Bonus-malus System Research Articles

Weekly dynamic motor insurance ratemaking with a telematics signals bonus-malus score

Abstract We present a dynamic pay-how-you-drive pricing scheme for motor insurance using telematics signals. More specifically, our approach allows the insurer to apply penalties to a baseline premium on the occurrence of events such as hard acceleration or braking. In addition, we incorporate a bonus-malus system (BMS) adapted for telematics data, providing a credibility component based on past telematics signals to the claim frequency predictions. We purposefully consider a weekly setting for our ratemaking approach to benefit from the signal’s high-frequency rate and to encourage safe driving via dynamic premium corrections. Moreover, we provide a detailed structure that allows our model to benefit from historical records and detailed telematics data collected weekly through an onboard device. We showcase our results numerically in a case study using data from an insurance company.

Cointegration Analysis of Crop Yields and Extreme Weather Factors Using Actuaries Climate Index with Application of Bonus–Malus System

This article analyzes the long-term temporal co-movement of the extreme weather variables in the Actuaries Climate Index (ACI) and crop yields, modeling their relationship using an error correction model (ECM). The analysis suggests that significant weather variables can serve as trigger parameters in the pricing framework of weather index crop insurance. To address the challenge of weather index crop insurance while preserving the advantages of a bonus–malus system (BMS), we propose a transition rule that distinguishes between damages caused by severe weather and those resulting from the policyholder’s decisions. Subsequently, we also explore the challenges of implementing such a new hybrid BMS for crop insurance where extreme weather outcomes are integrated into the classical BMS.

Modelling of Claim and Pricing of Motor Insurance Based on Bonus-Malus System Considering the Frequency and Severity of Claims

This article proposes new distributions for claim frequency and severity, specifically tailored for a bonus-malus system in automobile insurance. The mixed Poisson with weighted quasi Lindley distribution is recommended for modeling claim frequency, while the mixed exponential with weighted quasi Lindley distribution is suggested for modeling claim severity. To estimate insurance premiums, the Bayesian method is employed, incorporating both frequency and severity distributions. The study validates the proposed models using real data from an Australian insurance company, which includes 67,856 policies. The assessment of model adequacy indicates that the Poisson-weighted quasi Lindley distribution is a suitable fit for modeling claim frequency, while the exponential-weighted quasi Lindley distribution is appropriate for modeling claim severity. Overall, the results suggest that the proposed models offer optimal premium estimations, considering both claim frequency and severity, which can lead to fairer pricing and increased customer appeal during claim occurrences compared to conventional models.

On bayesian bonus-malus premium under linex loss function with applications

The majority of bonus-malus systems (BMS) determine the premium for each policyholder solely based on their claim frequency. Adopting this method would be unjust. A driver who files a claim for small damage cannot be subjected to the same penalties as a driver responsible for a claim involving significant damage. To achieve a balance in the portfolio between good and bad drivers, a new portion of the premium is computed as a compromise. To do this, the claim severity is created and assessed to provide a fair premium. The consideration was determined by taking into account both the quantity and magnitude of the claims. This work assumes that the frequency of claims follows a Poisson-Akash distribution, but the severity of claims follows a newly proposed distribution known as the Inverse-Gamma Lindley distribution. The bonus-malus premiums were computed using the Bayesian approach for both the frequency and severity of claims. The premium is computed using the asymmetric Linex loss function for both the frequency and severity of claims in the bonus-malus system. Illustrative instances utilizing an actual dataset for the application of BMS are presented, showcasing the use of claim frequency alone and the combination of claim frequency and claim severity. The R software is employed for these demonstrations. We concluded this paper with a discussion comparing the premiums obtained in each case.

