We consider models for analyzing the surplus of an insurance portfolio. Suppose an insurance business begins with a start-up capital, called the initial surplus. The insurance company receives premium payments and pays claim losses. The premium payments are assumed to be coming in at a constant rate. When there are claims, losses are paid out to policy holders. Unlike the constant premium payments, losses are random and uncertain, in both timing and amount. The net surplus through time is the excess of the initial capital and aggregate premiums received over the losses paid out. The insurance business is in ruin if the surplus falls to or below zero. The main purpose of this chapter is to consider the probability of ruin as a function of time, the initial surplus and the claim distribution. Ultimate ruin refers to the situation where ruin occurs at finite time, irrespective of the time of occurrence.

We first consider the situation in which premium payments and claim losses occur at discrete time. We derive recursive formulas for the probability of ultimate ruin given the initial surplus. These recursive formulas require the value of the probability of ultimate ruin when the start-up capital is zero. Formulas for the probability of ruin before fixed finite times are also derived. To obtain bounds for the probability of ultimate ruin, we introduce Lundberg's inequality. In the continuous-time set-up, we assume the claims follow a Poisson process.

Credibility models were first proposed in the beginning of the twentieth century to update predictions of insurance losses in light of recently available data of insurance claims. The oldest approach is the limited-fluctuation credibility method, also called the classical approach, which proposes to update the loss prediction as a weighted average of the prediction based purely on the recent data and the rate in the insurance manual. Full credibility is achieved if the amount of recent data is sufficient, in which case the updated prediction will be based on the recent data only. If, however, the amount of recent data is insufficient, only partial credibility is attributed to the data and the updated prediction depends on the manual rate as well.

We consider the calculation of the minimum size of the data above which full credibility is attributed to the data. For cases where the data are insufficient we derive the partial-credibility factor and the updating formula for the prediction of the loss. The classical credibility approach is applied to update the prediction of loss measures such as the frequency of claims, the severity of claims, the aggregate loss, and the pure premium of a block of insurance policies.

Learning objectives

1. Basic framework of credibility

2. The limited-fluctuation (classical) credibility approach

3. Full credibility

4. Partial credibility

5. Prediction of claim frequency, claim severity, aggregate loss, and pure premium

Some models assume that the failure-time or loss variables follow a certain family of distributions, specified up to a number of unknown parameters. To compute quantities such as the average loss or VaR, the parameters of the distributions have to be estimated. This chapter discusses various methods of estimating the parameters of a failure-time or loss distribution.

Matching moments and percentiles to the data are two simple methods of parametric estimation. These methods, however, are subject to the decisions of the set of moments or percentiles to be used. The most important estimation method in the classical statistics literature is perhaps the maximum likelihood estimation method. It is applicable to a wide class of problems: variables that are discrete or continuous, and data observations that are complete or incomplete. On the other hand, the Bayesian approach provides an alternative perspective to parametric estimation, and has been made easier to adopt due to the advances in computational techniques.

Parametric models can be extended to allow the distributions to vary with some attributes of the objects, such as the years of driving experience of the insured in predicting vehicle accident claims. This gives rise to the use of models with covariates. A very important model in the parametric estimation of models with covariates is Cox's proportional hazards model.

We also include in this chapter a brief introduction to the use of copula in modeling the joint distributions of several variables.

Given the assumption that a loss random variable has a certain parametric distribution, the empirical analysis of the properties of the loss requires the parameters to be estimated. In this chapter we review the theory of parametric estimation, including the properties of an estimator and the concepts of point estimation, interval estimation, unbiasedness, consistency, and efficiency. Apart from the parametric approach, we may also estimate the distribution functions and the probability (density) functions of the loss random variables directly, without assuming a certain parametric form. This approach is called nonparametric estimation. The purpose of this chapter is to provide a brief review of the theory of estimation, with the discussion of specific estimation methods postponed to the next two chapters.

Although the focus of this book is on nonlife actuarial risks, the estimation methods discussed are also applicable to life-contingency models. Specifically, the estimation methods may be used for failure-time data (life risks) as well as loss data (nonlife risks). In many practical applications, only incomplete data observations are available. These observations may be left truncated or right censored. We define the notations to be used in subsequent chapters with respect to left truncation, right censoring, and the risk set. Furthermore, in certain setups individual observations may not be available and we may have to work with grouped data. Different estimation methods are required, depending on whether the data are complete or incomplete, and whether they are individual or grouped.

Model construction and evaluation are two important aspects of the empirical implementation of loss models. To construct a parametric model of loss distributions, the parameters of the distribution have to be estimated based on observed data. Alternatively, we may consider the estimation of the distribution function or density function without specifying their functional forms, in which case nonparametric methods are used. We discuss the estimation techniques for both failure-time data and loss data. Competing models are selected and evaluated based on model selection criteria, including goodness-of-fit tests.

