Credibility theory is a branch of actuarial mathematics concerned with determining risk premiums.
[1] To achieve this, it uses mathematical models in an effort to forecast the (expected) number of insurance claims based on past observations.
Technically speaking, the problem is to find the best linear approximation to the mean of the Bayesian predictive density, which is why credibility theory has many results in common with linear filtering as well as Bayesian statistics more broadly.
, (i.e. the theoretical expected claims amount) for a particular employer in the coming year.
Bühlmann credibility works by looking at the Variance across the population.
Say we have a basketball team with a high number of points per game.
Also, their unusually high point totals greatly increases the variance of the population, meaning that if the league booted them out, they'd have a much more predictable point total for each team (lower variance).
So, this team is definitely unique (they contribute greatly to the Variance of the Hypothetical Mean).
So we can rate this team's experience with a fairly high credibility.
They often/always score a lot (low Expected Value of Process Variance) and not many teams score as much as them (high Variance of Hypothetical Mean).
You need to place a wager on the outcome after one is randomly drawn and flipped.
Although the approach can be formulated in either a frequentist or Bayesian statistical setting, the latter is often preferred because of the ease of recognizing more than one source of randomness through both "sampling" and "prior" information.
In a typical application, the actuary has an estimate X based on a small set of data, and an estimate M based on a larger but less relevant set of data.
The credibility estimate is ZX + (1-Z)M,[4] where Z is a number between 0 and 1 (called the "credibility weight" or "credibility factor") calculated to balance the sampling error of X against the possible lack of relevance (and therefore modeling error) of M. 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.
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.
However, if the portfolio is heterogeneous, it is not a good idea to charge a premium in this way (overcharging "good" people and undercharging "bad" risk people) since the "good" risks will take their business elsewhere, leaving the insurer with only "bad" risks.
These methods are used if the portfolio is heterogeneous, provided a fairly large claim experience.
has the following intuitive meaning: it expresses how "credible" (acceptability) the individual of cell
, while if the group were completely heterogeneous then it would be reasonable to set
For example, an actuary has an accident and payroll historical data for a shoe factory suggesting a rate of 3.1 accidents per million dollars of payroll.
She has industry statistics (based on all shoe factories) suggesting that the rate is 7.4 accidents per million.
With a credibility, Z, of 30%, she would estimate the rate for the factory as 30%(3.1) + 70%(7.4) = 6.1 accidents per million.