This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes of a customer in consecutive years forms a Markov chain, and most of the literature measures performance of the system in terms of the stationary characteristics of this Markov chain. However, the rate of convergence to stationarity may be slow in comparison to the typical sojourn time of a customer in the portfolio. We suggest an age-correction to the stationary distribution and present an extensive numerical study of its effects. An important feature of the modeling is a Bayesian view, where the Poisson rate according to which claims are generated for a customer is the outcome of a random variable specific to the customer.

In the classical actuarial model for bonus-malus systems in automobile insurance (Denuit

For a simple example of a bonus rule, consider the

Much of the discussion of the literature employs stationarity modeling, measuring characteristics of the system via the stationary distribution

Such stationary performance measures are only meaningful if the Markov chain

Bonus-malus systems may be seen as an example of

The Bayes premium enjoys the optimality property of minimizing the quadratic loss

Given the above optimality property, the Bayes premium can be viewed as the optimal fair choice of the insurer’s premium (it is often argued that the reason that bonus-malus systems are used instead in practice is that they are better understandable to the average customer who would not know about prior and posterior distributions). Nevertheless, as noted by Norberg [

The rule (

The paper is organized as follows. In

Motivated by the criticism of the traditional use of stationarity, we now assume that a customer stays in the portfolio only for a finite number of years

Much of the discussion of

We then need to specify what is meant by the ‘typical bonus level’ of a

Before stating the Theorem, we need some notation and assumptions. Let

A similar argument shows that the total times

It may be noted that expression (4) may be evaluated in closed analytical form. To this end, we need the

For the proof, see the

We have selected three rather different systems for our numerical studies. Doing so, our source has been the survey in Meyer [

The premiums

Trial distributions and their equilibrium distributions.

We selected four trial distributions for the distribution of the sojourn time

In the numerical calculations, the distributions were truncated at

Convergence of relativities, Ireland.

We have taken the distribution

Motivated by this assumption, we have in many of the illustrations selected four values of

The Irish system is very simple with

Bonus rules, Ireland.

6 | 100 | 5 | 6 | 6 |

5 | 90 | 4 | 6 | 6 |

4 | 80 | 3 | 6 | 6 |

3 | 70 | 2 | 5 | 6 |

2 | 60 | 1 | 4 | 6 |

1 | 50 | 1 | 3 | 6 |

The convergence speed to the stationary distribution is illustrated in two figures. The first,

Transient distributions, Ireland.

The shape of these figures may be understood from the transition rules. Consider for example a customer with

T.v. convergence rate, Ireland.

Similar remarks apply to other values of

The figures shows the fastest convergence rate among our three selected systems, and also that the rate is not that crucially depending on the value of

Stationary and age-corrected distributions, Ireland.

The Italian system is intermediate with

Finally the age-corrected distribution

Bonus rules, Italy.

18 | 200 | 17 | 18 | 18 | 18 | 18 |

17 | 175 | 16 | 18 | 18 | 18 | 18 |

16 | 150 | 15 | 18 | 18 | 18 | 18 |

15 | 130 | 14 | 17 | 18 | 18 | 18 |

14 | 115 | 13 | 16 | 18 | 18 | 18 |

13 | 100 | 12 | 15 | 18 | 18 | 18 |

12 | 94 | 11 | 14 | 17 | 18 | 18 |

11 | 88 | 10 | 13 | 16 | 18 | 18 |

10 | 82 | 9 | 12 | 15 | 18 | 18 |

9 | 78 | 8 | 11 | 14 | 17 | 18 |

8 | 74 | 7 | 10 | 13 | 16 | 18 |

7 | 70 | 6 | 9 | 12 | 15 | 18 |

6 | 66 | 5 | 8 | 11 | 14 | 17 |

5 | 62 | 4 | 7 | 10 | 13 | 16 |

4 | 59 | 3 | 6 | 9 | 12 | 15 |

3 | 56 | 2 | 5 | 8 | 11 | 14 |

2 | 53 | 1 | 4 | 7 | 10 | 13 |

1 | 50 | 1 | 3 | 6 | 9 | 12 |

Transient distributions, Italy.

