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Customer churn, which insurance companies use to describe the non-renewal of existing customers, is a widespread and expensive problem in general insurance, particularly because contracts are usually short-term and are renewed periodically. Traditionally, customer churn analyses have employed models which utilise only a binary outcome (churn or not churn) in one period. However, real business relationships are multi-period, and policyholders may reside and transition between a wider range of states beyond that of the simply churn/not churn throughout this relationship. To better encapsulate the richness of policyholder behaviours through time, we propose multi-state customer churn analysis, which aims to model behaviour over a larger number of states (defined by different combinations of insurance coverage taken) and across multiple periods (thereby making use of readily available longitudinal data). Using multinomial logistic regression (MLR) with a second-order Markov assumption, we demonstrate how multi-state customer churn analysis offers deeper insights into how a policyholder’s transition history is associated with their decision making, whether that be to retain the current set of policies, churn, or add/drop a coverage. Applying this model to commercial insurance data from the Wisconsin Local Government Property Insurance Fund, we illustrate how transition probabilities between states are affected by differing sets of explanatory variables and that a multi-state analysis can potentially offer stronger predictive performance and more accurate calculations of customer lifetime value (say), compared to the traditional customer churn analysis techniques.

The insurance industry is built on risk classification, grouping insureds into homogeneous classes. Through actions such as underwriting, pricing, and so forth, it differentiates, or discriminates, among insureds. Actuaries have responsibility for pricing insurance risk transfers and are intimately involved in other aspects of company actions and so have a keen interest in whether or not discrimination is appropriate from both company and societal viewpoints. This article reviews social and economic principles that can be used to assess the appropriateness of insurance discrimination. Discrimination issues vary by the line of insurance business and by the country and legal jurisdiction. This article examines social and economic principles from the vantage of a specific line of business and jurisdiction;these vantage points provide insights into principles. To sharpen understanding of the social and economic principles, this article also describes discrimination considerations for prohibitions based on diagnosis of COVID-19, the pandemic that swept the globe in 2020. Insurance discrimination issues have been an important topic for the insurance industry for decades and are evolving in part due to insurers’ extensive use of Big Data;that is, the increasing capacity and computational abilities of computers, availability of new and innovative sources of data, and advanced algorithms that can detect patterns in insurance activities that were previously unknown. On the one hand, the fundamental issues of insurance discrimination have not changed with Big Data;one can think of credit-based insurance scoring and price optimization as simply forerunners of this movement. On the other hand, issues regarding privacy and use of algorithmic proxies take on increased importance as insurers’ extensive use of data and computational abilities evolve. © 2021 The Author(s). Published with license by Taylor & Francis Group, LLC.

The dataset tracks 40,284 insurance clients over five years, between 2010 and 2015, who subscribed to both automobile and homeowners insurance. We have combined information on these customers. First, the characteristics including age, gender or driving experience, among others and dates of renewal for the two types of policies considered here. Note that we have only considered clients corresponding to persons and not commercial firms that can also underwrite home and motor insurance policies. Second, the policy data file for motor vehicle insurance consists of all vehicle insurance coverage including power, driving area or whether there is a second driver that drives the car occasionally. Third, the policy data file for homeowners insurance has information on the property such as value of the building (essentially the value of the home without any furniture, apparel and personal items), location and type of dwelling. Besides these three sources, we have access to data containing information on the number of claims and total cost of those claims per year and per policy type. So, for all policies that are in force, we finally have up to a five year record of the yearly cost of claims in the motor insurance and in the home coverage. If the customer does not renew one of those two policies or both, we do not have more information after this lapse occurs. After summarizing the data, we provide the usual marginal analysis, where we fit regression models using Tweedie distributions for claims and a logistic model for lapse. Data can be used for joint analysis of insurance policyholders with more than one product.

Insurance as a discipline has long embraced analytics, and market trends signal an even stronger relationship going forward. This article is a case study on the use of predictive analytics in the c...

