Machine Learning in P&amp;C Insurance: A Review for Pricing and Reserving

Christopher Blier-Wong; Hélène Cossette; Luc Lamontagne; Etienne Marceau

doi:10.3390/risks9010004

Machine Learning in P&C Insurance: A Review for Pricing and Reserving

Blier-Wong, Christopher;Cossette, Hélène;Lamontagne, Luc;Marceau, Etienne

2020-12-23 00:00:00

risks Review Machine Learning in P&C Insurance: A Review for Pricing and Reserving 1 1 2 1, Christopher Blier-Wong , Hélène Cossette , Luc Lamontagne and Etienne Marceau * École d’actuariat, Université Laval, Québec, QC G1V 0A6, Canada; christopher.blier-wong.1@ulaval.ca (C.B.-W.); helene.cossette@act.ulaval.ca (H.C.) Département d’informatique et de génie logiciel, Université Laval, Québec, QC G1V 0A6, Canada; luc.lamontagne@ift.ulaval.ca * Correspondence: etienne.marceau@act.ulaval.ca Abstract: In the past 25 years, computer scientists and statisticians developed machine learning algorithms capable of modeling highly nonlinear transformations and interactions of input features. While actuaries use GLMs frequently in practice, only in the past few years have they begun study- ing these newer algorithms to tackle insurance-related tasks. In this work, we aim to review the applications of machine learning to the actuarial science ﬁeld and present the current state of the art in ratemaking and reserving. We ﬁrst give an overview of neural networks, then brieﬂy outline applications of machine learning algorithms in actuarial science tasks. Finally, we summarize the future trends of machine learning for the insurance industry. Keywords: machine learning; ratemaking; reserving; property and casualty insurance; neural networks 1. Introduction The use of statistical learning models has been a common practice in actuarial science since the 1980s. The ﬁeld quickly adopted linear models and generalized linear models for ratemaking and reserving. The statistics and computer science ﬁelds continued to develop Citation: Blier-Wong, C.; Cossette, H.; more ﬂexible models, outperforming linear models in several research ﬁelds. To our Lamontagne, L.; Marceau, E. Machine knowledge and given the sparse literature on the subject, the actuarial science community Learning in P&C Insurance: A Review largely ignored these until the last few years. In this paper, we review nearly a hundred for Pricing and Reserving. Risks 2021, articles and case studies using machine learning in property and casualty insurance. 9, 4. https://dx.doi.org/10.3390/ A case study comparing machine learning models for ratemaking was conducted by risks9010004 Dugas et al. (2003), who compared ﬁve classes of models: linear regression, generalized linear models, decision trees, neural networks and support vector machines. From their Received: 30 October 2020 concluding remarks, we read, “We hope this paper goes a long way towards convinc- Accepted: 11 December 2020 ing actuaries to include neural networks within their set of modeling tools for ratemak- Published: 23 December 2020 ing.” Unfortunately, it took 15 years for this suggestion to be noticed. Recent events Publisher’s Note: MDPI stays neu- have sparked a spurge in the popularity of machine learning, especially in neural net- tral with regard to jurisdictional claims works. Frequently quoted reasons for this resurgence include introducing better activation in published maps and institutional functions, datasets composed of many more images, and much more powerful GPUs afﬁliations. LeCun et al. (2015). Machine learning algorithms learn patterns from data. Linear regression learns lin- ear relationships between features and a response variable, which may be too simple to reﬂect the real world. Generalized linear models (GLMs, also including logistic regression Copyright: © 2020 by the authors. Li- (LR)) add a link function to express a random variable’s mean as a function of a linear censee MDPI, Basel, Switzerland. This relationship between features. This addition enables the modeling of simple nonlinear article is an open access article distributed effects. For example, a logarithmic link function produces multiplicative relationships under the terms and conditions of the between input features and the response variable. While these models are simple, eas- Creative Commons Attribution (CC BY) ily explainable and have desirable statistical properties, they are often too restrictive to license (https://creativecommons.org/ learn complex effects. Property and casualty insurance (P&C) covers risks that result from licenses/by/4.0/). Risks 2021, 9, 4. https://dx.doi.org/10.3390/risks9010004 https://www.mdpi.com/journal/risks Risks 2021, 9, 4 2 of 26 the combination of multiple sources (causes), including behavioral causes. Rarely will a linear relationship be enough to model complex behaviors. Nonlinear transformations and interactions between variables could more accurately reﬂect reality. To include these effects in GLMs, the statistician must create these features by hand and include them in the model. For example, to model a 3rd degree polynomial of a variable x, we would need to 2 3 supplement x, x and x as new features. Creating these transformations and interactions by hand (a task called feature engineering) is tedious, so only simple transformations and interactions are usually tested by actuaries. A ﬁrst truly signiﬁcant advantage of recent machine learning models simpliﬁes the pre- vious drawback: recent models learn nonlinear transformations and interactions between variables from the data without manually specifying them. This is performed implicitly with tree-based models and explicitly with neural networks. The second advantage of machine learning is that many models exist for different types of feature formats. For instance, convolutional neural networks may model data where order or position is essential, like text, images, and time series of constant length. Recurrent neural networks may model sequential data like text and time series (think ﬁnancial data, telematics trips or claim payments). Most data created today is unstruc- tured, meaning it is hard to store in spreadsheets or other traditional support. Historical approaches to dealing with this data have been to structure them ﬁrst (actuaries aggregate individual reserves in structured triangles). Much individual information is lost when structuring (aggregating). Many machine learning models can take the unstructured data directly, opening possibilities for actuaries to better understand the problems, data and phenomenon they study. The ﬁeld of machine learning is expanding rapidly and shows great promise for use in actuarial science. The introduction of machine learning in actuarial science is recent and not neatly organized: when reviewing the literature, we identiﬁed independent and exclusive contributions. In this review, we analyze and synthesize the work conducted in this area. For each topic, we present the relevant literature and provide possible future directions for research. 1.1. Research Methodology We followed a structured methodology to search for contributions in this review. A three-pronged approach was used: 1. Query research databases (Google Scholar, ProQuest, SSRN, arXiv, ResearchGate) for a combination of machine learning keywords (machine learning, data science, decision tree (DT), classiﬁcation and regression trees (CART), neural network (NN) convolutional neural networks (CNN), recurrent neural networks (RNN), random forest (RF), gradient boosting (GBM/GBT/XGBoost), generalized additive model (GAM, GAMLSS), support vector machine (SVM, SVR, SVC), principal component analysis (PCA), autoencoders (AE), computer science) AND the subjects of interest in our review (actuarial science, general insurance, home insurance, auto insurance, P&C insurance, ratemaking, reserving). 2. Query actuarial journals (in no particular order, Risks, ASTIN Bulletin, Insurance: Mathematics and Economics (IME), Scandinavian Actuarial Journal (SAJ), Variance, North American Actuarial Journal (NAAJ), European Actuarial Journal (EAJ)). 3. For each pertinent article, we searched references therein for similar contributions. In the introduction, we included publications classiﬁed as overviews. References for books and lecture notes are also included. These overview publications are often not peer- reviewed and do not propose new modeling approaches but provide empirical evidence or strategic plans that set the stage for research. Overviews consist of white papers, case studies, reviews, surveys and reports if published in research journals or conference proceedings, sponsored by professional actuarial organizations or large insurance companies. Risks 2021, 9, 4 3 of 26 In the main review on reserving and ratemaking, we limited the contributions to articles in journals, conferences, and preprints. The time limit for research was August 2020. Although we do not have a beginning time limit, papers before 2015 are mainly included for historical context. This review considered papers that have analyzed the topic of machine learning in pricing or reserving. Priority was given to models that adapted machine learning models to a speciﬁc insurance task. Due to the unique structure of reserving data, most contributions for reserving ﬁt this criterion. We also included papers that have analyzed the topic of machine learning by justifying the use of a speciﬁc algorithm within a context or providing speciﬁc conclusions or model interpretations for the selected machine learning model. We read several papers where the authors proposed that a machine learning model could be used to perform a certain sub-task of a more signiﬁcant process. Unless the model’s choice was justiﬁed, we did not consider these papers as part of this review. We organize contributions with a thematic approach while using chronological order- ing within themes. 