Estimation, Evaluation, and Selection of Actuarial Models

Chapter 1
Introduction
For some time, the subjects of “Loss Models” and “Survival Models” have been part of the actuarial
syllabus. The former was most often associated with property and liability insurances while the
latter was most often associated with life and disability insurances. However, the goal of these two
subjects is the same — the determination of models for random phenomena. The purpose of this
Study Note is to present the two subjects in a unified context that indicates that either approach
may be appropriate for a given actuarial problem.
The difference between Parts 3 and 4 1 is reflected in the course names. Part 3 is “actuarial
models” and the objective is to be able to work with the most common actuarial models. Most such
models are random variables, such as the time to death or the amount of an automobile physical
damage claim. After mastering the content of Part 3, the student is left wondering where the
models come from. That is the purpose of Part 4, “actuarial modeling.”
One option is to simply announce the model. Your boss needs a model for basic dental payments.
You announce that it is the lognormal distribution with µ =5.1239 and σ =1.0345 (the many
decimal places are designed to give your announcement an aura of precision). When your boss, or
a regulator, or an attorney who has put you on the witness stand, asks you how you know that to
beso,itwilllikelynotbesufficient to answer that “I just know these things.” It may not even be
sufficient to announce that your actuarial friend at Gamma Dental uses that model.
An alternative is to collect some data and use it to formulate a model. Most distributional
models have two components. The first is a name, such as “Pareto.” The second is the set of values
of parameters that complete the specification. Matters would be simpler if modeling could be done
in that order. Most of the time we need to fix the parameters that go with a named model before we
can decide if we want to use that type of model. Thus, this note begins with parameter estimation,
the subject of Chapter 2. By the end of that Chapter, we will have a variety of estimation methods
at our disposal.
In Chapter 3 we learn how to evaluate the various methods and in particular learn how to
measure the accuracy of our estimator. Chapter 4 is devoted to model selection. Two basic
questions are answered. The first one is “Is our model acceptable for use?” while the second one
is “From our set of acceptable models, which one should we use?” Chapter 5 introduces some new
models that were not covered in Part 3.
1 The Casualty Actuarial Society refers to its education units as “exams” while the Society of Actuaries refers to
them as “courses.” To avoid making a decision, this Note will refer to units as “parts,” a term in favor when the
author was going through this process.
1