Multivariate discrete outcomes are common in a wide range of areas including insurance, finance, and biology. When the interplay between outcomes is significant, quantifying dependencies among interrelated variables is of great importance. Due to their ability to accommodate dependence flexibly, copulas are being applied increasingly. Yet, the application of copulas on discrete data is still in its infancy; one of the biggest barriers is the nonuniqueness of copulas, calling into question model interpretations and predictions. In this article, we study copula estimation with discrete outcomes in a regression context. As the marginal distributions vary with covariates, inclusion of continuous regressors expands the region of support for consistent estimation of copulas. Because some properties of continuous outcomes do not carry over to discrete outcomes, specification of a copula model has been a problem. We propose a nonparametric estimator of copulas to identify the “hidden” dependence structure for discrete outcomes and develop its asymptotic properties. The proposed nonparametric estimator can also serve as a diagnostic tool for selecting a parametric form for copulas. In the simulation study, we explore the performance of the proposed estimator under different scenarios and provide guidance on when the choice of copulas is important. The performance of the estimator improves as discreteness diminishes. A practical bandwidth selector is also proposed. An empirical analysis examines a dataset from the Local Government Property Insurance Fund (LGPIF) in the state of Wisconsin. We apply the nonparametric estimator to model the dependence among claim frequencies from different types of insurance coverage. Supplementary materials for this article are available online.

This is online supplement provides and extended abstract, a description of regulation by jurisdiction, several appendices, and an annotated bibliography.

The insurance industry is built on risk classification, grouping insureds into homogeneous classes. Through actions such as underwriting, pricing and so forth, it differentiates, or discriminates, among insureds. Actuaries have responsibility for pricing insurance risk transfers and are intimately involved in other aspects of company actions and so have a keen interest in whether or not discrimination is appropriate from both company and societal viewpoints. This paper reviews social and economic principles that can be used to assess the appropriateness of insurance discrimination. Discrimination issues vary by the line of insurance business and by the country and legal jurisdiction. This paper examines social and economic principles from the vantage of a specific line of business and jurisdiction; these vantage points provide insights into principles. To sharpen understanding of the social and economic principles, this paper also describes discrimination considerations for prohibitions based on diagnosis of COVID-19, the pandemic that swept the globe in 2020. Insurance discrimination issues have been an important topic for the insurance industry for decades and is evolving in part due to insurers' extensive use of *Big Data*, that is, the increasing capacity and computational abilities of computers, availability of new and innovative sources of data, and advanced algorithms that can detect patterns in insurance activities that were previously unknown. On the one hand, the fundamental issues of insurance discrimination have not changed with Big Data; one can think of credit-based insurance scoring and price optimization as simply forerunners of this movement. On the other hand, issues regarding privacy and use of algorithmic proxies take on increased importance as insurers' extensive use of data and computational abilities evolve.

Financial services industries, such as insurance, increasingly use data from their broad cross-section of customers and follow these customers over time. In other areas such as medicine, engineering, and communication systems, it is well known that following subjects over time may result in biased data, for example, the so-called ”dropout effect”. This paper introduces techniques to address dropout commonly encountered in the insurance domain. Specifically, in the insurance context, multivariate claims outcomes may be related to a customer’s dropout or decision to lapse a policy. Insurance claims outcomes are also naturally a hybrid with both discrete and continuous components, which adds complexity to model calibration. Decision makers in the insurance industry will find our work provides helpful guidance in integrating customer loyalty, especially with bundled coverages. This paper introduces a generalized method of moments technique to estimate dependence parameters where associations are represented using copulas. This is especially useful for large data sets. The paper describes how the joint model provides new information that insurers can use to better manage their portfolios of risks. An application to a Spanish insurer data set is presented.

In this article, I introduce a statistic for managing a portfolio of insurance risks. This tool is based on changes in the risk profile when changes in a risk parameter, such as a deductible, coins...

This article describes contributions of analytics and statistical methods to our understanding of insurance operations and markets. Specifically, it introduces insurance analytics, the foundations of the discipline, and the supporting literature. It also describes current trends in analytics. Insurance as a discipline has long embraced analytics, and market trends signal an even stronger relationship going forward.

