Machine learning based methods for ratemaking health care insurance

Abstract

In insurance, proposing an accurate premium that is adjusted to the insured risk profile allows companies to better manage their portfolios and to be more competitive. Machine learning methods have recently been adopted for various improvements in insurance ratemaking, especially in the automobile industry. These models are specifically used to mine potential data information and to build a predictive model for a variable of interest using explanatory variables. In this paper, we aim to provide a pricing method for ratemaking individual healthcare insurance contracts using machine learning algorithms that are applied to a Tunisian healthcare insurance portfolio. We start with a simple Classification and Regression Tree, and we work toward more advanced methods that are Random Forest, Extreme gradient boosting, Support Vector Regression, and Artificial Neural network regression model. The predictive performance of these non-parametric methods is compared with the standard generalized linear model. Our results showed the applicability of machine learning in the healthcare insurance market and that the XGBoost algorithm outperforms the predictive capacity of the classical generalized linear model.

Keywords

Machine Learning Pricing Healthcare Insurance Generalized Linear Models Predictive Models Insurance Claims

Introduction

Determining the right premium for a given insurance policy is one of the most important challenges for insurers. Dealing with this challenge involves putting research into practice by exploiting research advancements in operational contexts. The process of ratemaking healthcare contracts consists mainly of an accurate modelling of the frequency of claims produced by a policyholder as well as their costs. It is important that insurers performed the pricing models used to set premium to insure solvency of insurance companies. In fact, systematic under-pricing would lead to low profitability and possible losses.

In this regard, insurance companies are encouraged to seek more reliable actuarial pricing models to offer the best premium. Therefore, risk assessment and insurance premium determination are among the main challenges for insurers. Pricing in non-life insurance is typically operated within the framework of generalized linear models introduced by Nelder and Wedderburn (1972). With the current upswing of data analytics, the background of the present work lies in the growing advancement made in the research fields related to machine learning methods and their applications. In fact, in the present study we consider these techniques to develop a pricing process for individual healthcare insurance contracts that will be built from both frequency and severity of claims. Five machine learning methods for regression are considered: decision Tree known by CART model, Random Forest (RF), Extreme gradient boosting (XGBoost), support vector regression (SVR), and Artificial Neural Network (ANN).

The proposed approach is applied to a Tunisian Healthcare insurance portfolio and enable us to identify the factors explaining the claims rate and to propose a premium adjusted to the insured risk profile. In the Tunisian market of healthcare insurance, individuals demand complementary healthcare contracts to guarantee higher compensation for medical care. This lets the market today very competitive and makes premium pricing an issue for all insurers who are called upon to identify their consumers and segment their portfolios into classes of homogeneous risk (Blesa et al., 2020). According to specific characteristics, this research tends to find appropriate net premiums for each of the insureds with respect to their risk profile. In this work, we aim to investigate how can the integration of new techniques proceeding from machine learning improve the policyholder tariff over classical generalized linear models. In this sense, the predictive performance of applied machine learning methods is compared with the standard statistical learning model: the GLM. Our results showed that the extreme gradient boosting technique is consistently chosen as the best effective approach for frequency and severity of claim’s modelling; its higher flexibility greatly improves predictive power. It is important to mention that our methodology remains valid for application to any healthcare insurance database with explained and explanatories variables.

This article contributes to the existing literature by developing a complete tariff plan through the application of machine learning techniques for a real-life healthcare insurance portfolio. We propose an accurate ratemaking methodology for healthcare insurance policy. In contrast to the prevailing literature, our work takes a distinct approach by exclusively focusing on predicting both the frequency and the severity of individual health insurance claims. As a result, our study not only provides a tangible and real-world estimation of the individual pure premium in the realm of healthcare insurance but also extends the prior research by emphasizing the predictive accuracy of machine learning models. In particular we investigate how the tree-based pricing models XGBoost and Random Forest, the Support Vector Regression (SVR) and the Neural Network model perform in health care insurance ratemaking. Additionally, we conduct a comparative analysis between these machine learning models and the traditional actuarial approach using GLMs. Our study’s findings have important implications for marketing strategies, such as personalized pricing, which are critical for enhancing customer satisfaction and retention. In fact, when grouping similar risks together, the insurer can assess the risk characteristics of individuals within the same group and determine the appropriate adjusted risk premium based on the expected cost of claims and frequency. This ensures that the proposed prices are both adequate and competitive.

The remainder of this paper is structured as follows: section 2 is a review of previous works. Section 3 details the methodology with a brief theoretical background of GLM, CART, RF, XGBoost, SVR, and ANN regression models. Section 4 describes the dataset used in this study. Section 5 reports and discusses the empirical results of our analysis. Finally, section 6 concludes.

Literature review

The generalized linear models introduced by Nelder and Wedderburn (1972) are a parametric model that requires assuming the form of the conditional distribution of the explained variable as a function of the explanations. McCullagh and Nelder (1989) have pointed out the two main advantages of GLMs techniques. Firstly, these methods allow the deviation from the assumption of normality. Secondly, they allow the linear regression to be related to the dependent variable through a link function. These models have been successfully applied to solve actuarial problems (see Haberman & Renshaw, 1996). Indeed, considering a linear relationship between the variable to be explained and the explanatory variables reduces the ability of these models to capture the non-linear interactions between the loss factors. Additionally, GLMs method do not make it possible to model different effects for the same-named explanatory variable: the same coefficient applied for a continuous variable. Also, these models depend widely on the parametric law that will be specified. These facts can lead to a biased estimate of the insurance tariff (see De Jong & Heller, 2008; Kaas et al., 2008). The statistical machine learning models have the advantages of being non-parametric and nonlinear. Hence, thanks to their nature they are now suitable to address issues of pricing in non-life insurance. Recently, machine learning techniques have acquired increasing popularity in many fields such as genetics, rare disease detection, management, or engineering, but also in insurance field where a great interest is given by the pricing actuaries to these models. In parallel with these developments and around big data and data science methods, new sources of data are emerging and are available to insurers to allow a better knowledge of risk. In this context, teams of computer researchers continued to develop more flexible models, outperforming linear models in several research fields, and giving birth to the theory of statistical learning-machine learning, which was introduced for the first time by Vapnik (1998). These models acquired increasing popularity in business applications including for example fraud detection (Li et al., 2018), and pack classification tasks (Hang & Banks, 2019) and in insurance industry for claim analysis and modelling (Blier-Wong et al., 2020; Quan & Valdez, 2018).