1.2. Scope of This Review and Similar Work In this paper, we review the literature on machine learning in P&C insurance. Due to the early stage of research, the literature is mostly composed of white papers and case studies. We include the works that have been useful to set the stage for research. Addi- tionally, machine learning innovations are produced at a very high rate. This literature review contains some non-peer-reviewed works so that it is up to date with the current state-of-the-art. In the past few years, efforts similar to ours include white papers, comparative studies, surveys from industry, and lecture notes. In Table 1, we summarize the aspects treated in each contribution. Table 1. Overview papers on machine learning in actuarial science. Description Reference Methodologies/Approaches Book Frees et al. (2014a, 2014b) GLM, GAM Comparative study Dugas et al. (2003) GLM, DT, NN, SVM Comparative study Noll et al. (2018) GLM, DT, GBT, NN Comparative study Diana et al. (2019) GLM, RF, GBT, NN Comparative study Lee and Antonio (2015) GLM, GAM, NN, GBT, CART Comparative study Kašcelan ´ et al. (2016) SVR, Kernel LR Comparative study Fauzan and Murﬁ (2018) GBT, AdaBoost, RF, NN Comparative study Maynard et al. (2019) XGBoost, RF, LR, NN, Lecture notes Wuthrich and Buser (2019) GLM, GAM, NN, RF, GBM, SVM Lecture notes Denuit et al. (2019a, 2019b, 2019c) GLM, GAM, GBM, NN Report Bothwell et al. (2016) – Report Harej et al. (2017) NN Report Jamal et al. (2018) RF, NN, GBM Review Corlosquet-Habart and Janssen (2018) NN, RF, GBM, SVM Review Albrecher et al. (2019) – Review Grize et al. (2020) CART, NN, XGBoost Review Smietanka et al. (2020) – Review Richman(2020a, 2020b) NN Survey Rioux et al. (2019) – White paper Bruer et al. (2015) – White paper Panlilio et al. (2018) GLM, GBT, NN White paper Richman et al. (2019) NN Risks 2021, 9, 4 4 of 26 Recent interest in predictive modeling in actuarial science has emerged, and Frees et al. (2014a) presented a survey of early applications of such models. The premise is encouraging: as data becomes more abundant and machine learning models more robust, insurers should have the capacity to capture most heterogeneity represented by insured individuals and compute a premium that represents their individual risk in a more accurate way. We assume the reader is familiar with most statistical learning models such as GLMs, generalized additive models, random forests, gradient boosted machines, support vector machines and neural networks. Otherwise the reader is directed to Friedman et al. (2001) or Wuthrich and Buser (2019). The only model described in this review is the neural network since we believe it is underrepresented in actuarial science. Many white or review papers reﬂecting on the use of big data and machine learning in actuarial science are available, we highlight Richman (2020a, 2020b) (with a focus on deep learning). The ASTIN Big Data/Data Analytics working party published Bruer et al. (2015), composed of a collection of ideas concerning the direction of data analytics and big data, which our paper wishes to update ﬁve years later. Another work related to this paper but with a wider (but less speciﬁc) scope is Corlosquet-Habart and Janssen (2018), who collected high-level ideas of the use of big data in insurance. They present general machine learning techniques, while our goal is to present machine learning applications in actuarial science. In Grize et al. (2020), the authors present case studies on car insurance and home insurance pricing. We highlight the 6th section of that paper, enumerating several challenges for the insurance industry, including establishing a data-oriented company culture, continuing education and ethical concerns, including fairness and data ownership. We expand on the issue of fairness later in this paper. Professional organizations have also shown great interest in machine learning and big data by creating working parties and calls for papers. The working group, “Data Science” of the Swiss Association of Actuaries, has recently published a series of tutorials to offer actuaries an easy introduction to data science methods with actuarial use, see Noll et al. (2018), Ferrario et al. (2018), Schelldorfer and Wuthrich (2019), Ferrario and Hämmerli (2019) and others. The Casualty Actuarial Society (CAS) had a data and tech- nology working party who published a report Bothwell et al. (2016). The Institute and Faculty of Actuaries set up the Modelling, Analytics and Insights in Data working party and published their conclusions in Panlilio et al. (2018). The Society of Actuaries sponsored a survey of machine learning in insurance in Diana et al. (2019). The Society of Actuaries and the Canadian Institute of Actuaries sponsored a survey of predictive analytics in the Canadian life insurance industry Rioux et al. (2019). The Society of Actuaries also published a report on harnessing the new sources of data and the skills actuaries will need to deal with these new issues . Actuarial research journals have also been announcing special 3 4 issues on predictive analytics, for example, Variance and Risks. 1.3. Generalized Data on This Review To provide an overall view of the research, we provide generalized data of papers covered in the review. Figure 1 presents a breakdown of the 77 publications by year since 2015 (among the contributions in Tables 1–4). The increasing trend shows how current this subject is. Note that 2020 data are limited to August. We observe that pricing has started using machine learning before reserving because the research context is already familiar with pricing using generalized linear models. Reserving, being an unstructured source of data is less straightforward, but the number of publications using machine learning has increased in the past two years. https://www.soa.org/resources/research-reports/2019/big- data-future- actuary/. http://www.variancejournal.org/issues/archives/?fa=article_list&year=2018&vol=12&issue=1. Claim Models Taylor (2020), Machine Learning Asimit et al. (2020) and Finance, insurance and risk management (https://www.mdpi.com/journal/ risks/special_issues/Machine_Learning_Finance_Insurance_Risk_Management). Risks 2021, 9, 4 5 of 26 8 8 7 7 6 6 5 5 5 3 3 3 2 2 1 1 1 0 0 0 2000-2014 2015 2016 2017 2018 2019 2020 (aug) Overview Pricing Reserving Figure 1. Number of publications per year. Figure 2 presents the number of publications by source (among the contributions in Tables 2–4). Journals with a single paper were grouped in Other and mainly consisted of pricing. The categories of journals included in Other are business, statistics, expert systems, ﬁnance. 4 4 3 3 3 3 3 3 3 3 2 2 2 1 1 1 0 0 0 arXiv ASTIN CAS EAJ IME NAAJ Risks SAJ SSRN Other Pricing Reserving Figure 2. Number of publications by source. Finally, the distribution of model families is presented in Figure 3 (among the contribu- tions in Tables 2–4). In our experience and after analyzing the best models in competitions hosted on Kaggle, decision tree ensembles work best for structured problems, while neural networks work best for unstructured problems. This is in line with the breakdown of mod- els in this review: pricing uses structured data and boosting (XGBoost, GBT) is the most popular pricing framework, while reserving uses unstructured data (due to the triangular format of aggregated reserves or the time series format of individual reserves) and neural networks are the most popular for reserving models. We believe that GAMs are popular for pricing since actuaries are already familiar with generalized linear models, and GAMs are generalizations of GLMs. The remainder of the paper is organized as follows. In Section 2, we brieﬂy introduce neural networks and present two methods to estimate the parameters of a probability distribution. Section 3 covers machine learning applications to ratemaking, while Section 4 covers their applications to reserving. Section 5 concludes the review by summarizing the future trends and challenges using machine learning in insurance. Publications Publications Risks 2021, 9, 4 6 of 26 6 6 3 3 2 2 1 1 1 Neural networks CART Boosting GAM/GAMLSS Unsupervised SVM Pricing Reserving Figure 3. Number of publications by model. 2. Neural Networks In this section, we present a brief introduction to fully connected neural networks. We also present how to estimate the parameters of random variables with this model. Neural networks construct a function f such that f (x , q) = y , i = 1, . . . , n, where i i x corresponds to the features in the model, y is a response variable and q are model i i parameters. This function is built as a composition (aggregation) of functions (layers) f (x ) = f f f (x ), i = 1, . . . , n. (1) i 3 2 1 i In this case of 3 chained functions, f corresponds to the first layer, f to the second layer 1 2 and f to the third layer. Since we are not interested in the first or second layers’ output, we call these hidden layers. The last layer is called the output layer since this is the output of the classification or regression model. The number of chained functions is called the depth of the model. Each function is nonlinear; composing multiple functions produces a highly nonlinear model, thus having much flexibility to estimate the function f . 2.1. Basics and Notation Let x = (x , . . . , x ) 2 R be the p-dimensional features for observation i inputted i i1 i p into the neural network. We deﬁne the ﬁrst hidden layer as (1) (1) (1) (1) h = g z , j = 1, . . . , J , (2) j j with (1) (1) (1) (1) z = w x + b , j = 1, . . . , J , (3) å ik j k j j k=1 (1) (1) where J is the width of the ﬁrst hidden layer, g is a nonlinear function called the activation function. If the width is equal to 1 and the activation function g is the sigmoid function g(x) = (4) 1 + e (sometimes noted s(x)), we recognize the inverse link function in the logistic regression. However, we could use the hidden layer values as input variables in another function. The second hidden layer values are (2) (2) (2) (2) h = g z , j = 1, . . . , J , (5) j j with (2) (2) (2) (1) (2) (2) z = w h + b , j = 1, . . . , J , (6) j i j j j i=1 Publications Risks 2020, 1, 0 8 of 27 (sometimes noted σ(x)), we recognize the inverse link function in the logistic regression. However, we could use the hidden layer values as input variables in another function. The second hidden layer values are (2) (2) (2) (2) h = g z , j = 1, . . . , J , (5) j j with (2) (2) (2) (1) (2) (2) z = w h + b , j = 1, . . . , J , (6) j i j j j i=1 (2) (2) where J is the width of the second hidden layer, and g is the second activation function. We may then repeat this process for L layers, where values of the hidden layer are Risks 2021, 9, 4 7 of 26 (l) (l) (l) (l) h = g z , j = 1, . . . , J , 1 ≤ l < L, (7) j j (2) (2) where J is the width of the second hidden layer, and g is the second activation function. with We may then repeat this process for L layers, where values of the hidden layer are (l−1) (l) (l) (l−1) (l) (l) (l) (l) (l)(l) h = g z , j = 1, . . . , J , 1 l < L, (7) z = w h + b , j = 1, . . . , J , l < l ≤ L, (8) ∑ j j j i j j j i=1 with (l1) and (l) (l) ( l1) (l) (l) z = w h + b , j = 1, . . . , J , l < l L, (8) (L) j (L) i j j j (L) o = g z , j = 1, . . . , J , (9) i=1 and (L) (L) (L) (L) o = g z , j = 1, . . . , J , (9) where J is the output size. In Figure 4, we present the graphical diagram for a neural network with (1) (2) (L) two hidden layers. Here, J = J which is often the case in practice but does not need to be. Usually, where J is the output size. In Figure 4, we present the graphical diagram for a neural (1) (2) (L) (L) network with two hidden layers. Here, J = J which is often the case in practice J = 1 for regression, such that the model predicts a single value and g can be interpreted as (L) but does not need to be. Usually, J = 1 for regression, such that the model predicts a the GLM link function. In other cases, notably when the neural network predicts the parameters of (L) single value and g can be interpreted as the GLM link function. In other cases, notably (L) (L) when the neural network predicts the parameters of a probability distribution, J will a probability distribution, J will correspond to the number of parameters that deﬁne the random correspond to the number of parameters that deﬁne the random variable. We will return to variable. We will return to this in Section 2.2. this in Section 2.2. Input Hidden Hidden Output layer layer layer layer (1) (2) h h 1 1 . . . . . . (L) (1) (2) h h (1) (2) J J Figure 4. Graphical representation of a neural network. Figure 4. Graphical representation of a neural network. Along with the sigmoid function deﬁned in (4), popular choices of activation (nonlin- Along with the sigmoid earity) functions function are the deﬁned hyperbolic in tangent (4), popular (tanh), given choices by of activation (non-linearity) functions are the hyperbolic tangent (tanh), given by x x e e g(x) = tanh(x) = 2s(2x) 1 = (10) x x e + e x −x e − e and the Rectiﬁed Linear Unit (ReLU), deﬁned by g(x) = tanh(x) = 2σ(2x)− 1 = (10) x −x e + e g(x) = max(0, x). (11) and the Rectiﬁed Linear Unit (ReLU), deﬁned by We brieﬂy reviewed neural networks in this section, but interested readers may refer to Goodfellow et al. (2016) for a comprehensive overview of the ﬁeld. Neural networks may be used in regression tasks and classiﬁcation tasks. For regres- g(x) = max(0, x). (11) sion, there is a single output value representing the prediction. To better illustrate this idea, β Risks 2020, 1, 0 9 of 27 Risks 2020, 1, 0 9 of 27 We brieﬂy reviewed neural networks in this section, but interested readers may refer to Goodfellow et al. (2016) for a comprehensive overview of the ﬁeld. Risks 2021, 9, 4 8 of 26 We brieﬂy reviewed neural networks in this section, but interested readers may refer to Neural networks may be used in regression tasks and classiﬁcation tasks. For regression, there Goodfellow et al. (2016) for a comprehensive overview of the ﬁeld. is a single output value representing the prediction. To better illustrate this idea, we present the link Neural networks may be used in regression tasks and classiﬁcation tasks. For regression, there between neural networks and GLMs. The prediction formula for a GLM is is a single output value representing the prediction. To better illustrate this idea, we present the link we present the link between neural networks and GLMs. The prediction formula for a between neural networks and GLMs. The prediction formula for a GLM is GLM is −1 ! y ˆ= E[Y ] = g x β + β . (12) i i ∑ i j j 0 −1 j=1 y ˆ = E[Y ] = g x b + b . (12) y ˆ= E[Y ] = g x β + β . (12) i i 0 å i j j 0 i i ∑ i j j j=1 j=1 A neural network with no hidden layer and one output neuron corresponds to a GLM. This A neural network with no hidden layer and one output neuron corresponds to a GLM. A neural network with no hidden layer and one output neuron corresponds to a GLM. This process is shown in Figure 5. In neural network graph notation, each node (other than in the input This process is shown in Figure 5. In neural network graph notation, each node (other than process is shown in Figure 5. In neural network graph notation, each node (other than in the input layer) implicitly contains an activation function, omitting to draw a node for g. Each arrow between layer) implicitly in contains the input an layer) activation implicitly function, contains omitting anto activation draw a node function, for g. Each omitting arrowto between draw a node for g. nodes has a weight, and the bias is also assumed. nodes has a weight, Each arr and ow thebetween bias is also nodes assumed. has a weight, and the bias is also assumed. x 1 x x 1 x 1 x −1 g y ˆ y ˆ −1 3 g y ˆ y ˆ . .. . . . x (a) (b) (a) (b) Figure 5. Visualizing a GLM in a neural network graph diagram. (a) Graph in GLM notation. (b) Graph (a) (b) Figure 5. Visualizing a GLM in a neural network graph diagram. (a) Graph in GLM notation. (b) Graph in NN notation. in NN notation. Figure 5. Visualizing a GLM in a neural network graph diagram. (a) Graph in GLM notation. (b) Graph Therefore, a neural network with many hidden layers may be viewed as stacked in NN notation. Therefore, a neural network with many hidden layers may be viewed as stacked GLMs. GLMs. Each hidden layer adds nonlinearity and can learn complex functions and non- Each hidden layer adds non-linearity and can learn complex functions and non-linear interactions Therefore, a neural network with many hidden layers may be viewed as stacked GLMs. linear interactions between input values. We may interpret the output layer as a GLM between input values. We may interpret the output layer as a GLM on transformed input variables, on transformed input variables, where the model learns the necessary transformations, Each hidden layer adds non-linearity and can learn complex functions and non-linear interactions where the model learns the necessary transformations, performing automatic feature engineering. performing automatic feature engineering. between input values. We may interpret the output layer as a GLM on transformed input variables, A signiﬁcant drawback of neural networks is that they are black boxes and offer minimal A signiﬁcant drawback of neural networks is that they are black boxes and offer theoretical guarantees. In order to perform risk management, we also need a probability distribution. where the model learns the necessary transformations, performing automatic feature engineering. minimal theoretical guarantees. In order to perform risk management, we also need a The next subsection presents how to estimate parameters of a probability distribution with A signiﬁcant drawback of neural networks is that they are black boxes and offer minimal probability distribution. The next subsection presents how to estimate parameters of a neural networks. theoretical guarantees. In order to perform risk management, we also need a probability distribution. probability distribution with neural networks. The next subsection presents how to estimate parameters of a probability distribution with 2.2. Estimating Probability Distribution Parameters with Neural Networks neural networks. 2.2. Estimating Probability Distribution Parameters with Neural Networks Most data scientists ﬁtting neural networks for regression use a mean squared error loss function, and the output of the Most network data is scientists the expected ﬁtting value neural of thenetworks response variable. for regression The two use drawbacks a mean to squared error 2.2. Estimating Probability Distribution Parameters with Neural Networks this approach are that (1) the mean squared error assumes a normal distribution, and (2) there is no loss function, and the output of the network is the expected value of the response variable. Most data scientists ﬁtting neural networks for regression use a mean squared error loss function, way to quantify variability. Instead of directly predicting the outcome, we propose estimating the The two drawbacks to this approach are that (1) the mean squared error assumes a normal random variable parameters directly, surmounting these drawbacks. and the output of the network is the expected value of the response variable. The two drawbacks to distribution, and (2) there is no way to quantify variability. Instead of directly predicting Let us ﬁrst consider a discrete response variable and assume that the Poisson distribution is this approach are that the(1) outcome, the mean wesquar propose ed err estimating or assumes the a random normalvariable distribution, parameters and (2) dir ther ectly e ,is surmounting no appropriate, as in Fallah et al. (2009). Let n represent the number of observations in the training dataset. these drawbacks. way to quantify variability. Instead of directly predicting the outcome, we propose estimating the The output of the neural network is the intensity parameter λ , i = 1, . . . , n. The exponential function Let us ﬁrst consider a discrete response variable and assume that the Poisson distribu- random variable parameters directly, surmounting these drawbacks. tion is appropriate, as in Fallah et al. (2009). Let n represent the number of observations Let us ﬁrst consider a discrete response variable and assume that the Poisson distribution is in the training dataset. The output of the neural network is the intensity parameter appropriate, as in Fallah et al. (2009). Let n represent the number of observations in the training dataset. l , i = 1, . . . , n. The exponential function is the logical choice for the ﬁnal activation func- The output of the neural network is the intensity parameter λ , i = 1, . . . , n. The exponential function (L) tion g such that the intensity parameter is positive. The loss function is the negative log-likelihood, proportional to y ln l l . å i i i i=1 See Figure 6a for a graphical representation of a network with one hidden layer. n Risks 2020, 1, 0 10 of 27 Risks 2020, 1, 0 10 of 27 (L) (L) is the logical choice for the ﬁnal activation function g such that the intensity parameter is positive. is the logical choice for the ﬁnal activation function g such that the intensity parameter is positive. The loss function is the negative log-likelihood, proportional to The loss function is the negative log-likelihood, proportional to − y ln λ − λ . − y ln λ − λ . ∑ i i i ∑ i i i Risks 2021, 9, 4 i=1 9 of 26 i=1 See Figure 6a for a graphical representation of a network with one hidden layer. See Figure 6a for a graphical representation of a network with one hidden layer. Input Hidden Output Input Hidden Output Input Hidden Output Input Hidden Output layer layer layer layer layer layer layer layer layer layer layer layer x x x x 1 1 1 1 (1) (1) (1) (1) h h h h 1 1 1 1 x x x x 2 2 2 2 .. .. y ˆ ˆ x x x x y 3 λ 3 3 λ 3 . . . . . . . . (1) (1) (1) (1) h h h h . . . J . J J J . . . . . . . . x x x