This paper synthesizes and extends the literature on multivariate two-part regression modelling, with an emphasis on actuarial applications. To illustrate the modelling, we use data from the US Medical Expenditure Panel Survey to explore expenditures that come in two parts. In the first part, zero expenditures correspond to no payments for health care services during a year. For the second part, a positive expenditure corresponds to the payment amount, a measure of utilization. Expenditures are multivariate, the five components being (i) office-based, (ii) hospital outpatient, (iii) emergency room, (iv) hospital inpatient, and (v) home health expenditures. Not surprisingly, there is a high degree of association among expenditure types and so we utilize models that account for these associations. These models include multivariate binary regressions for the payment type and generalized linear models with Gaussian copulas for payment amounts.As anticipated, the strong associations among expenditure types allow us to establish significant model differences on an in-sample basis. Despite these strong associations, we find that commonly used statistical measures perform similarly on a held-out validation sample. In contrast, out-of-sample risk measures used by actuaries reveal differences in the association among expenditure types.

Welfare economics uses Lorenz curves to display skewed income distributions and Gini indices to summarize the skewness. This article extends the Lorenz curve and Gini index by ordering insurance risks; the ordering variable is a risk-based score relative to price, known as a relativity. The new relativity-based measures can cope with adverse selection and quantify potential profit. Specifically, we show that the Gini index is proportional to a correlation between the relativity and an out-of-sample profit (price in excess of loss). A detailed example using homeowners insurance demonstrates the utility of these new measures. INTRODUCTION In today's world of readily available information and interest in business analytics, insurers seek to utilize nontraditional information in their ratemaking and under-writing processes. Table 1 shows some nontraditional rating variables that are being considered and used by some insurers. To see how insurers can use new information in the rate making process, we first consider the traditional premium. To set notation, we wish to compare the distribution of an insurance loss y to a price P in a portfolio of risks. For each risk in the portfolio, we assume that the analyst has available a set of known exogenous risk characteristics x upon which both the loss and price distributions depend. We emphasize the dependence of the price on the risk characteristics x through the notation P = P(x). Ignoring expenses, in an efficient marketplace, the price P will be close to the cost y. This is a difficult proposition for the marketplace to ensure because: * y is random with a distribution of outcomes. * The distribution of y is complex. To illustrate this complexity, in typical homeowners insurance data described in the "Homeowners Example" section, 94 percent of the losses are zeros (corresponding to no claims) and when losses are positive, the distribution tends to be right skewed and thick tailed. * There are many different sets of insureds, corresponding to a variety of [x.sub.i]s. * There are many different contract variations, corresponding to different deductibles, limits, coverages, riders, and so forth. One point of view is that the premium should be the expected loss. This viewpoint is supported in the context of (1) many independent contracts and (2) a competitive marketplace. We suppose that the insurer introduces a new risk-based score, denoted by S([x.sub.i]). This score may be based on a new rating algorithm, for example, from a modern statistical approach such as the generalized linear model. Alternatively or additionally, it may be based on new information, for example, a new variable such as in Table 1. To assess this score, it is common (e.g., Werner and Modlin, 2010) for insurers to examine a relative premium R([x.sub.i]) = S([x.sub.i])/P([x.sub.i]), known as a relativity. Suppose that the score S is a desirable approximation of the expected loss. Then, if the relativity is small, we can expect a small loss relative to the premium. This is a profitable policy but is also one that is susceptible to potential raiding by a competitor (adverse selection). In contrast, if the relativity is large, then we can expect a large loss relative to the premium. This is one where better loss control measures, for example, renewal underwriting restrictions, can be helpful. Using these relativities, this article shows how to form portfolios of policies and compare losses to premiums to assess profitability. This is the purpose of the ordered Lorenz curve. If the insurer can form profitable portfolios, then the competition may also be able to do so, inviting potential raiding by competing firms. To provide protection, the insurer may use the scores: * to form the basis of a new rating algorithm (pricing), or * as part of the underwriting guidelines, either at policy initiation or renewal. …