In the past few years, researchers have begun studying these newer algorithms to tackle insurance-related tasks. The results obtained show that thanks to machine learning algorithms, it is possible to extract dependency structures between the data which until then remained undetected by classical regression tools. This results in a significant reduction in the error between the value predicted by the model and the measured value. Baudry and Robert (2019) have used the Extra-Trees algorithm for individual claims reserving. Neumann et al. (2019) applied five machine learning methods of classification to predict a car driver’s decision about taking a replacement car after a vehicle accident happens. They showed that the gradient boosting models outperform the classical logistic regression method and other machine learning models. In their work, Castellani et al. (2021) have applied deep learning networks and support vector regressors to the valuation of the solvency capital requirement of participating life insurance policies. Yang et al. (2018) have proposed a gradient tree-boosting algorithm and applied it to Tweedie compound Poisson models for auto insurance pure premiums. Their findings indicate that the new method is superior to the existing methods in the sense that it generates more accurate premium predictions. Wuthrich and Buser (2019) considered Adaboost, gradient boosting, regression tree boosting, GLM and generalized additive model for a suitable non-life insurance actuarial pricing model. Their experimental results indicate that the generalized additive model was able to outperform the generalized linear model for non-log-linear components. In the research of Grize et al. (2020), the authors have studied first a comparison of retention models for household insurance and then a dynamic pricing problem for online motor insurance. They have shown the advantages but also some of the difficulties of applying ML tools in practice. Furthermore, Staudt and Wagner (2021) have examined the predictive performance of the generalized linear model, the generalized additive model and the random forest model using a dataset of a Swiss collision insurance portfolio. Empirical results indicate that the generalized additive model has the best overall performance. A case study comparing machine learning techniques with generalized linear models to perform pricing optimization in car insurance is conducted by Spedicato et al. (2018). They have given both the advantages and disadvantages induced using these computer tools. More recently, Henckaerts et al. (2020) have focused on tree-based methods to develop full tariff plans built from both the frequency and severity of claims. Results show that boosted trees outperform the classical GLMs. Using Indonesia automobile Insurance dataset, Rustam and Ariantari (2018) show the capacity of the support vector machines model to classify policyholders satisfactorily (see also Dey et al., 2021). Huang and Meng (2019) have employed support vector machines, random forests, XGBoost and artificial neural networks for risk classification and Poisson GLM for claim frequency prediction. Based on automobile dataset from insurance company in China, results for risk classification problems show that XGBoost model has the highest accuracy of prediction and driving behavior variables play crucial roles in pricing automobile insurance.

In health insurance, Hartman et al. (2020) have analysed how to best predict which members will be high-cost next year by comparing the predictive accuracy of logistic regression to extreme gradient boosting using the Health Care Cost Institute data. They have found that the added flexibility of the extreme gradient boosting improves predictive power. Moreover, Rampal et al. (2020) introduced an Ensemble regression model to predict future health insurance premiums. The accuracy of this model was compared to four other regression models: Decision Table, Gaussian Process, Linear Regression, and Zero R. Their findings illustrated that the Ensemble regression model outperformed the others, establishing itself as the most accurate predictor of health insurance premiums. Similarly, Kaushik et al. (2022) applied an artificial neural network (ANN) model to forecast individuals' health insurance costs based on their characteristics. They conducted a comparative analysis of the predictive performance of the ANN-based regression model against a linear regression model. The empirical results demonstrated an impressive accuracy rate of 92.72% for the ANN-based regression model, highlighting its effectiveness in predicting health insurance costs. These studies align with our approach of employing advanced machine learning techniques to enhance the predictive accuracy of health insurance premium models. By incorporating similar methodologies, our research further validates the use of non-traditional models over conventional methods. Specifically, our study’s use of XGBoost, a robust ensemble learning algorithm, echoes the findings of Rampal et al. (2020), underscoring the superior performance of ensemble models in this domain. Moreover, our results complement those of Kaushik et al. (2022), demonstrating the efficacy of sophisticated machine learning models, such as ANN and SVR, in accurately forecasting health insurance costs. In light of these findings, our study extends the application of machine learning models to a real-life health insurance portfolio, reinforcing the practical utility and generalizability of these advanced methods.

Methodology

In this section, we provide a brief description of technical methods that will be considered in this work. In addition, we present two popular metrics used by the financial industry to measure the performance of regression type models: the mean squared error and the relative mean squared error.

Generalized linear models

The generalized linear models, introduced in the work of Nelder and Wedderburn (1972), are extensions of the simple linear model by relaxing the assumptions of normality and constant variance. They restrict themselves to the case of the exponential family, which groups together classical laws such as Poisson, Gamma, Normal, Binomial. These models make it possible to estimate an endogenous variable $Y$ (to be explained or response variable) depending on one or more exogenous variables $X_{i}$ (explanatory variables). Conventionally and in the scientific bibliography (Charpentier, 2014), we have two discrete distributions which will be used to model the frequency of claims in this paper: the Poisson distribution and the negative binomial distribution. Then, to model the average severity, we will apply the Gamma or the Log-Normal laws. The main purpose of the GLM is to predict the expected values of the dependent variables through conditional means, given independent observations. To do this, there are searched the parameters $β_{0}, . . ., β_{p}$ , through a monotonous and differentiable function $(g)$ of the dependent variable mean $μ_{i}$ $for individual i$ written as a linear combination of the explanatory variables $X_{j}, j = 1, \dots, p :$

g (μ_{i}) = β_{0} + \sum_{j = 1}^{p} β_{j} X_{ij} = η_{i}

(1)

Classification and regression trees

Classification and Regression decision Trees are introduced by Breiman et al. (1984) and constitute a major family of data mining methods. They are classified among non-parametric supervised models, i.e., they do not impose any hypothesis on the data distribution. They make it possible to obtain both explanatory and predictive models from a database. These algorithms are used to group the observation of the target variable $Y$ using a series of questions called segmentation criteria based on the values taken by the explanatory variables $X = (X_{1}, X_{2} . . ., X_{p})$ of this observation. So, each node of the CART tree corresponds to a segmentation criterion. Thus, their objective is to build a partitioning of groups of data as homogeneous as possible according to the variable to be predicted. The structure of this recursive learning method is like a tree with, at each node representing tests performed by the algorithm, a new branching until arriving at leaves, representing a full class. Therefore, the process to build a CART tree contains two stages: the determination of the saturated tree on the train set, pruning it and then building the optimal tree.