x n n n n (a) (b) (a) (b) (a) (b) Figure 6. Examples of neural network architectures for EF distributions. (a) An approach for non-linear Figure 6. Examples of neural network architectures for EF distributions. (a) An approach for non-linear Figure 6. Examples of neural network architectures for exponential family (EF) distributions. (a) An approach for nonlinear Poisson regression. (b) The approach proposed by Denuit et al. (2019a). Poisson regression. (b) The approach proposed by Denuit et al. (2019a). Poisson regression. (b) The approach proposed by Denuit et al. (2019a). Another Another method method to to estimate estimate parameters parameters of of exponential exponential family family (EF) (EF) distributions distributions with with neural neural Another method to estimate parameters of exponential family (EF) distributions with networks is presented in Denuit et al. (2019a). Exponential family distributions have a probability networks is presented in Denuit et al. (2019a). Exponential family distributions have a probability neural networks is presented in Denuit et al. (2019a). Exponential family distributions have density function of the form density function of the form a probability density function of the form y θ − a(θ ) y θ − a(θ ) i i i y q a(q ) i i i i i i f (y |x ) = exp + c(y , φ) , i = 1, . . . , n (13) f (y |x ) = expf (y jx ) = exp+ c(y , φ) , i+=c1, (y. ,.f . ,)n , i = 1, . . . , n (13) (13) Y Y i i i i i Y i i i i 0 0 0 0 0 0 with g(μ ) = a (θ ) = x β. The mean and variance are respectively given by with g(μ ) = a (θ ) = with x βg . ( The m ) mean = a (qand ) =variance x b. Thear mean e respectively and variance given ar by e respectively given by i i i i i i i 0 0 E[Y ] = a (θ ) E[Y ] = a (q ) E[Y ] = a (θ ) i i i i i i and and and Var(Y ) = fa (q ). Var(Y ) = φa (θ ). Var(Y ) = φa (θ ). i i i i i i The loss function is the unscaled deviance. In this approach, the neural network is The loss function is the unscaled deviance. In this approach, the neural network is designed to The loss function is the unscaled deviance. In this approach, the neural network is designed to designed to estimate only 0 the mean parameter y ˆ = a (q ), see Figure 6b. For distributions i i estimate only the mean parameter y ˆ= a (θ ), see Figure 6b. For distributions with two parameters estimate only the mean parameter y ˆ= a (θ ), see Figure 6b. For distributions with two parameters i i i i with two parameters (gamma, normal), we obtain the second parameter using the method (gamma, normal), we obtain the second parameter using the method of moments method with the (gamma, normal), we obtain the second parameter using the method of moments method with the 2 of moments method with the c statistic: χ χ statistic: statistic: n n 2 1 y − y ˆ 1 ((y − y ˆ)) i i 2 i i n φ = 1 , (y y ˆ ) φ = , ∑ i i ˆ 00 n n−− m m f a a=((θθ )) , ii å ii= =1 1 00 n m a (q ) i=1 where n is the number of observations used to train the model, and m is the number of parameters where n is the number of observations used to train the model, and m is the number of parameters where n is the number of observations used to train the model, and m is the number of in the model. We note that neural networks often have a very high number of parameters, so the in the model. We note that neural networks often have a very high number of parameters, so the parameters in the model. We note that neural networks often have a high number of denominator denominator n n−− m m may may be be lar large ge (or (or negative negative if if n n < < m m). ). parameters, so the denominator n m may be large (or negative if n < m). The difference between models in Figure 6a,b is that the ﬁrst model estimates the parameter of the The difference between models in Figure 6a,b is that the ﬁrst model estimates the parameter of the The difference between models in Figure 6a,b is that the ﬁrst model estimates the Poisson distribution, while the second model predicts the mean of the random variable. We note that Poisson distribution, while the second model predicts the mean of the random variable. We note that parameter of the Poisson distribution, while the second model predicts the mean of the due to the non-convexity of loss functions in neural networks, the solutions will be different unless the due to the non-convexity of loss functions in neural networks, the solutions will be different unless the random variable. We note that due to the nonconvexity of loss functions in neural networks, predicted parameter corresponds to the random variable’s mean. predicted parameter corresponds to the random variable’s mean. the solutions will be different unless the predicted parameter corresponds to the random variable’s mean. For distributions outside the exponential family or when the number of parameters in the neural network is high, another technique is preferable to estimate distribution param- eters. We generalize the neural network presented in Fallah et al. (2009). In this approach, the output of the neural network corresponds to the parameters of the distribution, and the loss function is the negative log-likelihood (NLL) of observations ln f (y jx ). å i i i=1 Risks 2020, 1, 0 11 of 27 Risks 2020, 1, 0 11 of 27 For distributions outside the exponential family or when the number of parameters in the neural For distributions outside the exponential family or when the number of parameters in the neural network network is is high, high, another another technique technique is is pr preferable eferable to to estimate estimate distribution distribution parameters. parameters. W W e egeneralize generalizethe the neural neural network network pr presented esented in in Fallah Fallah et et al. al. ((2009 2009). ). In In this this appr approach, oach, the the output output of of the the neural neuralnetwork network corresponds to the parameters of the distribution, and the loss function is the negative log-likelihood corresponds to the parameters of the distribution, and the loss function is the negative log-likelihood Risks 2021, 9, 4 10 of 26 (NLL) of observations (NLL) of observations n n − ln f (y |x ). − ln f (y |x ). ∑ i i ∑ i i ii= =1 1 In Figure 7a, we present a negative binomial neural network, where the output of the In Figure 7a, we present a negative binomial neural network, where the output of the network is In Figure 7a, we present a negative binomial neural network, where the output of the network is network is the two parameters of the model. The r parameter must be positive, so this the two parameters of the model. The r parameter must be positive, so this parameter ’s activation the two parameters of the model. The r parameter must be positive, so this parameter ’s activation parameter ’s activation function could be the exponential function. The p parameter has a function function could could be be the the exponential exponential function. function. The The p p parameter parameter has has a a[[0, 0,1 1]] domain domain so so that that we we can canuse use [0, 1] domain so that we can use a sigmoid activation function. The Tweedie distribution, a a sigmoid sigmoid activati activation on function. function. The The T Tweedie weedie distribution, distribution, important important in in actuarial actuarial science, science,can canalso alsobe be important in actuarial science, can also be trained using a neural network. The output trained trained using using a a neural neural ne network. twork. The The output output of of the the network network corr corresponds esponds to to the the distribu distribution’s tion’s thr three ee of the network corresponds to the distribution’s three parameters, see Figure 7b. Since parameters, see Figure 7b. Since the domain for every parameter is the positive real numbers, we can parameters, see Figure 7b. Since the domain for every parameter is the positive real numbers, we can the domain for every parameter is the positive real numbers, we can use the exponential use the exponential activation function. use the exponential activation function. activation function for each output neuron. Input Hidden Output Input Hidden Output Input Hidden Output Input Hidden Output layer layer layer layer layer layer layer layer layer layer layer layer x x x x 1 1 1 1 (1) (1) (1) (1) h h h h 1 1 1 1 x x x x 2 2 2 2 λλ rr .. .. x x x x 3 3 3 3 αα p p .. .. .. .. ((1 1)) ((1 1)) h h h h . . . . ββ JJ JJ .. .. .. .. x x x x n n n n ((a a)) ((b b)) (a) (b) Figure Figure 7. 7. Examples Examples of of alternate alternate distributions distributions in in neural neural networks networks with withthe theNLL NLLappr approach. oach. ((a a))Negative Negative Figure 7. Examples of alternate distributions in neural networks with the NLL approach. (a) Negative binomial neural binomial binomial neural neural network. network. ((b b)) T Tweedie weedie neural neural network. network. network. (b) Tweedie neural network. 3. 3. Pricing Pricing with with Machine Machine Learning Learning 3. Pricing with Machine Learning This This section section pr provides ovides an an overview overview of of machine machine learning learning techniques techniques for for actuarial actuariala apriori prioripricing pricing This section provides an overview of machine learning techniques for actuarial a (also (also r referr eferred ed to to as as ratemaking ratemaking or or setting setting tarif tariffs). fs). T The he objective objective in in pricing pricing is is to to pr predict edict futur future e costs costs priori pricing (also referred to as ratemaking or setting tariffs). The objective in pricing associated associated with with a a new new customer customer’s ’s insurance insurance contract contract with with no no claim claim history history information information for for this this is to predict future costs associated with a new customer ’s insurance contract with no customer. Since GLMs are current practice, we do not cover contributions using this method. We ﬁrst customer. Since GLMs are current practice, we do not cover contributions using this method. We ﬁrst claim history information for this customer. Since GLMs are current practice, we do not present pricing with conventional features, followed by neural pricing. These contributions are present pricing with conventional features, followed by neural pricing. These contributions are cover contributions using this method. We ﬁrst present pricing with conventional features, summarized summarized in in T Table able 2 2.. Then, Then, we we pr present esent a a brief brief overview overview of of telematics telematics pricing pricing with withmachine machinelearning learning followed by neural pricing. These contributions are summarized in Table 2. Then, we and and conclude conclude with with an an outlook outlook on on the the pricing pricing literatur literature. e. Contributions Contributions for for conventional conventionalpricing pricingand and present a brief overview of telematics pricing with machine learning and conclude with an telematics telematics pricing pricing usually usually apply apply the the methods methods to to auto auto insurance insurance datasets. datasets. outlook on the pricing literature. Contributions for conventional pricing and telematics pricing usually apply the methods to auto insurance datasets. 