihood underpinning the chain ladder method. Adjusted cash flow estimates were obtained as constrained maximum likelihood estimates. The statistical derivation of the new method is provided in the generalised linear model framework. A related approach in the literature, combining unconstrained and constrained maximum likelihood estimates, is presented in the same framework and compared theoretically. A data illustration is described using a motor portfolio from a Greek insurer.This research was funded by the programme for Economic Modelling, Oxford and European Research Council, grant AdG 694262, and the Spanish Ministry of Economy and Competitiveness, grant MTM2016-76969P, which includes support from the European Regional Development Fund (ERDF)

The valuation of life contingencies: A symmetrical triangular fuzzy approximation

This paper extends the framework for the valuation of life insurance policies and annuities by Andrés- Sánchez and González-Vila (2012, 2014) in two ways. First we allow various uncertain magnitudes to be estimated by means of fuzzy numbers. This applies not only to interest rates but also to the amounts to be paid out by the insurance company. Second, the use of symmetrical triangular fuzzy numbers allows us to obtain expressions for the pricing of life contingencies and their variability that are closely linked to standard financial and actuarial mathematics. Moreover, they are relatively straightforward to compute and understand from a standard actuarial point of view

Fuzzy Markovian Bonus-Malus Systems in Non-Life Insurance

Markov chains (MCs) are widely used to model a great deal of financial and actuarial problems. Likewise, they are also used in many other fields ranging from economics, management, agricultural sciences, engineering or informatics to medicine. This paper focuses on the use of MCs for the design of non-life bonus-malus systems (BMSs). It proposes quantifying the uncertainty of transition probabilities in BMSs by using fuzzy numbers (FNs). To do so, Fuzzy MCs (FMCs) as defined by Buckley and Eslami in 2002 are used, thus giving rise to the concept of Fuzzy BMSs (FBMSs). More concretely, we describe in detail the common BMS where the number of claims follows a Poisson distribution under the hypothesis that its characteristic parameter is not a real but a triangular FN (TFN). Moreover, we reflect on how to fit that parameter by using several fuzzy data analysis tools and discuss the goodness of triangular approximates to fuzzy transition probabilities, the fuzzy stationary state, and the fuzzy mean asymptotic premium. The use of FMCs in a BMS allows obtaining not only point estimates of all these variables, but also a structured set of their possible values whose reliability is given by means of a possibility measure. Although our analysis is circumscribed to non-life insurance, all of its findings can easily be extended to any of the abovementioned fields with slight modifications.University of Barcelon

Machine Learning in Insurance

Machine learning is a relatively new field, without a unanimous definition. In many ways, actuaries have been machine learners. In both pricing and reserving, but also more recently in capital modelling, actuaries have combined statistical methodology with a deep understanding of the problem at hand and how any solution may affect the company and its customers. One aspect that has, perhaps, not been so well developed among actuaries is validation. Discussions among actuaries’ “preferred methods” were often without solid scientific arguments, including validation of the case at hand. Through this collection, we aim to promote a good practice of machine learning in insurance, considering the following three key issues: a) who is the client, or sponsor, or otherwise interested real-life target of the study? b) The reason for working with a particular data set and a clarification of the available extra knowledge, that we also call prior knowledge, besides the data set alone. c) A mathematical statistical argument for the validation procedure

Life settlement pricing with fuzzy parameters

Existing literature asserts that the growth of life settlement (LS) markets, where they exist, is hampered by limited policyholder participation and suggests that to foster this growth appropriate pricing of LS transactions is crucial. The pricing of LSs relies on quantifying two key variables: the insured's mortality multiplier and the internal rate of return (IRR). However, the available information on these parameters is often scarce and vague. To address this issue, this article proposes a novel framework that models these variables using triangular fuzzy numbers (TFNs). This modelling approach aligns with how mortality multiplier and IRR data are typically provided in insurance markets and has the advantage of offering a natural interpretation for practitioners. When both the mortality multiplier and the IRR are represented as TFNs, the resulting LS price becomes a FN that no longer retains the triangular shape. Therefore, the paper introduces three alternative triangular approximations to simplify computations and enhance interpretation of the price. Additionally, six criteria are proposed to evaluate the effectiveness of each approximation method. These criteria go beyond the typical approach of assessing the approximation quality to the FN itself. They also consider the usability and comprehensibility for financial analysts with no prior knowledge of FNs. In summary, the framework presented in this paper represents a significant advancement in LS pricing. By incorporating TFNs, offering several triangular approximations and proposing goodness criteria of them, it addresses the challenges posed by limited and vague data, while also considering the practical needs of industry practitioners