International practice of the bonus-malus system

Purpose of the research – evaluation of the eff ectiveness of the "bonus-malus" in the class of compulsory insurance of civil liability of vehicle owners and the identification of the conceptual foundations of compulsory insurance of civil liability inherent in the experience of foreign countries, as well as the development of proposals to improve the rules of this insurance. Methodology – the methodological basis of the research was the works and publications of foreign economic scientists, a set of general scientifi c and private methods of scientific cognition (analytical, systematic method, analysis, synthesis, comparison, etc.). Originality / value – in order to ensure a balance with favorable values of compulsory civil liability insurance, methodological approaches to the modernization of the bonus-malus system are provided and determined by the expediency and possibility of use in the insurance market of kazakhstan. Findings – it provides for the use of optimal ways of differentiating insurance tariff s by domestic insurance companies, taking into account the degree of risk and interests of policyholders of various categories, and maintaining the availability of tariffs for a wide range of consumers of insurance services.

Obavezno osiguranje vlasnika motornih vozila od odgovornosti između slobode ugovaranja i prinudne kartelizacije

The European Union enacted directives on motor vehicle liability insurance to accomplish the goals of free movement of persons and equal protection of third parties against damages caused by using motor vehicles. Serbia has been harmonising its legislation with the EU insurance acquis for several decades. Regarding motor vehicle liability insurance, market liberalisation has been prevented by the Law on Compulsory Insurance in Transport – LCIT 2009. LCIT authorised the Association of Insurers of Serbia to determine minimal premium prices with the consent of the National Bank of Serbia – NBS and stipulated mandatory Bonus-Malus System – BMS. Effectively, the NBS laid down minimal premium prices applicable within the BMS grades and conditions for the application of bonuses and maluses. The author analyses EU legislation and the case law of the Court of Justice related to fixing insurance premiums, mandatory BMS and competition law issues in the motor vehicle liability insurance market. The BMS in several EU member states has been compared with the BMS in Serbia. Competition in the motor vehicle liability insurance market in Serbia has been prevented as a consequence of the fixed premium prices and mandatory BMS. The author has also analysed the negative effects of the current legal framework on the motor vehicle liability insurance market structure, pointing out the static market concentration level and the substantial share of motor vehicle liability insurance premiums in the total insurers’ income. It is essential to amend LCIT without delay to liberalise the Serbian motor vehicle liability insurance market.

Underwriting information and insurers' profitability: Evidence from automobile physical damage insurance in Taiwan

The asymmetric learning hypothesis describes that an insurer can earn a higher profit by possessing underwriting information on repeat customers that is not available to other insurers. This study investigates whether this advantage is still sustained under a complete information-sharing mechanism. We employ data on automobile physical damage insurance in Taiwan to analyze whether insurers still enjoy an information advantage for repeat customers and whether they earn higher profits on staying policies. The findings reveal that switching and repeat customers do not affect insurers' profitability, suggesting that insurers do not have an information advantage over repeat customers; therefore, a complete information-sharing mechanism can eliminate asymmetric information among insurers. Moreover, the availability of abundant underwriting information enables insurers to more accurately assess customers' risk levels, which increases insurers' profitability. Furthermore, the results indicate that the bonus-malus system does not effectively reduce risks associated with policyholders with high claim frequencies because the degree of the increase in premium is lower than the increase in claim amounts, which is attributable to institutional factors. Our study implies the necessity of designing a more effective bonus-malus system in addition to underwriting screening.

Equilibrium reporting strategy: Two rate classes and full insurance

AbstractWe propose a multiperiod insurance model under a bonus–malus system with two rate classes and consider an insured who has purchased full insurance for her losses. To explore the potential advantage of underreporting her insurable losses, the insured follows a barrier strategy and only reports lossses above the barrier to the insurer. We obtain a unique equilibrium declaration strategy in closed form for a risk‐neutral insured who maximizes her expected wealth, and in semiclosed form for a risk‐averse insured who maximizes her expected exponential utility of wealth, both over an exogenous random horizon. We find that the equilibrium barriers for the two classes are equal and strictly greater than zero, offering a theoretical explanation for the underreporting of insurable losses, a form of ex post moral hazard. Finally, we consider the case of three rate classes and show, through numerical examples, that the equilibrium barriers are not equal.

