Additional Aspects of One-year Premium Risk and Emergence Pattern of Ultimate Loss Based on Conditional Distribution

We study the relation between one-year premium risk and ultimate premium risk. We provide additional insight and information to the work by Delong and Szatkowski (2019) and investigate new properties of the one-year premium risk vs. the ultimate premium risk. In practice, the one-year risk is usually related to the ultimate risk by using a so-called emergence pattern formula introduced by England et al. (2012) and Bird, Cairns (2011). We follow the work by Delong and Szatkowski (2019), in which they propose an extension of the emergence pattern formula based on the conditional distribution of the best estimate of the ultimate loss given the ultimate loss. We focus on the emergence pattern formula from the Hertig’s Lognormal model. We find parameters for which the emergence pattern formula from England et al. (2012), Bird, Cairns (2011) under and overestimates the true one-year risk and calculate the misestimation errors. We also study the modified version of the emergence pattern formula from England et al. (2012) and Bird, Cairns (2011) - the so-called multiplicative lognormal emergence pattern formula with an arbitrary ultimate. We discuss the key parameter for the distribution of the one-year risk. We compare the skewness and the kurtosis of the one-year and the ultimate losses. We also present an example where the key parameter of the distribution of the one-year risk is not unique