T.v. convergence rate, Italy.

Stationary and age-corrected distributions, Italy.

The German system is rather elaborate. It has a large number of classes,

Bonus rules, Germany.

29 | 245 | 25 | 29 | 29 | 29 | 29 |

25 | 100 | 24 | 26 | 29 | 29 | 29 |

20 | 55 | 19 | 23 | 26 | 27 | 29 |

15 | 40 | 14 | 21 | 25 | 27 | 29 |

10 | 35 | 9 | 17 | 24 | 26 | 29 |

5 | 30 | 4 | 16 | 22 | 24 | 29 |

1 | 30 | 1 | 15 | 22 | 24 | 29 |

A quite special feature of the German system is the very high initial class, 26, meaning that a customer at earliest can reach the lowest premium level in class 1 after 25 years! This clearly shows up in the following

Transient distributions, Germany.

T.v. convergence rate, Germany.

Stationary and age-corrected distributions, Germany.

We believe the differentiation between high and low values of

Population averaged convergence rates: population mean 0.1.

Such averaging is done in

These figures show essentially the same behavior as for the intermediate values 0.08 and 0.16 of

Population averaged age-corrected distributions: population mean 0.1.

As a comparison, similar figures have been produced for the substantially smaller population mean 0.05, see

Population averaged convergence rates: population mean 0.05.

Population averaged age-corrected distributions: population mean 0.05.

We now turn to the influence of finite customer sojourn times on the Bayes premium, proceeding as follows. For each of the three selected bonus systems and of the four trial sojourn time distributions, we first compute our age-corrected alternatives

The results are in the following three

Relativities, Ireland.

Relativities, Italy.

Relativities, Germany.

When interpreting the figures, we first note that it does not contradict financial equilibrium that for a given country, one set of relativities is below the other. For example, all relativities corresponding to one of our four trial sojourn time distributions (colored graphs) are below the given relativities (solid black graph). But the explanation is simply that one set of relativities should be weighted with the age corrected distribution and the other with the stationary distribution, and the age corrected distributions have a region of importance which is more shifted towards high classes.

We next note that the two distributions

Analogous with the stationarity-based definition (

We see a considerable difference between the two stationarity-based average premiums (solid black and dotted black) for Ireland and Italy, whereas they appear almost identical for Germany. The age-corrected average premiums are again quite different, and exhibit somewhat similar behavior as the relativities in

The

The

The

The ideal fairness criterion for a Bayesian premium rule is that the premium for a

In this paper, we have inspected how reasonable it is to view bonus-malus system via the stationary distribution, as is usually done. The conclusion is that in many cases the transient distributions are quite far from the stationary ones, and that this has considerable consequences on the computation of such quantities as Bayesian relativities and average premiums.

We do not necessarily insist that our trial distributions for the sojourn time in the portfolio have the relevant time span. A motor insurance may be terminated for example just if the insured gets a new car. In that case, he will typically continue with a new policy in the same company, but not enter in the same level

Examples of numerical studies of special bonus-malus systems are, for example, in Lemaire [

Of further classical references in the bonus-malus area not cited elsewhere in the text, we mention in particular (in chronological order) Grenander [

The author declares no conflict of interest.

For ease of notation, we suppress the dependency on

A different way to arrive at distribution (4) as the relevant bonus class distribution in a model with finite sojourn times of customers is to ‘sieve customers one-by-one through the system’. By this we mean that we consider a sequence of

This follows simply by noting that the instances

It should be noted, however, that

We here suggest a model which incorporates several features not covered by the basic bonus-malus model consider in the body of the paper.

We assume that a customer is characterized by a random mark

In addition to potentially influencing the Poisson parameter, the

In the first, we take the state space for the

Further examples, not spelled out in detail, are

These references have as their main theme not the Bayes premium but rather the

Once this rule is chosen, one can also assert what is the optimal initial bonus level

For convenience, the dependence on

To be strict, one needs also to define some ordering of customers. These matters are to our mind of formal nature rather than intrinsically difficult, and so we omit the details.

This value is also compatible with the data in a recent study of a Greek portfolio, Tsougas [