In insurance and related industries including healthcare, it is common to have several outcome measures that the analyst wishes to understand using explanatory variables. For example, in automobile insurance, an accident may result in payments for damage to one’s own vehicle, damage to another party’s vehicle, or personal injury. It is also common to be interested in the frequency of accidents in addition to the severity of the claim amounts. This paper synthesizes and extends the literature on multivariate frequency-severity regression modeling with a focus on insurance industry applications. Regression models for understanding the distribution of each outcome continue to be developed yet there now exists a solid body of literature for the marginal outcomes. This paper contributes to this body of literature by focusing on the use of a copula for modeling the dependence among these outcomes; a major advantage of this tool is that it preserves the body of work established for marginal models. We illustrate this approach using data from the Wisconsin Local Government Property Insurance Fund. This fund offers insurance protection for (i) property; (ii) motor vehicle; and (iii) contractors’ equipment claims. In addition to several claim types and frequency-severity components, outcomes can be further categorized by time and space, requiring complex dependency modeling. We find significant dependencies for these data; specifically, we find that dependencies among lines are stronger than the dependencies between the frequency and average severity within each line.

This work proposes a new copula class that we call the MGB2 copula. The new copula originates from extracting the dependence function of the multivariate GB2 distribution (MGB2) whose marginals follow the univariate generalized beta distribution of the second kind (GB2). The MGB2 copula can capture non-elliptical and asymmetric dependencies among marginal coordinates and provides a simple formulation for multi-dimensional applications. This new class features positive tail dependence in the upper tail and tail independence in the lower tail. Furthermore, it includes some well-known copula classes, such as the Gaussian copula, as special or limiting cases. To illustrate the usefulness of the MGB2 copula, we build a trivariate MGB2 copula model of bodily injury liability closed claims. Extended GB2 distributions are chosen to accommodate the right-skewness and the long-tailedness of the outcome variables. For the regression component, location parameters with continuous predictors are introduced using a nonlinear additive function. For comparison purposes, we also consider the Gumbel and t copulas, alternatives that capture the upper tail dependence. The paper introduces a conditional plot graphical tool for assessing the validation of the MGB2 copula. Quantitative and graphical assessment of the goodness of fit demonstrate the advantages of the MGB2 copula over the other copulas.

This article extends the standard two-part model for predicting health care expenditures to the case where multiple events may occur within a one-year period. The first part of the extended model represents the frequency of events, such as the number of inpatient hospital stays or outpatient visits, and the second part models expenditure per event. Both component models also use independent variables that consist of an individual’s demographic and access characteristics, socioeconomic status, health status, health insurance coverage, employment status, and industry classification. The second part of the model also includes a variable representing the number of events to predict the expenditure per event, thus capturing dependencies between the first and second parts. This article introduces closed-form predictors of annual total expenditures and demonstrates how to create simulated predictive distributions for individuals and groups. The data for this study are from the Medical Expenditure Panel ...

Individuals, corporations and government entities regularly exchange financial risks y at prices Π. Comparing distributions of risks and prices can be difficult, particularly when the financial risk distribution is complex. For example, with insurance, it is not uncommon for a risk distribution to be a mixture of 0’s (corresponding to no claims) and a right-skewed distribution with thick tails (the claims distribution). However, analysts do not work in a vacuum, and in the case of insurance they use insurance scores relative to prices, called “relativities,” that point to areas of potential discrepancies between risk and price distributions. Ordering both risks and prices based on relativities, in this article we introduce what we call an “ordered” Lorenz curve for comparing distributions. This curve extends the classical Lorenz curve in two ways, through the ordering of risks and prices and by allowing prices to vary by observation. We summarize the ordered Lorenz curve in the same way as the classic Lor...

