Credibility using a loss function from Spline theory

Current formulas in credibility theory often calculate net premium as a weighted sum of the average experience of the policyholder and the average experience of the entire collection of policyholders. Because these formulas are linear, they are easy to use. Another advantage of linear formulas is that the estimate changes a fixed amount per change in claim experience; if an insurer uses such a formula, then the policyholder can predict the change in premium. On the other hand, Venter (1990) points out that in some cases, the loss of accuracy makes a linear formula undesirable. We apply decision theory to develop a credibility formula that minimizes a loss function that is a linear combination of a squared-error term and a second-derivative term. The squared-error term measures the accuracy of the estimator, while the second-derivative term constrains the estimator to be close to linear. An actuary may balance the sometimes conflicting goals of accuracy and linearity by changing a single parameter in the loss function. Our loss function is similar to one used in spline theory, although in a different context.