Mathematical foundation of the replicating portfolio approach

In the last few years, the first theoretical foundations for replicating portfolios – probably the most prevailing technique for risk capital calculation in life insurance – have been given in a series of papers by Beutner, Pelsser and Schweizer. In these papers, the asymptotic behaviour of replicating portfolios concerning the approximation of the terminal value (TVL) and the fair value distribution of the liabilities (FVL) has been investigated in detail. We complement this line of research by providing results on approximations based on a finite number of replicating instruments. We do so by providing the link between the approximation error of the TVL distribution, the FVL distribution and the error in the resulting risk capital figure, either value at risk or some coherent risk measure. We further allow for a variety of practically relevant formulations of the replication problem, including cash flow matching approaches. In contrast to the existing literature, all our results apply to approaches both under the risk-neutral and the real-world measure. Our strongest bounds are due to the observation that in discrete time, the measure change from the real-world to the risk-neutral measure can be both bounded below and above by a suitable constant in the first period.