Gradient Boosting in Motor Insurance Claim Frequency Modelling

Modelling claim frequency and claim severity are topics of great interest in property-casualty insurance for supporting underwriting, ratemaking, and reserving actuarial decisions. This paper investigates the predictive performance of Gradient Boosting with Decision Trees as base learners to model the claim frequency in motor insurance using a private cross-country large insurance dataset. The Gradient Boosting algorithm combines many weak base learners to tackle conceptual uncertainty in empirical research. The findings show that the Gradient Boosting model is superior to the standard Generalised Linear Model in the sense that it provides closer predictions in the claim frequency model. The finding also shows that Gradient Boosting can capture the nonlinear relation between the claim counts and feature variables and their complex interactions being, thus, a valuable tool for feature engineering and the development of a data-driven approach to risk management

Modelling Motor Insurance Claim Frequency and Severity Using Gradient Boosting

Clemente, C., Guerreiro, G. R., & Bravo, J. M. (2023). Modelling Motor Insurance Claim Frequency and Severity Using Gradient Boosting. Risks, 11(9), 1-20. [163]. https://doi.org/10.3390/risks11090163 ---This research was funded by national funds through the FCT—Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020—Center for Mathematics and Applications—(G.R. Guerreiro) and grants UIDB/04152/2020—Centro de Investigação em Gestão de Informação (MagIC) and UIDB/00315/2020—BRU-ISCTE-IUL—(J.M. Bravo).Modelling claim frequency and claim severity are topics of great interest in property-casualty insurance for supporting underwriting, ratemaking, and reserving actuarial decisions. Standard Generalized Linear Models (GLM) frequency–severity models assume a linear relationship between a function of the response variable and the predictors, independence between the claim frequency and severity, and assign full credibility to the data. To overcome some of these restrictions, this paper investigates the predictive performance of Gradient Boosting with decision trees as base learners to model the claim frequency and the claim severity distributions of an auto insurance big dataset and compare it with that obtained using a standard GLM model. The out-of-sample performance measure results show that the predictive performance of the Gradient Boosting Model (GBM) is superior to the standard GLM model in the Poisson claim frequency model. Differently, in the claim severity model, the classical GLM outperformed the Gradient Boosting Model. The findings suggest that gradient boost models can capture the non-linear relation between the response variable and feature variables and their complex interactions and thus are a valuable tool for the insurer in feature engineering and the development of a data-driven approach to risk management and insurance.publishersversionpublishe

Open markov type population models: From discrete to continuous time

Funding Information: Funding: For the second author, this work was done under partial financial support of RFBR (Grant n. 19-01-00451). For the first and third author this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020 financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was funded by the insurance company Fidelidade.We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.publishersversionpublishe

Preliminary results on confidence intervals for open bonus malus

This work was partially supported by Financiamento Base 2009 ISFL-1-297 from FCT/MCTES/PT.Considering open portfolios, we analyze bonus–malus systems (BMS) under a realistic approach, as we already did in Guerreiro and Mexia (Discuss. Math. Probab. Stat. 24(2):197–213, 2004). Using stochastic vortices model we are now able to predict long-run distribution through confidence intervals.authorsversionpublishe

The multi-compartment si(Rd) model with regime switching: An application to covid-19 pandemic

Grant No. 19-01-00451 UID/MAT/00297/2020We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading of the infections, such as a total lockdown. This model is parametrised by three parameters: the basic reproduction number, the mortality rate of the infected, and the duration of the disease. We discuss a particular example of application to Portuguese COVID-19 data in two short periods just after the start of the epidemic in 4 March 2020, with the first two cases dated that day. We propose a simple and effective method for the estimation of the main parameters of the disease, namely, the basic reproduction number and the mortality rate of the infected. We correct these estimated values to take into account the asymptomatic non-diagnosed members of the population. We compare the outcome of the model in the cases of the existence, or not, of a regime switch, and under three different scenarios, with a remarkable agreement between model and data deaths in the case of our basis scenario. In a final short remark, we deal with the existence of symmetries for the proposed model.publishersversionpublishe

Calibration of transition intensities for a multistate model: Application to long-term care

UID/MAT/00297/2020We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.publishersversionpublishe

Ruin probabilities and capital requirement for open automobile portfolios with a bonus-malus system based on claim counts

For a large motor insurance portfolio, on an open environment, we study the impact of experience rating in finite and continuous time ruin probabilities. We consider a model for calculating ruin probabilities applicable to large portfolios with a Markovian Bonus‐Malus System (BMS), based on claim counts, for an automobile portfolio using the classical risk framework model. New challenges are brought when an open portfolio scenario is introduced. When compared with a classical BMS approach ruin probabilities may change significantly. By using a BMS of a Portuguese insurer, we illustrate and discuss the impact of the proposed formulation on the initial surplus required to target a given ruin probability. Under an open portfolio setup, we show that we may have a significant impact on capital requirements when compared with the classical BMS, by having a significant reduction on the initial surplus needed to maintain a fixed level of the ruin probabilityinfo:eu-repo/semantics/publishedVersio

