Chapter Preview. Many insurance datasets feature information about frequency, how often claims arise, in addition to severity, the claim size. This chapter introduces tools for handling the joint distribution of frequency and severity. Frequency-severity modeling is important in insurance applications because of features of contracts, policyholder behavior, databases that insurers maintain, and regulatory requirements. Model selection depends on the data form. For some data, we observe the claim amount and think about a zero claim as meaning no claim during that period. For other data, we observe individual claim amounts. Model selection also depends on the purpose of the inference; this chapter highlights the Tweedie generalized linear model as a desirable option. To emphasize practical applications, this chapter features a case study of Massachusetts automobile claims, using out-of-sample validation for model comparisons.

How Frequency Augments Severity Information

At a fundamental level, insurance companies accept premiums in exchange for promises to indemnify a policyholder on the uncertain occurrence of an insured event. This indemnification is known as a claim. A positive amount, also known as the severity, of the claim, is a key financial expenditure for an insurer. One can also think about a zero claim as equivalent to the insured event not occurring. So, knowing only the claim amount summarizes the reimbursement to the policyholder. Ignoring expenses, an insurer that examines only amounts paid would be indifferent to two claims of 100 when compared to one claim of 200, even though the number of claims differs.

Chapter Preview. This chapter presents regression models where the random variable is a count and compares different risk classification models for the annual number of claims reported to the insurer. Count regression analysis allows identification of risk factors and prediction of the expected frequency given characteristics of the risk. This chapter details some of the most popular models for the annual number of claims reported to the insurer, the way the actuary should use these models for inference, and how the models should be compared.

Introduction

In the early 20th century, before the theoretical advances in statistical sciences, a method called the minimum bias technique was used to find the premiums that should be offered to insureds with different risk characteristics. This technique's aim was to find the parameters of the premiums that minimize their bias by using iterative algorithms.

Instead of relying on these techniques that lack theoretical support, the actuarial community now bases its methods on probability and statistical theories. Using specific probability distributions for the count and the costs of claims, the premium is typically calculated by obtaining the conditional expectation of the number of claims given the risk characteristics and combining it with the expected claim amount. In this chapter, we focus on the number of claims.

Chapter Preview. Linear modeling, also known as regression analysis, is a core tool in statistical practice for data analysis, prediction, and decision support. Applied data analysis requires judgment, domain knowledge, and the ability to analyze data. This chapter provides a summary of the linear model and discusses model assumptions, parameter estimation, variable selection, and model validation around a series of examples. These examples are grounded in data to help relate the theory to practice. All of these practical examples and exercises are completed using the open-source R statistical computing package. Particular attention is paid to the role of exploratory data analysis in the iterative process of criticizing, improving, and validating models in a detailed case study. Linear models provide a foundation for many of the more advanced statistical and machine-learning techniques that are explored in the later chapters of this volume.

Introduction

Linear models are used to analyze relationships among various pieces of information to arrive at insights or to make predictions. These models are referred to by many terms, including linear regression, regression, multiple regression, and ordinary least squares. In this chapter we adopt the term linear model.

Linear models provide a vehicle for quantifying relationships between an outcome (also referred to as dependent or target) variable and one or more explanatory (also referred to as independent or predictive) variables.

Chapter Preview. Generalized additive models (GAMs) provide a further generalization of both linear regression and generalized linear models (GLM) by allowing the relationship between the response variable y and the individual predictor variables xj to be an additive but not necessarily a monomial function of the predictor variables xj. Also, as with the GLM, a nonlinear link function can connect the additive concatenation of the nonlinear functions of the predictors to the mean of the response variable, giving flexibility in distribution form, as discussed in Chapter 5. The key factors in creating the GAM are the determination and construction of the functions of the predictor variables (called smoothers). Different methods of fit and functional forms for the smoothers are discussed. The GAM can be considered as more data driven (to determine the smoothers) than model driven (the additive monomial functional form assumption in linear regression and GLM).

Motivation for Generalized Additive Models and Nonparametric Regression

Often for many statistical models there are two useful pieces of information that we would like to learn about the relationship between a response variable y and a set of possible available predictor variables x1, x2, …, xk for y: (1) the statistical strength or explanatory power of the predictors for influencing the response y (i.e., predictor variable worth) and (2) a formulation that gives us the ability to predict the value of the response variable ynew that would arise under a given set of new observed predictor variables x1,new, x2,new, …, xk,new (the prediction problem).

Chapter Preview. In the actuarial context, fat-tailed phenomena are often observed where the probability of extreme events is higher than that implied by the normal distribution. The traditional regression, emphasizing the center of the distribution, might not be appropriate when dealing with data with fat-tailed properties. Overlooking the extreme values in the tail could lead to biased inference for rate-making and valuation. In response, this chapter discusses four fat-tailed regression techniques that fully use the information from the entire distribution: transformation, models based on the exponential family, models based on generalized distributions, and median regression.

Introduction

Insurance ratemaking is a classic actuarial problem in property-casualty insurance where actuaries determine the rates or premiums for insurance products. The primary goal in the ratemaking process is to precisely predict the expected claims cost which serves as the basis for pure premiums calculation. Regression techniques are useful in this process because future events are usually forecasted from past occurrence based on the statistical relationships between outcomes and explanatory variables. This is particularly true for personal lines of business where insurers usually possess large amount of information on policyholders that could be valuable predictors in the determination of mean cost.

The traditional mean regression, though focusing on the center of the distribution, relies on the normality of the response variable.

Chapter Preview. This chapter provides an introduction to transition modeling. Consider a situation where an individual or entity is, at any time, in one of several states and may from time to time move from one state to another. The state may, for example, indicate the health status of an individual, the status of an individual under the terms of an insurance policy, or even the “state” of the economy. The changes of state are called transitions. There is often uncertainty associated with how much time will be spent in each state and which state will be entered on each transition. This uncertainty can be modeled using a multistate stochastic model. Such a model may be described in terms of the rates of transition from one state to another. Transition modeling involves the estimation of these rates from data.

Actuaries often work with contracts involving several states and financial implications associated with presence in a state or transition between states. A life insurance policy is a simple example. A multistate stochastic model provides a valuable tool to help the actuary analyze the cash flow structure of a given contract. Transition modeling is essential to the creation of this tool.

This chapter is intended for practitioners, actuaries, or analysts who are faced with a multistate setup and need to estimate the rates of transition from available data. The assumed knowledge – only basic probability and statistics as well as life contingencies – is minimal.

Introduction

Life and pension insurance are arrangements for which payment streams are determined by states occupied by individuals, for example active, retired, disabled and so on. These contracts typically last for a long time, up to half a century and more. A simple example is an arrangement where an account is first built up and then harvested after a certain date. At first glance this is only a savings account, but insurance can be put into it by including randomness due to how long people live. When such accounts are managed for many individuals simultaneously, it becomes possible to balance lifecycles against one another so that short lives (for which savings are not used up fully) partially finance long ones. There is much sense in this. In old age benefits do not stop after an agreed date, but may go on until the recipient dies.

Many versions and variants of such contracts exist. Benefits may be one-time settlements upon retirement or death of a policy holder or distribute over time as a succession of payments. Traditional schemes have often been defined benefit, where economic rights after retirement determine the contributions that sustain them. Defined contributions are the opposite. Now the pension follows from the earlier build-up of the account. Whatever the arrangement, the values created depend on investment policy and interest rate and also on inflation (which determines the real worth of the pension when it is put to use).