Credibility theory

Credibility theory is a branch of actuarial mathematics concerned with determining risk premiums. 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.

For example, in group health insurance an insurer is interested in calculating the risk premium, ${\displaystyle RP}$, (i.e. the theoretical expected claims amount) for a particular employer in the coming year. The insurer will likely have an estimate of historical overall claims experience, ${\displaystyle x}$, as well as a more specific estimate for the employer in question, ${\displaystyle y}$. Assigning a credibility factor, ${\displaystyle z}$, to the overall claims experience (and the reciprocal to employer experience) allows the insurer to get a more accurate estimate of the risk premium in the following manner:

The credibility factor is derived by calculating the maximum likelihood estimate which would minimise the error of estimate. Assuming the variance of ${\displaystyle x}$ and ${\displaystyle y}$ are known quantities taking on the values ${\displaystyle u}$ and ${\displaystyle v}$ respectively, it can be shown that ${\displaystyle z}$ should be equal to:

Therefore, the more uncertainty the estimate has, the lower is its credibility.