# A Data Science Approach to Risk Assessment for Automobile Insurance Policies

Patrick Hosein
The University of the West Indies, St. Augustine, Trinidad
<EMAIL_ADDRESS>
###### Abstract

In order to determine a suitable automobile insurance policy premium one needs
to take into account three factors, the risk associated with the drivers and
cars on the policy, the operational costs associated with management of the
policy and the desired profit margin. The premium should then be some function
of these three values. We focus on risk assessment using a Data Science
approach. Instead of using the traditional frequency and severity metrics we
instead predict the total claims that will be made by a new customer using
historical data of current and past policies. Given multiple features of the
policy (age and gender of drivers, value of car, previous accidents, etc.) one
can potentially try to provide personalized insurance policies based
specifically on these features as follows. We can compute the average claims
made per year of all past and current policies with identical features and
then take an average over these claim rates. Unfortunately there may not be
sufficient samples to obtain a robust average. We can instead try to include
policies that are“similar” to obtain sufficient samples for a robust average.
We therefore face a trade-off between personalization (only using closely
similar policies) and robustness (extending the domain far enough to capture
sufficient samples). This is known as the Bias-Variance Trade-off. We model
this problem and determine the optimal trade-off between the two (i.e. the
balance that provides the highest prediction accuracy) and apply it to the
claim rate prediction problem. We demonstrate our approach using real data.

_Keywords_ Risk $\cdot$ Motor Insurance $\cdot$ Machine Learning $\cdot$
Premium Pricing $\cdot$ Claims Prediction

##  1 Introduction

Traditionally insurance companies have determined automobile policy premiums
using rate tables computed by Actuaries Hassani et al. (2020). Today, however,
the vast amount of data collected in electronic form can now be used to
determine more suitable premiums for a given policy since such data can be
used to better predict risk Errais (2019). Furthermore, by using data from
present and past customers, the predictions are better suited for the
particular environment in which the insurance company operates. This form of
personalized policies benefit the customer (who pays an amount more in line
with their risk) as well as the insurance company (which can now better ensure
that it can safely cover claims costs from risky policies). The typical
approach is straightforward. For a given new customer, one can use historical
data of past and present customers with similar characteristics (features) to
better estimate the risk level of the new customer and then use this to
determine a premium for their policy. This is similar to recommender systems
used by companies such as Netflix. In that case movies are recommended to an
individual based on movies that were enjoyed by customers with similar
characteristics (collaborative filtering). In the case of insurance, one must
recommend a policy that is both desirable to the customer (through
personalization) and profitable to the insurance provider.

##  2 Related Work and Contributions

Many past papers have focused on Recommender systems for insurance companies
where one of a small number of insurance products is offered. In Qazi et al.
(2020, 2017) they used historical data of existing and past customers to
determine the most suitable policy for a new customer. In this case a
relatively small number of insurance products are available and hence the
number of customers who have been using a specific product will be
sufficiently high so that the sample size is not an issue when computing the
recommendation. The paper Kanchinadam et al. (2018) also addresses the same
problem but focuses on speeding up the computation of the recommendations. The
papers He et al. (2018); Bian et al. (2018) do address personalized auto
insurance premiums but they focus on using Telematics data to do so. Such
devices are not available from all insurance companies and hence has limited
applicability. The authors in Esfandabadi et al.  use Fuzzy Logic to come up
with a rule based approach to risk. In our case we use a data science approach
and focus on personalization while using traditionally available data.

Several papers also focus on risk assessment. In general, few customers make
claims during a year. Furthermore, the claims that are made vary widely from
minor incidents (such as a scratched bumper) to major ones (such as when a car
has to be written off because it cannot be repaired). This results in a large
variation in the average annual claims made by a customer making it difficult
to predict. Therefore papers generally focus on predicting either, severity
(the expected claim value given that a claim is made, Qazi et al. (2017); Su
and Bai (2020)) or frequency (the expected number of claims made per year, Liu
et al. (2014); David and Jemna (2015)). Another typical metric is the loss
ratio which is the ratio of the claims made to the premium charged (Guelman
(2012); Zhang and Dukic (2013). We focus on a more direct measure which is the
average total value of claims made per year for a given policy (shortened to
simply claims rate) which can be thought of as the product of severity and
frequency and hence captures both metrics. As mentioned before, predicting
claims rate can be challenging because of its high variance. A significant
number of samples are required for a good estimate but as one tries to achieve
greater personalization the number of available samples decreases. We
investigate the optimal trade-off between these objectives.

Note one potential issue of Recommender Systems is the following. The
Recommender system chooses the most appropriate product for the customer but
this may not be a very profitable product for the company and so this trade-
off must be taken into account (see Hosein et al. (2019) for a more detailed
discussion on this issue). In our case we need not worry about this issue
since we are focusing on providing the most suitable (unique) product and the
premium is then determined to achieve an acceptable profit for each unique
policy.

Our contribution is in the analysis of the trade-off between personalization
and robustness. Instead of a finite number of products from which to choose
for customer offerings, we provide a unique (personalized) product to each
customer. Furthermore, we only take into account demographic and other data
collected from each policy but do not consider Telematics data. Naturally our
approach can include such data as well. Therefore we are (a) using a new model
for risk (based on claim rate), (b) obtaining the best trade-off between
personalization and robustness, (c) using the proposed approach for feature
importance and selection and (d) demonstrating how the results obtained can be
interpreted so that one can explain to the customer the reason for the
provided premium.

