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 et al. (2015), Pelsser and Schweizer (2015) and Beutner et al. (2013). We complement this mainly asymptotic line of research on the approximation of the aggregated terminal value distribution of the liabilities under the risk neutral measure by several fundamental results concerning the overall effectiveness of the replicating portfolio approach. We first prove that both replication by terminal value and by cash flow matching are consistent with the aim to obtain an accurate approximation not only to the aggregated terminal value distribution, but, more importantly, to an accurate approximation of the distribution of the fair value of liabilities (FVL) after one period. In contrast to the existing literature, our results are not of asymptotic nature but provide exact bounds on the error of the approximation of the FVL distribution and apply to both the risk neutral and the real world measure. We further provide the missing link between the error in the FVL distribution and the error in the resulting risk capital figure, by providing explicit bounds on the latter in terms of the former. One important mathematical tool in our analysis is 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