Measuring the impact of a bonus-malus system in finite and continuous time ruin probabilities for large portfolios in motor insurance

Motor insurance is a very competitive business where insurers operate with quite large portfolios, often decisions must be taken under short horizons and therefore ruin probabilities should be calculated in finite time. The probability of ruin, in continuous and finite time, is numerically evaluated under the classical Cram´er-Lundberg risk process framework for a large motor insurance portfolio, where we allow for a posteriori premium adjustments, according to the claim record of each individual policyholder. Focusing on the classical model for bonus-malus systems we propose that the probability of ruin can be interpreted as a measure to decide between different bonus-malus scales or even between different bonus-malus rules. In our work the required initial surplus can also be evaluated. We consider an application of a bonus-malus system for motor insurance to study the impact of experience rating in ruin probabilities. For that we used a real commercial scale of an insurer operating in the portuguese market, and we also work various well known optimal bonus-malus scales estimated with real data from that insurer. Results involving these scales are discussedinfo:eu-repo/semantics/publishedVersio

Clinical factors influencing long-term survival in a real-life cohort of early stage non-small-cell lung cancer patients in Spain

Funding Information: This work was supported in part by Centro de Matematica e Aplicaçoes, UID (MAT/00297/2020), Portuguese Foundation of Science and Technology. Acknowledgments Publisher Copyright: Copyright © 2023 Torrente, Sousa, Guerreiro, Franco, Hernández, Parejo, Sousa, Campo-Cañaveral, Pimentão and Provencio.Background: Current prognosis in oncology is reduced to the tumour stage and performance status, leaving out many other factors that may impact the patient´s management. Prognostic stratification of early stage non-small-cell lung cancer (NSCLC) patients with poor prognosis after surgery is of considerable clinical relevance. The objective of this study was to identify clinical factors associated with long-term overall survival in a real-life cohort of patients with stage I-II NSCLC and develop a prognostic model that identifies features associated with poor prognosis and stratifies patients by risk. Methods: This is a cohort study including 505 patients, diagnosed with stage I-II NSCLC, who underwent curative surgical procedures at a tertiary hospital in Madrid, Spain. Results: Median OS (in months) was 63.7 (95% CI, 58.7-68.7) for the whole cohort, 62.4 in patients submitted to surgery and 65 in patients submitted to surgery and adjuvant treatment. The univariate analysis estimated that a female diagnosed with NSCLC has a 0.967 (95% CI 0.936 - 0.999) probability of survival one year after diagnosis and a 0.784 (95% CI 0.712 - 0.863) five years after diagnosis. For males, these probabilities drop to 0.904 (95% CI 0.875 - 0.934) and 0.613 (95% CI 0.566 - 0.665), respectively. Multivariable analysis shows that sex, age at diagnosis, type of treatment, ECOG-PS, and stage are statistically significant variables (p1) while adjuvant chemotherapy is a good prognostic variable (HR<1). The prognostic model identified a high-risk profile defined by males over 71 years old, former smokers, treated with surgery, ECOG-PS 2. Conclusions: The results of the present study found that, overall, adjuvant chemotherapy was associated with the best long-term OS in patients with resected NSCLC. Age, stage and ECOG-PS were also significant factors to take into account when making decisions regarding adjuvant therapy.publishersversionpublishe

Results from the Clarify Study

Centro de Matemática e Aplicações, UID (MAT/00297/2020), Portuguese Foundation of Science and Technology. Publisher Copyright: © 2022 by the authors.Background: Artificial intelligence (AI) has contributed substantially in recent years to the resolution of different biomedical problems, including cancer. However, AI tools with significant and widespread impact in oncology remain scarce. The goal of this study is to present an AI-based solution tool for cancer patients data analysis that assists clinicians in identifying the clinical factors associated with poor prognosis, relapse and survival, and to develop a prognostic model that stratifies patients by risk. Materials and Methods: We used clinical data from 5275 patients diagnosed with non-small cell lung cancer, breast cancer, and non-Hodgkin lymphoma at Hospital Universitario Puerta de Hierro-Majadahonda. Accessible clinical parameters measured with a wearable device and quality of life questionnaires data were also collected. Results: Using an AI-tool, data from 5275 cancer patients were analyzed, integrating clinical data, questionnaires data, and data collected from wearable devices. Descriptive analyses were performed in order to explore the patients’ characteristics, survival probabilities were calculated, and a prognostic model identified low and high-risk profile patients. Conclusion: Overall, the reconstruction of the population’s risk profile for the cancer-specific predictive model was achieved and proved useful in clinical practice using artificial intelligence. It has potential application in clinical settings to improve risk stratification, early detection, and surveillance management of cancer patients.publishersversionpublishe