When calculating the risk margins of a company with multiple Lines of Business-typically, a quantile in the right tail of an aggregate loss, assumptions about the dependence structure between the different Lines are crucial. Many current multivariate reserving methodologies focus on aggregated claims information, typically in the format of claim triangles. This aggregation is subject to some inefficiencies, such as possibly insufficient data points, and potential elimination of useful information. This inefficiency is particularly problematic for the estimation of dependence. So-called 'micro-level models', on the other hand, utilise more granular levels of observations. Such granular data lend themselves naturally to a stochastic process modelling approach. However, the literature interested in the incorporation of a dependency structure with a micro-level approach is still scarce.In this paper, we extend the literature of micro-level stochastic reserving models to the multivariate context. We develop a multivariate Cox process to model the joint arrival process of insurance claims in multiple Lines of Business. This allows for a dependency structure between the frequencies of claims. We also explicitly incorporate known covariates, such as seasonality patterns and trends, which may explain some of the relationship between two insurance processes (or at least help tease out those relationships). We develop a filtering algorithm to estimate the unobservable stochastic intensities. Model calibration is illustrated using real data from the AUSI data set.

This monograph presents a time-dynamic model for multivariate claim counts in actuarial applications. Inspired by real-world claim arrivals, the model balances interesting stylized facts (such as dependence across the components, over-dispersion and the clustering of claims) with a high level of mathematical tractability (including estimation, sampling and convergence results for large portfolios) and can thus be applied in various contexts (such as risk management and pricing of (re-)insurance contracts). The authors provide a detailed analysis of the proposed probabilistic model, discussing its relation to the existing literature, its statistical properties, different estimation strategies as well as possible applications and extensions. Actuaries and researchers working in risk management and premium pricing will find this book particularly interesting. Graduate-level probability theory, stochastic analysis and statistics are required.

This thesis studies the applications of Pascal mixture models in three closely related topics in insurance and risk management. The first topic is on the modeling of correlated frequencies of operational risk (OR) losses from financial institutions. We propose a copula-free approach for modeling correlated frequencies using an Erlang-based multivariate mixed Poisson distribution. Many properties possessed by this class of distributions are investigated and a tailor-made generalized expectation-maximization (EM) algorithm is derived for fitting purposes. The applicability of the proposed distribution is illustrated in an OR management context, where this class is used to model the OR loss. The accuracy of the proposed approach is analyzed using a modified real operational loss data set. The second topic is about multivariate count regression with application in modeling correlated claim frequencies. We propose a multivariate Pascal mixture regression model as an alternative to understand the association between multivariate count response variables and their covariates. We examine the many properties possessed by this class of regression. A generalized EM algorithm is derived for fitting purposes, which also provides the standard errors of the regression coefficients which are useful for inference. Its applicability is demonstrated by fitting an automobile insurance claim count data set. The third topic is about modeling and predicting the number of incurred but not reported (IBNR) claims in Property Casualty (P) insurance. We model the claim arrival process together with the reporting delays as a marked Cox process whose intensity function is governed by a hidden Markov chain. The associated reported claim process and IBNR claim process remain to be marked Cox processes with easily convertible intensity functions and marking distributions. Closed-form expressions for both the autocorrelation function (ACF) and the distributions of the numbers of reported claims and IBNR claims are derived. A generalized EM algorithm is obtained to estimate the model parameters. The proposed model is examined through simulation studies and is also applied to a real insurance claim data set. We compare the predictive distributions of our model with those of the over-dispersed Poisson model (ODP), a stochastic model that underpins the widely used chain-ladder method.

Quality and quantity of social science data is continually improving, from large publicuse survey microdata to private industry data. This wealth of data allows researchers to ask more complex questions about interdependencies of social and economic processes and behavior. This dissertation presents methods for models that address interdisciplinary research questions about the association structure of multiple outcomes of similar or disparate types, e.g. count and duration outcomes. The proposed models and methods address associations of multiple outcomes through correlated unobserved subject-specific effects. Chapter 2 presents a semiparametric method for estimating the marginal response and association parameters in a random effects multivariate longitudinal count model. In the context of the generalized estimating equations (GEE) framework, we use a specific form of the covariance matrix of the response vector based on a model that induces dependence over time and outcomes using random effects. This moment based method is robust to distributional misspecification and reduces the computational burden associated with a high-dimensional joint distribution by avoiding parametric assumptions on the response and unobserved effects. Through a simulation study we compare finite sample robustness properties of this semiparametric method with a pseudo-likelihood approach that imposes distributional assumptions. Both of these methods are then used to analyze a dataset of insurance claim counts for three types of coverage over time. The economic significance of these results is presented in Chapter 3. Chapter 4 presents a Gaussian variational approximation (GVA) approach for estimation of a joint multivariate longitudinal count and multivariate duration random effects model. GVA proposes an approximate posterior distribution of the random effects to obtain a closed form lower bound of the marginal likelihood. GVA estimators are obtained by maximizing the variational lower bound, which coincides with minimizing the Kullback-Leibler distance between the random effects posterior distribution and the assumed approximate posterior distribution. This approach circumvents the computationally complex, high-dimensional integral associated with the marginal distribution of a joint longitudinal and duration model. Through a simulation study we compare finite sample properties of the variational approximation approach with comparable univariate and multivariate two-stage plug-in approaches. These methods are then used to analyze a dataset of insurance claim counts and policy duration for three types of coverage over time.

Starting from the question: “What is the accident risk of an insured?”, this paper considers a multivariate approach by taking into account three types of accident risks and the possible dependence between them. Driven by a real data set, we propose three trivariate Sarmanov distributions with generalized linear models (GLMs) for marginals and incorporate various individual characteristics of the policyholders by means of explanatory variables. Since the data set was collected over a longer time period (10 years), we also added each individual's exposure to risk. To estimate the parameters of the three Sarmanov distributions, we analyze a pseudo-maximumlikelihood method. Finally, the three models are compared numerically with the simpler trivariate Negative Binomial GLM.

Diese Dissertation stellt innovative Pricing- und Hedging-Modelle für eine breite Klasse von Versicherungsprodukten vor. Eine wichtige Neuerung im Hinblick auf die existierende Literatur ist dabei das Anwenden F-doppelt stochastischer Markovketten, was die Ausarbeitung der Formeln anhand stochastischer Intensitätsprozesse ermöglicht. Für die Prämienbestimmung für Arbeitslosigkeitsversicherungsprodukte werden die Intensitätsprozesse durch mikro- und makroökonomische stochastische Kovariablenprozesse generiert, um Einflüsse und Abhängigkeitsstrukturen innerhalb von Arbeitsmärkten zu untersuchen. Als Preisregel wird die „Real-World“-Preisformel des Benchmark-Ansatzes gewählt. Für die Bestimmung optimaler Hedgingstrategien werden quadratische Hedging-Methoden auf eine breite Klasse von Versicherungsprodukten, u.a. Lebensversicherungsprodukten, angewandt. Die Lösungen werden dabei anhand der Galtchouk-Kunita-Watanabe-Zerlegung jeweiligen der Schadenprozesse bestimmt.