Computer simulation using random numbers is an important tool in analyzing complex problems for which analytical answers are difficult to obtain. We discuss methods of generating random numbers suitable for various continuous and discrete distributions. We also consider the use of simulation for the estimation of the mean squared error of an estimator and the p-value of a hypothesis test, as well as the generation of asset-price paths.

This part of the book is about two important and related topics in modeling insurance business: measuring risk and computing the likelihood of ruin. In Chapter 4 we introduce various measures of risk, which are constructed with the purpose of setting premium or capital. We discuss the axiomatic approach of identifying risk measures that are coherent. Specific measures such as Value-at-Risk, conditional tail expectation, and the distortion-function approach are discussed. Chapter 5 analyzes the probability of ruin of an insurance business in both discrete-time and continuous-time frameworks. Probabilities of ultimate ruin and ruin before a finite time are discussed. We show the interaction of the initial surplus, premium loading, and loss distribution on the probability of ruin.

Having discussed models for claim frequency and claim severity separately, we now turn our attention to modeling the aggregate loss of a block of insurance policies. Much of the time we shall use the terms aggregate loss and aggregate claim interchangeably, although we recognize the difference between them as discussed in the last chapter. There are two major approaches in modeling aggregate loss: the individual risk model and the collective risk model. We shall begin with the individual risk model, in which we assume there are n independent loss prospects in the block. As a policy may or may not have a loss, the distribution of the loss variable in this model is of the mixed type. It consists of a probability mass at point zero and a continuous component of positive losses. Generally, exact distribution of the aggregate loss can only be obtained through the convolution method. The De Pril recursion, however, is a powerful technique to compute the exact distribution recursively when the block of policies follow a certain set-up.

On the other hand, the collective risk model treats the aggregate loss as having a compound distribution, with the primary distribution being the claim frequency and the secondary distribution being the claim severity. The Panjer recursion can be used to compute the distribution of the aggregate loss if the claim-frequency distribution belongs to the (a, b, 0) class and the claimseverity distribution is discretized or approximated by a discrete distribution.

In this part of the book we discuss actuarial models for claim losses. The two components of claim losses, namely claim frequency and claim severity, are modeled separately, and are then combined to derive the aggregate-loss distribution. In Chapter 1, we discuss the modeling of claim frequency, introducing some techniques for modeling nonnegative integer-valued random variables. Techniques for modeling continuous random variables relevant for claim severity are discussed in Chapter 2, in which we also consider the effects of coverage modifications on claim frequency and claim severity. Chapter 3 discusses the collective risk model and individual risk model for analyzing aggregate losses. The techniques of convolution and recursive methods are used to compute the aggregate-loss distributions.

While the classical credibility theory addresses the important problem of combining claim experience and prior information to update the prediction for loss, it does not provide a very satisfactory solution. The method is based on arbitrary selection of the coverage probability and the accuracy parameter. Furthermore, for tractability some restrictive assumptions about the loss distribution have to be imposed.

Bühlmann credibility theory sets the problem in a rigorous statistical framework of optimal prediction, using the least mean squared error criterion. It is flexible enough to incorporate various distributional assumptions of loss variables. The approach is further extended to enable the claim experience of different blocks of policies with different exposures to be combined for improved forecast through the Bühlmann–Straub model.

The Bühlmann and Bühlmann–Straub models recognize the interaction of two sources of variability in the data, namely the variation due to between-group differences and variation due to within-group fluctuations. We begin this chapter with the set-up of the Bühlmann credibility model, and a review of how the variance of the loss variable is decomposed into between-group and within-group variations. We derive the Bühlmann credibility factor and updating formula as the minimum mean squared error predictor. The approach is then extended to the Bühlmann–Straub model, in which the loss random variables have different exposures.

In this chapter we consider the Bayesian approach in updating the prediction for future losses. We consider the derivation of the posterior distribution of the risk parameters based on the prior distribution of the risk parameters and the likelihood function of the data. The Bayesian estimate of the risk parameter under the squared-error loss function is the mean of the posterior distribution. Likewise, the Bayesian estimate of the mean of the random loss is the posterior mean of the loss conditional on the data.

In general, the Bayesian estimates are difficult to compute, as the posterior distribution may be quite complicated and intractable. There are, however, situations where the computation may be straightforward, as in the case of conjugate distributions. We define conjugate distributions and provide some examples for cases that are of relevance in analyzing loss measures. Under specific classes of conjugate distributions, the Bayesian predictor is the same as the Bühlmann predictor. Specifically, when the likelihood belongs to the linear exponential family and the prior distribution is the natural conjugate, the Bühlmann credibility estimate is equal to the Bayesian estimate. This result provides additional justification for the use of the Bühlmann approach.

