Delong, Ł., Szatkowski, M., 2019, One-year premium risk and emergence pattern of ultimate loss based on conditional distribution
Working Paper, 04-09-2019
We study the relation between one-year premium risk and 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 postulate to define the emergence pattern of the ultimate loss based on the conditional distribution of the best estimate of the ultimate loss given the ultimate loss, where the conditional distribution is derived from the multivariate distribution of the claims development process. We start with investigating Gaussian Incremental Loss Ratio, Hertig's Lognormal and Over-Dispersed Poisson claims development models. We derive the true emergence pattern formulas in these models and prove that they are different from the emergence pattern postulated by England et al. (2012), Bird, Cairns (2011). We assume that the risk is measured with Value-at-Risk. We identify that the true one-year risk can be significantly under and overestimated if the emergence pattern formula from England et al. (2012), Bird, Cairns (2011) is applied. We show that the ratio of the true one-year risk to the ultimate risk varies across the claims development models and depends on the confidence level. We prove that the one-year risk is lower than the ultimate risk only if a suficiently high confidence level is used. Moreover, in a general claims development model we illustrate that the one-year risk can be higher than the ultimate risk at all high confidence levels and the distributions of the one-year risk and the ultimate risk can have different tail behaviour.