Allocating Capital to Time: Introducing Credit Migration for Measuring Time-Related Risks

This paper deals with the problem of assessing the risk related to time. On one hand, c}lassical regulatory rules for capital allocation of long-tailed insurance risks do not ask insurance companies to hold solvency capital early in the process. However, this may underestimate the risk of a deterioration of the credit state of the company until the time when the capital is needed. On the other hand, actually implemented actuarial capital management strategies of companies can often be interpreted as implicitly allocating that capital earlier than demanded. We propose a framework for quantifying risks associated with time. Using it, we evaluate strategies for capital allocation as a function of the time to ultimate, aiming to effectively manage long-tail business without impeding its growth. We model the impact of exogenous credit migration risk and the financial repercussions of overlooking it. We evaluate six different strategies, including the costs associated with potential company bankruptcy until the settlement of long-term claims. A detailed numerical implementation is provided through a simple example of a stand-alone heavy-tailed insurance risk anticipated in the future. We estimate a Markov chain credit migration model using insurance market data and analyze and interpret the liability values resulting from the various capital management strategies discussed in this paper. It appears that the actuarial practice of early capital raising is costly, even with appropriate penalties for avoided credit risk, unless the company's initial credit rating is poor. In such cases, purchasing protection through a credit derivative could be more efficient, provided such products are available in the market.