Credibility theory

Credibility theory is a branch of actuarial science. It was developed originally as a method to calculate the risk premium by combining the individual risk experience with the class risk experience.

When an insurance company calculates the premium it will charge, it divides the policy holders into groups. For example it might divide motorists by age, sex, and type of car; a young man driving a fast car being considered a high risk, and an old woman driving a small car being considered a low risk. The division is made balancing the two requirements that the risks in each group are sufficiently similar and the group sufficiently large that a meaningful statistical analysis of the claims experience can be done to calculate the premium. This compromise means that none of the groups contains only identical risks. The problem is then to devise a way of combining the experience of the group with the experience of the individual risk the better to calculate the premium. Credibility theory provides a solution to this problem.

For actuaries, it is important to know credibility theory in order to calculate a premium for a group of insurance contracts. The goal is to set up an experience rating system to determine next year's premium, taking into account not only the individual experience with the group, but also the collective experience.

There are two extreme positions: One is to charge the same premium to everyone, estimated by the overall mean $\overline{X}$ of the data. This makes sense only if the portfolio is homogeneous, which means that all risks cells have identical mean claims. However, if the portfolio is not homogeneous, it is not a good idea to charge premium in this way, since the "good" risks will take their business elsewhere (overcharging "good" people and undercharging "bad" risk people), leaving the insurer with only bad risks. This is an example of adverse selection.

The other way around is to charge to group j its own average claims, being $\overline{X_j}$ as premium charged to the insured. These methods are used if the portfolio is heterogeneous, provided a fairly large claim experience. To compromise these two extreme positions, we take the weighted average of these two extremes:

$C = z_j\overline{X_j} + (1 - z_j) \overline{X}\,$

z_j has the following intuitive meaning: it expresses how "credible" (acceptability) the individual of cell j is. If it is high, then use higher z_j to attach a larger weight to charging the $\overline{X_j}$, and in this case, z_j is called a credibility factor, such a premium charged is called a credibility premium.

If the group were completely homogeneous then it would be reasonable to set z_j = 0, while if the group were completely heterogeneous then it would be reasonable to set z_j = 1. Using intermediate values is reasonable to the extent that both individual and group history are useful in inferring future individual behavior.

References

• Behan, Donald F. (2009) "Statistical Credibility Theory", Southeastern Actuarial Conference, June 18, 2009