Learning objectives

1. Bayesian inference and estimation

2. Prior and posterior pdf

3. Bayesian credibility

4. Conjugate prior distribution

5. Linear exponential distribution

6. Bühlmann credibility versus Bayesian credibility

Bayesian inference and estimation

The classical and Bühlmann credibility models update the prediction for future losses based on recent claim experience and existing prior information.

In this chapter we discuss some applications of Monte Carlo methods to the analysis of actuarial and financial data. We first re-visit the tests of model misspecification introduced in Chapter 13. For an asymptotic test, Monte Carlo simulation can be used to improve the performance of the test when the sample size is small, in terms of getting more accurate critical values or p-values. When the asymptotic distribution of the test is unknown, as for the case of the Kolmogorov–Smirnov test when the hypothesized distribution has some unknown parameters, Monte Carlo simulation may be the only way to estimate the critical values or p-values.

The Monte Carlo estimation of critical values is generally not viable when the null hypothesis has some nuisance parameters, i.e. parameters that are not specified and not tested under the null. For such problems, the use of bootstrap may be applied to estimate the p-values. Indeed, bootstrap is one of the most powerful and exciting techniques in statistical inference and analysis. We shall discuss the use of bootstrap in model testing, as well as the estimation of the bias and mean squared error of an estimator.

The last part of this chapter is devoted to the discussion of the simulation of asset-price processes. In particular, we consider both pure diffusion processes that generate lognormally distributed asset prices, as well as jump–diffusion processes that allow for discrete jumps as a random event with a random magnitude.

The main focus of this chapter is the estimation of the distribution function and probability (density) function of duration and loss variables. The methods used depend on whether the data are for individual or grouped observations, and whether the observations are complete or incomplete.

For complete individual observations, the relative frequency distribution of the sample observations defines a discrete distribution called the empirical distribution. Moments and df of the true distribution can be estimated using the empirical distribution. Smoothing refinements can be applied to the empirical df to improve its performance. We also discuss kernel-based estimation methods for the estimation of the df and pdf.

When the sample observations are incomplete, with left truncation and/or right censoring, the Kaplan–Meier (product-limit) estimator and the Nelson–Aalen estimator can be used to estimate the survival function. These estimators compute the conditional survival probabilities using observations arranged in increasing order. They make use of the data set-up discussed in the last chapter, in particular the risk set at each observed data point. We also discuss the estimation of their variance, the Greenwood formula, and interval estimation.

For grouped data, smoothing techniques are used to estimate the moments, the quantiles, and the df. The Kaplan–Meier and Nelson–Aalen estimators can also be applied to grouped incomplete data.

Learning objectives

1. Empirical distribution

2. Moments and df of the empirical distribution

3. Kernel estimates of df and pdf

4. Kaplan–Meier (product-limit) estimator and Nelson–Aalen estimator

5. Greenwood formula

6. Estimation based on grouped observations

This book is on the theory, methods, and empirical implementation of nonlife actuarial models. It is intended for use as a textbook for senior undergraduates. Users are assumed to have done one or two one-semester courses on probability theory and statistical inference, including estimation and hypothesis testing. The coverage of this book includes all the topics found in Exam C of the Society of Actuaries (Exam 4 of the Casualty Actuarial Society) as per the 2007 Basic Education Catalog. In addition, it covers some topics (such as risk measures and ruin theory) beyond what is required by these exams, and may be used by actuarial students in general.

This book is divided into four parts: loss models, risk and ruin, credibility, and model construction and evaluation. An appendix on the review of statistics is provided for the benefit of students who require a quick summary. Students may read the appendix prior to the main text if they desire, or they may use the appendix as a reference when required. In order to be self contained, the appendix covers some of the topics developed in the main text.

Some features of this book should be mentioned. First, the concepts and theories introduced are illustrated by many practical examples. Some of these examples explain the theory through numerical applications, while others develop new results. Second, several chapters of the book include a section on numerical computation using Excel. Students are encouraged to use Excel to solve some of the numerical exercises.

Every statistician and data analyst has to make choices. The need arises especially when data have been collected and it is time to think about which model to use to describe and summarise the data. Another choice, often, is whether all measured variables are important enough to be included, for example, to make predictions. Can we make life simpler by only including a few of them, without making the prediction significantly worse?

In this book we present several methods to help make these choices easier. Model selection is a broad area and it reaches far beyond deciding on which variables to include in a regression model.

Two generations ago, setting up and analysing a single model was already hard work, and one rarely went to the trouble of analysing the same data via several alternative models. Thus ‘model selection’ was not much of an issue, apart from perhaps checking the model via goodness-of-fit tests. In the 1970s and later, proper model selection criteria were developed and actively used. With unprecedented versatility and convenience, long lists of candidate models, whether thought through in advance or not, can be fitted to a data set. But this creates problems too. With a multitude of models fitted, it is clear that methods are needed that somehow summarise model fits.