The insurance industry is known to have high operating expenses in the financial services sector. Insurers, investors and regulators are interested in models to understand the behavior of expenses. However, the current practice ignores skewness, occasional negative values as well as their temporal dependence. Addressing these three features, this paper develops a longitudinal model of insurance company expenses that can be used for prediction, to identify unusual behavior, and to measure firm efficiency. Specifically, we use a three-parameter asymmetric Laplace density for the marginal distribution of insurers’ expenses in each year. Copula functions are employed to accommodate their temporal dependence. As a function of explanatory variables, the location parameter allows us to analyze an insurer’s expenses in light of the firm’s characteristics. Our model can be interpreted as a longitudinal quantile regression. The analysis is performed using property–casualty insurance company data from the National Association of Insurance Commissioners of years 2001–2006. Due to the long-tailed nature of insurers’ expenses, two alternative approaches are proposed to improve the performance of the longitudinal quantile regression model: rescaling and transformation. Predictive densities are derived that allow one to compare the predictions for individual insurers in a hold-out-sample. Both predictive models are shown to be reasonable with the rescaling method outperforming the transformation method. Compared with standard longitudinal models, our model is shown to be superior in identifying insurers’ unusual behavior.

What types of households own life insurance? Who owns term life and who owns whole life insurance? These are questions of great interest to insurers that operate in a highly competitive market. To answer these questions, we jointly examine household demand of two types of insurance, term and whole life, using data from the Survey of Consumer Finances, a probability sample of the U.S. population. We model both the frequency and the severity of demand for insurance, building on the work of Lin and Grace by using explanatory variables that they developed. For the frequency portion, the household decisions about whether to own term and whole life insurance are modeled simultaneously with a bivariate probit regression model. Given ownership of life insurance by a household, the amounts of insurance are analyzed using generalized linear models with a normal copula. The copula permits the bivariate modeling of insurance amounts for households who own both term and whole life insurance, about 20% of our ...

For the construction of a fair tariff structure in automobile insurance, insurers classify the risks that they underwrite. The idea behind this risk classification is to subdivide the portfolio into classes of risks with similar profiles. While some insurers may have sufficient historical data, several others may not have significant volume of experience data in order to produce reliable claims predictions to help enhance their risk classification systems. A database containing a pooled experience of several insurers thereby helps to produce a more fair, reliable, and equitable premium structure for all risks concerned. Research and analysis of such "intercompany" insurance experience data is lacking in both the actuarial and statistical literature. Its benefits goes beyond the insurer; reinsurers (i.e. insurers of insurers) together with regulators also benefit from statistical models of this type of data because they typically deal with analyzing the experience of a collection of insurers. In this paper, we use multilevel models to analyze the data on claim counts provided by the General Insurance Association of Singapore, an organization consisting of most of the general insurers in Singapore. Our data comes from the financial records of automobile insurance policies followed over a period of nine years. The multilevel nature of the data is due to the following: a certain vehicle is observed over a period of years and is insured by a particular insurance company under a certain ‘fleet’ policy. Fleet policies are umbrella-type policies issued to customers whose insurance covers more than a single vehicle with a taxicab company being a typical example. We show how intercompany data lead to a priori premiums and a posteriori corrections to these initial premiums. Specific focus is made in understanding the intercompany effects using various count distribution models (Poisson, negative binomial, zero-inflated and hurdle Poisson). The performance of these various models is compared; we also investigated how to use the historical claims of a company, fleet and/or vehicle in order to correct for the premium initially set.

This paper demonstrates actuarial applications of modern statistical methods that are applied to detailed, micro-level automobile insurance records. We consider 1993-2001 data consisting of policy and claims files from a major Singaporean insurance company. A hierarchical statistical model, developed in prior work (Frees and Valdez (2008)), is fit using the micro-level data. This model allows us to study the accident frequency, loss type and severity jointly and to incorporate individual characteristics such as age, gender and driving history that explain heterogeneity among policyholders.Based on this hierarchical model, one can analyze the risk profile of either a single policy (micro-level) or a portfolio of business (macro-level). This paper investigates three types of actuarial applications. First, we demonstrate the calculation of the predictive mean of losses for individual risk rating. This allows the actuary to differentiate prices based on policyholder characteristics. The nonlinear effects of coverage modifications such as deductibles, policy limits and coinsurance are quantified. Moreover, our flexible structure allows us to “unbundle” contracts and price more primitive elements of the contract, such as coverage type. The second application concerns the predictive distribution of a portfolio of business. We demonstrate the calculation of various risk measures, including value at risk and conditional tail expectation, that are useful in determining economic capital for insurance companies. Third, we examine the effects of several reinsurance treaties. Specifically, we show the predictive loss distributions for both the insurer and reinsurer under quota share and excess-of-loss reinsurance agreements. In addition, we present an example of portfolio reinsurance, in which the combined effect of reinsurance agreements on the risk characteristics of ceding and reinsuring company are described.

