An application of Sigmoid and Double-Sigmoid functions for dynamic policyholder behaviour

The growing relevance of risk-based valuations of insurance contracts has stimulated the extension of the traditional deterministic lapse rate models towards a dynamic modelling. A popular dynamic model uses deterministic lapse rates as base rates and dynamic adjustment factors, generally assuming a relationship between lapses and one or more economic factors to describe policyholder behaviour. This relationship is generally represented by an S-Shaped function. This implies a monotonic increase in lapse rate by increasing the economic variable, usually set equal to a “market spread” between a benchmark rate and the policy crediting rate. In this paper, we assume a different policyholder behaviour, based on the assumption that the policyholder does not modify his/her behaviour for small values of the market spread. Hence, for a better description of such behaviour, the double-sigmoid function appears to be more adequate. The double-sigmoid function is obtained as a combination of two logits in their sum or product. Theoretical features and practical applications of the model are discussed.