Multivariate stochastic loss reserving with common shock approaches

Les provisions techniques (ou « réserves ») constituent habituellement l'un des plus importants passifs au bilan d'un assureur IARD (Incendie, Accidents et Risques Divers). Conséquemment, il est crucial pour les assureurs de les estimer avec précision. En outre, un assureur IARD opère généralement dans plusieurs lignes d'affaires dont les risques ne sont pas parfaitement dépendants. Il en résulte un « bénéfice de diversification » qu'il est primordial de considérer pour son effet sur les réserves et le capital. Il est donc essentiel de prendre en compte la dépendance entre lignes d'affaires dans l'estimation des réserves. L'objectif de cette thèse est de développer de nouvelles approches d'évaluation des réserves pour des portefeuilles d'assurance avec lignes d'affaires dépendantes. Pour ce faire, nous explorons la technique de choc commun, une méthode populaire de modélisation de la dépendance qui offre plusieurs avantages, tels qu'une structure de dépendance explicite, une facilité d'interprétation et une construction parcimonieuse des matrices de corrélation. Afin de rendre les méthodes utiles à la pratique, nous incorporons au modèle des caractéristiques réalistes et souhaitables. Motivés par la richesse de la famille de distributions de probabilité Tweedie, laquelle recoupe les distributions Poisson, amma et bien d'autres, nous introduisons un cadre commun de choc Tweedie avec dépendance entre lignes d'affaires. Les propriétés attrayantes de ce cadre sont étudiées, y compris la flexibilité de ses marges, ses moments ayant une forme analytique et sa capacité d'inclure des masses à 0. Pour surmonter la complexité de la structure distributionnelle Tweedie, nous utilisons une approche bayésienne dans l'estimation des paramètres du modèle, que nous appliquons à un ensemble de données réelles. Nous formulons des remarques sur les caractéristiques pratiques de notre cadre. Les données sur les provisions techniques pour sinistres sont asymétriques par nature. C'est-à-dire, les montants de réclamations qu'on retrouve dans différentes cases d'un même triangle et entre différents triangles peuvent varier considérablement. Ceci s'explique car, habituellement, le nombre de sinistres est plus élevé dans les premières périodes de développement. Nous tenons compte explicitement de cette caractéristique dans nos modèles de chocs communs en y incluant un ajustement parcimonieux. Des illustrations utilisant des données théoriques (simulées) et réelles sont présentées. Enfin, dans la dernière partie de cette thèse, nous élaborons un cadre dynamique avec des facteurs évolutifs tenant compte des tendances de développement des sinistres pouvant changer avec le temps. Une dépendance entre années civiles est introduite à l'aide de chocs communs. Nous formulons également une méthode d'estimation adaptée à la structure des données de provisionnement des sinistres, que nous illustrons à l'aide de données réelles.Outstanding claims liability is usually one of the largest liabilities on the balance sheet of a general insurer. Therefore, it is critical for insurers to accurately estimate their outstanding claims. Furthermore, a general insurer typically operates in multiple business lines whose risks are not perfectly dependent. This results in ``diversification benefits", the consideration of which is crucial due to their effects on the aggregate reserves and capital. It is then essential to consider the dependence across business lines in the estimation of outstanding claims. The goal of this thesis is to develop new approaches to assess outstanding claims for portfolios of dependent lines. We explore the common shock technique for model developments, a very popular dependence modelling technique with distinctive strengths, such as explicit dependence structure, ease of interpretation, and parsimonious construction of correlation matrices. We also aim to enhance the practicality of our approaches by incorporating realistic and desirable model features. Motivated by the richness of the Tweedie distribution family which covers Poisson distributions, gamma distributions and many more, we introduce a common shock Tweedie framework with dependence across business lines. Desirable properties of this framework are studied, including its marginal flexibility, tractable moments, and ability to handle masses at 0. To overcome the complex distributional structure of the Tweedie framework, we formulate a Bayesian approach for model estimation and perform a real data illustration. Remarks on practical features of the framework are drawn. Loss reserving data possesses an unbalanced nature, that is, claims from different positions within and between loss triangles can vary widely as more claims typically develop in early development periods. We account for this feature explicitly in common shock models with a parsimonious common shock adjustment. Theoretical and real data illustrations are performed using the multivariate Tweedie framework. Finally, in the last part of this thesis, we develop a dynamic framework with evolutionary factors to account for claims development patterns that change over time. Calendar year dependence is introduced using common shocks. We also formulate an estimation approach that is tailored to the structure of loss reserving data and perform a real data illustration