In this paper, we consider “heavy-tailed” data, that is, data where extreme values are likely to occur. Heavy-tailed data have been analyzed using flexible distributions such as the generalized beta of the second kind, the generalized gamma and the Burr. These distributions allow us to handle data with either positive or negative skewness, as well as heavy tails. Moreover, it has been shown that they can also accommodate cross-sectional regression models by allowing functions of explanatory variables to serve as distribution parameters. The objective of this paper is to extend this literature to accommodate longitudinal data, where one observes repeated observations of cross-sectional data. Specifically, we use copulas to model the dependencies over time, and heavy-tailed regression models to represent the marginal distributions. We also introduce model exploration techniques to help us with the initial choice of the copula and a goodness-of-fit test of elliptical copulas for model validation. In a longitudinal data context, we argue that elliptical copulas will be typically preferred to the Archimedean copulas. To illustrate our methods, Wisconsin nursing homes utilization data from 1995 to 2001 are analyzed. These data exhibit long tails and negative skewness and so help us to motivate the need for our new techniques. We find that time and the nursing home facility size as measured through the number of beds and square footage are important predictors of future utilization. Moreover, using our parametric model, we provide not only point predictions but also an entire predictive distribution.

This work describes statistical modeling of detailed, microlevel automobile insurance records. We consider 1993–2001 data from a major insurance company in Singapore. By detailed microlevel records, we mean experience at the individual vehicle level, including vehicle and driver characteristics, insurance coverage, and claims experience, by year. The claims experience consists of detailed information on the type of insurance claim, such as whether the claim is due to injury to a third party, property damage to a third party, or claims for damage to the insured, as well as the corresponding claim amount. We propose a hierarchical model for three components, corresponding to the frequency, type, and severity of claims. The first model is a negative binomial regression model for assessing claim frequency. The driver’s gender, age, and no claims discount, as well as vehicle age and type, turn out to be important variables for predicting the event of a claim. The second is a multinomial logit model to predict ...

When there exist omitted effects, measurement error, and/or simultaneity in multilevel models, explanatory variables may be correlated with random components, and standard estimation methods do not provide consistent estimates of model parameters. This paper introduces estimators that are consistent under such conditions. By employing generalized method of moments (GMM) estimation techniques in multilevel modeling, the authors present a series of estimators along a robust to efficient continuum. This continuum depends on the assumptions that the analyst makes regarding the extent of the correlated effects. It is shown that the GMM approach provides an overarching framework that encompasses well-known estimators such as fixed and random effects estimators and also provides more options. These GMM estimators can be expressed as instrumental variable (IV) estimators which enhances their interpretability. Moreover, by exploiting the hierarchical structure of the data, the current technique does not require additional variables unlike traditional IV methods. Further, statistical tests are developed to compare the different estimators. A simulation study examines the finite sample properties of the estimators and tests and confirms the theoretical order of the estimators with respect to their robustness and efficiency. It further shows that not only are regression coefficients biased, but variance components may be severely underestimated in the presence of correlated effects. Empirical standard errors are employed as they are less sensitive to correlated effects when compared to model-based standard errors. An example using student achievement data shows that GMM estimators can be effectively used in a search for the most efficient among unbiased estimators.

The recent development and availability of sophisticated computer software has facilitated the use of predictive modeling by actuaries and other financial analysts. Predictive modeling has been used for several applications in both the health and property and casualty sectors. Often these applications employ extensions of industry-specific techniques and do not make full use of information contained in the data. In contrast, we employ fundamental statistical methods for predictive modeling that can be used in a variety of disciplines. As demonstrated in this article, this methodology permits a disciplined approach to model building, including model development and validation phases. This article is intended as a tutorial for the analyst interested in using predictive modeling by making the process more transparent. This article illustrates the predictive modeling process using State of Wisconsin nursing home cost reports. We examine utilization of approximately 400 nursing homes from 1989 to 2001...

