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 which postulates a linear relation between both risks. We define the true emergence pattern of the ultimate loss for the one-year premium risk based on a conditional distribution of the ultimate loss derived from a multivariate distribution of the claims development process. We investigate three models commonly used in claims reserving and prove that the true emergence pattern formulas are different from the linear emergence pattern formula used in practice. We show that the one-year risk, when measured by VaR, can be under and overestimated if the linear emergence pattern formula is applied. We present two modifications of the linear emergence pattern formula. 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.

中文翻译：

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基于条件分布的一年期溢价风险和最终损失的应急模式

我们研究了一年保费风险和最终保费风险之间的关系。实际上，一年期风险有时通过使用所谓的出现模式公式与最终风险相关，该公式假定两种风险之间存在线性关系。我们根据索赔开发过程的多元分布得出的最终损失的条件分布，定义了一年期保费风险最终损失的真实出现方式。我们研究了通常用于索赔准备中的三种模型，并证明了真实的出现方式公式与实际使用的线性出现方式公式不同。我们显示，如果采用线性出现模式公式，则通过VaR衡量的一年风险可以低估或高估。我们介绍了线性出现模式公式的两个修改。这些修改使我们能够超越在第一部分中研究的索赔开发模型，并可以对最终损失进行任意分配。