A Neural Network Linking Process

A novel method of integrating multiple neural networks into one large network via a process referred to as a neural network linking process is proposed. Neural networks are commonly trained to solve a specific problem for an encapsulated problem domain. A single network can undertake simple classification or generalisation problems. Dividing them into sub-problems, which in turn are solved by a sub-network, can disentangle more complicated classification or generalisation problems. A controller generally combines sub-network results. A controller can be, for instance, a gating network, voting system or a mathematical combiner. In each case, every sub-network is used as a separate unit and is not interconnected to any other sub-network. However, with the linking process a novel method for linking trained sub-networks into one large network by maintaining the knowledge of each individual sub-network is introduced. Furthermore, the linked network will utilize a stimulus process in order to distinguish the type of sub-problem to be solved, by largely retaining the accuracy of the sub-network, as well as being one step closer to the biological reality

Debiasing Cyber Incidents – Correcting for Reporting Delays and Under-reporting

This research addresses two key problems in the cyber insurance industry – reporting delays and under-reporting of cyber incidents. Both problems are important to understand the true picture of cyber incident rates. While reporting delays addresses the problem of delays in reporting due to delays in timely detection, under-reporting addresses the problem of cyber incidents frequently under-reported due to brand damage, reputation risk and eventual financial impacts. The problem of reporting delays in cyber incidents is resolved by generating the distribution of reporting delays and fitting modeled parametric distributions on the given domain. The reporting delay distribution was found to be non-stationary and bimodal. While non-stationarity was handled by generating the monthly reporting delay distribution over the rolling two-year moving window, the bimodal aspect required an optimization algorithm to compute the parameters. The modeled parametric distribution is further extended to infinite domain to obtain the complete overview of the incidents occurred but not yet reported. The complete modeled parametric distribution provides the correction factors showing an increasing trend in recent months rather than a decline as observed from reported incidents. The correction of reporting delays is computed for the US market. The study is further extended to highlight how reporting delays vary from industry to industry. Four different industries of US companies were compared within US market: Finance and Insurance, Educational Services, Health Care and Social Assistance, and Public Administration. The comparative study showed the corrections for reporting delays in the overall US market and by industry, with specific emphasis on the four distinct industries. The problem of under-reporting in cyber incidents is addressed in context of population characteristics. The proposed solution computes the large variations in under-reporting as a function of the three variables - revenue, incident type, and industry. Three different incident types–hacking, social engineering, and ransomware-- and five industries– Retail Trade, Manufacturing, Finance and Insurance, Professional Scientific Technical Services, and Wholesale Trade– were studied. The research highlighted that there is a need to address under-reporting by incident types and by industry

Fuzzy Sets in Business Management, Finance, and Economics

This book collects fifteen papers published in s Special Issue of Mathematics titled “Fuzzy Sets in Business Management, Finance, and Economics”, which was published in 2021. These paper cover a wide range of different tools from Fuzzy Set Theory and applications in many areas of Business Management and other connected fields. Specifically, this book contains applications of such instruments as, among others, Fuzzy Set Qualitative Comparative Analysis, Neuro-Fuzzy Methods, the Forgotten Effects Algorithm, Expertons Theory, Fuzzy Markov Chains, Fuzzy Arithmetic, Decision Making with OWA Operators and Pythagorean Aggregation Operators, Fuzzy Pattern Recognition, and Intuitionistic Fuzzy Sets. The papers in this book tackle a wide variety of problems in areas such as strategic management, sustainable decisions by firms and public organisms, tourism management, accounting and auditing, macroeconomic modelling, the evaluation of public organizations and universities, and actuarial modelling. We hope that this book will be useful not only for business managers, public decision-makers, and researchers in the specific fields of business management, finance, and economics but also in the broader areas of soft mathematics in social sciences. Practitioners will find methods and ideas that could be fruitful in current management issues. Scholars will find novel developments that may inspire further applications in the social sciences