A Bayesian Credibility Formula for IBNR Counts

A formula for IBNR counts is derived as the credibility weighted average of three standard actuarial estimates: (formula can be found in paper) Here LDF denotes the age-to-ultimate development factor. The credibility weights vary by age of development in a methodical fashion rejecting prior belief in the reporting pattern and the estimate of ultimate. To derive the formula, IBNR is modelled as a parametrically dependent random variable. Bayes Theorem leads to a natural revision of the prior distribution of the parameters based on the data to date. Using the best least squares linear approximation to the true Bayesian estimate, and performing some algebraic manipulations, the credibility formula is obtained. While the formula could be applied in many ways, for demonstration purposes a fully automatic procedure is applied to three hypothetical triangles of data.