Construction of a saturated binary tree

A classification or regression tree begins at the root (initial node) where the entire sample is located. Then, each node is defined by the choice of an explanatory variable; and of a division criterion which induces a partition into two or more classes as homogeneous as possible leading to two new nodes that we will try to divide into two new ones (subclasses). The classification process is, therefore, recursive until the separation no longer improves the prediction. i.e., until the terminal node (a leaf) is obtained.

Pruning the optimal tree

The saturated tree tends to produce a very high number of leaves; the previous model is, then, particularly exposed to the risk of over-learning. This phenomenon results in a very high number of parameters in the standard regression, an excess against which it is possible to fight thanks to the automatic selection of the variables or the penalization. To avoid this risk, we use a cross-validation technique called pruning. This technique was introduced by Breiman et al. (1984). Its main objective is to arrive at a less precise and complex model on the learning base but whose ability to predict values on other bases is stable and robust. It consists of expanding the previously constructed tree to the maximum and then removing branches and nodes whose splitting does not significantly improve the tree on the test basis. Indeed, the sum of the quadratic errors is used as an adjustment criterion for a quantitative target variable, as is the case for our modelling.

Random forest

The Random Forest methods were introduced by Breiman (2001). The principle of these methods is to retain the advantages of the CART method while mitigating the effect of over-learning of the model and therefore the complexity of the pruning phase. These algorithms are known for their robustness and their flexibility. They are based on a sampling technique. A bootstrap is performed on the training sample containing p predictor variables. We proceed to the creation of $N$ samples. On the other hand, a step is added between each node of the tree rather than based on all the explanatory variables; we randomly draw m from these variables. On each of these $N$ samples, we build the $N$ different trees without pruning. Generally, if we denote by p the total number of explanatory variables, we will choose $m \sim \sqrt{p}$ for a classification tree or $m \sim \frac{p}{3}$ for a regression tree (Breiman, 2001; Fang et al., 2016; Varian, 2014). This makes it possible to reduce the correlation between each tree, and therefore to have much lower variance of the tree.

Extreme gradient boosting (XGBoost)

XGBoost is proposed by Chen and Guestrin (2016) as an alternative approach for predicting a response variable given certain exogenous variables. Data scientists claim that this algorithm is the fastest and the best performing boosting algorithm. We have chosen this model because it combines the strengths of regression trees model and boosting model (Elith et al., 2008). The general principle behind boosting is that it builds $D$ decision trees (CART) one by one. So, each classification and regression model is trained using the errors made by the previously trained tree.

Support vector regression

Support vector regression, introduced by Vapnik (1995), is a highly effective technique used for solving nonlinear regression problems. In recent years, this algorithm has become well known as one of the most effective machine learning methods. Its main objective is to minimize the generalized error bound. This generalization error bound is the combination of the training error and a regularization term that controls the complexity of the hypothesis space (Basak et al., 2007). It gives us the flexibility to define how much error is acceptable in our model and will find an appropriate line (or hyperplane in higher dimensions) to fit the data. Nevertheless, the hyperparameters of the SVR model should be carefully determined because inappropriate choice of SVR parameters may lead under-fitting or to over-fitting (Lin et al., 2005). Smola and Schölkopf (2004) have given an overview of the basic idea underlying support vector machines for regression and function estimation.

Artificial neural network regression model

Regression ANNs are able to handle non-linear representations and easily learn rich representations by mapping inputs to outputs. These algorithms are flexible in the way they can be used, particularly for pattern recognition, time series and image processing. Notably, they outperform traditional statistical models in terms of speed, simplicity, and capacity (see e.g. Hastie et al. (2009) for additional insights). In the current study, a neural network, featuring one hidden layer, was adapted to predict healthcare insurance premiums. The stopping criterion, crucial for optimal performance, was chosen by identifying the number of neurons in the hidden layer that minimizes errors on the test set. The most effective configuration was found, with a network comprising seven neurons in the hidden layer yielding the minimum error.

Performance metrics

Quadratic prediction error

Several prediction models could be applied to estimate the pure premium of the contracts in the portfolio. It is, therefore, necessary to define performance criteria to compare models and their results. To do this, we will study the criteria of the quadratic prediction error also called “Mean Squared Error”, and “the Relative Mean Squared Error” (MSE/MEAN). The MSE measure quantifies the gap between the predicted model and the perfect model.

Its formula is defined in the literature as follows:

M S E = \frac{1}{n} \sum_{i = 1}^{n} {(\hat{f} (x_{i}) - y_{i})}^{2},

where

n

is the sample size and

\hat{f} (x)

is a prediction function fitted on the set

x_{i} = {x_{i, j}}_{j = 1}^{p}

with p the number of the explanatory variables and

y

is the observed response variable.

Data

Data description

The database used for testing the methodology in this paper corresponds to the care expenses contracts related to specialist consultations gathered throughout one year from a Tunisian insurance company.

The database contains information about the insured population, all medical benefits received by these same insured persons and the various insured enterprises, indicating their type and their sector of activity, thus specifying the establishment service of each enterprise was created to provide both claims and beneficiary information with a total of 228343 contracts. Figure 1 shows the distribution of the average annual costs as a function of the number of individuals (left) and the distribution of the average annual costs as a function of the number of individuals with at least one claim (right). Table 1 below gives more details of the claim amount distribution. Before the modelling, we removed both upper and lower outliers which represent 1.26% of the total amount of claims (see Table 1). In our investigation, both visual and statistical methods have been employed to identify outliers. Our primary aim was to discern and exclude data that exhibit anomalies or deviate significantly from the overall distribution of the dataset. The scatter plot serves as a visual aid in visualizing data point distribution. This technique facilitates the identification of a threshold beyond which data is deemed as outliers. In our case, we set the upper threshold at 140,000. For the statistical analysis, we opted for the modified Z-score method to determine the lower threshold. Consequently, the lower threshold is determined to be 20,000. Thus, 25.15% of policyholders had a claim for an amount between 30,000 and 50,000 TND. These claims contribute to 91.35% of the total claims expense. The bounds of the intervals in Table 1 are chosen with reference to the histogram of claims average costs (right).

