In modern actuarial risk management, the temporal clustering of severe claims is as critical as their cumulative financial magnitude. This study investigates a stochastic risk process that terminates upon the occurrence of a run of consecutive claims exceeding a predefined critical threshold. First, using recursive conditioning techniques, we derive the exact moment generating function for the total severity of exceedances accumulated prior to termination, providing an explicit probability density function for the exponential case when . Second, we determine the exact cumulative distribution and probability mass functions for the maximum number of consecutive non-exceedances observed between two critical claims via first-order linear difference equations. The derived analytical expressions bridge the theory of runs and practical risk management, offering direct tools for dynamic solvency monitoring and operational stress testing without relying on asymptotic approximations
Kalkan et al. (Wed,) studied this question.