Why should we look beyond traditional bonus-malus schemes?

The no-claims discount incentives issued for traditional car insurance generally leads to the accumulation of policyholders occupying the best rating and class enjoying the highest discount. This suggests potential inefficiency in classifying policyholders according to their actual underlying risks.

Combining this with changing macro-environments and decreased vehicle traffic nationwide, this snowstorm of clashing factors presents the window of opportunity to rethink existing policy structures. Investigating improvements to the traditional bonus-malus system (BMS) provides an alternative to possibly allow for better categorisation of risks inherent to the drivers.

Recall that the actual premium charged to a policyholder is usually expressed as the base premium times the ‘relativity’: (1) the base premium depends on the observable characteristics via regression; and (2) a ‘relativity’ is a multiplier attached to a given rating class the policyholder occupies in a year (e.g. 0.7 means a 30% discount while 1.5 means a 50% loading). The use of relativities serves as a posteriori pricing mechanism based on the policyholder’s claim history. A study in [1] funded by the Casualty Actuarial Society into finding the optimal relativity associated with each rating class provides a flexible framework that spreads policyholders across different classes. The study extends upon the classical BMS and introduces two major improvements to the existing structure.

  1. Long memory transition rule: Instead of immediately moving a policyholder to a better rating class if he/she has a single claim-free year, a policyholder needs to have consecutive claim-free years to improve his/her rating class. This helps differentiate between consistently careful drivers and temporarily good drivers.

  2. Frequency-severity dependence: Dependence between claim frequencies and claim amounts can be significant but is often ignored in the literature. Its incorporation leads to a better reflection of the underlying risks.


A data set was utilised to compare the traditional BMS with the newly proposed BMS.

Understanding the features of the extended model

In the traditional framework for BMS, a policyholder’s rating class immediately improves by one level after a single claim-free year but worsens by h levels for every claim made, and this is commonly known as the -1/ + h transition rule. In the modified system, an additional penalty component ‘pen‘ is added to the mix. In essence, a new policyholder will start with a penalty of zero, and for every at-fault claim that is reported the rating class worsens by h levels as per traditional calculations.

However, once a claim has been lodged in a policy year, it will take 1 + pen number of consecutive claim-free years for the policyholder to improve his/her rating class by one. That is, a penalty period is activated once a claim is filed so that the policyholder cannot have his/her premium reduced for some years. In this way, the conditions for a policyholder to improve his/her rating class become more stringent, thereby avoiding an extraordinary portion of policyholders from staying at the best rating class.

In determining the optimal relativities, the classical system typically relies on the modelling of the frequency of claims only. However, recent research including [1] has found significant dependence between frequency and severity. In[1], it is assumed that frequency and severity are driven by different unobservable risk parameters that are dependent, and the optimal relativities are obtained by minimising the mean squared difference between the true (unobservable) risk and the actual premium charged.

Why should general insurance companies be interested in this new approach?

The numerical results in [1] have shown that the extended BMS using more claim history in the transition rule is successful in shifting a certain portion of policyholders away from the best rating class, leading to better classification of the policyholders. The new approach has its advantageous capabilities as it satisfies policyholder expectations by calculating premiums that better reflect their true risk, thereby creating an equitable scheme. However, the implications of [1] in profitability are less obvious. Indeed, for a given rating class, the optimal relativity (and hence premium) in the modified BMS can be lower than that in the traditional BMS. Such apparent loss of premium income can possibly be compensated by having more policyholders occupying worse rating classes in the modified BMS, but the net effect may differ from portfolio to portfolio.

Nevertheless, in cases where the premium income for the portfolio increases, a business dilemma that should be considered is the balance between achieving underwriting profitability and the ease of policy functionality. Adding complexity into the transition rule may deter consumers from the product, but this can be mitigated by communicating the key attractive features associated with the modified BMS: lower optimal relativities in the modified BMS means a lower premium as long as a policyholder does not transit to a worse rating class. Incorporating this applied system into widespread commercial use will require collective action that is tackled from a top-down approach and coordinated at an industry level.

How should insurers remain competitive in a changing environment?

In a post-pandemic landscape, a revised approach to tackle the differentiation of drivers presents a stimulating challenge that is marked by a backdrop of decreasing traffic volumes and a reduction in collision frequencies. This has only heightened driver demand for alternative types of cover such as usage-based insurance (UBI) as fewer trips and commutes are needed.

Moreover, insurers are seeing an uptick in recovery costs for claims due to several factors:

  • delays in global supply chain logistics, with extended wait times for the delivery of parts that spill into prolonged car hire fees;


  • added protection and costs concerning vehicle inspection and sanitisation that are mandatory before repairs begin; and


  • reduced opportunity to negotiate volume discounts with auto repairers as with significant downsizing in the number of claims.


The modified BMS provides a solid framework of premium calculations that insurers may seriously consider as an alternative. While implementation of this system would require large upfront costs in research and design, the collective work of actuaries has the potential to revamp longstanding industry practices and lift the conventional standards towards a more efficient and optimal system.

[1] Ahn, J.Y., Cheung, E.C.K., Oh, R. and W. (2021) Optimal relativities in a modified bonus-malus system with long memory transition rules and frequency-severity dependence, Variance, accepted. (Preprint available at https://arxiv.org/abs/2106.00911.)


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