Table 1.

Distribution of claim expense.

Claim charge	% observations	% of total amount	Average
0	72	0	0
[20 000, 30 000]	2,73	7,34	28 806,29
[30 000, 50 000]	25,15	91,35	38 872,59
[50 000, 140 000]	0,12	1,31	65 450.60

The setting of the explanatory variables

In this section, we focus on the settings necessary to consider the explanatory variables that have a significant impact on frequency and severity of claims. In fact, when constructing generalized linear models for insurance data, we usually use categorical instead of continuous variables. For example, we might use the gender factor, which is by nature a categorical variable, or even view continuous variables, such as age, as categorical variables by grouping their values. This specificity leads to the need for special treatment. So, we proceed to a classification into categories for each variable as shown in Table 2.

Table 2.

Definition of variables.

Variable	Definition
Dependency	The type of customer. Principal code as 0, Spouse code as 1, Child code as 2
Gender	Gender of customer. Male code as 0, Female code as 1
Age	Age band of customers. Age in [0,6] code as 1, age in [6,25] code as 2, age in [25,46] code as 0, age in [46,61] code as 3, and age in [+61] code as 4
Region	The region of the enterprise to which each customer is attached. Tunis code as 0, Ariana code as 1, Sfax code as 2, Bizerte code as 3, Ben Arous/Nabeul/Sousse code as 4, and others code as 5
Sector	The sector of activity of the company to which each customer is attached. Industry code as 0, services code as 1, finance code as 2, Transport code as 3, and others code as 4
Type of sector	The type of sector of the company to which each customer is attached. Public code as 0, Private code as 1

In this study, we define a reference modality for each explanatory variable, representing the most prevalent category within our portfolio for that specific variable. Consequently, the intercept in the GLM regression model, often referred to us as the reference individual, corresponds to all these chosen reference modalities and therefore, the consumption of any individual will thus be interpreted as overconsumption or underconsumption compared to the reference individual. It’s essential to note that the reference modality is coded as 0. The terms of the reference individual (Intercept) are as follows: Gender: Male; Age: [25, 46[; Dependency: Principal; Type: Public; Region: Tunis; Sector: Industry.

To mitigate the risk of multicollinearity among the modalities of each independent variable, we conducted an Analysis of Variance (ANOVA) test, with the results outlined in Table 3. The statistical method ANOVA effectively dissects observed variance data into distinct components, thereby enhancing the scope for subsequent analyses.

Table 3.

Analysis of Variance test results.

Variable	VIF
Dependency	2.3288
Gender	1.83136
Age	1.74577
Region	1.27025
Type	1.19577
Sector	2.04403

Results from Table 3 show that all the VIF are less than 10. Furthermore, the correlation matrix, as presented in Table 4, indicates that all explanatory variables are weakly correlated. Hence there is no significant evidence of multicollinearity problem.

Table 4.

Correlation matrix.

	Gender	Age	Dependency	Region	Type	Sector
Gender	1	−0.0791	0.1548	−0.0021	0.0030	−0.0038
Age	−0.0791	1	−0.1347	−0.0311	−0.0432	−0.0443
Dependency	0.1548	−0.1347	1	−0.0048	−0.0213	0.0150
Region	−0.0021	−0.0311	−0.0048	1	−0.0230	0.1422
Type	0.0030	−0.0432	−0.0213	−0.0230	1	0.1310
Sector	−0.0038	−0.0443	0.0150	0.1422	0.1310	1

Therefore, the average cost and the average frequency are modeled according to the following six explanatory variables: Dependency, Gender, Age, Region, Type of sector and Sector which are retained after carrying out the likelihood tests on the data processed. Figure 2 displays the distribution of claim’s frequency by Gender (Male, Female), and by type of beneficiary (Principal, Spouse, and Child). And Figure 3 exposes the distribution of Average severity and Frequency of claims by Age band ([0,6[, [6,25[, [25,46[, [46,61[, and [+61[).

According to Figure 2, we can say that the most important claim frequency is mainly for women that is in the order of 90924 (57%), while that for men is in the order of 68024 (43%) claims. Also, we point out that the largest number and severity of claims principally come from people aged between 25 and 45 years old (See Figure 3). However, because of the small number of elderly people in the portfolio studied, about 7%, the frequency of claims that must be high is low. Additionally, the highest number of claims is for members. In fact, the frequency of claims for spouses is lower than that for children, which can be explained by the fact that the number of children insured is large.

Empirical results and discussion

In this section, we provide details of the numerical results of the various methods described in Section 4. Indeed, the handling of observations that did not make a claim is as follows: all observations, including the 72% that did not make a claim, were included in the claim frequency modelling. This is essential to accurately estimate the probability of a policyholder filing a claim. By including all observations, we ensure that our model captures the full spectrum of policyholder behavior, from those who do not file claims to those who do. For the claim cost modelling, only observations with at least one claim were included. This approach is standard practice as the objective here is to model the cost associated with claims. Including observations without claims would not contribute useful information to the cost estimation process and could potentially skew the results.

Coefficient estimates for the GLM model

To model the frequency and severity of claims, it is fundamental to treat the adequacy of empirical laws to theoretical laws to apply generalized linear models. As widely known in the actuarial literature (Charpentier, 2014),we have two discrete distributions which can possibly model the frequency of claims in non-life insurance: the Poisson distribution and the negative binomial distribution. And for the average cost, we have the Gamma and the Log Normal distributions. We carry out below a graphic adjustment of the theoretical law to the empirical law.