##  3 Problem Formulation and Assumptions

We formulate a model for this problem and then develop an algorithm for its
solution. Our objective is as follows, given policy information for a new or
renewal automobile policy (i.e., information about the drivers, cars, etc.),
predict the expected total amount in claims that will have to be paid out to
this customer over the subsequent year of the policy. This prediction will be
based on several factors but correlates with the risk associated with the
drivers and cars on the policy. This information can then be used to determine
an appropriate premium for the policy. Traditionally this computation is
performed using risk tables but independent of the specific historical data of
the company’s customers. Here we use historical data of the provider’s
customers to make the prediction. This is more appropriate since the
parameters used in the risk tables may have been developed based on a
different customer base (country) and so unsuitable for the one under
consideration.

###  3.1 New versus Renewal Policies

Note that we need to distinguish between a new policy, for which only customer
provided data is available, and renewals, for which information about the
customer since the start of their policy is available. We develop a model that
can be applied to both new and renewal policies. In the case of renewal
policies, the historical data of the policy is included in the training set.
The proposed approach therefore automatically includes the past claim
information of the policy (since it is now included in the training set).
Therefore we assume that all policies that are at least one year old are
included in the training set used for parameter determination. In this way
recent information is included in the predictions. Note that this means there
is no need for an accident penalty or a no-claim discount since these
adjustments are implicit.

###  3.2 Quantity versus Currency of Data

The more data that is used for predictions, the more accurate the prediction.
However, as we increase the dataset by going further back in time we will be
using outdated information (e.g., automobile models, cost of repairs, etc.).
We manage this as follows. As the cost of claims increases (with time), the
claim rate of a policy will also increase. The prediction we get from using
outdated information will therefore be lower than what would actually occur.
We therefore scale predictions as follows. We predict the total claims for the
previous year and we then use a scaling factor to ensure that the total
predicted claims equals the total actual claims. This scaling factor is then
included when making new predictions. This scale factor computation is
repeated every year so that the total predicted claims for the upcoming year
will be close to the actual total claims for the year.

###  3.3 Comprehensive versus Third-Party Policies

There are two types of policies, Comprehensive, in which the company has to
pay for repairs to the customer’s car even if they were at fault, and Third-
Party, in which the company only pays for repairs to the other involved party
in the accident (i.e., the third party). Note that the risk behaviour (and
claims requests) of Third-Party versus Comprehensive policy customers may be
different but the approach we use has the ability to extract the relevant
information. We therefore make predictions using the combined dataset (i.e.
policies of both types) but include the type of policy as a feature. Note that
the features for both types of policies are the same except that, for
Comprehensive policies, there is also the Sum Insured (based on the value of
car) feature. This value is set to zero for Third Party policies but the same
model can be used for both policy types.

###  3.4 Multi-Car versus Single-Car Policies

For each policy we must predict the total annual claims for the policy which
may have multiple drivers and/or cars. Note that a premium is charged per car
and the sum of these forms the policy premium. Our model uses the primary car
and primary driver of that car as the sample for that policy (and ignores all
other drivers/cars). This means that the prediction is made for a single
driver/car pair and this can be repeated for each car on the policy to
determine the total claim rate for the policy.

##  4 Dataset Description and Preparation

The policy data used for this study spans a period of 5 years. No confidential
information is disclosed and all monetary values are normalized. It consists
of data collected from past and existing customers. Each policy record
consists of policy information, information for each driver on the policy,
information for each vehicle on the policy and information on each claim made
on the policy since its inception. Some of this information is not relevant
for our purposes (e.g., Vehicle Identification Number) and is ignored. Certain
features must be derived from the information provided. For example, the
policy lifetime is computed as the difference between the termination date and
start date (if terminated) or the difference between the present date and the
start date (if currently active). Note that the metric of concern is the
average claim rate for a driver/car pair. For each policy we determine the
total value of all claims made (by the primary driver) and divide by the total
lifetime of the policy (in years) to obtain the claim rate.

Our objective is to predict the claim rate and use this claim rate to
determine a suitable price. In order to do this we focus only on the primary
driver and their associated car for each policy. This happens to be the
majority of cases so we do not lose too much information. For this driver we
compute the claim rate based on accidents in which they were involved. We
remove features that were mostly empty or corrupt and also placed filters to
remove anomalous data such as drivers over the age of 85. The data that was
finally used for the problem is provided in Table 1. POL is the policy number
which is used as a unique identifier for the policy. CLR is the average claims
per year computed for the primary driver and their associated car for the
policy. TOC is the type of policy (customer) which we also use as a feature.
SIV is the sum insured value of the primary vehicle of the policy and this
value is zero for Third Party policies. All other features are described in
the Table.

