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

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

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We study the relation between one-year premium risk and ultimate premium risk. In practice, the one-year risk is sometimes 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 true emergence pattern of the ultimate loss X_n for the one-year premium risk based on the conditional distribution of the best estimate of the ultimate loss given the ultimate loss BE_1 |X_n where the conditional distribution is derived from the multivariate distribution of the claims development process (X_1,…,X_n ) and the definition of the best estimate of the ultimate loss after the first year BE_1. We investigate three claims development models commonly used in claims reserving. 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 identify that the true one-year premium risk, when measured with Value-at-Risk, can be under and overestimated if the emergence pattern formula from England et al. (2012), Bird, Cairns (2011) is applied. We present two modifications of the emergence pattern formula from England et al. (2012), Bird, Cairns (2011). These modifications allow us to go beyond the claims development models investigated in the first part and work with an arbitrary distribution of the ultimate loss.

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