3.1. Conventional Pricing 3.1. Conventional Pricing Generalized linear models aim to establish a relationship between variables and the response by Generalized linear models aim to establish a relationship between variables and the response by Table 2. Summary of contributions in pricing. combining a link function and a response distribution. This relationship is determined by the GLM combining a link function and a response distribution. This relationship is determined by the GLM Reference Models scor score, e, the the linear linear r relationship elationship between between variables variables and and rregr egression ession weights. weights. The The linear linearrrelationship elationshipmay may be be too too r restrictive estrictive to to model model the the r response esponse distribution distribution adequately adequately .. GAMs GAMs and and neural neural networks networks of offer fer Christmann (2004) LR, SVR Denuit and Lang (2004) GAM solutions solutions by by adding adding ﬂexibility ﬂexibility to to the the scor score e function. function. Another Another popular popular appr approach oach for for pricing pricingis isusing using Paglia and Phelippe-Guinvarc’h (2011) CART tree-based methods, which often surpass other algorithms for regression tasks. Since a priori pricing is tree-based methods, which often surpass other algorithms for regression tasks. Since a priori pricing is Guelman (2012) GBT a straightforward regression task, actuaries may use most regression models like GLMs, tree-based a straightforward regression task, actuaries may use most regression models like GLMs, tree-based Liu et al. (2014) SVC models and neural networks. models and neural networks. Klein et al. (2014) GAMLSS Sakthivel and Rajitha (2017) NN Henckaerts et al. (2018) GAMLSS Quan and Valdez (2018) DT Yang et al. (2018) GBT Lee and Lin (2018) Boosting Risks 2021, 9, 4 11 of 26 Table 2. Cont. Reference Models Yang et al. (2019) NN Wüthrich and Merz (2019) GLM, NN Fontaine et al. (2019) GLM Diao and Weng (2019) CART Wüthrich (2019) NN So et al. (2020) Adaboost Zhou et al. (2020) GBT Henckaerts et al. (2020) GBT 3.1. Conventional Pricing Generalized linear models aim to establish a relationship between variables and the response by combining a link function and a response distribution. This relationship is determined by the GLM score, the linear relationship between variables and regression weights. The linear relationship may be too restrictive to model the response distribution adequately. GAMs and neural networks offer solutions by adding ﬂexibility to the score function. Another popular approach for pricing is using tree-based methods, which often surpass other algorithms for regression tasks. Since a priori pricing is a straightforward regression task, actuaries may use most regression models like GLMs, tree-based models and neural networks. We start this section by presenting the frequency models. In Christmann (2004), the probability of ﬁling m claims is modeled by 1 Pr( M > 0jx ), m = 0 Pr( M = mjx ) = Pr( M = mj M > 0jx ) Pr( M > 0jx ), m > 0, i i where Pr( M > 0) is predicted with a logistic regression and Pr( M = mj M > 0) is modeled with a support vector regressor. Another method to perform frequency modeling is by considering frequency not as a discrete random variable but as a class to predict. This approach is used in Liu et al. (2014), using the support vector classiﬁer. In a similar spirit, So et al. (2020) presented a multiclass Adaboost algorithm to classify the number of claims ﬁled in a period. This new algorithm is capable of handling class imbalance (a large proportion of zero claims). We note that these approaches treat a regression task as a classiﬁcation task, classifying discrete counts instead of a discrete probability distribution. This approach is rare in practice, but see Chapter 4 of Deng (1998) or Salman and Kecman (2012) for examples where this technique works well. The main disadvantage of the three frequency models is that predictions are determin- istic, meaning that a single value is predicted instead of a distribution. This approach is not frequently used in actuarial science since a distribution is useful for diagnosing model accuracy and calculating other actuarial metrics. The remaining papers in this subsection deal with total costs associated with an insur- ance contract. Generalized additive models in insurance academia were ﬁrst studied by Denuit and Lang (2004) and revisited with GAMLSS in Klein et al. (2014). This model was also used in actuarial ratemaking by Henckaerts et al. (2018), who employed generalized additive models to discover nonlinear relationships in continuous and spatial risk factors. Then, these ﬂexible functions are binned into categorical variables and used as a variable in a GLM. GAMs also appear in telematics pricing in Boucher et al. (2017), who explored the nonlinear relationship between distance driven or driving duration and claim frequency. A modiﬁcation of the regression tree is presented in Paglia and Phelippe-Guinvarc’h (2011) to adjust for exposures different than one. Instead of dividing total claims by the exposure to return to the unit exposure base, the offset is incorporated in the deviance function, which served as a splitting criterion. Risks 2021, 9, 4 12 of 26 Gradient boosting is applied to prediction of at-fault accident loss costs in Guelman (2012). Multivariate decision trees are applied in Quan and Valdez (2018) to model the joint distribution of response variables in multiple coverages. Extensions to random forests and gradient boosting are also presented. The TDboost algorithm is presented in Yang et al. (2018). This algorithm uses gradient boosting to estimate the parameters of a Tweedie distribution. As opposed to the XGBoost framework, explicit update formulas are established from the proﬁle likelihood. Important interactions are identiﬁed using partial dependence plots. A zero-inﬂated variant is presented in Zhou et al. (2020). Then, Henckaerts et al. (2020) compared GLM, GAM, trees and gradient boosting machine to predict future costs associated with an insurance contract. Many methods of variable importance and interpretability are applied, a crucial step in insurance pricing. To extract interactions, they applied Friedman et al. (2008). In Lee and Lin (2018), a boosting algorithm is presented. The original gradient boosting is based on three actions: a basis, a regression and an adjustment. The delta boosting machine is proposed, combining the regression and adjustment steps. Therefore, the algorithm is said to be computationally efﬁcient. Algorithms for distributions in the Tweedie family are presented, and the model is applied to car insurance claims data. Then, Diao and Weng (2019) presented the regression tree credibility model, a tree- based method for pricing with credibility theory. The classic Bühlmann–Straub credibility premium is applied within each tree terminal node. To our knowledge, this is the only contribution using machine learning for a posteriori pricing. A model capable of performing variable selection, such as elastic-net, for multiple response variables with a Tweedie distribution, is presented in Fontaine et al. (2019). A multitask regression model selects variables useful for many regression tasks, so useless variables are less likely to be retained. The proposed algorithm updates weights via the proximal gradient descent scheme to update model coefﬁcients. Mixture of experts or model averaging are other ﬂexible approaches to insurance pricing. Since these methods are not machine learning but statistical, we do not investigate further but highlight Fung et al. (2019a, 2019b); Hu et al. (2018, 2019); Jurek and Zakrzewska (2008); Pocu ˇ ca ˇ et al. (2020); Richman and V. Wüthrich (2020); Ye et al. (2018). See Fung et al. (2020) for an application in reserving. 3.2. Neural Pricing Early attempts of applying neural networks for insurance pricing include Lowe and Pryor (1996), Pelessoni and Picech (1998), Speights et al. (1999) and Francis (2001). Neural networks are also used in Chapados et al. (2002), who compared statistical learning models for estimating the pure premium. They also compared support vector regression but concluded that their predictive performance is not good since data are asymmetric, and the model must be overﬁtted to learn something useful. A fairness criterion is then deﬁned to ensure the pure premium does not systematically discriminate against a speciﬁc group in the population. A neural network is used to predict a posteriori frequency in Sakthivel and Rajitha (2017). The model input is historical claim frequency for a contract, along with the credibility factor and estimated annual claims frequency calculated using Bayesian credibility theory. The model output is the estimated annual claims frequency for the following year. The adversarial variational Bayes method is used to model a Tweedie distribution with mixed models in Yang et al. (2019). Parameters of this distribution are optimized using adversarial variational Bayes, minimizing the Kullback–Leibler divergence with a variance reduction technique to stabilize gradients. An interesting model proposition is the Combined Actuarial Neural Net (CANN) approach of Wuthrich (2019); Wüthrich and Merz (2019), where a neural network is used to model nonlinear relationships that were not captured by a simpler model. An example CANN model could be starting with a GLM to estimate G L M g (E[Y]) = Xb. (14) Risks 2021, 9, 4 13 of 26 Then, the GLM coefficients b are estimated. This task is repeated with a neural network to estimate N N [L1] g (E[Y]) = b + W h . (15) L L Finally, the two regression functions are added C AN N [L] [L] [L1] g (E[Y]) = Xb + b + W h . (16) We can interpret the GLM as a skip connection to the fully-connected neural network. In reality, the GLM predicts the pure premium, and the neural network adjusts for nonlinear transformations and interactions that the GLM could not capture. We can therefore interpret the original GLM parameters, partially explaining the ﬁnal prediction. While neural network predictions may be accurate at the policyholder level, it may not be at the portfolio level. This goes against an actuarial pricing principle Casualty Actuarial Society, and Committee on Ratemaking Principles (1988), which states, “A rate provides for all costs associated with the transfer of risk,” meaning that equity among all insured in the portfolio is maintained. Wüthrich (2019) proposed two methods that address this issue. The ﬁrst uses a GLM step to the gradient descent method’s early stopped solution because GLMs are unbiased at the portfolio level. The other is to apply a penalty term in the loss function. Machine learning models require large datasets. Publicly available datasets for insur- ance pricing are few, making research with complex models difﬁcult. Generating synthetic datasets keeping the risk characteristics but removing conﬁdential information is therefore important. Kuo (2019b) presented a model based on CTGAN to synthesize tabular data, and Côté et al. (2020) compared additional approaches. 