Table 1: Policy Features used for Analysis Feature | Description
---|---
POL | policy identification number
CLR | annual claim rate (total claims divided by policy lifetime)
ADR | city of home address
COV | were drivers continuously insured over the last 5 years? (y/n)
SEX | gender of driver
AGE | age of driver
MST | marital status of driver
USE | type of use (business, work or pleasure)
WRK | whether car is used for work (y/n)
NAF | the number of at-fault accidents over the last 5 years
DAF | number of years primary driver has been free of Claims
NNF | the number of not-at-fault accidents over last 5 years of driver
MAK | car manufacturer
VYR | model year
BDY | body type
YCF | the number of years the Primary Car has been claim free
NCC | engine size of car (in CC)
TOC | type of policy (Comprehensive or Third Party)
SIV | sum insured value

##  5 Proposed Model

The model we propose is unique in that (a) the metric of concern is claim rate
and (b) we use a novel solution approach rather than the traditional
approaches. We do not present a full comparison with other Machine Learning
approaches in this paper since our intent is to introduce the model. Future
papers will include detailed comparisons with state of the art Machine
Learning algorithms.

###  5.1 Definition of Distance Metric

In this section we describe the approach used for predicting the annual
financial claims per year (henceforth called claim rate) for a given policy.
We denote the set of features that we consider by the set $\mathbf{F}$.
Features include information such as age, gender, etc., as well as information
about their associated vehicle such as model, body type, etc. We denote the
set of samples by $\mathbf{S}$ where a sample is a policy and includes
features for the associated driver/car pair. One way to predict the claim rate
is to find the expected value of the claim rates of all existing policies with
identical features. However, there may be none or very few of such policies.
We must therefore include policies with features that are nearby and include
them in the average.

In order to find “close” policies we need to define a distance metric between
pairs of categories of a given feature and then use some measure (e.g.,
Euclidean Distance) to define the distance between two policies. We define
this distance as follows. For each category $v$ of feature $f$ let $C(f,v)$
denote the claim rate averaged over all policies that has a value $v$ for
feature $f$. For example, for the feature gender ($f=gender$) with members $m$
and $f$, let $C(gender,male)$ denote the average claim rate over all male
drivers and let $C(gender,female)$ denote the average claim rate over all
female drivers. We define the distance between these two categories of this
feature by $|C(gender,male)-C(gender,female)|$. In general, if we had several
feature categories then the distance between any two of them will be computed
in this manner. Therefore if the test policy has a male driver then their
gender distance from another policy with a male driver is 0 while for a female
driver it would be $|C(gender,male)-C(gender,female)|$. Note that the same
computation is done for numerical features such as age. For example, the
distance between a 48 year old and a 30 year old is given by
$|C(age,48)-C(age,30)|$. By doing this we maintain the same measurement unit
(claim rate) for all distances. If the 48 year old is a male and the 30 year
old is a female then the Euclidean distance is used (i.e. the root of the sum
of the squares of the gender and age feature distances).

###  5.2 Claim Rate Prediction

If there were several existing policies with the exact feature values as the
test policy then one could obtain a good estimate on the claim rate for the
test policy by taking the average of claim rates over all policies with the
same features. However, in general there may not be sufficient samples (or
none) to obtain an estimate with sufficient confidence and so we need to
include nearby samples as well. The more nearby samples we use the more robust
the estimate but the less personalized since included samples are further
away. This in turn leads to lower prediction accuracy. We take a weighted
average of claims of all policies where the weight is inversely proportional
to the Euclidean distance between the policies.

Suppose we wish to predict the claim rate for some test policy and denote the
distance between this policy and some training policy $s$ by $d_{s}$. We use a
weight $(1+d_{s})^{-\kappa}$ for $\kappa\geq 0$ when taking into account the
claim rate of sample $s\in\mathbf{S}$. However we need to have a normalizing
factor $\alpha$. The predicted claim rate $c$ for the test sample is therefore
given by

$c(\kappa)\equiv\sum_{s\in\mathbf{S}}\alpha\frac{c_{s}}{(1+d_{s})^{\kappa}}$
(1)

where $c_{s}$ is the claim rate of policy $s$. If all policies had the same
claim rate then the predicted claim rate should also have this value and hence
we must have

$c\equiv\sum_{s\in\mathbf{S}}\alpha\frac{c}{(1+d_{s})^{\kappa}}$ (2)

and hence

$\alpha=\left(\sum_{s\in\mathbf{S}}\frac{1}{(1+d_{s})^{\kappa}}\right)^{-1}$
(3)

and so we have the predicted claim rate for the test policy as

$c(\kappa)\equiv\frac{\sum_{s\in\mathbf{S}}\frac{c_{s}}{(1+d_{s})^{\kappa}}}{\sum_{s\in\mathbf{S}}\frac{1}{(1+d_{s})^{\kappa}}}$
(4)

The pseudo-code for this computation is provided in Algorithm 1.

