Bonus-Malus System¶

Definition¶

A bonus-malus system (BMS) assigns each policyholder to a rating level, updates that level using claim history, and charges a base premium multiplied by a level-specific premium relativity. It specifies the levels, initial level, transition rules and premium scale.

Interpretation¶

Claim-free years can earn lower premiums and reported claims can trigger higher premiums. The system uses observed experience to distinguish underlying risks that are not fully captured by initial rating factors.

State diagram of the two-level bonus-malus scheme: levels with premiums c and 2c, green arrows for a claim-free year (N_t = 0) and red arrows for a year with claims (N_t at least 1).

The figure draws the two-level scheme of the Example. The next level depends only on whether the year's claim count $N_t$ is zero, not on the current level: a claim-free year (green, probability $\mathrm{e}^{-\lambda}$ at a fixed rate $\lambda$ ) leads to premium $c$ , and a year with at least one reported claim (red, probability $1-\mathrm{e}^{-\lambda}$ $e^{- λ}$ ) leads to $2c$ . The two levels, the arrows of the transition rule and the premium relativities $1$ and $2$ are the ingredients the definition lists.

The Poisson distribution models the annual claim count at fixed claim rate $\lambda$ . In particular, its zero-claim probability $\mathrm{e}^{-\lambda}$ determines how often a claim-free-year bonus applies.

Key Results¶

Let $L_t$ be level in policy year $t$ and $r_l$ the premium relativity for level $l$ .
With random claim rate $\Lambda$ , the notes model annual claims conditionally as $N_t\mid\Lambda=\lambda\sim\operatorname{Pois}(\lambda)$ $Pois (λ)$ .
A transition rule $T(l,n)$ maps the current level and reported annual count to next year's level.
Conditioning on a fixed claim rate can give a Markov chain. Mixing over unobserved claim rates does not automatically preserve the Markov property.

Examples¶

A two-level scheme charges $c$ after a claim-free year and $2c$ after a year with claims. Its next level depends on whether the reported count is zero.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 12

Aliases: BMS, experience rating classes, bonus-malus scheme, no-claims discount system, bonus-malus Keywords: premium relativities, rating level, transition rule, claim-free discount, malus after a claim, two-level scheme Symbols: lambda

Last updated: 17 October 2026 21:53