From Figure 4 above, we stress that the negative binomial law seems the most adequate for frequency modelling and from Figure 5, it seems that the Gamma distribution better adjusts the average severity. To confirm these results, we proceed to the Chi-square and the Kolmogorov-Smirnov tests, as presented in Table 5. Results indicate that the p values associated with Poisson and Negative Binomial distributions are null. Therefore, we base ourselves on the mean square error criteria to make our decision. The latter has a lower value for the negative binomial distribution. Additionally, it can be noted that the Gamma distribution gives a Kolmogorov-Smirnov statistic lower than that given by the log-normal distribution. Also, the AIC and BIC criteria have smaller values for the Gamma distribution. We will therefore use the Negative Binomial law for frequency modelling and the Gamma distribution for the average cost modelling.

Table 5.

Chi-square and Kolmogorov-Smirnov tests results.

	Poisson	Negative binomial
Sum of residues	0	0.02
Chi-square statistic	47 467.61	234.01
p-value	0	0
Mean squared error	136 035 455	348 112.53
	Gamma	Log normal
KS statistics	0.2628069	0.2867501
AIC	1 403 501	1 425 892
BIC	1 403 520	1 425 910

The regression estimates for the negative binomial and Gamma distributions for frequency and average cost modelling are summarized in Table 6.

Table 6.

Regression estimates for frequency and average Cost modelling on train data.

	Negative binomial GLM			Gamma GLM
	Estimate	Std error	Pr(>\|z\|)	Estimate	Std error	Pr(>\|z\|)
Intercept	−3.35616	0.05820	<2E-16^***	10.54715	0.00190	<2E-16^***
Gender_1	−0.42159	0.01236	<2E-16^***	0.00674	0.00144	2.94E-06^***
Age_1	−0.22648	0.02492	<2E-16^***	−0.05052	0.00626	7.26E-16^***
Age_2	3.51004	0.04514	<2E-16^***	−0.00374	0.00599	0.53188
Age_3	2.19412	0.04279	<2E-16^***	−0.00190	0.00172	0.27088
Age_4	−0.23162	0.02501	<2E-16^***	0.01495	0.00277	6.76E-08^***
Dependency_1	3.38645	0.03697	<2E-16^***	−0.00496	0.00181	0.00614^**
Dependency_2	3.13436	0.03669	<2E-16^***	−0.01152	0.00606	0.05751
Region_1	0.51718	0.02984	<2E-16^***	0.00195	0.00271	0.47144
Region_2	0.64041	0.03640	<2E-16^***	−0.01389	0.00344	5.26E-05^***
Region_3	0.76413	0.03809	<2E-16^***	−0.06126	0.00354	<2E-16^***
Region_4	0.02870	0.04107	0.48471	−0.03224	0.00255	<2E-16^***
Region_5	0.81945	0.03442	<2E-16^***	−0.04948	0.00350	<2E-16^***
Type_1	−0.14569	0.01787	<2E-16^***	0.02218	0.00214	<2E-16^***
Sector_1	0.01273	0.02597	0.62403	0.03147	0.00168	<2E-16^***
Sector_2	0.01621	0.02597	0.53248	0.01895	0.00211	<2E-16^***
Sector_3	−0.21290	0.02695	<2E-16^***	−0.01307	0.00233	2.13E-08^***
Sector_4	0.03528	0.03016	0.24214	0.03218	0.00290	<2E-16^***
2*Loglikelihood	−322 390			−893 520
AIC	322 428			893 559
BIC	322 617			893 724

***, **, * indicates statistical significance at 0.001, 0.01 and 0.05 level respectively.

The coefficient estimates in the output indicates that, with respect to the Negative Binomial distribution and the modalities employed for frequency modeling, all p-values are less than 5% except “Rgion_4”, “Sector_1”, “Sector_2” and “Sector_4”. Consequently, we would say that these covariates are not statistically significant predictors variables for frequency modelling at 5% significance level. In addition, we observe that the “Age_2”, “Age_3”, “Dependency_2” and “Region_1” are not significant predictors variables for average severity modelling since their associated p-values with the Gamma distribution being less than 5% significance level.

Results of machine learning algorithms

In this article, we use a cross-validation scheme. First, we tune the parameters in the algorithm under study on train data (70%). Second, we evaluate the predictive performance of the algorithms on test data (30%).

Hyper parameters

We present in Table 7 below an overview of the parameters used by the different Machine Learning techniques.

Table 7.

Tuning Parameters for Machine Learning algorithms.

Model	Hyperparameters
CART	Complexity parameter $c p$
Random forest	Number of trees $T$
	Number of split candidates $m$
	Warm_start [TRUE, FALSE]
XGBoost	min_sample_split
	Booster [gbtree, gblinear, dart]
	Tree method [auto, exact, approx., hist, gpu_hist]
	learning_rate
SVR	Epsilon is the error sensitivity parameter
	Cost is the regularization parameter
	Gamma is the inverse of the support vector influence radius
ANN	Default hyperparameters associated with the neuralnet() function of the NeuralNet R package

To define the optimal dimension of the CART tree, an adjustment criterion depending on a penalty parameter α was defined. This setting allows for a compromise between the complexity of the structure of the tree and the fit to the original data. Initially, a zero-penalty coefficient is chosen. To pruning the saturated tree, it is necessary to determine the optimal penalization coefficient. The « plotcp » function of R software makes it possible to represent the curve of the prediction errors as a function of the penalty coefficient: the penalty coefficient to be chosen for pruning is therefore, that which minimizes the prediction error thus defined. The optimal trees for frequency and average severity modelling built using the “rpart” command of R software are displayed in Appendix A (see Figures 7 and 8).

For the random forest model, the idea is to retain the advantages of the CART method while overcoming its limitations, mainly the effect of over-fitting the model and therefore, the complexity of the pruning phase. It makes it possible to constitute a forest of $N$ trees for which, at each node, only $m$ explanatory variables can be used as a subdivision criterion. Having 7 explanatory variables, the number of variables available at each node for segmentation was defined by the option « mtry » = 2. Then, we set the number of iterations to 500 and we look for the optimal number of iterations $B^{*}$ which minimizes our error. Hence, we retain the random forest model with 150 iterations to model the frequency and 80 iterations to model the average cost.

