## Mathematical Foundation of the Replicating Portfolio Approach

• Due to the Solvency II directive life insurance companies are required to quantify the risk of the distribution of their market consistent embedded value (MCEV) one year ahead in time. One of the prevailing techniques currently applied is the construction of a static replicating portfolio. With the only notable exception by Beutner et al. [2015], research has so far solely focused on how well replicating portfolios work in empirical studies. In this paper we give a mathematical justification for the use of replicating portfolios. We prove that both replication by terminal value and by cash flow matching is consistent with the aim to obtain an accurate approximation to the MCEV distribution. In contrast to Beutner et al. [2015], our results are not of asymptotic nature but provide exact bounds on the MCEV model error. We further complete the final step to link the MCEV model error to the risk capital figure to obtain upper bounds on the inaccuracy in the final risk capital. OneDue to the Solvency II directive life insurance companies are required to quantify the risk of the distribution of their market consistent embedded value (MCEV) one year ahead in time. One of the prevailing techniques currently applied is the construction of a static replicating portfolio. With the only notable exception by Beutner et al. [2015], research has so far solely focused on how well replicating portfolios work in empirical studies. In this paper we give a mathematical justification for the use of replicating portfolios. We prove that both replication by terminal value and by cash flow matching is consistent with the aim to obtain an accurate approximation to the MCEV distribution. In contrast to Beutner et al. [2015], our results are not of asymptotic nature but provide exact bounds on the MCEV model error. We further complete the final step to link the MCEV model error to the risk capital figure to obtain upper bounds on the inaccuracy in the final risk capital. One important mathematical tool in our analysis is the observation that in finite time, the measure change from the real world to the risk neutral measure provided by the FTAP can be both bounded below and above in the first period.