Algorithm 1 Pseudo-code for proposed Algorithm to predict test sample claim
rate $c(\kappa)$

1:$\mathbf{F}\equiv\text{\sf set of features}$

2:$\mathbf{S}\equiv\text{\sf set of training samples}$

3:$\mathbf{v}_{f}\equiv\text{\sf set of categories for feature
$f\in\mathbf{F}$}$

4:$\kappa>0$ tuning parameter

5:$X_{sf}\in\mathbf{v}_{f}\equiv\text{\sf category of feature $f\in\mathbf{F}$
of training sample $s\in\mathbf{S}$}$

6:$x_{f}\in\mathbf{v}_{f}\equiv\text{\sf category of feature $f\in\mathbf{F}$
for test sample}$

7:$c_{s}\equiv\text{claim rate for sample $s\in\mathbf{S}$}$

8:$c(\kappa)\equiv\text{predicted claim rate for test sample using parameter
$\kappa$}$

9:$\displaystyle\bar{c}\leftarrow\frac{1}{|\mathbf{S}|}\sum_{s\in\mathbf{S}}c_{s}\;\;\;\text{(average
claim rate over all training samples)}$

10:for each $f\in\mathbf{F}$ do

11: for each $v\in\mathbf{v}_{f}$ do

12: $\mathbf{z}\equiv\\{s\in\mathbf{S}\;|\;X_{sf}=v\\}$

13: $\displaystyle
C(f,v)\leftarrow\frac{1}{|\mathbf{z}|}\sum_{s\in\mathbf{z}}c_{s}\;\;\;\text{(average
claim rate over samples where feature $f$ has value $v$)}$

14: end for

15:end for

16:for each $s\in\mathbf{S}$ do

17: $\displaystyle
d_{s}\leftarrow\left(\sum_{f\in\mathbf{F}}\left(C(f,x_{f})-C(f,X_{sf})\right)^{2}\right)^{\frac{1}{2}}$

18: $d_{s}\leftarrow\frac{d_{s}}{\bar{c}}$

19:end for

20:$\displaystyle
c(\kappa)=\frac{\sum_{s\in\mathbf{S}}\frac{c_{s}}{(1+d_{s})^{\kappa}}}{\sum_{s\in\mathbf{S}}\frac{1}{(1+d_{s})^{\kappa}}}$
(predicted claim rate for test policy)

###  5.3 Computing the Optimal value of $\kappa$

Next we determine the optimal value of $\kappa$. For an existing policy $s$ we
have the actual claim rate $c_{s}$. Note that we can predict a claim rate for
this sample (in which case the sample must be removed from the training set)
and we denote this predicted value by $\hat{c}(\kappa)$. We introduce the hat
to distinguish this predicted value with the actual value (which has no hat).
Note that we use 5-Fold cross validation and hence 80% of the samples are used
for training (computing the average claim rates $C(f,v)$) while the other 20%
are used for testing (and determination of the accuracy). Note that when
$\kappa=0$ then $\hat{c}(\kappa)=\bar{c}$ and so the prediction is simply the
average over all (training) samples. As $\kappa$ is increased, close samples
are weighted more heavily but the average becomes less robust and hence the
error will eventually start increasing again. Therefore the optimal $\kappa$
lies somewhere in between (see Figure 4 for an example of this relationship).
We therefore will find $\kappa$ that minimizes the Mean Absolute Error (MAE)
of the prediction. For convenience we will normalize this by the MAE if one
used the average claim rate over all policies, $\bar{c}$, as the predictor.
One can think of this case as making a prediction without features. Therefore
we will compare the error of the prediction made with features with the error
of the prediction made without features. Let us denote the test set by
$\mathbf{T}$ then we compute the normalized error over the test samples as

$E(\kappa)=\frac{\sum_{t\in\mathbf{T}}|\hat{c}_{t}(\kappa)-c_{t}|}{\sum_{t\in\mathbf{T}}|\bar{c}-c_{t}|}$
(5)

If the predictor is the same as averaging over all policies (i.e.,
$\hat{c}_{t}(\kappa)=\bar{c}$) then this ratio is 1. However if, by adding
features, the MAE of the predictor is decreased then this ratio drops below 1.
Therefore this metric provides an indication of prediction performance using
features when compared to prediction performance without using features and
hence demonstrates the benefit of the feature-based approach. We then find the
$\kappa$ value that optimizes the predictor as

$\kappa^{*}=\operatorname*{argmin}_{\kappa}E(\kappa)$ (6)

This value is then used to obtain the optimal prediction as
$c_{t}^{*}=\hat{c}_{t}(\kappa^{*})$.

###  5.4 Feature Importance

Consider a single feature. We know from the previous section that, as $\kappa$
is increased then $E(\kappa)$ should initially decrease before increasing once
again. If this does not occur then the feature does not capture sufficient
information to be useful for predictions. One can therefore use the value of
$E(\kappa)$ evaluated at the optimal $\kappa$ for that feature alone as an
indication of importance. In fact, even if we used a fixed value of $\kappa$
for each feature the corresponding value of $E(\kappa)$ is an indication of
relative importance with lower values indicating more importance. For example,
in Figure 1 we plot $E(\kappa)$ as a function of $\kappa$ for two features DAF
(years claim free for the driver) and YCF (years claim free for the car). For
DAF the minimum error occurs at $\kappa=8$ while for YCF it occurs at
$\kappa>20$. However, at $\kappa=10$ we find that the respective values
provide a good representation of the optimal value and hence can be used to
compare the two features. Also note that here we clearly see that risk depends
primarily on the driver with the car playing a minor role.