Variable importance

The fundamental concept behind Permutation Variable Importance (PVI) of Breiman et al. (1984) is to assess the increase in the deviance of a model after permuting the values of a specific covariate on the model’s performance. In other words, the importance of the variable is quantified by measuring the rise in the deviance of the model when the information provided by an explanatory variable is eliminated. A variable is considered important if the random mixing of its values results in an augmented model deviance, indicating that the model heavily depends on that specific feature for accurate predictions. Conversely, a covariate is not important if the act of shuffling its values yields minimal or negligible increase in the deviance of the model. In general, if a variable has a more substantial impact on one model’s performance compared to another, it suggests that the variable is more critical or influential in the former model. This difference in importance can arise due to various reasons such as the difference in the structure and the complexity of the models. A variable might be more influential in a complex model that can capture intricate patterns, while a simpler model may not give the same weight to that variable.

In this subsection, we conduct a comparative analysis of variable importance plots for both GLM and machine learning algorithms to identify which variable matter for predicting frequency and severity of claims. Figure 6 illustrates the variable importance in the predicting frequency and severity of claims. The left column displays the variable importance for frequency models, while the right column showcases the variable importance for severity models. Upon examination of frequency models, we observe a strikingly similar ranking across GLM, CART, RF, and XGBoost methods. Notably, the risk factors “Age” and “Dependency” emerge as relevant risk factors to these models. However, in the ANN regression model, variables such as “Age,” “Region,” and “Sector” seem far less relevant. Shifting focus to severity models, “Region” and “Sector” variables demonstrate heightened importance in GLM, CART, RF, and XGBoost models. Interestingly, the Region risk factor exhibits no discernible impact on frequency and severity predictions when employing the SVR model.

Prediction performance

Several prediction models could be applied to estimate the pure premium of the contracts in the portfolio. It is, therefore, necessary to define performance criteria to compare models and their results. To do this, we consider the Mean Squared Error (MSE) and the Relative Mean Squared Error (RMSE). The smaller the MSE, the better the model discerns between high- and low-cost members. We compare in Table 8 and Table 9 the predictions in terms of

M S E

and relative MSE from a logistic regression to those from machine learning for frequency and average severity modelling. The empirical average frequency is equal to 0.696 and the empirical average severity is equal to 38 098.59 TND.

Table 8.

Evaluation of $M S E$ and $R M S E$ metrics for Frequency modelling.

Metric	MSE		RMSE
Models	${M S E}_{T r a i n - d a t a}$	${M S E}_{T e s t - d a t a}$	${R M S E}_{T r a i n - d a t a}$	${R M S E}_{T e s t - d a t a}$
GLM (negative binomial)	2.994	2.810	4.30	4.03
CART	2.235	2.211	3.21	3.18
Random forest	2.236	2.181	3.21	3.13
XGBoost	2.237	2.153	3.21	3.09
SVR	2.437	2.363	3.50	3.39
ANN	2.407	2.339	3.46	3.36

Table 9.

Evaluation of $M S E$ and $R M S E$ metrics for Average severity modelling.

Metric	MSE		RMSE
Models	${M S E}_{T r a i n - d a t a}$	${M S E}_{T e s t - d a t a}$	${R M S E}_{T r a i n - d a t a}$	${R M S E}_{T e s t - d a t a}$
GLM (Gamma)	25 951 692	27 964 624	681.17	734.01
CART	27 500 773	25 652 085	721.83	673.31
Random forest	27 278 135	25 502 631	715.99	669.38
XGBoost	26 764 698	25 486 324	702.51	668.96
SVR	28 033 997	25 754 490	735.83	675.99
ANN	27 893 728	26 956 259	732.15	707.54

Regarding performance of the methods on test data, presented in Table 8 and Tables 9 and it can be clearly seen that all machine learning methods significantly outperform GLM in predicting frequency and severity of claims based on MSE metric. Indeed, XGBoost model shows the highest performance on test data with the lowest MSE value of 2.153 for frequency modelling and lowest MSE value of 25 486 324 for average severity modelling as compared to other models. This result is also validated by comparing the RMSE values of this model. However, when examining machine learning methods against each other, both SVR and ANN models exhibit the highest MSE and RMSE values for both frequency and severity in claims modeling. This can be explained by the fact that the SVR and ANN algorithms do not perform very well when the target classes are overlapping.

In specific figures, the MSE for GLM is 27 964 624 and for ANN is 26 956 259, while XGBoost records 25 486 324 on test data for average cost modeling. Hence XGBoost is 10% better than GLM and 6% better than ANN model. Additionally, the standard error on the coefficients for GLM, averaging at “1%”, underscores the significance of the observed differences. It is worth noting that these differences are not as pronounced in the training data, with a mere 2% difference in MSE, and both SVR and ANN models underperforming compared to GLM. Moreover, when categorizing variables into multiple categories, regression trees tend to lose information, and this makes it less effective in making predictions. Besides, while growing the trees, the random forest model is used to add additional randomness to the model by searching for the best risk factor among a random subset of risk factors. This model is known for its flexibility. But the extreme gradient boosting model is highly flexible. It can handle missing data and large datasets efficiently. Further, it has several hyperparameters that can be tuned to improve model performance by minimizing prediction errors. That is why the MSE for extreme gradient boosting technique is much better than for CART and RF models. Furthermore, it can be observed that the MSE and RMSE on the test set for the GLM model in claim frequency and severity modelling are distinctly greater than those on the training set, indicating a drop in predictive accuracy when applied to new data. GLMs are parametric models that can fit training data well, particularly when the relationship between the predictors and the response variable is adequately captured by the linear assumptions of the model. This often results in relatively high-performance metrics for the training dataset. While GLMs can fit training data effectively, there is a potential risk of overfitting, where the model captures noise and specific patterns in the training data that do not generalize well to unseen data. Several studies, including those of McCullagh and Nelder (1989), Harrell (2001) and Babyak (2004), address the issue of overfitting in generalized linear models. This overfitting might not be immediately apparent from the training performance alone but can manifest as reduced performance on test dataset. To sum up, it can be confidently asserted that the premium predicted by the XGBoost model aligns more closely with the empirical premium.

Analysis of the pure premium

We expose in Table 10 the pure premiums calculated by the different models used in this paper.

Table 10.

Pure premium estimated on the train data.