3.3. Telematics Pricing Telematics data is one of the ﬁrst unstructured data sources to be extensively used in insurance pricing since it provides a better exposure base. This data source enables Pay-As- You-Drive insurance, where the premium is a multiple of the vehicle (in the distance or duration). A more recent innovation is the Pay-how-you-drive insurance model, where surcharge or discounts are applied based on driving behavior. An early survey of telematics use in insurance companies is presented in Yao and Katz (2013). Telematics data are highly voluminous and unstructured, two situations where ﬂexible algorithms have high predictive performance. For this reason, many models for telematics pricing are based on machine learning. Table 3 presents a summary of telematics models using machine learning. Table 3. Summary of contributions in telematics pricing. Reference Models Boucher et al. (2017) GAM Wüthrich (2017) k-means Gao and Wüthrich (2018) PCA, AE Gao et al. (2018) GAM Gao et al. (2019) GAM Pesantez-Narvaez et al. (2019) LR, GBT Gao and Wüthrich (2019) CNN Narwani et al. (2020) LR, GBT, k-means Gao et al. (2020) CNN 3.3.1. Pay-as-You-Drive A pure premium should represent the expected loss for an insurance contract to an exposure. An example of such exposure could be the value of the insured good, since if the value of materials doubles, the insurance contract should, in principle, also double. When Risks 2021, 9, 4 14 of 26 pricing a contract, we must define an exposure base. According to Werner and Modlin (2010), a good exposure base should • be proportional to the expected loss; • be practical (objective and inexpensive to obtain and verify); • consider preexisting exposure base established within the industry. Indeed, car insurance has, for a long time, been based on car-years insured. While reported kilometers-driven is often used as a rating variable, it is not always used as an exposure base since it is expensive to verify (and simple for the insured to provide false information). The use of telematics data to investigate alternative exposure bases was done in Boucher et al. (2013), who showed that the relationship between annual driven kilometers and the frequency of claims was not linear, concluding there is a reduced risk of accidents as a result of experience. The relationship has been modeled individually with the use of GAMs in Boucher et al. (2017), so these variables may not be used as exposure bases (offsets, in a linear or additive model). However, they must be considered a rating variable to capture the pure premium relationship fully. Then, Verbelen et al. (2018) combines policy information with telematics information with a GAM model and the models which include both types of data have a better regression performance on most criteria. Telematics data have also enabled different statistical distribution assumptions based on the usage of the vehicle. For example, Ayuso et al. (2014) analyzed the time between frequency claims based on distance. 3.3.2. Pay-How-You-Drive In traditional non-life insurance ratemaking, an issue with providing an adequate premium for the risk is that losses associated with contracts may take time to manifest. For example, a hazardous driver may be fortunate and have no accident, while a skilled and alert diver may experience bad luck and get involved in an accident rapidly. It was impossible to differentiate between the driving styles and many years of experience (along with credibility theory), leading an actuary to price a contract for the risk accurately. The use of variables describing the driving context (for instance, road type or acceleration tendencies) is a groundbreaking solution since this data may rapidly provide insights on driving behavior. Thus, individuals with the same classical actuarial attributes may be priced differently according to the insured driving style through the data collected from telematics devices Weidner et al. (2017). We are currently in a stage where the velocity and volume of telematics data are hard to manage since actuaries are not historically equipped with the computer science skills associated with dealing with such data. Additionally, the little available data are sometimes not accompanied by claims data, meaning it is challenging to validate if a driving style is riskier or safer. For example, one could think that an individual with hard breaking tendencies represents an increased probability of an accident. However, this individual may have higher reﬂexes enabling him to make such adjustments rapidly. To our knowledge, some insurance companies consider hard breaks as accidents. The reason is that hard breaks occur during accidents or near-misses, so any insured performing a hard break had or almost had an accident. This hypothesis increases accident frequency in the dataset, potentially leading to better model performance. Since the study of telematics data is in its infancy, such expert knowledge hypotheses must be made. However, there have been some efforts to summarize driving behaviors in the actuarial literature. Weidner et al. (2016) presents a method to build a driving score based on pattern recognition and Fourier analysis. Then, a solution to the small number of observations was proposed by Weidner et al. (2017), who used a Random Waypoint Principle model to generate stochastic simulations of trip data, under constraints, such as speed limits, acceleration and brake performance. A clustering analysis based on medians of speed, acceleration and deceleration is performed to create classes of trips. Then, the few trip data available can be associated with each class. Risks 2021, 9, 4 15 of 26 Another approach to summarize driving styles was proposed by Wüthrich (2017) through the use of velocity–acceleration heatmaps. These heatmaps are generated for different speed buckets to represent the velocity and acceleration tendencies of drivers. Then, the k-means algorithm is applied to create similar clusters of drivers. Note that contrarily to Weidner et al. (2017), these heatmaps are generated by drivers and not by trips, meaning a change in drivers is an issue with this approach. This idea is followed by Gao and Wüthrich (2018), who use Principal Component Analysis and autoencoders to project the velocity- acceleration heatmaps in a two-dimensional vector that can be used as part of rating variables. In Pesantez-Narvaez et al. (2019), logistic regression is compared to gradient boost- ing to predict the occurrence of a claim using telematics data. They conclude that the higher training difﬁculty associated with XGBoost makes this approach less attractive than logistic regression. A study by Gao and Wüthrich (2019) examined the use of convolution neural networks in classifying a driver based on the telematics data (speed, angle and acceleration) of short trips (180 s). Classiﬁcation results were encouraging, but only three drivers could be modeled at once due to the small volume of data. As previously stated, clustering or classiﬁcation models are hard to evaluate since researchers may have access to telematics data but not the claims data associated with these trips. Therefore, it is up to the actuaries’ domain knowledge to adapt the insurance premium based on identiﬁed clusters’ characteristics. Additionally, the models proposed do not contextualize the data, meaning the surrounding conditions that caused the driving patterns (speed limits, weather, trafﬁc, road conditions) are not used when classifying the drivers. More recently, Gao et al. (2018) and Gao et al. (2019) proposed a GAM Poisson regression model that provides early evidence of the explanatory power of telematics driving data. Then, Gao et al. (2020) provided empirical evidence that using velocity- acceleration heatmaps in convolutional neural networks improves pricing. The clustering and creation of driver scores is a branch of transportation research (see Castignani et al. (2015); Chen and Chen (2019); Ferreira et al. (2017); Wang et al. (2020)). We highlight Narwani et al. (2020), who presented a driver score based on a classiﬁcation of the presence of claim and k-means clustering with similar drivers. 3.4. Outlook on Pricing Telematics pricing is a new problem in actuarial science since data are only available recently. As the industry begins to share data with researchers, there is enormous potential for this technology in the pricing literature. There is a gap in the literature for feature engineering with pay-how-you-drive models. In our conversations with insurers and after surveying the literature, expert priors (like velocity-acceleration heatmaps or number of hard breaks) are the preferred methods used in practice and research to model telematics trips. We believe that recurrent neural networks will provide more insights into driving patterns but have not found any literature on the subject. Actuarial science researchers could also look at autonomous driving models — sometimes based on deep reinforcement learning — to model driver behavior. Finally, actuaries must be conscious of the debate on insurance fairness in science and technology studies (STS), political science and sociology of markets. Insurance is becoming individualized or personalized due to increased predictive abilities (machine learning) and more data (telematics, wearable technology and smartphone sensors). This questions the role of insurance in society, transitioning from pooling risks (wealth redistribution) to the transfer of individual risks by paying a premium proportional to the actual hazard. See Barry (2019); Keller et al. (2018); Frezal and Barry (2019); Barry and Charpentier (2020); Cevolini and Esposito (2020) for an overview of this debate. 