Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

50 Actuarial Modelling of Claim Counts
most cases). Actuaries use regression techniques to predict the expected number of claims
knowing some information about the policyholders, vehicles and types of contract. It is worth
mentioning that even with all the covariates included here, there still remain substantial risk
differentials between individual drivers (due to hidden characteristics, like temper and skill,
aggressiveness behind the wheel, knowledge of the highway code, etc.). Random effects
are added to the linear predictor on the score scale to take this residual heterogeneity into
account, reconciling the two approaches (i)–(ii) mentioned above.
In nonlife business, the pure premium is the expected cost of all the claims that
policyholders will file during the coverage period (under the assumption of the Law of Large
Numbers). The computation of this premium relies on a statistical model incorporating all
the available information about the risk. The technical tariff aims to evaluate as accurately as
possible the pure premium for each policyholder via regression techniques. It is well-known
that market premiums may differ from those computed by actuaries; see, e.g., Coutts (1984)
for a discussion. In that respect, the overall market position of the company compared to its
competitors with regard to growth and pricing is crucial.
Sometimes, motor ratemaking is performed on panel data. Bringing several observation
periods together has some advantages: it increases the sample size and avoids granting too
much importance to a single calendar year (during which the particular weather conditions
could have increased or decreased the number of traffic accidents, for instance). However,
this induces some dependence in the data, since observations relating to the same policyholder
across time are expected to be correlated. The analysis of correlated data with Poisson
marginals arising from repeated measurements can be performed with the help of Generalized
Estimating Equations (GEEs). GEEs provide a practical method with reasonable statistical
efficiency to analyse such panel data. GEEs also give initial values for maximum likelihood
procedures in models for longitudinal data.
2.1.2 Risk Sharing in Segmented Tariffs
The following discussion is inspired by the paper by De Wit & Van Eeghen (1984).
Consider a portfolio of n policies from motor third party liability insurance. The random
variable Y i models a quantity of actuarial interest for policy i (for instance the amount of a
claim, the aggregate claim amount in one period or the number of accidents at fault reported
by policyholder i during one period). In order to explain the outcomes of Y i , the actuary has
observable covariates Xi
T = X i1 X i2 at his disposal (e.g., age, gender and occupation
of policyholder i, the place where he resides, type and use of his car). However, Y i also
depends on a sequence of unknown characteristics Zi
T = Z i1 Z i2 (e.g., annual mileage,
accuracy of judgment, aggressiveness behind the wheel, drinking behaviour, etc.). Some of
these quantities are unobservable, others cannot be measured in a cost efficient way.
The ‘true’ premium for policyholder i is EY i X i Z i . It is the function g of X i and Z i
that is the ‘closest’ to Y i , in the sense that EY i − gX i Z i 2 is minimum for gX i Z i =
EY i X i Z i . If the insurer charges EY i X i Z i to policyholder i, then the policyholders pay
premiums that absorb the inter-individual variations (that is, the variations of the premiums
due to the modifications in personal characteristics X i and Z i , the magnitude of which are
quantified by V [ EY i X i Z i ] ). The company covers the purely random intra-individual risks
(that is, the random fluctuations of Y i , which are quantified by the variance E [ VY i X i Z i ]
of the outcomes of Y i once the personal characteristics X i and Z i have been fixed). Risk