Life insurance liabilities are in general a function of various risk sources such as equity, interest rate, or mortality. It is common to measure the total risk by advanced stochastic models. However, the question of how to allocate the randomness of life insurance liabilities to different risk sources, and how to quantify and compare the individual risk contributions, is rarely discussed. Nevertheless, in order to be able to take adequate risk management strategies including product modifications, insurance companies need to assess the relative importance of each risk source.

Simple approaches such as sensitivity analyses with respect to model parameters usually lack any comparability between the different risk sources. In literature, the most common solution to this risk allocation problem is the variance decomposition approach which consecutively decomposes the risk by means of conditional expectations. Unfortunately, the results of this decomposition depend on the sequence of conditioning and are restricted to the variance as risk measure. Furthermore, most methods proposed in literature are only applicable at a certain point in time or state of the considered contract, although the risk decomposition clearly changes over time. Additionally, the natural requirement that the sum of the single risk components should be equal to the total insurance liabilities is often neglected.

In this paper, we first propose a decomposition method mainly based on the martingale representation theorem and show how this method provides a dynamic allocation of the total risk to the different risk sources over time. A comparison of the proposed method to those mentioned above demonstrates how this dynamic decomposition overcomes all discussed drawbacks. Second, we apply the mentioned decomposition approaches to several types of annuity conversion options and quantify the respective equity, interest and mortality risks. We show that different product designs imply significantly different risk decompositions and derive valuable insights for risk managers.

Keywords: Life insurance liabilities, risk management, risk decomposition, stochastic modeling of financial and mortality risk