AUTOMOBILE BONUS-MALUS SYSTEM MODELLING USING MACHINE LEARNING ALGORITHMS

Third-party liability is the most important sub-category in automobile insurance, that is why actuaries seek always to design the ideal price list by classifying the insureds into homogeneous classes. However, the heterogeneity persists in the priori tariffication. For that, actuaries use the Bonus Malus system to redistribute the cost of claims more equitably between insureds by rewarding good insureds with a bonus and penalizing bad insureds with a Malus. Nevertheless, the classical approach used in the conception of Bonus Malus systems is limited to the parametric methods that need to make the hypothesis of the number of claims distribution and don’t consider the cost of claims. In this direction, this paper seeks to avoid this issue by using machine learning algorithms, in response to offering a fair Bonus Malus System. Two models of posteriori tariffication will be built. In addition, three algorithms will be used, in occurrence, the CART Classification And Regression Tree method, SVM the Support Vector Machine for regression, and KNN the K-Nearest Neighbor. The suggested models take into account, not only the number of claims but also the importance of the cost of claims. A numerical illustration shows the flexibility of posteriori premiums calculated by our models in relation to the risk levels. This work is a start for new actuarial research which seeks to use artificial intelligence in the design of bonus malus systems.

Improving an existing proxy-based approach for floodplain denitrification assessment to facilitate decision making on restoration

Excess nitrogen (N) from agricultural sources is a major contributor to the water pollution of rivers in Europe. Floodplains are of tremendous importance as they can permanently remove nitrate (NO3) from the environment by releasing reactive N to the atmosphere in its gaseous forms (N2O, N2) during denitrification. However, the quantitative assessment of this ecosystem function is still challenging, particularly on the national level. In this study, we modeled the potential of NO3-N removal through microbial denitrification in soils of the active floodplains of the river Elbe and river Rhine in Germany. We combined laboratory measurements of soil denitrification potentials with straightforward modelling data, covering the average inundation duration from six study areas, to improve an existing Germany-wide proxy-based approach (PBAe) on NO3-N retention potential. The PBAe estimates this potential to be 30–150 kg NO3-N ha−1 yr−1. However, with soil pH and Floodplain Status Category identified as essential parameters for the proxies, the improved PBA (PBAi) yields a removal potential of 5–480 kg N ha−1 yr−1. To account for these parameters, we applied scaling factors using a bonus-malus system with a base value of 10–120 N ha−1 yr−1. Upscaling the determined proxies of the PBAi to the entire active floodplains of the river Elbe and river Rhine results in similarly high NO3-N retention sums of ~7000 t yr−1 in spite of very different retention area sizes, strengthening the argument for area availability as the primary objective of restoration efforts. Although PBAs are always subject to uncertainty, the PBAi enables a more differentiated spatial quantification of denitrification because local key controlling parameters are included. Hence, the PBAi is an innovative and robust approach to quantify denitrification in floodplain soils, supporting a better assessment of ecosystem services for decision-making on floodplain restoration.

Claim modeling and insurance premium pricing under a bonus-malus system in motor insurance

Claim modeling and insurance premium pricing under a bonus-malus system in motor insurance

An alternative approach to bonus malus

Under the assumptions of an open portfolio, i.e., considering that a policyholder can transfer his policy to another insurance company and the continuous arrival of new policyholders into a portfolio which can be placed into any of the bonus classes and not only in the starting class, we developed a model (Stochastic Vortices Model) to estimate the Long Run Distribution for a Bonus Malus System. These hypothesis render the model quite representative of the reality. With the obtained Long Run Distribution, a few optimal bonus scales were calculated, such as Norberg’s (1979), Borgan, Hoem’s and Norberg’s (1981), Gilde and Sundt’s (1989) and Andrade e Silva’s (1991). To compare our results, since this was the rst application of the model, we used the Classic Model for Bonus Malus and the Open Model developed by Centeno and Andrade e Silva (2001). The results of the Stochastic Vortices and the Open Model are highly similar and quite different from those of the Classic Model. Besides this the distribution of policyholders in the various bonus classes was derived assuming that the entrances followed adequate stochastic models.

