ABSTRACT
Credibility theory in general insurance is essentially a form of experiencerating that attempts to use the data in hand as well as the experience of others in determining rates and premiums. An interesting and early historical example of the use of credibility theory deals with the setting of premium rates for employers to cover for workers compensation in the early 20th century [51]. The challenge (to a casualty actuary) was often to balance the claims experience of a particular employer with that of all employers having similar working practices and conditions in determining premiums for insurance coverage. In this chapter, we address the challenge of trying to estimate expected
future claim numbers and/or total aggregate claims for a portfolio of policies on the basis of rather limited sample or current information x, but where other collateral (and possibly useful) information is also at hand. Let us assume there is crucial parameter of interest denoted by θ, which, for example, may be the annual claim rate or a related expected aggregate claims total. Often there is other (in some cases considerable) collateral or prior information from business or portfolios of a somewhat similar nature, which might be useful in estimating θ. Let us denote by θˆs an estimate of θ based on the sample information x, and by θˆc an estimate of θ based on the available collateral information. In the situation where θ is a mean, then θˆs might be the sample mean x¯ and θˆc some prior estimate (say µ0) of this mean. In this type of situation, a key question is often “How might we combine the two (sample and collateral) sources of information to get a good estimate of θ, and in particular how much weight or credibility Z should our estimate put on the sample estimator θˆs?” Surely the value of Z should both be an increasing function of the amount of sample information which we might acquire over time, and also take account of the relative values of the sample and collateral information available. A credibility estimate of θ is a linear combination of the sample estimator
θˆs and the collateral estimate θˆc of the form
Z θˆs + (1− Z) θˆc (for example Z x¯+ (1− Z)µ0), (5.1)
estimator θˆs. The expression given by (5.1) is often called the credibility premium formula. Traditionally, there has been an emphasis on only using estimates θˆs of the form∑n
1 ajxj (i.e., linear in the observations) in the credibility premium formula, and although such estimates have considerable appeal there is no theoretical reason why other sample estimates cannot be used.