Statistical methodology for handling omitted variables is presented in a multilevel modeling framework. In many nonexperimental studies, the analyst may not have access to all requisite variables, and this omission may lead to biased estimates of model parameters. By exploiting the hierarchical nature of multilevel data, a battery of statistical tools are developed to test various forms of model misspecification as well as to obtain estimators that are robust to the presence of omitted variables. The methodology allows for tests of omitted effects at single and multiple levels. The paper also introduces intermediate-level tests; these are tests for omitted effects at a single level, regardless of the presence of omitted effects at a higher level. A simulation study shows, not surprisingly, that the omission of variables yields bias in both regression coefficients and variance components; it also suggests that omitted effects at lower levels may cause more severe bias than at higher levels. Important factors resulting in bias were found to be the level of an omitted variable, its effect size, and sample size. A real data study illustrates that an omitted variable at one level may yield biased estimators at any level and, in this study, one cannot obtain reliable estimates for school-level variables when omitted child effects exist. However, robust estimators may provide unbiased estimates for effects of interest even when the efficient estimators fail, and the one-degree-of-freedom test helps one to understand where the problem is located. It is argued that multilevel data typically contain rich information to deal with omitted variables, offering yet another appealing reason for the use of multilevel models in the social sciences.

This paper develops credibility predictors of aggregate losses using a longitudinal data framework. For a model of aggregate losses, the interest is in predicting both the claims number process as well as the claims amount process. In a longitudinal data framework, one encounters data from a cross-section of risk classes with a history of insurance claims available for each risk class. Further, explanatory variables for each risk class over time are available to help explain and predict both the claims number and claims amount process. For the marginal claims distributions, this paper uses generalized linear models, an extension of linear regression, to describe cross-sectional characteristics. Elliptical copulas are used to model the dependencies over time, extending prior work that used multivariate t -copulas. The claims number process is represented using a Poisson regression model that is conditioned on a sequence of latent variables. These latent variables drive the serial dependencies among claims numbers; their joint distribution is represented using an elliptical copula. In this way, the paper provides a unified treatment of both the continuous claims amount and discrete claims number processes. The paper presents an illustrative example of Massachusetts automobile claims. Estimates of the latent claims process parameters are derived and simulated predictions are provided.

Multilevel models are proven tools in social research for modeling complex, hierarchical systems. In multilevel modeling, statistical inference is based largely on quantification of random variables. This paper distinguishes among three types of random variables in multilevel modeling—model disturbances, random coefficients, and future response outcomes—and provides a unified procedure for predicting them. These predictors are best linear unbiased and are commonly known via the acronym BLUP; they are optimal in the sense of minimizing mean square error and are Bayesian under a diffuse prior.

Employee termination and retirement probabilities affect the valuation of employee benefit plans and thus are of concern to actuaries. To provide timely experience for the profession, the Society of Actuaries’ Non-Mortality Decrement Task Force organized a data collection effort. Thirty-two contributors provided over 1.7 million life years of pension plan turnover data for years 1994–2000. This article summarizes the results of this data collection effort. Traditionally, the most important determinants of termination and retirement are age, a proxy for attachment to the workforce, and service, a measure of attachment to a firm. This article documents the importance of these traditional quantities using current data and provides tables so that actuaries may quantitatively assess their importance when valuing pension plans. For the middle working years, ages 25–55, we find female termination probabilities are higher than males, although the differences are smaller than has been true historically. T...

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes. This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specific...

Credibility ratemaking is a technique used in pricing health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since the time of Mowbray's (1914) contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of mixed linear models. This article extends this approach to credibility by formally introducing collateral information through the use of Bayesian methods. Specifically, we derive credibility estimators and mean square errors for normal hierarchical linear models. We provide intuition for the credibility estimators by establishing the link between these estimators and homogeneous and inhomogeneous estimators that appear in non-Bayesian credibility theory.

