Estimating an Implied Paid Tail Factor: An Application of Shepard’s Method

By Devan A. Griffith

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This paper proposes a method to derive paid tail factors using incurred tail factors and historical payout patterns. Traditionally, a ratio of paid-to-incurred losses—and its reciprocal, the conversion factor—may be used to convert payments at a specific maturity to incurred losses, prior to attaching an incurred tail factor. The implied paid tail factor would be the product of the incurred tail factor and the selected conversion factor. However, relying on averages of historical ratios of paid-to-incurred losses for conversion may lead to unreasonable ultimate loss predictions. Shepard’s method of inverse-distance weighting provides a distribution-free prediction of the conversion factor, utilizing historical payout patterns that most closely resemble the year being projected to the point of tail attach-ment. This application should more reliably incorporate information from case reserves when deriving implied paid tail factors. In this paper, the application of Shepard’s method is compared to traditional conversion factor approaches (using various averaging techniques). In addition, the theoretical basis for the method and an example of the potential impact on ultimate loss projections are presented.

Keywords Shepard, inverse-distance weighting, paid development, chain-ladder, paid-to-incurred conversion, payout pattern, tail factor


Griffith, Devan A., "Estimating an Implied Paid Tail Factor: An Application of Shepard’s Method," Variance 13:1, 2020, pp. 80-92.

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Variance (ISSN 1940-6452) is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submissions by non-CAS members is encouraged.