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Credibility theory often utilizes the following basic formula to calculate a credibility-weighted estimate:
Estimate = Z * [Mean of current observation] + (1-Z) * [Prior mean],
where 0<=Z<=1.
Z is the credibility ascribed to the mean of the current observation. The prior mean may be based on prior observed data or some type of benchmark. The question is – how do we determine Z?
Three approaches are commonly used:
- Under the Limited Fluctuation (LF) approach, also referred to as classical credibility or American credibility, it is assumed that the current observation of data includes independent trials and that the Central Limit Theorem holds. In order to limit the effect that random fluctuations may have on the estimate, a confidence interval approach is used to determine how much data is required in the current observation in order to assign full credibility (i.e. Z=1), and how partial credibility (i.e. Z<1) should be calculated with smaller amounts of data.
- With the Greatest Accuracy approach, also referred to as Bühlmann’s approach or least squares credibility, the objective is to minimize the square of the error between the estimate and the true expected value of the quantity being estimated. The credibility Z increases asymptotically toward 1 as the amount of data in the current observation increases, but also depends on an analysis of variance – how is the total variance comprised between the expected value of the underlying process variance and the variance of the hypothetical means? As the former increases, Z decreases, but as the latter increases, Z increases.
- Bayesian Analysis utilizes Bayes’ Theorem, conditional probabilities, and distribution hypotheses to develop the estimate. It can be shown that estimates using the Greatest Accuracy approach provide the best “least squares fit” to Bayesian estimates.
Each credibility approach includes subjective elements, and the actuary is often free to choose the approach that he or she deems appropriate for the situation and data. Key considerations include the existence or reliability of the prior mean, the reliability of distribution hypotheses, the desired accuracy of the estimate both in overall terms and with respect to sub-classifications, regulatory or industry conventions, computational convenience, and ease of communication to technical and non-technical audiences.
In our recent work with VA industry studies, we have found that the LF approach provides a reasonable balance of technical sophistication, computational convenience, and ease of communication. Importantly, we have also found that with appropriate choice of parameters, the LF approach produces results that are reasonably consistent with our intuition and that of our clients. We believe that these attributes are particularly important in this type of situation, where we are trying to use the current observation to develop highly credible industry benchmarks where none existed before.
Hence, we view the LF approach as a natural starting point, and we will likely continue to use it in our ongoing VA industry studies, but this does not preclude further refinement or utilization of other credibility approaches as circumstances warrant. In the next article, we will share details about how we have applied the LF approach to our VA industry studies.
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