A Multivariate Claim Count Model For Applications In Insurance

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 dissertation contributes to the risk and insurance literature by expanding our understanding of insurance claims modeling, deductible ratemaking, and the insurance risk retention problem. In the claims modeling part, a data-driven approach is taken to analyze insurance losses using statistical methods. It is often common for an analyst to be interested in several outcome measures depending on a large set of explanatory variables, with the goal of understanding both the average behavior, and the overall distribution of the outcomes. The use of multivariate analysis has an advantage in a broad context, and the literature on multivariate regression modeling is extended with a focus on dependence among multiple insurance lines. In this process, a deductible is an important feature of an insurance policy to consider, because it may influence the frequency and severity of claims to be censored or truncated. Standard textbooks have approached deductible ratemaking using models for coverage modification, utilizing parametric loss distributions. In practice, regression could be used with explanatory variables including the deductible amount. The various approaches to deductible ratemaking are compared in this dissertation. Ultimately, an insurance manager would be interested in understanding the influence of a retention parameter change to the risk of a portfolio of losses. The retention parameter may be deductible, upper limit, or coinsurance. This dissertation contributes to the statistics and actuarial literature by introducing and applying the 01-inflated negative binomial frequency model (a frequency model for observations with an inflated number of zeros and ones), and illustrating how discrete and continuous copula methods can be empirically applied to insurance claims analysis. In the process, the dissertation provides a comparison among various deductible analysis procedures, and shows that the regression approach has an advantage in problems of moderate size. Finally, the dissertation attempts to broaden our understanding of the risk retention problem within a constrained optimization framework, and demonstrates the quasiconvexity of the objective function in this problem. The dissertation reveals that the loading factor of a reinsurance premium has a risk measure interpretation, and relates to the risk measure relative margins (RMRM). Concepts are illustrated using the Wisconsin Local Government Property Insurance Fund (LGPIF) data.

Predictive modeling uses data to forecast future events. It exploits relationships between explanatory variables and the predicted variables from past occurrences to predict future outcomes. Forecasting financial events is a core skill that actuaries routinely apply in insurance and other risk-management applications. Predictive Modeling Applications in Actuarial Science emphasizes life-long learning by developing tools in an insurance context, providing the relevant actuarial applications, and introducing advanced statistical techniques that can be used to gain a competitive advantage in situations with complex data. Volume 2 examines applications of predictive modeling. Where Volume 1 developed the foundations of predictive modeling, Volume 2 explores practical uses for techniques, focusing on property and casualty insurance. Readers are exposed to a variety of techniques in concrete, real-life contexts that demonstrate their value and the overall value of predictive modeling, for seasoned practicing analysts as well as those just starting out.

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.

Consider two different portfolios which have claims triggered by the same events. Their corresponding collective model over a fixed time period is given in terms of individual claim sizes $(X_i,Y_i), i ge 1$ and a claim counting random variable $N$. In this paper we are concerned with the joint distribution function $F$ of the largest claim sizes $(X_{N:N}, Y_{N:N})$. By allowing $N$ to depend on some parameter, say $ theta$, then $F=F( theta)$ is for various choices of $N$ a tractable parametric family of bivariate distribution functions. We present three applications of the implied parametric models to some data from the literature and a new data set from a Swiss insurance company. Furthermore, we investigate both distributional and asymptotic properties of $(X_{N:N,Y_{N:N})$

This collection of articles addresses the most modern forms of loss reserving methodology: granular models and machine learning models. New methodologies come with questions about their applicability. These questions are discussed in one article, which focuses on the relative merits of granular and machine learning models. Others illustrate applications with real-world data. The examples include neural networks, which, though well known in some disciplines, have previously been limited in the actuarial literature. This volume expands on that literature, with specific attention to their application to loss reserving. For example, one of the articles introduces the application of neural networks of the gated recurrent unit form to the actuarial literature, whereas another uses a penalized neural network. Neural networks are not the only form of machine learning, and two other papers outline applications of gradient boosting and regression trees respectively. Both articles construct loss reserves at the individual claim level so that these models resemble granular models. One of these articles provides a practical application of the model to claim watching, the action of monitoring claim development and anticipating major features. Such watching can be used as an early warning system or for other administrative purposes. Overall, this volume is an extremely useful addition to the libraries of those working at the loss reserving frontier.

This paper focuses on the estimation and application aspects of the Erlang Count Logit-weighted Reduced Mixture of Experts model (EC-LRMoE), which is a fully flexible multivariate insurance claim frequency regression model proposed in Fung et al. (2018a). We first prove the identifiability property of the proposed model to ensure that it is a suitable candidate for statistical inference. An Expectation-Conditional-Maximization (ECM) algorithm is developed for efficient model calibrations. Three simulation studies are performed so that the effectiveness of the proposed ECM algorithm and the versatility of the proposed model can be examined. The applicability of the EC-LRMoE is shown through fitting an European automobile insurance dataset. Since the dataset contains several complex features, we find it necessary to adopt such a flexible model. Apart from showing excellent fitting results, we are able to interpret the fitted model in an insurance perspective and to visualize the relationship between policyholders' information and their risk level. Finally, we demonstrate how the fitted model may be useful for insurance ratemaking.

Because of its many advantages, the use of decision trees has become an increasingly popular alternative predictive tool for building classification and regression models. Its origins date back for about five decades where the algorithm can be broadly described by repeatedly partitioning the regions of the explanatory variables and thereby creating a tree-based model for predicting the response. Innovations to the original methods, such as random forests and gradient boosting, have further improved the capabilities of using decision trees as a predictive model. In addition, the extension of using decision trees with multivariate response variables started to develop and it is the purpose of this paper to apply multivariate tree models to insurance claims data with correlated responses. This extension to multivariate response variables inherits several advantages of the univariate decision tree models such as distribution-free feature, ability to rank essential explanatory variables, and high predictive accuracy, to name a few. To illustrate the approach, we analyze a dataset drawn from the Wisconsin Local Government Property Insurance Fund (LGPIF) which offers multi-line insurance coverage of property, motor vehicle, and contractors' equipments. With multivariate tree models, we are able to capture the inherent relationship among the response variables and we find that the marginal predictive model based on multivariate trees is an improvement in prediction accuracy from that based on simply the univariate trees.

The accurate estimation of outstanding liabilities of an insurance company is an essential task. This is to meet regulatory requirements, but also to achieve efficient internal capital management. Over the recent years, there has been increasing interest in the utilisation of insurance data at a more granular level, and to model claims using stochastic processes. So far, this so-called 'micro-level reserving' approach has mainly focused on the Poisson process.In this paper, we propose and apply a Cox process approach to model the arrival process and reporting pattern of insurance claims. This allows for over-dispersion and serial dependency in claim counts, which are typical features in real data. We explicitly consider risk exposure and reporting delays, and show how to use our model to predict the numbers of Incurred-But-Not-Reported (IBNR) claims. The model is calibrated and illustrated using real data from the AUSI data set.