A priori ratemaking using bivariate Poisson regression models
∗
Llu´ıs Berm´udez i Morata
†
Departament de Matem`atica Econ`omica, Financera i Actuarial.
Risc en Finances i Assegurances-IREA. Universitat de Barcelona.
September 22, 2008
Abstract
In automobile insurance, it is useful to achieve a priori ratemaking by resorting to gene-
ralized linear models, and here the Poisson regression model constitutes the most widely
accepted basis. However, insurance companies distinguish between claims with or without
bodily injuries, or claims with full or partial liability of the insured driver. This paper exa-
mines an a priori ratemaking procedure when including two different types of claim. When
assuming independence between claim types, the premium can be obtained by summing the
premiums for each type of guarantee and is dependent on the rating factors chosen. If the
independence assumption is relaxed, then it is unclear as to how the tariff system might
be affected. In order to answer this question, bivariate Poisson regression models, suitable
for paired count data exhibiting correlation, are introduced. It is shown that the usual
independence assumption is unrealistic here. These models are applied to an automobile
insurance claims database containing 80,994 contracts belonging to a Spanish insurance
company. Finally, the consequences for pure and loaded premiums when the independence
assumption is relaxed by using a bivariate Poisson regression model are analysed.
JEL classification: C51; IM classification: IM11; IB classification: IB40.
Keywords: Bivariate Poisson regression models, Zero-inflated models, Automobile insurance,
Bootstrap methods, A priori ratemaking.
∗
Acknowledgements. The author wishes to acknowledge the support of the Spanish Ministry of Education and
FEDER grant SEJ 2007-63298. The author is grateful for the valuable suggestions from the participants in the
12th International Congress on Insurance: Mathematics and Economics in Dalian on July 16-18, 2008.
†
Corresponding Author. Departament de Matem`atica Econ`omica, Financera i Actuarial, Universitat
de Barcelona, Diagonal 690, 08034-Barcelona, Spain. Tel.: +34-93-4034853; fax: +34-93-4034892; e-mail:
lbermudez@ub.edu
1
1 Introduction
Designing a tariff structure for insurance is one of the main tasks for actuaries. Such pricing
is particularly complex in the branch of automobile insurance because of highly heterogeneous
portfolios. A thorough review of ratemaking systems for automobile insurance, including the
most recent developments, can be found in Denuit et al. (2007).
One way to handle this problem of heterogeneity in a portfolio -referred to as tariff segmenta-
tion or a priori ratemaking- involves segmenting the portfolio in homogenous classes so that all
insured parties belonging to a particular class pay the same premium. This procedure ensures
that the exact weight of each risk is fairly distributed within the portfolio. In the case of auto-
mobile insurance, in order to group the policies in homogenous classes, a series of classification
variables are used (i.e., age, sex and place of residence of driver or horsepower, class and use of
the vehicle). These variables are called a priori ratemaking variables, since their values can be
determined before the insured party begins to drive.
If all the factors influencing a risk could be identified, measured and introduced in the tariff
system, then the classes defined would be homogenous. However, this is not that case as there
are important risk factors that are not considered in the a priori tariff. Some examples are
especially difficult to quantify, such as a driver’s reflexes, his or her aggressiveness, or knowledge
of the Highway Code, among others. As a result, tariff classes can be quite heterogeneous.
Hence, the idea has arisen of considering individual differences in policies within the same class
by using an a posteriori mechanism, i.e., fitting an individual premium based on the experience
of claims for each insured party. This concept has received the name of a posteriori tariff,
experience rating or the bonus-malus system.
Here, only the first step in pricing is studied, the a priori ratemaking. In short, the classi-
fication or segmentation of risks involves establishing different classes of risk according to their
nature and probability of occurrence. For this purpose, factors are determined in order to classify
each risk, and it is statistically tested that the probability of a claim depends on these factors,
and hence, their influence can be measured. A priori classification based on generalized linear
models is the most widely accepted method; see e.g. Dionne and Vanasse (1989), Haberman
2
and Renshaw (1996), Pinquet (1999), Berm´udez et al. (2001) and Boucher and Denuit (2006)
for applications in the actuarial sciences, and Mc Cullagh and Nelder (1989) or Dobson (1990)
for a general overview of the statistical theory.
The most commonly used generalized linear model for this tariff system is the Poisson re-
gression model and its generalizations (Denuit et al., 2007). Introduced by Dionne and Vanasse
(1989) in the context of automobile insurance, the model can be applied if the number of claims
for each individual policy observation is known. Although it is possible to use the total number
of claims as the response variable, the nature of automobile insurance policies (covering diffe-
rent risks) is such that the response variable is the number of claims for each type of guarantee.
Therefore, a premium is obtained for each class of guarantee as a function of different factors.
Then, assuming independence between types of claim, the total premium is obtained from the
sum of the expected number of claims of each guarantee.
Here, two different types of guarantee are assumed: third-party liability automobile insurance
and the rest of guarantees. Following the usual methodology, assuming independence between
types, the premium paid by the policyholder is obtained by summing the premiums for each
type of guarantee and this depends on the rating factors. However, the question remains as
to whether the independence assumption is realistic. When this assumption is relaxed, it is
interesting to see how the tariff system might be affected.