Model	Estimated frequency	Estimated cost (TND)	Estimated premium	Empirical premium
GLM	0.749	38095.41	28 533.46	26 516.623
CART	0.698	38095.32	26 590.53	26 516.623
Random forest	0.698	38095.27	26 590.49	26 516.623
XGBoost	0.696	38098.37	26 516.46	26 516.623
SVR	0.695	38089.76	26 472.38	26 516.623
ANN	0.697	38095.32	26 552.43	26 516.623

The empirical average frequency is equal to 0.696 and the empirical average severity is equal to 38 098.59. Comparing the calculated pure premiums by the different models studied from Table 10, we find that the generalized linear model overestimates the average pure premium. While the ANN model adeptly approximates the pure premium, closely aligning with the empirical average, it presents a challenge in terms of interpretability as it is difficult to identify the effect of a variable on the final result. Additionally, the extreme gradient boosting model estimates the pure premium very close to the empirical average. This model facilitates the effective identification of risks, enabling the precise determination of premiums for policyholders that align with their individual risk profiles. This not only enhances the accuracy of premium pricing but also promotes fairness, ensuring that policyholders are charged in a manner that reflects their specific risk factors. Ultimately, the use of this model contributes to a more equitable and reasonable approach to premium pricing in the insurance industry. In summary, the experiments based on Tunisian healthcare insurance dataset indicate that the application of machine learning techniques based on a set of selected features outperforms the classical GLM, and the proposed methodology is fundamentally adaptable to different health insurance databases. Also, in the pricing process of individual healthcare insurance contracts, the most important features for frequency and severity of claim’s modelling are Age (Kelly & Nielson, 2006), Dependency and Region. In our work, we aimed to make the best use of the provided data to offer valuable insights into health insurance premium pricing based on the available features in the current dataset. However, we note that the inclusion of additional variables such that smoking and drinking could further enhance the accuracy and comprehensiveness of our pricing model. Furthermore, the pricing of individual health insurance contracts is based on the risk profile of each individual. Indeed, estimates of the average frequency and average cost for each individual, conditional on their specific characteristics, serve as indicators of their respective risk levels. Consequently, individuals with a higher risk profile will have a higher individual pure premium and will therefore pay more.

Conclusion

The constant evolution of the healthcare market as well as the capture of new low-risk clients challenge insurers to develop new methodologies for pricing and cultivating a profitable portfolio. In our study, we presented the different pricing methods for individual healthcare contracts. Indeed, during modelling by generalized linear models, an alternative method to the traditional frequency-cost approach was favored. We assumed a negative binomial distribution for modelling the frequency of complaints and a Gamma distribution for modelling the average severity. However, these models do not make it possible to bring out all the information on the insured risk even though they impose both constraints on the structure of the modeled risk and the interactions between explanatory variables of the risk. To overcome this limitation, we used non-parametric models from machine learning algorithms. The modeling process consists of subdividing the database into a train and a test sample to optimize the results and limit the phenomena of over-learning. The results provided are, therefore, more robust. The comparative analysis of the different models showed that the gradient boosting algorithms, particularly the XGBoost, were deemed to be the most satisfactory. In conclusion, our study highlights the importance of integrating AI into health insurance pricing. Our empirical finding reveals the effective predictive power of ML models compared to the classical statistical learning models (GLM) in the case of large-scale databases. The dynamic and adaptive nature of these innovative methods facilitates the identification of complex patterns in datasets, enabling continuous learning and adjustment to align premiums with policyholders' risk profiles. However, the challenges facing researchers and practitioners, who are actively shaping the future landscape of healthcare pricing, include issues such as model fairness and ethical considerations (see e.g. Char et al., 2020). Particularly in the field of data-driven decision making, it is very important to prevent potential discriminatory outcomes generated by AI models. Addressing these challenges is crucial to promote the responsible and ethical use of AI techniques, thereby contributing to the quest for transparency and unbiased ratemaking. Our findings have potential implication in health insurance market highlighting how advanced predictive models can improve customer segmentation and provide personalized pricing.

Footnotes

Acknowledgements

We are grateful to the anonymous referees whose valuable suggestions and remark allowed us to greatly improve the original version of this paper. We would like to thank Mr Ksouri Najed actuary and director of ERM partner which is a consulting firm in Risk Management and Actuarial in the insurance, finance and industry sector based in Tunisia for providing us with the database used in this study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Amal Ben Hamida

Manel Kacem

Data availability statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to restrictions related to protecting the proprietary information of the insurance company. The data contain confidential and sensitive information related to the company’s operations.

Appendix

Figure 7.

The optimal tree for Frequency modeling.

Figure 8.

The optimal tree for Average Severity modeling.

References

Babyak

M. A.

(2004). What you see may not be what you get: A brief, nontechnical introduction to overfitting in regression-type models. Psychosomatic Medicine, 66(3), 411–421. https://doi.org/10.1097/01.psy.0000127692.23278.a9

Basak

Pal

Patranabis

D. C.

(2007). Support vector regression neural information processing–Letters and reviews.

Baudry

Robert

C. Y.

(2019). A machine learning approach for individual claims reserving in insurance. Applied Stochastic Models in Business and Industry, 35(5), 1127–1155. https://doi.org/10.1002/asmb.2455

Blesa

Íñiguez

Moreno

Ruiz

(2020). Use of open data to improve automobile insurance premium rating. International Journal of Market Research, 62(1), 58–78. https://doi.org/10.1177/1470785319862734

Blier-Wong

Cossette

Lamontagne

Marceau

(2020). Machine learning in P&C insurance: A review for pricing and reserving. Risks, 9(1), 4. https://doi.org/10.3390/risks9010004

Breiman

(2001). Random forests. Machine Learning, 45(1), 5–32. https://doi.org/10.1023/a:1010933404324

Breiman

Friedman

Stone

C. J.

Olshen

R. A.

(1984). Classification and regression trees. Taylor & Francis.

Castellani

Fiore

Marino

Passalacqua

Perla

Scognamiglio

Zanetti

(2021). Machine learning techniques in nested stochastic simulations for life insurance. Applied Stochastic Models in Business and Industry, 37(2), 159–181. https://doi.org/10.1002/asmb.2607

Char

D. S.

Abràmoff

M. D.

Feudtner

(2020). Identifying ethical considerations for machine learning healthcare applications. The American Journal of Bioethics, 20(11), 7–17. https://doi.org/10.1080/15265161.2020.1819469

10.

Charpentier

(2014). Computational actuarial science with R. CRC Press.