On monotone likelihood ratio of stationary probabilities in bonus-malus systems

Abstract Bonus-malus system is an often used risk management tool in the insurance industry, and it is usually modeled with Markov chains. Under mild conditions it can be stated that the bonus-malus system converges to a unique stationary distribution in the long run. The maximum likelihood ratio property is a well-known statistical concept and we define it for the stationary distribution of a bonus-malus system. For two special cases we could justify it algebraically. For other cases we describe a numerical method with which we can test this property in any case. With the help of the described method, we checked this property for cases that appear in actuarial practice.

Discrete-time risk models with surplus-dependent premium corrections

This paper studies discrete-time risk models with insurance premiums adjusted according to claims experience. The premium correction mechanism follows the well-known principle in the non-life insurance industry, the so-called bonus-malus system. The bonus-malus framework that we study here extends the current literature by allowing the premium correction rules to vary according to the current surplus level of the insurance company. The main goal of this paper is to evaluate the risk of ruin for the insurer who implements the proposed bonus-malus system. Two premiums correction principles are examined: by aggregate claims or by claim frequency. Further, the Parisian type of ruin is also considered, where the premium adjustment rules are different in positive- and negative-surplus environment.

Bonus-Malus Premiums Based on Claim Frequency and the Size of Claims

The bonus-malus system (BMS) is one of the most widely used tools in merit-rating automobile insurance, with the primary goal of ensuring that fair premiums are paid by all policyholders. The traditional BMS is dependent only on the claim frequency. Thus, an insured person who makes a claim with a small severity is penalized unfairly compared to an individual who makes a large severity claim. This study proposes a model for estimating the bonus-malus premium by employing a limit value (monetary unit) which distinguishes claim size into small and large based on claim frequency and claim severity distributions. This assists in determining the penalties for policyholders with claim sizes falling above and below the limit value. The number of claims is assumed to follow a Poisson distribution, and the total number of claims with a size greater than the limit value is considered a binomial distribution. The underlying risk of each policyholder is assumed to follow a beta Lindley distribution and is referred to as the prior distribution. Each policyholder’s claim size is also assumed to follow a gamma distribution, with the Lindley distribution considered as the prior distribution. Bonus-malus premiums are calculated following the Bayesian method. Practical examples using an actual data set are provided, and the results generated are compared to those produced using the traditional Poisson binomial-exponential beta model. This methodology provides a more equitable mechanism for penalizing policyholders in the portfolio.

Updating Bonus–Malus Indexing Mechanism to Adjust Long-Term Health Insurance Premiums

Economic shocks, high inflation, longevity, and new emerging technologies make the long-term health care insurance challenging for insurers. To overcome this problem, an indexing mechanism has been employed to update predicted premiums based on the new information in hand. Such indexing mechanisms have thus far failed to consider the available policyholder’s risk experience at its updating time. This article employs the well-known bonus–malus system to introduce a bonus–malus indexing mechanism that takes into account the policyholder’s risk experience in its updating mechanism. More precisely, it uses the bonus–malus system’s idea to update the premium of each policyholder based upon her or his risk experience as well as updated inflation. The theoretical foundation of this approach has been developed and its practical implementation is shown through a simulation study.