$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$0.700$$0.750$$0.800$$0.850$$0.900$$0.950$$1.000$Years
Claims Free (car)Increasing ImportanceYears Claims Free
(Driver)$\kappa$$E(\kappa)$ Figure 1: $E(\kappa)$ of two Features to
demonstrate Relative Importance

We therefore use this approach to determine which features are important and
hence should be included in the analysis. We use a value of $\kappa=10$ and,
using a single feature at a time, we compute $E(10)$. The resulting values are
provided in Figure 2. The features represented in red have normalized errors
greater than 1.

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$1.1$DAFNAFNNFTOCYCFVYRBDYCOVNCCMAKUSEMSTWRKADRSEXAGESIVDAF+NAFDAF+NAF+NNFDAF+NAF+NNF+TOCDAF+NAF+NNF+TOC+YCFDAF+NAF+NNF+TOC+YCF+VYRDAF+NAF+NNF+TOC+YCF+BDYDAF+NAF+NNF+TOC+YCF+COVDAF+NAF+NNF+TOC+YCF+COV+NCCDAF+NAF+NNF+TOC+YCF+COV+MAKDAF+NAF+NNF+TOC+YCF+COV+USEDAF+NAF+NNF+TOC+YCF+COV+USE+MSTDAF+NAF+NNF+TOC+YCF+COV+USE+WRKDAF+NAF+NNF+TOC+YCF+COV+USE+ADRDAF+NAF+NNF+TOC+YCF+COV+USE+SEXDAF+NAF+NNF+TOC+YCF+COV+USE+SEX+AGEDAF+NAF+NNF+TOC+YCF+COV+USE+SEX+SIV$E(10)$
Figure 2: Normalized MAE for proposed predictor computed for each feature

###  5.5 Feature Selection

Now that we know which features are important we focus on which of them should
be included in the model. We do this as follows. Starting from the most
important feature (lowest value for $E(10)$) we add one feature at a time and
again compute $E(10)$ for the combination. If the performance metric decreases
(i.e., better results) then we keep it and repeat. If the performance metric
increases then we remove the recently added feature and repeat. Note that,
some features may have low importance when considered in isolation but
together with other features (such as type of policy) their value increases.
The results of this process are provided in the lower part of Figure 2. A
brown bar indicates that the addition of a feature resulted in a loss of
performance and hence the feature should be removed.

The final features to be used include number of years since the driver was
last in an accident, the number of at-fault accidents by the driver over the
last five years and the number of not-at-fault accidents over the last five
years. Each of these is a strong indicator of risk. In the case of renewals we
would have actual claim rate values but for new customers these three features
(even without financial information) correlates well with claim rate. The type
of policy feature is also needed since it helps to distinguish the two types
of policies. Note that we could do the analysis separately for each type of
policy but the increase in sample size by combining the two types provides
better overall results. Only one car feature was found to be sufficiently
beneficial and that was the number of years since the last claim was made on
the car. However this feature is far less important than the driver features
that were included indicating that what really matters is the driver on the
policy and not the car. Whether the driver was continually insured over the
last 5 years (i.e., mature driver), the type of use (personal versus business)
as well as the gender of the driver were also found to be useful (but far less
so than the others).

Note that we had expected certain features (like age) to be beneficial but
they were not. In Figure 3 we provide a histogram of the average claim rate by
age (in blue). We see that there is a weak dependency on age but because of
the large variations from year to year (because of limited data), the
dependency is not sufficiently robust. Next we predicted the claim rate for
each age using the approach described previously. We found that the optimal
value of $\kappa$ was 2 with a normalized MAE of $E(2)=0.9996$ which indicates
that limited personalization was possible. We then used this value to find
optimal claim rate values for each age. In Figure 3 we provide the histogram
of the original claim rates (blue) and the filtered claim rates (red). We note
that the red claim rates are each close to unity and hence provides little
differentiation. This is why this feature does not provide much benefit for
predictions.

$20$$25$$30$$35$$40$$45$$50$$55$$60$$65$$70$$75$$80$$85$$0$$1$$2$$3$$4$$5$$6$
Figure 3: Histogram of Claim Rate versus Age for Original (blue) and filtered
(red) Cases

###  5.6 Parameter Optimization

We now have the set of features to be included in the model. Next we find the
optimal value of $\kappa$ for this combination of features. This value will
then be used for making predictions. In Figure 4 (brown curve) we plot
$E(\kappa)$ as a function of $\kappa$. We find the optimal value to be
$\kappa^{*}=8$ with $E(8)=0.63$ and hence once can reduce the MAE obtained
with no features by 37% by using the using the 8 chosen features. We also note
that, although $E$ increases with $\kappa$ beyond the optimal point, the
increase is very gradual so the error remains nearly constant for a wide range
of values and so the approach is robust with respect to $\kappa$.

We believe that if we had performed feature selection using each policy type
separately that we would get the same features. We therefore used these
features and determined $E(\kappa)$ for Third-Party policies only and also for
Comprehensive policies only. These are also plotted in Figure 4. We find that
the accuracy for Third-Party only samples is close to that of the case of
using both Third-Party and Comprehensive samples. This is primarily due to the
fact that there are 50% more Third-Party samples than Comprehensive samples.
Therefore the Third-Party samples are more useful to the Comprehensive
predictions than the other way around. Also note that all three cases are
optimal at $\kappa=8$.