11.

Chen

Guestrin

(2016). Xgboost: A scalable tree boosting system. Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, 785–794.

12.

De Jong

Heller

G. Z.

(2008). Generalized linear models for insurance data. Cambridge Books.

13.

Dey

A. K.

Lyubchich

Gel

Y. R.

(2021). Modeling Weather-induced Home insurance risks with support vector machine regression. ArXiv Preprint ArXiv:2103.08761.

14.

Elith

Leathwick

J. R.

Hastie

(2008). A working guide to boosted regression trees. Journal of Animal Ecology, 77(4), 802–813. https://doi.org/10.1111/j.1365-2656.2008.01390.x

15.

Fang

Jiang

Song

(2016). Customer profitability forecasting using big data analytics: A case study of the insurance industry. Computers & Industrial Engineering, 101(1), 554–564. https://doi.org/10.1016/j.cie.2016.09.011

16.

Grize

Fischer

Lützelschwab

(2020). Machine learning applications in nonlife insurance. Applied Stochastic Models in Business and Industry, 36(4), 523–537. https://doi.org/10.1002/asmb.2543

17.

Haberman

Renshaw

A. E.

(1996). Generalized linear models and actuarial science. Journal of the Royal Statistical Society: Series D (The Statistician), 45(4), 407–436. https://doi.org/10.2307/2988543

18.

Hang

Banks

(2019). Machine learning applied to pack classification. International Journal of Market Research, 61(6), 601–620. https://doi.org/10.1177/1470785319841217

19.

Harrell

F. E.

(2001) Regression modeling strategies: with applications to linear models, logistic regression, and survival analysis (608). Springer.

20.

Hartman

Owen

Gibbs

(2020). Predicting high-cost health insurance members through boosted trees and oversampling: An application using the HCCI database. North American Actuarial Journal, 25(1), 53–61. https://doi.org/10.1080/10920277.2020.1754242

21.

Hastie

Tibshirani

Friedman

J. H.

Friedman

J. H.

(2009) The elements of statistical learning: data mining, inference, and prediction (2). Springer.

22.

Henckaerts

Côté

M.-P.

Antonio

Verbelen

(2020). Boosting insights in insurance tariff plans with tree-based machine learning methods (pp. 1–31). North American Actuarial Journal.

23.

Huang

Meng

(2019). Automobile insurance classification ratemaking based on telematics driving data. Decision Support Systems, 127(4), 113156. https://doi.org/10.1016/j.dss.2019.113156

24.

Kaas

Goovaerts

Dhaene

Denuit

(2008) Modern actuarial risk theory: using R (128). Springer Science & Business Media.

25.

Kaushik

Bhardwaj

Dwivedi

A. D.

Singh

(2022). Machine learning-based regression framework to predict health insurance premiums. International Journal of Environmental Research and Public Health, 19(13), 7898. https://doi.org/10.3390/ijerph19137898

26.

Kelly

Nielson

(2006). Age as a variable in insurance pricing and risk classification. The Geneva Papers on Risk and Insurance - Issues and Practice, 31(2), 212–232. https://doi.org/10.1057/palgrave.gpp.2510082

27.

Yan

Liu

(2018). A principle component analysis-based random forest with the potential nearest neighbor method for automobile insurance fraud identification. Applied Soft Computing, 70(1), 1000–1009. https://doi.org/10.1016/j.asoc.2017.07.027

28.

Lin

P.-T.

S.-F.

Lee

T.-T.

(2005). Support vector regression performance analysis and systematic parameter selection. Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2(2), 877–882. https://doi.org/10.1109/IJCNN.2005.1555968

29.

McCullagh

Nelder

J. A.

(1989). Generalized linear models (2nd ed.). Chapman & Hall.

30.

Nelder

J. A.

Wedderburn

R. W. M.

(1972). Generalized linear models. Journal of the Royal Statistical Society: Series A, 135(3), 370–384. https://doi.org/10.2307/2344614

31.

Neumann

Ł.

Nowak

R. M.

Okuniewski

Wawrzyński

(2019). Machine learning-based predictions of customers’ decisions in car insurance. Applied Artificial Intelligence, 33(9), 817–828. https://doi.org/10.1080/08839514.2019.1630151

32.

Quan

Valdez

E. A.

(2018). Predictive analytics of insurance claims using multivariate decision trees. Dependence Modeling, 6(1), 377–407. https://doi.org/10.1515/demo-2018-0022

33.

Rampal

Singh

Kaur

Singh

(2020). An ensemble model to predict health insurance premium using machine learning. Dogo Rangsang Research Journal, 10(08).

34.

Rustam

Ariantari

N. P. A. A.

(2018). Support Vector Machines for classifying policyholders satisfactorily in automobile insurance. Journal of Physics: Conference Series, 1028(1), 12005. https://doi.org/10.1088/1742-6596/1028/1/012005

35.

Smola

A. J.

Schölkopf

(2004). A tutorial on support vector regression. Statistics and Computing, 14(3), 199–222. https://doi.org/10.1023/b:stco.0000035301.49549.88

36.

Spedicato

G. A.

Dutang

Petrini

(2018). Machine learning methods to perform pricing optimization. A comparison with standard GLMs. Variance, 12(1), 69–89.

37.

Staudt

Wagner

(2021). Assessing the performance of random forests for modeling claim severity in collision car insurance. Risks, 9(3), 53. https://doi.org/10.3390/risks9030053

38.

Vapnik

(1995). The nature Stat. Learning theory. Springer-Verlag.

39.

Vapnik

(1998). The support vector method of function estimation. In Nonlinear modeling (pp. 55–85). Springer.

40.

Varian

H. R.

(2014). Big data: New tricks for econometrics. The Journal of Economic Perspectives, 28(2), 3–28. https://doi.org/10.1257/jep.28.2.3

41.

Wuthrich

M. V.

Buser

(2019). Data analytics for non-life insurance pricing (pp. 16–68). Swiss Finance Institute Research. Paper.

42.

Yang

Qian

Zou

(2018). Insurance premium prediction via gradient tree-boosted Tweedie compound Poisson models. Journal of Business & Economic Statistics, 36(3), 456–470. https://doi.org/10.1080/07350015.2016.1200981