Distributional effects from policies for reduced CO2-emissions from car use in 2030

• Distributional effects of car-related taxes are estimated. • Focus on fuel taxes and bonus-malus. • People in rural areas are more hurt by car-related taxes than urban residents. • Fuel taxes are slightly less regressive than bonus-malus scheme. We analyze the distributional effects, from car use and car choice adaptation to three car-related policy instruments intended to reduce CO 2 emissions in Sweden to 2030: fossil-fuel taxes, a bonus-malus system for new cars, and mandated biofuel blending. The results show that even with a fast introduction of plug-in electric vehicles, many fossil cars remain, which has important distributional consequences. The fuel tax and the bonus-malus scenarios impose further burdens compared to the reference scenario. The fraction of each population group incurring substantial welfare losses is higher the lower the income. In 2030, the highest-income group can avoid some adaptation costs by their car choice. In all scenarios rural areas bear the largest burden, smaller urban areas the second largest burden, and the largest urban areas the smallest burden. Thus, individuals with high incomes and inhabitants in urban areas appear to have more opportunities to adapt and avoid welfare losses.

Motivational Bonus-System Based On Pavement Installation Temperatures Measurement By Thermographic System (TGS Pavement) In Estonia

Paving is one of the most crucial stage in the matter of road lifespan, since it is the surface layer that has stand up to all the external factors (weather, traffic etc.). Insufficient pavement density caused by thermal segregation during paving works can reduce the lifespan significantly, especially in Estonian climate (freeze-thaw cycles). Modern technology offers different solutions to reduce the risk of low quality in asphalt production and road paving works. Mobile asphalt plant, feeder and thermo-isolated trailers are some piece of equipment, that contractor can use to level up the minimal required quality requirements. The question is, when to use those and which to use? Moreover, is there any possibility to motivate the contractors to put in some extra effort? In Estonia, motivational bonus-system has been established to encourage innovation and reward the extra effort that has been made for quality improvements. The methodology is based on years of experience gained in different researches and pilot-projects. There are no strict rules for the road paving equipment in the methodology – for example contractor can choose himself either the feeder or/and thermo-insulated trailers are used on not. The main requirement is that the temperatures of entire paving process (surface layer) has been measured and analyzed by special thermographic system. Current presentation discusses the symbiosis of bonus-malus system and development of special thermographic system (TGS Pavement) as a multifunctional tool in asphalt paving in Estonia.

Bonus-Malus System in Iran: An Empirical Evaluation

The aim of this paper is to represent the Bonus-Malus System (BMS) of Iran, which is a mandatory scheme based on Insurance act num ber 56. We examine the current Iranian BMS, using various criteria such as elasticity and time of convergence to steady state with respect to the claim frequency as well as financial balance. We also find the closed form of stationary distribution of the Iranian BMS that plays a key role in study of BMSs. Moreover, we compare the results with the German and Japan BMS. Finally we give some hints that can be used to improve the performance of the current Iranian BMS.

Designing a Bonus-Malus system reflecting the claim size under the dependent frequency–severity model

In the auto insurance industry, a Bonus-Malus System (BMS) is commonly used as a posteriori risk classification mechanism to set the premium for the next contract period based on a policyholder's claim history. Even though the recent literature reports evidence of a significant dependence between frequency and severity, the current BMS practice is to use a frequency-based transition rule while ignoring severity information. Although Ohet al.[(2020). Bonus-Malus premiums under the dependent frequency-severity modeling.Scandinavian Actuarial Journal2020(3): 172–195] claimed that the frequency-driven BMS transition rule can accommodate the dependence between frequency and severity, their proposal is only a partial solution, as the transition rule still completely ignores the claim severity and is unable to penalize large claims. In this study, we propose to use the BMS with a transition rule based on both frequency and size of claim, based on the bivariate random effect model, which conveniently allows dependence between frequency and severity. We analytically derive the optimal relativities under the proposed BMS framework and show that the proposed BMS outperforms the existing frequency-driven BMS. Later, numerical experiments are also provided using both hypothetical and actual datasets in order to assess the effect of various dependencies on the BMS risk classification and confirm our theoretical findings.