$5$$5.5$$6$$6.5$$7$$7.5$$8$$8.5$$9$$9.5$$10$$10.5$$11$$0.630$$0.635$$0.640$$0.645$$0.650$$0.655$$0.660$$0.665$$0.670$$\kappa$$E(\kappa)$ComprehensiveThird-
PartyComprehensive and Third-Party Figure 4: $E(\kappa)$ as a function of
$\kappa$ for Selected Features

##  6 Illustrative Examples of Predictions

We have now determined the features to be used and the optimal value of
$\kappa$. In this section we will consider various policy scenarios and
predict the resulting claim rate to demonstrate the dependence on the
features. Although we use both Comprehensive and Third Party policies in our
model we will illustrate using a Comprehensive policy and normalize claim
rates with respect to the average over all Comprehensive Policies. In Table 2
we provide various scenarios to illustrate that the model provide reasonable
outputs. The top table starts with a low risk policy and features are changed
one at a time that results in increased claim rates. The bottom table starts
with a high risk policy and features are adjusted one at a time in order to
lower the claim rate.

There is one outstanding case that provided unexpected results. In the lower
table when we reduce the number of not-at-fault accidents from 1 to zero we
expect a decrease in the claim rate but instead we found that it increased. We
investigated this in detail. We found that the provided data has some
inconsistencies. There were many cases where the number of not-at-fault
accidents and at-fault accidents were 0 but the driver indicated that they
made a claim within the last five years. This of course is inconsistent. This
lead to a lot of claims listed under NNF=0 when they should be listed under
NNF=1 or above. We believe this to be the reason for the result obtained. Our
intent was not to make any adjustments to the given data for this paper to
avoid any appearance of data tweaking. However, in the future, we will
investigate what happens when we adjust the data to make it more consistent
while justifying any changes made.

Table 2: Predictions for Sample Cases starting with Low Risk Case (top) and
High Risk Case (bottom)

DAF | NAF | NNF | TOC | YCF | COV | USE | SEX | Claim Rate
---|---|---|---|---|---|---|---|---
15 | 0 | 0 | CM | 15 | Y | Business | M | 0.07
6 | 0 | 0 | CM | 15 | Y | Business | M | 0.63
2 | 1 | 0 | CM | 15 | Y | Business | M | 0.93
15 | 0 | 1 | CM | 15 | Y | Business | M | 0.68
15 | 0 | 0 | TP | 15 | Y | Business | M | 0.03
15 | 0 | 0 | CM | 0 | Y | Business | M | 0.20
15 | 0 | 0 | CM | 15 | N | Business | M | 0.07
15 | 0 | 0 | CM | 15 | Y | Private | M | 0.06
15 | 0 | 0 | CM | 15 | Y | Business | F | 0.07

DAF | NAF | NNF | TOC | YCF | COV | USE | SEX | Claim Rate
---|---|---|---|---|---|---|---|---
1 | 1 | 1 | CM | 1 | N | Private | F | 6.04
5 | 1 | 1 | CM | 1 | N | Private | F | 1.53
6 | 0 | 1 | CM | 1 | N | Private | F | 0.56
1 | 1 | 0 | CM | 1 | N | Private | F | 7.59
1 | 1 | 1 | TP | 1 | N | Private | F | 4.04
1 | 1 | 1 | CM | 15 | N | Private | F | 5.55
1 | 1 | 1 | CM | 1 | Y | Private | F | 5.97
1 | 1 | 1 | CM | 1 | N | Business | F | 6.00
1 | 1 | 1 | CM | 1 | N | Private | M | 6.04

##  7 Interpreting Prediction

Once a predicted claim rate is computed then this information can be used to
compute a premium. The premium will take into account the operational costs of
the company as well as the desired profit margin. This is another interesting
area of research but is outside the scope of this paper. Once a premium is
computed it is important to explain the reason for the amount (i.e,
interpretability). The operational cost and profit is independent of the
customer so the only customer dependent factor is the predicted claim rate. We
can determine the influence of each feature on this claim rate and this
information can be used to explain the decision made. We do this as follows.
Consider any feature $f$ and let $v$ represent the category value of this
feature for the new policy. We can use the model, with only feature $f$, to
determine the predicted claim rate for anyone in category $v$. Let us denote
this predicted claim rate of this feature by $\tilde{C}(f,v)$. Note that this
is not the same as the average claim rate over all training samples with
category value $v$ which we previously denoted by $C(f,v)$. Let us explain
with the feature gender. If $\kappa=0$ then $\tilde{C}({\sf
gender,male})=\tilde{C}({\sf gender,female})=\bar{c}$. However as $\kappa$
goes to infinity then $\tilde{C}({\sf gender,male})$ approaches $C({\sf
gender,male})$ and $\tilde{C}({\sf gender,female})$ approaches $C({\sf
gender,female})$. For positive values of $\kappa$, $\tilde{C}(f,v)$ will lie
between $\bar{c}$ and $C(f,v)$.

