Insurance pricing under ambiguity

An actuarial model is typically selected by applying statistical methods to empirical data. The actuary employs the selected model then when pricing or reserving an individual insurance contract, as the selected model provides complete knowledge of the distribution of the potential claims. However, the empirical data are random and the model selection process is subject to errors, such that exact knowledge of the underlying distribution is in practice never available. The actuary finds her- or himself in an ambiguous position, where deviating probability measures are justifiable model selections equally well. This paper employs the Wasserstein distance to quantify the deviation from a selected model. The distance is used to justify premiums and reserves, which are based on erroneous model selections. The method applies to the Net Premium Principle, and it extends to the well-established Conditional Tail Expectation and to further, related premium principles. To demonstrate the relations and to simplify the computations, explicit formulas for the Conditional Tail Expectation for standard life insurance contracts are provided.