In this study, a bivariate Poisson regression model is introduced. Holgate (1964) provided a
practical basis for the bivariate Poisson distribution but its use has been largely ignored, mainly
because of computational difficulties. Therefore, only a few applications can be found, for
example, Jung and Winkelmann (1993) used a bivariate Poisson regression in a labour mobility
study and Karlis and Ntzoufras (2003) modelled sports data. For a comprehensive review of the
bivariate Poisson distribution and its applications (especially multivariate regression), the reader
should see Kocherlakota and Kocherlakota (1992, 2001) and Johnson, Kotz and Balakrishnan
(1997).
One early application of the bivariate Poisson distribution in the actuarial literature is des-
cribed in Cummins and Wiltbank (1983). In ruin theory, some applications of this distribution
are also to be found, for example Partrat (1994), Ambagaspitiya (1999), Walhin and Paris (2000)
and Centeno (2005). Cameron and Trivedi (1998) studied the relationship between type of health
insurance and various responses that measure the demand for health care by using a bivariate
3
Poisson regression. In addition, two studies related to fitting purposes should also be quoted,
albeit that no factors are considered. First, Vernic (1997) carried out a comparative study
with the bivariate Poisson distribution based on data related to natural events insurance and
third-party liability automobile insurance. Second, Walhin (2003) compared bivariate Hofmann
and bivariate Poisson distributions by fitting a data set for accidents sustained by members of
a sample of 122 shunters in two consecutive 2-year periods. However, in a ratemaking context,
bivariate Poisson regression models have not been used to model claim counts that depend on
the usual rating factors.
In the next section, the model used here is defined. This model is based on the bivariate
Poisson regression model, which is appropriate for modelling paired count data that exhibit
correlation. In Section 3 the database obtained from a Spanish insurance company is described.
In Section 4 the results are summarised. Finally, some concluding remarks are given in Section
5.
2 Bivariate Poisson regression models
Let N
1
and N
2
be the number of claims for third-party liability and for the rest of guarantees
respectively and N = N
1
+N
2
. The usual methodology to obtain the a priori premium under the
assumption of independence between types of claims can be described as follows. First, the model
assumed is N
1
∼ P oisson(λ
1
) and N
2
∼ P oisson(λ
2
) independently, and λ
1
and λ
2
depend on
a number of rating factors associated with the characteristics of the car, the driver and the use of
the car. Second, with λ
1
and λ
2
estimated for each policyholder and following the net premium
principle, the total net premium
1
( π ) is obtained as π = E[N] = E[N
1
] + E[N
2
] = λ
1
+ λ
2
.
However, an amount inflates the net premium to ensure that the insurer will not, on average,
lose money. Many well-known premium principles can be applied for this purpose. Here the
variance premium principle is used. This principle builds on the net premium by including a
risk loading that is proportional to the variance of the risk. Under the above assumptions,
the variance is equal to the expected value, and the total loaded premium ( π
∗
) is equal to
π
∗
= E[N ] + αV [N] = (1 + α)(E[N
1
] + E[N
2
]) .
In bivariate Poisson regression models, the independence assumption is relaxed. The model
1
Assuming the amount of the expected claim equals one monetary unit.
4
can be defined as follows. Let us consider independent random variables X
i
(i = 1, 2, 3) to
be distributed as Poisson with parameters λ
i
respectively. Then the random variables N
1
=
X
1
+ X
3
and N
2
= X
2
+ X
3
follow jointly a bivariate Poisson distribution:
(N
1
, N
2
) ∼ BP (λ
1
, λ
2
, λ
3
).
This is the so-called trivariate reduction method that leads to the bivariate Poisson distribution.
Its joint probability function is given by:
P (N
1
= n
1
, N
2
= n
2
) = e
−(λ
1
+λ
2
+λ
3
)
λ
n
1
1
n
1
!
λ
n
2
2
n
2
!
min(n
1
,n
2
)
X
i=0
n
1
i
n
2
i
i!
λ
3
λ
1
λ
2
i
. (1)
The bivariate Poisson distribution defined above presents several interesting and useful pro-
perties. First, it allows for positive dependence between the random variables N
1
and N
2
which
is what we expect for these types of claims
2
. Moreover Cov(N
1
, N
2
) = λ
3
and therefore λ
3
is
a measure of this dependence. Obviously, if λ
3
= 0 the two random variables are independent
and the bivariate Poisson distribution reduces to the product of two independent Poisson dis-
tributions, referred to as a double Poisson distribution (Kocherlakota and Kocherlakota, 1992).
Second, the marginal distributions for N
1
and N
2
are Poisson with E[N
1
] = λ
1
+ λ
3
and
E[N
2
] = λ
2
+ λ
3
.
Hence, the total net premium can be obtained with π = E[N] = E[N
1
] + E[N
2
] = λ
1
+ λ
2
+
2λ
3
. The variance necessary to obtain the loaded premium is now V [N ] = λ
1
+ λ
2
+ 4λ
3
. Since
λ
3
is expected to be positive, the relaxation of the independence assumption leads to a variance
greater than the expected value. Overdispersion has often been observed when modelling claim
counts in automobile insurance data (Denuit et al., 2007).
Let us assume that N
1j
and N
2j
denote the random variables indicating the number of
claims of each type of guarantee for the jth policyholder. If covariates are introduced to model
λ
1
, λ
2
and λ
3
, a bivariate Poisson regression model can be defined with the following scheme:
(N
1j
, N
2j
) ∼ BP (λ
1j
, λ
2j
, λ
3j
),
log(λ
1j
) = x
0
1j
β
1
,
log(λ
2j
) = x
0
2j
β
2
,
log(λ
3j
) = x
0
3j
β
3
, (2)
2
In case of negatively correlated claims (not considered here) it would be necessary a more general specification.
5