In longitudinal data analysis, as with cross-sectional regression, correction for heteroscedasticity is important for accurate hypothesis tests and confidence intervals. Using empirical standard errors to adjust for heteroscedasticity is common practice. In this article, we investigate alternative empirical standard errors that have better finite sample properties than those commonly used. We review the properties of an estimator that is based on the deletion, or jackknife, principle. We also consider an estimator based on standardized residuals that is unbiased in many circumstances. Our Monte Carlo simulations show that both alternatives have better small-sample properties than the usual empirical standard errors, regardless as to whether heteroscedasticity is present. Both alternatives behave well in terms of size and power even when the covariance matrix is completely misspecified. We recommend the use of the alternative empirical standard errors whenever there is a reason to suspect that heteroscedasticity is present.

This paper shows how to forecast using a class of linear mixed longitudinal, or panel, data models. Forecasts are derived as special cases of best linear unbiased predictors, also known as BLUPs, and hence are optimal predictors of future realizations of the response. We show that the BLUP forecast arises from three components: (1) a predictor based on the conditional mean of the response, (2) a component due to time-varying coefficients, and (3) a serial correlation correction term. The forecasting techniques are applicable in a wide variety of settings. This article discusses forecasting in the context of marketing and sales. In particular, we consider a data set of the Wisconsin State Lottery, in which 40 weeks of sales are available for each of 50 postal codes. Using sales data as well as economic and demographic characteristics of each postal code, we forecast sales for each postal code.

Forecasts of labor force participation rates are important components of government projections of social insurance as well as projections of other aspects of a nation’s economy. However, the current practice is to provide only deterministic projections. In contrast, this paper shows how to model the uncertainty of labor participation and provide stochastic projections. To accommodate a wide variety of applications, labor force participation rates are highly disaggregated; this paper considers rates separated by age, sex, marital status and the presence of young children in the family. Instead of using traditional multivariate time series models, this paper introduces hierarchical and longitudinal data models to forecast labor force participation rates. Forecasting based on longitudinal data models gives desirable results. Not only are the forecasts comparable to those produced by government projections but we can also supplement these projections with prediction intervals. These prediction intervals help users understand the forecast uncertainty, thus more completely portraying the reliability of the projections.

Predictive modeling involves the use of data to forecast future events. It relies on capturing relationships between explanatory variables and the predicted variables from past occurrences and exploiting this to predict future outcomes. Forecasting future financial events is a core actuarial skill - actuaries routinely apply predictive-modeling techniques in insurance and other risk-management applications. This book is for actuaries and other financial analysts who are developing their expertise in statistics and wish to become familiar with concrete examples of predictive modeling. The book also addresses the needs of more seasoned practising analysts who would like an overview of advanced statistical topics that are particularly relevant in actuarial practice. Predictive Modeling Applications in Actuarial Science emphasizes lifelong learning by developing tools in an insurance context, providing the relevant actuarial applications, and introducing advanced statistical techniques that can be used by analysts to gain a competitive advantage in situations with complex data.

Chapter Preview. Predictive modeling involves the use of data to forecast future events. It relies on capturing relationships between explanatory variables and the predicted variables from past occurrences and exploiting them to predict future outcomes. The goal of this two-volume set is to build on the training of actuaries by developing the fundamentals of predictive modeling and providing corresponding applications in actuarial science, risk management, and insurance. This introduction sets the stage for these volumes by describing the conditions that led to the need for predictive modeling in the insurance industry. It then traces the evolution of predictive modeling that led to the current statistical methodologies that prevail in actuarial science today.

The omission of important variables is a well-known model specification issue in regression anal- ysis and mixed linear models. The author considers longitudinal data models that are special cases of the mixed linear models; in particular, they are linear models of repeated observations on a subject. Models of omitted variables have origins in both the econometrics and biostatistics literatures. The author describes regression coefficient estimators that are robust to and that provide the basis for detecting the influence of certain types of omitted variables. New robust estimators and omitted variable tests are introduced and illustrated with a case study that investigates the determinants of tax liability.