The metric $I_{f}\equiv\tilde{C}(f,v)/\bar{c}$ will be used to represent the
impact of feature $f$ (of a policy with value $v$ for the feature) where
$\bar{c}$ is the average claim rate. For this exercise we only use Third-Party
samples to better explain the approach. If $I_{f}<1$ then the feature is
causing a reduction of the claim rate otherwise it is causing an increase in
the claim rate. Note that all values are being computed using
$\kappa=\kappa^{*}$ and normalized with respect to the average claim rate for
Third-Party policies.

For the policy we chose we have the following information. The driver got into
an accident and made a claim 9 years ago ($I_{DAF}>1$). They have no at-fault
accidents over the last 5 years ($I_{NAF}<1$). They have had 1 not-at-fault
accidents over the last five years ($I_{NNF}>1$). They have a Third-Party
Policy (hence $I_{TOC}=1$). Their car was last in an accident 18 years ago
($I_{YCF}<1$). They have been continuously insured over the last five years
($I_{COV}<1$). This is their private vehicle ($I_{USE}<1$). The driver is Male
($I_{SEX}<1$ but almost 1). The predicted claim rate (which was impacted by
the various features) has $I_{CLR}=1.0$. We provide this information visually
in Figure 5. Hence the provider can explain to the customer the specific
reasons for the premium of their policy.

Figure 5: Contribution of each Feature to
PredictionDAFNAFNNFTOCYCFCOVUSESEXCLR$0$$0.5$$1$$1.5$$2$$2.5$$3$

##  8 Some Analytical Results

###  8.1 Expected value of Prediction

Consider the predicted claim rate $\hat{c}$ for some test sample. If we assume
that the training samples have an average claim rate $\bar{c}$ then we will
show that the expected value of $\hat{c}$ is $\bar{c}$. This would mean that
the sum of predicted claim rates approaches the sum of the actual claim rates
as the number of test samples increases. This ensures that, at the end of the
year, the total claims that are predicted is close to the total actual claims
that were made.

######  Lemma 1.

If the training samples have a mean claim rate of $\bar{c}$ then the expected
value of the prediction for a test sample is equal to $\bar{c}$.

###### Proof.

Recall that the predicted value for a given $\kappa$ is given by

$\hat{c}(\kappa)=\frac{\sum_{s\in\mathbf{S}}\frac{c_{s}}{(1+d_{s})^{\kappa}}}{\sum_{s\in\mathbf{S}}\frac{1}{(1+d_{s})^{\kappa}}}$

We need to take the expectation of the right hand side. Now note that the
feature values for a particular set of training samples is fixed and only the
claim rate varies. Also note that the feature values of the test sample is
also fixed. This means that $d_{s}$ (for sample $s$) is fixed given the
specific test and training samples. This means that the denominator is
constant and so we have

${\cal E}[\hat{c}(\kappa)]=\frac{\sum_{s\in\mathbf{S}}\frac{{\cal
E}[c_{s}]}{(1+d_{s})^{\kappa}}}{\sum_{s\in\mathbf{S}}\frac{1}{(1+d_{s})^{\kappa}}}$

Using the fact that ${\cal E}[c_{s}]=\bar{c}$ we obtain ${\cal
E}[\hat{c}(\kappa)]=\bar{c}$. ∎

###  8.2 Limiting values of $c(\kappa)$

We have seen that $\hat{c}(\kappa=0)=\bar{c}$. In this section we compute the
limit of $\hat{c}(\kappa)$ as $\kappa$ tends to $\infty$. We then take the
expected value of this limit.

######  Lemma 2.

If the training samples have a mean claim rate of $\bar{c}$ then

${\cal E}[\lim_{\kappa\to\infty}c(\kappa)]=\bar{c}$ (7)

###### Proof.

We first compute the limit of the predicted value, $\hat{c}(\kappa)$ as
$\kappa$ tends to infinity. There are two cases to consider. If one or more
training samples have identical feature values as the test sample (i.e.,
$d_{s}=0$) then $\hat{c}_{s}$ is the average of these values. However, if none
of the training samples have identical features then both the numerator and
denominator of $\hat{c}(\kappa)$ tend to zero as $\kappa$ tends to infinity so
we instead do the following. Denote the training sample with the smallest
distance from the test sample by $s^{\prime}$. Let us multiply top and bottom
of the equation for $\hat{c}(\kappa)$ by $(1+d_{s^{\prime}})^{\kappa}$ to
obtain

$\lim_{\kappa\to\infty}c(\kappa)=\lim_{\kappa\to\infty}\frac{c_{s^{\prime}}+\sum_{s\in\mathbf{S}|s\neq
s^{\prime}}c_{s}\left(\frac{1+d_{s^{\prime}}}{1+d_{s}}\right)^{\kappa}}{1+\sum_{s\in\mathbf{S}|s\neq
s^{\prime}}\left(\frac{1+d_{s^{\prime}}}{1+d_{s}}\right)^{\kappa}}$ (8)

Now note that since $d_{s^{\prime}}<d_{s}$ for all samples $s$ then, as
$\kappa$ goes to infinity, the summations go to zero and hence

$\lim_{\kappa\to\infty}\hat{c}(\kappa)=c_{s^{\prime}}$ (9)

If multiple samples are at this distance $d_{s^{\prime}}$ then the numerator
constant would be the sum of these samples and the denominator would be the
number of them and hence the limit is the average of the claim rates of these
samples. Since $c_{s^{\prime}}$ is a sample from the training set space then
its expected value is $\bar{c}$ and hence

${\cal E}[\lim_{\kappa\to\infty}\hat{c}(\kappa)]=\bar{c}$ (10)

∎

One should note the following. As the number of training samples increases,
more training samples will be available close to the test sample and hence the
optimal value of $\kappa$ will increase. In the limit, the predicted value
becomes the true mean for the features of the test sample thus achieving
precise personalization.

