Pricing in General Insurance

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27

Multilevel Factors and Smoothing

In our treatment of generalised linear models (GLMs) in Chapter 26, we have come across

two different types of rating factors:

a. Continuous variables such as the annual mileage or the driver’s age, which may or

may not be related monotonically to the modelled variable (e.g. the claim frequency)

b. Categorical variables (i.e. variables with a limited number of possible values) that

have in general no inherent order: examples are sex, occupation, car model and

geographical location

Categorical variables are straightforward to model when the number of levels is small,

such as is the case with sex. However, they pose a particular challenge if the number of

levels is very large and there is no obvious way of ordering them, such as is the case for car

models* or postcodes. In this case, the calibration of a GLM becomes problematic because,

for some of the values (in our example, some of the car models), loss data may be scanty.

Also, the lack of a natural ordering means that there is no obvious way of merging catego-

ries together (as we can do with ages) to obtain categories with more data.

Following Ohlsson and Johansson (2010), we call these factors multilevel factors (MLF),

and we describe them as nominal variables (i.e. with no particular ordering) that have too

many levels to be analysed by standard GLM techniques.

Traditional approaches to MLFs are as follows:

1. Reduce the number of categories by merging together categories that are thought

to be similar, in a heuristic fashion. For example, you could reduce the number of

car models drastically by considering only the make of the car (e.g. BMW, Vauxhall,

Fiat) rather than the full car model code, or use the Association of British Insurers’

(ABI) grouping, which classies all vehicles into one of 50 groups with similar

characteristics (Institute and Faculty of Actuaries [IFoA] 2010). Analogously, post-

codes at, say, town or borough level could be merged together. This is of course

not ideal because a lot of useful information is discarded, and is not the modern

approach to MLFs. However, it has traditionally been used when not enough com-

puter power and adequate statistical techniques were available.

2. Find the underlying factors that explain the difference between the different cat-

egories captured by a MLF, and use these as factors as inputs to the GLM instead

of the original MLF. For example, in the case of car models, one can replace the car

model code with factors such as the vehicle’s cubic capacity, year of manufactur-

ing, fuel type, body type and so on, and calibrate the GLM based on these under-

lying factors. The factors can be found by factor analysis (a statistical technique) or

by using the factors already available in vehicle classication systems such as that

*

Ohlsson and Johansson (2010) estimate 2500 car model codes in Sweden, and this estimate can probably be

transferred to other countries as well.