Bonus-Malus Transition Matrix¶

Definition¶

At a fixed claim rate $\lambda$ , let $p_{ij}(\lambda)=\mathbb{P}(L_{t+1}=j\mid L_t=i,\lambda)$ $i, λ)$ . The transition matrix $\boldsymbol{P}(\lambda)=(p_{ij})$ has nonnegative entries and each row sums to one. For row-vector level probabilities $\boldsymbol{a}_t$ , the next-year distribution is $\boldsymbol{a}_{t+1}=\boldsymbol{a}_t\boldsymbol{P}$ $a_{t} P$ .

Interpretation¶

The bonus-malus system supplies the rule assigning next year's class from the current state and claim count. The entries above are conditional probabilities given the current state; the matrix product adds their contributions to the next-year class probabilities.

State diagram of the three-state bonus-malus chain: a claim (probability p) sends every state to state 1, a claim-free year (probability q) moves 1 to 2 to 3, next to the matrix P(lambda).

Each arrow in the figure is one entry of $\boldsymbol{P}(\lambda)$ from the Example, with $p=1-\mathrm{e}^{-\lambda}$ and $q=\mathrm{e}^{-\lambda}$ $e^{- λ}$ . Red arrows are years with a claim: from every state they lead to state 1, which fills the first column with $p$ . Green arrows are claim-free years: state 1 moves to 2, while states 2 and 3 move to 3. The arrows leaving state $i$ give row $i$ , and their probabilities sum to $p+q=1$ $1$ .

Pick a starting state and move the year $t$ and the claim rate $\lambda$ : $\boldsymbol{a}_t=\boldsymbol{a}_0\boldsymbol{P}^t$ reaches the stationary law $(p,pq,q^2)$ after two years from any start. The default $\lambda=0.2231\approx\log1.25$ gives $p=0.2$ $0.2$ .

The state must store enough history for the transition rule. For a two-year claim-free discount, a simple high/low premium label may omit the information needed for next year.

Key Results¶

Under a deterministic rule $T(i,n)$ and Poisson count, $\displaystyle p_{ij}(\lambda)=\sum_{n\colon T(i,n)=j}\mathrm{e}^{-\lambda}\lambda^n/n!$ $n : T (i, n) = j \sum e^{- λ} λ^{n} / n!$ .
At a fixed rate with independent yearly claims, $\boldsymbol{a}_t=\boldsymbol{a}_0\boldsymbol{P}^t$ $a_{0} P^{t}$ .
The notes' three states are a claim in the latest year, one claim-free year after a claim, and at least two claim-free years.
Matrix orientation must match the row-vector convention. The general Markov chain framework is formalized in Year 3 Stochastic Models; the transition calculation here uses the conditional probabilities defined above.

Examples¶

With $p=1-\mathrm{e}^{-\lambda}$ and $q=\mathrm{e}^{-\lambda}$ $e^{- λ}$ , the three-state matrix has rows $(p,q,0)$ , $(p,0,q)$ and $(p,0,q)$ . These rows each sum to one.

Prerequisites¶

Stationary Bonus-Malus Distribution and Premium

Sources¶

Risk Insurance · Lecture 13

Aliases: rating-class transition probabilities, BMS Markov chain, BMS transition matrix, bonus-malus Markov chain Keywords: p_ij(λ), row-stochastic matrix, a_{t+1} = a_t P, three-state scheme, transition rule T(i, n) Symbols: lambda, sum

Last updated: 14 October 2026 08:21