4. Reserving with Machine Learning This section reviews the contributions to the actuarial science literature that use machine learning for reserving. While the link between regression and a priori ratemaking Risks 2021, 9, 4 16 of 26 is straightforward, it is not for reserving. Thus, actuaries must modify the machine learning algorithm to ﬁt reserving data or modify reserving data to ﬁt structured datasets for regression. Reserving is a time series forecasting problem, see Benidis et al. (2020); Lim and Zohren (2020) for recent reviews on forecasting with machine learning. We say that the total reserve (Tot Res) is the sum of the reported but not settled (RBNS) reserve and the incurred but not reported (IBNR) reserve. We also note that the overdispersed Poisson model (ODP) and chain ladder (CL) can model total, RBNS or IBNR. An overview of contributions using machine learning for reserving with aggregate (Agg) and individual (Ind) data is presented in Table 4. Table 4. Summary of contributions in reserving. Description Reference Approaches Type ODP England and Verrall (2001) GAM Agg ODP Spedicato et al. (2014) GAMLSS Agg IBNR Lopes et al. (2012) SVR Agg CL Wüthrich (2018b) NN Agg Tot Res Kuo (2019a) RNN Agg ODP Gabrielli et al. (2019) NN Agg ODP Gabrielli (2020b) NN Agg — Mulquiney (2006) NN Ind Tot Res Wüthrich (2018a) CART Ind RBNS Llaguno et al. (2017) — Ind RBNS Lopez et al. (2016) CART Ind Tot Res Baudry and Robert (2019) CART Ind Simulation Gabrielli and Wüthrich (2018) NN Ind Tot Res Pigeon and Duval (2019) GBT Ind RBNS Lopez et al. (2019) CART Ind RBNS Kuo (2020) RNN Ind RBNS Lopez and Milhaud (2020) CART Ind RBNS Gabrielli (2020a) NN Ind RBNS De Felice and Moriconi (2019) CART Ind CL Carrato and Visintin (2019) k-means Ind Tot Res Delong et al. (2020) NN Ind RBNS Crevecoeur and Antonio (2020) GBM Ind RBNS Lopez and Milhaud (2020) CART Ind 4.1. Aggregate Reserving Reserving data is unstructured: since the number of payments and the time until a claim closes are unknown at the time of reporting (or at a certain valuation date), we may not store individual claims neatly in spreadsheets. For this reason, actuaries classically aggregate claim information in two ways: 1. aggregation of multiple claims at the portfolio level or other grouping types if the actuary believes that development patterns are heterogeneous within the portfolio; 2. aggregation of continuous-time into interval time, usually yearly, quarterly or monthly. The usual aggregate reserve strategy is to estimate loss development factors (LDFs) to project the reserve at a particular period to the next period. The LDFs are determined as averages of age-to-age factors of observations split in accident year/development year triangles. Let C (x) be the incremental claims amount for accident year i = 1, . . . , I and devel- i,j opment year j = 1, . . . , J, where x are optional regression parameters. Let f be the loss development factor for period j. The Mack chain ladder assumption is E[C (x)] = f (x)C (x). i,j j1 i,j1 Risks 2021, 9, 4 17 of 26 Many nonparametric models for aggregate reserving have been proposed, such as GAMs England and Verrall (2001). The incremental paid claims C is modelled with a i,j GAM, where E[C ] = m , Var(C ) = jm , i,j i,j i,j i,j and ln m = u + dt + c + s (i) + s (j) + s (ln j), i,j i,j q q q i j j where u are offsets, dt represents an inﬂation term. Smooth spline terms are s (i) for i,j q accident years and s (j) + s (ln j) for development years. An extension to GAMLSS q q j j (with distribution other than the one-parameter exponential family) is then presented in Spedicato et al. (2014). In Lopes et al. (2012), a two-stage procedure is proposed for estimating IBNR values. The ﬁrst step consists of calculating chain ladder estimates of IBNR values, and a section step applies SVR and Gaussian process regression to residuals of the ﬁrst model. 4.2. Neural Aggregate Reserving An extension of chain ladder reserving is offered in Wüthrich (2018b), who models development factors ln f (x) with a shallow neural network. The loss function is a j1 weighted square loss (with positive observed claims) given by Ij C (x) i,j L = C (x) f (x) , j = 1, . . . , J. j å å i,j1 j1 s C (x) j1 i,j1 i=1 x:C (x)>0 i,j1 When ignoring attributes from x, the loss function becomes the Mack CL model. In Gabrielli et al. (2019), the cross-classiﬁed over-dispersed Poisson reserving model is gen- eralized to neural networks. This enables more ﬂexibility, including the joint modeling of claims triangles across different lines of business. This idea is expanded to the joint development of claim counts and claim amounts in Gabrielli (2020b). A more general ODP model is presented in Lindholm et al. (2020), which uses regression functions like GBM and NN. Recurrent neural networks are neural networks capable of dealing with sequential data. Therefore, they are well suited for reserving tasks. This model is examined in Kuo (2019a) for aggregate triangles. Aggregate loss experience data for subsequent is fed to a recurrent neural network layer. Company information is fed to an embedding layer. Both layers are combined with fully-connected layers to predict claims outstanding and paid loss. 4.3. Individual Reserving With individual reserving, actuaries may observe individual claims (removing the aggregation within the portfolio). This is also called triangle-free reserving or granular reserving. The advantage is twofold. First, the reserving model may depend on the claim’s characteristics that may impact its development pattern, for instance, line of business, injury part, and age of the claimant. Second, actuaries may model individual events within a claim. For instance, in discrete time individual reserving, predicted values may include • claim status (open, close, reopen), a classiﬁcation task; • activity status (presence of claim or change in case reserve indicator during the period), a classiﬁcation task; • individual payment value or change in case reserve value conditional on the presence of claim during the period, a regression task; • involvement of lawyers or doctors, a classiﬁcation task. Some individual reserving models also deal with claims continuously (removing the aggregation within periods). Since individual reserving is useful for following individ- Embeddings are vectorial representations of data created with deep neural networks to compress high dimensional data, categorical data or unstructured data. Risks 2021, 9, 4 18 of 26 ual claims, these models usually focus on RBNS claims and use aggregate methods for IBNR claims. Machine learning methods have rapidly become a methodology of choice for the analysis of individual reserves. The use of neural networks for individual reserving dates back to Mulquiney (2006), extending the previous state-of-the-art GLM reserving models. See Taylor (2019) for a recent review of reserving models. Individual reserving brings up new challenges for actuaries. First, this approach requires dealing with two types of data. In Taylor et al. (2008), the notion of static variables and dynamic variables is brought up. Static variables remain constant over the claim settlement process, while dynamic variables may change over time. For example, the gender of the client will most likely remain the same, while the client’s age will evolve for claims spanning over one year. Another example of dynamic variables is the claims paid and a variable indicating if a claim is open or closed. Reserving models need to deal with dynamic variables since we try to model payments over time, and variables often change in time. The paper goes on to propose a few parametric individual loss reserving models. Public individual claims data may be difﬁcult to obtain for researchers. In this situa- tion, simulation offers a great way to generate anonymized individual claims histories and attributes. Such a model is offered in Gabrielli and Wüthrich (2018) , who train a neural network to predict individual claim histories based on a risk portfolio. For every claim, we have individual characteristics that models may use as input variables. A sequence of claim amounts and closed/open status for each claim is available for every development year (for a maximum of 12 years). This simulation machine produces observations at the individual level but time-aggregated to periods of one year (for this reason, continuous models are not appropriate for this type of data). Many of the contributions of this section use this simulation machine as applications of individual reserving models. A ﬂexible method for applying machine learning techniques in individual claims reserving is proposed in Wüthrich (2018a). Only regression trees are considered in the paper, and only the number of payments is modelled, although actuaries may scale the approach to other applications. Regression trees are used to model a claim indicator and a close indicator, using variables from initial claim information and past payments. Llaguno et al. (2017) expand this model by removing the reliance on dynamic variables with clustering, and De Felice and Moriconi (2019) consider frequency and severity components. One problem in reserving is that claims that take more time to develop are usually more expensive (and short settling times are usually associated with smaller claim amounts). When building a reserving model with a particular valuation date, we include a higher proportion of smaller claims than reality. The complete claims history of short settling times is included in the dataset, but only partial claim histories of longer developing claims. This is a problem of right censoring, and Lopez et al. (2016) presents a modiﬁed weighted CART algorithm to take this into account. Lopez et al. (2019) use weighted CART as an extension of Wüthrich (2018a). See also Lopez and Milhaud (2020) for an alternate approach to loss reserving using the weighted CART. The gradient boosting algorithm is applied in Pigeon and Duval (2019), using indi- vidual reserving claim histories to predict the total payment. The paper provides multiple approaches for dealing with incomplete (undeveloped) data. Since complete claim histo- ries are needed to