The last passage time before ruin: Theory and applications in liquidation risk management

In response to challenges posed by emerging risks such as climate change, practitioners are increasingly aware of the need for a more forward-looking approach to insurance solvency risk management, which requires not only the identification of risks but also timely intervention. However, determining when to implement risk mitigation is often complex, as it involves balancing insolvency prevention against the potential costs and consequences of such actions. In this paper, we provide insights into the timing of risk mitigation before it is too late by studying the last time a Lévy insurance risk process is above a certain threshold before ruin. In the theoretical part, we first derive the joint Laplace transform of the last passage time and the remaining time until ruin. We then study an optimal prediction problem of approximating the last passage time before ruin with a stopping time under the L1 distance, showing that the optimum occurs when the risk process first drops below a certain level. The stopping boundary is independent of the initial surplus level, and we provide an explicit characterization of this boundary. These theoretical results fill a gap in the literature, where last passage times are typically analyzed over an infinite time horizon or an independent exponential time horizon. By focusing on the dynamics of risk processes up to ruin, our findings offer interesting insights into liquidation risk management. These are discussed in the application part, where we develop a framework to endogenously determine financial distress and rehabilitation levels under contemporary regulations. We further analyze the liquidation time under Chapter 7 and Chapter 11 of the U.S. Bankruptcy Code. Numerical examples and an empirical study using real data are presented to illustrate the practical implications of our results.