Attributes of Territorial Ratemaking in Insurance: Practice Questions and Solutions

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of Basic Ratemaking, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5. Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 11, pp. 185-189.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S5-84-1.

(a) According to Werner and Modlin, p. 185, what are the two steps of the territorial ratemaking process?

(b) Describe two main challenges of the territorial ratemaking process.

Solution S5-84-1.

(a) According to Werner and Modlin, p. 185, the two steps of the territorial ratemaking process are as follows:
1. "Establishing territorial boundaries";
2. "Determining rate relativities for the territories".

(b) According to Werner and Modlin, p. 185, two main challenges of the territorial ratemaking process are as follows:
1. Location is often strongly correlated with other rating variables, such as amount of insurance (based on the value of a house), and this significantly distorts any univariate analysis of territorial experience.
2. If small geographic units are analyzed as part of a company's territorial analysis, the problem of high-dimensionality arises; this is essentially the scarcity of data in each individual territorial unit.

Problem S5-84-2.
(a)Name three kinds of geographical units that are used by many insurers in classifying territories for rating purposes.

(b) For each of your answers in part (a), name an advantage and a disadvantage involved in using the geographical unit.

(c) Actual experience by territory reflects both signal and noise components. The signal components can be geographic or non-geographic. List two possible non-geographic signal components and two possible geographic signal components.

Solution S5-84-2. This problem is based on the discussion in Werner and Modlin, p. 186, of defining the geographic unit.

(a) Three kinds of geographical units that are used by many insurers in classifying territories for rating purposes are (1) zip codes/postal codes, (2) counties, and (3) census blocks. Other valid answers may also be possible.

(b)
1. Zip codes/postal codes are the most readily available of the territorial units, but they also change over time.
2. Counties typically do not change over time and are also readily available, but they are quite large and include many heterogeneous risks.
3. Census blocks typically do not change over time, "but require a process to map insurance policies to the census blocks" (Werner and Modlin, p. 186).

(c) The following signal components are mentioned by Werner and Modlin, p. 186:

Geographic signal components:
1. Density of population;
2. Weather indices;
3. Crime rates.

Non-geographic signal components:
1. Age of house;
2. Amount of insurance;
3. Number of employees.

The non-geographic components may vary by territory but are not dependent on the territory as such; they are still reflected in the signal given by actual experience.

Any two items from each category would suffice as an answer. Other valid answers may also be possible.

Problem S5-84-3.

(a) What are two major disadvantages of using univariate techniques in developing a geographic estimator for each unit used in territorial ratemaking?

(b) How does the development of a multivariate model such as a generalized linear model (GLM) help overcome these disadvantages?

Solution S5-84-3. This problem is based on the discussion in Werner and Modlin, p. 186

(a) The following are the two major disadvantages of univariate techniques in developing a geographic estimator:

1. The univariate techniques reflect both the signal and the noise components of actual experience. This is particularly important when the geographic units used are small, such that there is substantial volatility in the limited experience data for each unit.

2. The univariate estimator is often biased because of the correlation between location and non-geographic factors.

(b) Multivariate models can help overcome the disadvantages from part (a) as follows:

1. A multivariate model's design can enable it to isolate the noise from the signal if it is not over-fitted or under-fitted to the observed data.

2. Because non-geographic predictors are incorporated into a multivariate model, the model can capture and identify interactions between geographic and non-geographic variables that are defined within the model. If the geographic variables within the model do not fully capture the signal, then there will also be some geographic residual variation in the observed experience.

Problem S5-84-4.

(a) Describe distance-based spatial smoothing.

(b) Name one advantage and one disadvantage of distance-based spatial smoothing.

(c) Describe adjacency-based spatial smoothing.

(c) Name one advantage and one disadvantage of adjacency-based spatial smoothing.

Solution S5-84-4.

(a) Distance-based spatial smoothing weights "the information from one geographic unit with the information from all nearby geographic units based on the distance from the primary unit and some measure of credibility. The influence of nearby areas is deemed to diminish with increasing distance" (Werner and Modlin, p. 187).

(b) Distance-based spatial smoothing "has the advantage of being easy to understand and implement" (Werner and Modlin, p. 187). Its disadvantages include the implicit assumption that the same amount of distance corresponds to the same effect of risk, irrespective of differences between geographical areas - e.g., urban versus rural areas. Another disadvantage is that natural geographic boundaries are not taken into account by this method.

(c) Adjacency-based spatial smoothing "weights the information from one geographic unit with the information estimators of rings of adjacent units (i.e., immediately adjacent units get more weight than the units adjacent to adjacent units, etc)" (Werner and Modlin, p. 187).

(d) Adjacency-based spatial smoothing can more effectively account for urban versus rural differences and differences based on natural or artificial geographic boundaries than can distance-based spatial smoothing. It is still vulnerable to over-smoothing by the actuary; by using this technique, or any smoothing technique, the actuary runs the risk of concealing actual geographically based variation that meaningfully affects loss experience (Werner and Modlin, p. 187).

Problem S5-84-5.

(a) Complete the following statement by filling in the blanks: "The purpose of clustering in territorial ratemaking is to maximize heterogeneity ______ groups and to minimize heterogeneity ______ groups."

(b) Briefly discuss how quartile methods of clustering work.

(c) Werner and Modlin, p. 188, discuss the following three similarity methods of clustering:

1. Average linkage method;
2. Centroid method;
3. Ward's clustering method.

Match each feature below to the method that exhibits that feature:
(i) Clusters that have the same number of observations are produced.
(ii) Clusters with smaller variances are joined.
(iii) Outliers are identified with relative ease.

Solution S5-84-5. This problem is based on the discussion of clustering by Werner and Modlin, p. 188.

(a) "The purpose of clustering in territorial ratemaking is to maximize heterogeneity between groups and to minimize heterogeneity within groups."

(b) Quartile methods of clustering "create clusters based on either equal numbers of observations (such as geographic units) or equal weights (such as exposure)" (Werner and Modlin, p. 188).

(c)
Feature (i) is characteristic of 3. Ward's clustering method.
Feature (ii) is characteristic of 1. Average linkage method.
Feature (iii) is characteristic of 2. Centroid method.

See other sections of The Actuary's Free Study Guide for Exam 5.