train the model, underdeveloped claims are completed using aggregate techniques such as Mack or Poisson GLM. Bootstrap is applied to complete triangles, so the variance of ﬁnal reserves isn’t underestimated. Variables are used in the model, such as age, but not in the gradient boosted tree, only as variables in the Poisson GLM. The case study in this paper is useful for practitioners since many hypotheses are compared and validated. A creative approach for individual claims reserving was proposed by Baudry and Robert (2019). Although machine learning contributions are not the focus of the paper, the See also https://github.com/kasaai/simulationmachine for a user-friendly package. Risks 2021, 9, 4 19 of 26 train and test database building provides future researchers with the opportunity to deal with individual claims data with many kinds of machine learning models. Another approach to reserving is clustering observations into homogeneous groups. Carrato and Visintin (2019) explains how to use the chain ladder method for individual data. They then propose clustering observations based on static variables like the line of business and dynamic variables like payment sequence. Then, they construct a linear chain ladder model for each cluster. Finally, we highlight Crevecoeur and Antonio (2020), who present a hierarchical framework for working with individual reserves. The likelihood function for RBNS is decomposed into temporal dimensions (chronological events) and event dimensions (called update vectors, composed of distributions for a payment indicator, a closed indicator and a payment size). The framework allows for any modeling technique at each layer, so actuaries may use machine learning algorithms to model the three event types (the paper uses GLMs and GBMs). Additionally, many aggregate reserving models can be restated as hierarchical reserving models. 4.4. Neural Individual Reserving The simulation machine is used in Gabrielli et al. (2019). When only one aggregated claims triangle is available, a machine learning algorithm cannot be trained. To create many triangles, individual claim histories are split in a train and test dataset, and aggregated triangles are build using the subset of claim histories. They then apply a neural network to predict the total reserve. An individual claim reserving model is presented in Delong et al. (2020). The reserving task is broken down into six steps, and a neural network is trained for each task: modeling IBNR counts, payment status process of RBNS claims, an indicator of RBNS recovery claims, expected claim and recovery payments of RBNS claims, an indicator of IBNR with no payment, and claim amounts of nonzero IBNR claims. In Gabrielli (2020a), the RBNS prediction task is separated into sub-networks. For each possible development period, a sub-network predicts the type of payment (classiﬁcation task) and the mean parameter of a log-normal distribution for the amount of payment. This network leverages parameter sharing, a principle of multitask learning that generalizes features learned in the network. An individual claims model for RBNS claims using recurrent neural networks is introduced in Kuo (2020). The author uses an encoder LSTM for past cash ﬂows and claim status sequences and a decoder LSTM to generate a paid loss distribution. Also, a Bayesian neural network at the output of the decoder enables uncertainty quantiﬁcation. 4.5. Outlook on Reserving While several researchers proposed models for aggregate reserving with machine learning, most of these approaches build separate runoff triangles for every set of variables (or cluster of similar attributes). When actuaries aggregate reserving data in development triangles, they lose individual development characteristics. Simple models like the chain ladder are often sufﬁcient for large risk portfolios. Individual reserving may beneﬁt much more from modern machine learning methods. There are three main approaches to individual reserving with machine learning. The ﬁrst uses the framework introduced by Wüthrich (2017) and uses past payments as attributes to the model. The second, headed by Mario Wüthrich and Andrea Gabrielli, construct complex fully-connected neural network architectures developed using in-depth knowledge of the reserving problem (domain knowledge). The problem with using these complex architectures in machine learning is that they tend not to generalize well to other tasks. For an actuary to implement these models in practice, they need to have a high understanding of neural networks and of the reserving problem, a combination of skills that is currently rare. Thus, we believe models with simpler architectures that learn development patterns from data will be more feasible in practice. This third approach, Risks 2021, 9, 4 20 of 26 headed by Kevin Kuo, treat the reserving problem as a time series problem and use recurrent neural networks. According to the authors, the main problem with individual reserving research with machine learning is that researchers do not compare models (or are only compared to the chain ladder model). However, many models have a publicly available code. Therefore, it is hard for practitioners to determine the best model to implement and determine which technique is state-of-the-art. Since most researchers use the same simulation machine from Gabrielli and Wüthrich (2018), we hope this changes. 5. Conclusions This paper reviewed the literature on pricing and reserving for P&C insurance. Insurance ratemaking with machine learning and traditional structured insurance data is straightforward since the regression setup is natural. Since actuaries use GLMs for insurance pricing, the leap to GAMs, GBMs or neural networks is natural. The next step for the ratemaking literature is to incorporate novel data sources in ratemaking with neural networks. Insurers already collect telematics data, and works on these datasets use novel machine learning algorithms as predictive models. Other novel sources of data are mainly unstructured, meaning they do not ﬁt neatly in a spreadsheet. Examples include images, geographic data, textual data and medical histories. Other sources of data could be structured but of large size. See Blier-Wong et al. (2020) and Blesa et al. (2020) for use of open data and Pechon et al. (2019) to select potential features. Most reserving approaches ﬁt into three main approaches: a generic framework using past payments as attributes in the model, modifying the chain-ladder to incorporate more ﬂexible relationships, and using recurrent neural networks. In our experience, the second approach is favorable for actuaries that have in-depth knowledge of their book of business to construct the network architectures. If there is sufﬁcient data, the third approach with recurrent neural networks offer more modeling ﬂexibility and enhance their understanding of the claim development process. The RNN approach is successful in ﬁnance (see, e.g., Giles et al. (1997); Maknickiene ˙ et al. (2011); Oancea and Ciucu (2014); Roman and Jameel (1996); Rout et al. (2017); Wang et al. (2016)) but not popular in actuarial science for the moment. We also identiﬁed three overall challenges: explainability, prediction uncertainty and discrimination. Machine learning models learn complex nonlinear transformations and interactions between variables. Although establishing a cause and effect relationship is not required in practice LaMonica et al. (2011), regulatory agencies could require the proof of a causal relationship to include a variable. See Henckaerts et al. (2020) and Kuo and Lupton (2020) for studies of variable importance in actuarial science. Quantifying the variability of predictions is vital for solvability and risk management purposes. Models like GBMs and neural networks usually ignore process and parameter variance. Due to the bias–variance tradeoff (increasing model ﬂexibility usually also increases prediction uncertainty), actuaries should beware of being seduced by better predictive performance if they ignore the resulting increase in prediction variance (feature signiﬁcance). Studying this uncertainty could also lead to omitting a feature in a model. Some regulatory agencies may prohibit using a protected attribute like sex, race or age in a model. A simple approach in practice is anticlassiﬁcation, which consists of simply removing the protected attributes. However, proxy features in the dataset could reconstruct the effect of using the protected attribute. See Lindholm et al. (2020) for a discrimination-free approach to ratemaking. Author Contributions: C.B.-W.: literature review, writing—original draft preparation. E.M. is his supervisor and was responsible for project administration, funding, supervision, review and editing. H.C. and L.L. are cosupervisors and were responsible for supervision, review and editing. All authors have read and agreed to the published version of the manuscript. Risks 2021, 9, 4 21 of 26 Funding: This research was funded by the Natural Sciences and Engineering Research Council of Canada grant number (Cossette: 04273, Marceau: 05605) and by the Chaire en actuariat de l’Université Laval grant number FO502323. Acknowledgments: We thank the three anonymous referees for their useful comments that have helped to improve this paper. We would also like to thank Jean-Thomas Baillargeon for valuable conversations. Conﬂicts of Interest: The authors declare no conﬂict of interest. Abbreviations The following abbreviations are used in this manuscript: AVB adversarial variational Bayes CART classiﬁcation and regression trees DT decision tree EF exponential family GAM generalized additive model GBM gradient boosting machine GBT gradient boosted trees GLM generalized linear model knn k-nearest neighbour LDA linear discriminant analysis LR logistic regression NB naïve Bayes NLL negative log-likelihood NN neural network P&C property and casualty RF random forest RNN recurrent neural network SVM support vector machine SVR support vector regression SVC support vector classiﬁer References Albrecher, Hansjörg, Antoine Bommier, Damir Filipovic, ´ Pablo Koch-Medina, Stéphane Loisel, and Hato Schmeiser. 2019. Insurance: Models, digitalization, and data science. European Actuarial Journal 9: 349–60. [CrossRef] Asimit, Vali, Ioannis Kyriakou, and Jens Perch Nielsen. 2020. Special issue “Machine Learning in Insurance”. Risks 8: 54. 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Machine Learning in P&C Insurance: A Review for Pricing and Reserving

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