1. of an insurance premium. Insured’s pose varying level

1.      INTRODUCTION

 

       ESTIMATING AUTO INSURANCE PREMIUM USING
GENERALISED LINEAR MODELS

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1.1   
BACKGROUND INFORMATION AND PROBLEM
STATEMEENT

      The basic role of
insurance is to pool fortuitous losses together, provide financial protection,
offering a means of transferring the risk of losses in exchange of an insurance
premium. Insured’s pose varying level of risks to an insurer. It is rational that
the rates charged correspond to individual risk levels.

       The non-uniform rates charged are emphasized
by the non-homogenous nature of insurance portfolio which brings about the phenomenon of
anti-selection. This basically implies charging same rates for the entire
portfolio, meaning that the unfavorable risks are also insured at a lower rate
and as a negative effect, it discourages insuring medium risks. According to
Ohlsson and Johansen (2010), the idea is that if an insurance company charges
too high a premium for some policies, these will be lost to a competitor with a
fairer price. Suppose that a company charges too little for young drivers and
too much for old drivers; then they will tend to loose old drivers to
competitors while attracting young drivers; this adverse selection will result
in economic loss both ways: by losing profitable and gaining underpriced
policies. Therefore, non-life-insurance pricing techniques are mainly employed to
combat the phenomenon of anti-selection by dividing the insurance portfolio
into sub-portfolios based on certain influence variables. Every sub-division thus
will contain policyholders with identical risk profile that will be charged the
same reasonable tariff

         A
method usually employed to estimate the premium involves combining the
conditional expectation of the claim frequency with the expected cost of
claims, considering the observable risk characteristics. The process of
evaluating risks in order to determine the insurance premium is performed by
the actuaries, which over time proposed and applied different statistical
methods.  Most often a distinction is
made between the overall premium level of an insurance portfolio and the
question of how the total required premium should be divided between the policyholders.
The overall premium level is
based on considerations on predicted future costs (reserves) and the cost of
capital (expected profit), as well as the market situation. Historically, these
calculations have not involved much statistical analysis. On the other hand,
the question of how much to charge each individual given the premium level
involves the application of rather advanced statistical models.

 

Statistical models suggest a simple summary of data in terms
of the major systematic effects together with a summary of the nature and
magnitude of the random variation. Regression analysis plays a key role in
statistics as one of its most powerful and widely used techniques for analyzing
models and predicting future trends. However, the simple linear regression is
not always the best tool due to the following reasons. First of all, the
dependent variable of interest may have a non-continuous distribution implying
that the predicted values should also follow the respective distribution.
However, since the simple linear regression is based on the assumption that the
response variable follows a normal distribution only, it may not be the best
model to analyze the model which follows the normal distribution. In addition,
when the effect of the predictors on the dependent variable is not linear in
nature the simple linear regression model is inadequate.

     The Generalized
Linear Models (GLMs) is an extension of the linear modeling process to a wider
class of problems involving the relationship between a response and one or more
explanatory variables. In this context, linear regression used to evaluate the
effect of explanatory variables on the event of interest, has been replaced
starting with 1980 by the GLMs. These models according to Michaela (2013) allow
modeling a non-linear behavior and a non-Gaussian distribution of residuals. This
property is very useful for the analysis of non-life insurance, where claim
frequency and claim cost follow an asymmetric density that is clearly
non-Gaussian. The establishment of the GLMs has brought an improvement in the
quality of risk predictive modeling techniques given the nature of risks to
give a fair tariff.  

              The main
objective of this paper is to apply the GLMs in order to assess the premiums
applied to each insured, in an equitable and reasonable manner.

       

1.2 SCOPE OF STUDY

The next section presents a review of the literature concerning
the application of the GLMs in non-life insurance pricing. Section 3 describes
the research methodology employed in this paper. Each subsection of this part probes
the estimation methods of the claim frequency and cost of claims, leading to
the calculation model of the pure premium. Section 4 is presents a study on an
auto-insurance branch data in order to explain briefly the risk factors that
enable dividing the insurance portfolio in premium classes and how to obtain
the associated or corresponding premium. Section 5 presents the conclusions of
the study.

 

 

 

 

2.      LITERATURE REVIEW

         Historically,
the Gaussian linear regression model proposed by Legendre and Gauss in the 19th
century, limited the application of actuarial science models in explaining
realistic and possible risk occurrences. The model proposed mainly to quantify
the impact of exogenous variables over the phenomenon of interests, has taken
lead in econometrics but the application of this model in insurance has been
found to be difficult. According to Michaela (2013), the linear model implied a
series of hypothesis that are not compatible with the reality imposed by
frequency and cost of damages generated by risk occurrences. The Gaussian
probability density, the linearity of the predictor and homoscedasticity are
the most relevant assumptions of the model.

       The increasing
complexity of statistical methods and development of wide risk contests meant
that actuaries had to find models that explained risk occurrences as realistic
as possible. A great development in non-life insurance pricing, the Minimum
Bias procedure was introduced by Bailey and Simon (1960). The method defines
randomly the link between the explanatory variables, the risk levels and the
difference between predicted values and observed ones.

Once these variables are established, an iterative algorithm
can be used to calculate the coefficient associated with each risk level using
the minimizing distance criterion. 
According to Michaela (2013), the algorithm has been found subsequently
to be a particular case of the Generalized Linear Models although it was
created outside a recognized statistical framework.

         The
implementation merits of these models, both in statistics and actuarial science
are attributed to British actuaries from City University, John Nelder and
Robert Wedderburn (1972). They prove that the generalization of the linear
modeling allows the deviation from the assumption of normality, extending the
Gaussian model to a particular family of distributions, known as the
exponential family. Members belonging to this family of distributions include,
but not limited to the Binomial, Normal, Poisson and the Gamma distributions.  Regression models where the response variable
is distributed as a member of the exponential family share the same
characteristics. In contrary to the classical normal linear regression, there
are less restrictions here: in addition to the wide gamma of possible response
distributions, the variance need not be constant (heteroscedasticity is
allowed) and the relation between the fitted values and the predictors need not
be linear.      

       The GLM Models have the advantage of a
theoretical framework that enables the application of statistical tests in
order to evaluate the fitting of models comparing to the fitting of models.
Nelder and Wedderburn (1972) also suggest that the estimation of parameters is
done using the maximum likelihood procedure, so that the parameter estimates
are obtained through iterative an algorithm. The contribution of Nelder in
propounding and completing the GLMs theory continues while collaborating with
Irish statistician Peter McCullagh, whose paper (1989) offers extensive information
on the iterative algorithm and the asymptotic properties (properties true when
the sample size becomes large) of the parameter estimation.

         The complexity
and abundance of papers have been remarkable since the establishment of the
GLMs principles. Many authors and scientists have successfully highlighted,
developed and improved the assumptions imposed by the practical applications of
the models in non-life insurance. Among the precursors of the GLMs approach as
the main statistical tool in determining the insurance pricing is noted Jean
Lemaire (1985). Based on these models, he aims to estimate the probability of
risk occurrence in auto insurance, to establish the insurance premium and also
to measure the effectiveness if the models used to estimate it. In this field,
a significant contribution also goes to Arthur Charpentier and Michel Denuit
(2005) who have successfully covered, in a modern view, all the aspects of
insurance mathematics. Recent studies also reveal the contribution of Jong and
Zeeler (2008), Kaas and al. (2009), Frees ( 2010) and Ohlsson and Johansen
(2010), who have highlighted the particularities of the GLMs in non-life
insurance risk modeling.