##  9 Conclusions and Future Work

We presented a new model for automobile insurance risk assessment and
demonstrated its effectiveness using real data. We showed how feature
importance can be computed, how features can be selected and how model
parameters are optimized. Finally we demonstrated how the model can be used in
practice and results interpreted. Note that this approach can be applied to
any regression problem and its performance will improve as the variance of the
target metric decreases.

Future work will include application of the model to other problems as well as
on model improvements and increased computational efficiency. We also plan to
investigate how recursive improvement of distance values can be used to
increase accuracy (see Hosein (2021) for an example of this improvement).
Finally we plan to investigate properties of the model such as whether
$E(\kappa)$ is convex when its gradient at $\kappa=0$ is negative.

## References

  * Bian et al. (2018) Bian Y, Yang C, Zhao JL, Liang L (2018) Good drivers pay less: A study of usage-based vehicle insurance models. Transportation Research Part A: Policy and Practice 107:20–34, DOI https://doi.org/10.1016/j.tra.2017.10.018, URL https://www.sciencedirect.com/science/article/pii/S096585641730561X
  * David and Jemna (2015) David M, Jemna DV (2015) Modeling the frequency of auto insurance claims by means of poisson and negative binomial models. Analele stiintifice ale Universitatii “Al I Cuza” din Iasi Stiinte economice/Scientific Annals of the” Al I Cuza”
  * Errais (2019) Errais E (2019) Pricing insurance premia: a top down approach. Annals of Operations Research pp 1–16
  * Esfandabadi et al. (0) Esfandabadi ZS, Ranjbari M, Scagnelli SD (0) Prioritizing risk-level factors in comprehensive automobile insurance management: A hybrid multi-criteria decision-making model. Global Business Review 0(0):0972150920932287, DOI 10.1177/0972150920932287, URL https://doi.org/10.1177/0972150920932287, https://doi.org/10.1177/0972150920932287
  * Guelman (2012) Guelman L (2012) Gradient boosting trees for auto insurance loss cost modeling and prediction. Expert Systems with Applications 39(3):3659–3667
  * Hassani et al. (2020) Hassani H, Unger S, Beneki C (2020) Big data and actuarial science. Big Data Cogn Comput 4:40
  * He et al. (2018) He B, Zhang D, Liu S, Liu H, Han D, Ni LM (2018) Profiling driver behavior for personalized insurance pricing and maximal profit. In: 2018 IEEE International Conference on Big Data (Big Data), pp 1387–1396, DOI 10.1109/BigData.2018.8622491
  * Hosein (2021) Hosein P (2021) On the prediction of automobile insurance claims: The personalization versus confidence trade-off. In: 2021 IEEE International Conference on Technology Management, Operations and Decisions (ICTMOD), IEEE, pp 1–6
  * Hosein et al. (2019) Hosein P, Rahaman I, Nichols K, Maharaj K (2019) Recommendations for long-term profit optimization. In: ImpactRS@ RecSys
  * Kanchinadam et al. (2018) Kanchinadam T, Qazi M, Bockhorst J, Morell MY, Meissner K, Fung G (2018) Using discriminative graphical models for insurance recommender systems. In: 2018 17th IEEE International Conference on Machine Learning and Applications (ICMLA), pp 421–428, DOI 10.1109/ICMLA.2018.00069
  * Liu et al. (2014) Liu Y, Wang BJ, Lv SG (2014) Using multi-class adaboost tree for prediction frequency of auto insurance. Journal of Applied Finance and Banking 4(5):45
  * Qazi et al. (2017) Qazi M, Fung GM, Meissner KJ, Fontes ER (2017) An insurance recommendation system using bayesian networks. In: Proceedings of the Eleventh ACM Conference on Recommender Systems, Association for Computing Machinery, New York, NY, USA, RecSys ’17, p 274–278, DOI 10.1145/3109859.3109907, URL https://doi.org/10.1145/3109859.3109907
  * Qazi et al. (2020) Qazi M, Tollas K, Kanchinadam T, Bockhorst J, Fung G (2020) Designing and deploying insurance recommender systems using machine learning. WIREs Data Mining and Knowledge Discovery 10(4):e1363, DOI https://doi.org/10.1002/widm.1363, URL https://wires.onlinelibrary.wiley.com/doi/abs/10.1002/widm.1363, https://wires.onlinelibrary.wiley.com/doi/pdf/10.1002/widm.1363
  * Su and Bai (2020) Su X, Bai M (2020) Stochastic gradient boosting frequency-severity model of insurance claims. PloS one 15(8):e0238000
  * Zhang and Dukic (2013) Zhang Y, Dukic V (2013) Predicting multivariate insurance loss payments under the bayesian copula framework. Journal of Risk and Insurance 80(4):891–919