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Risks 2017, 5(1), 13; doi:10.3390/risks5010013

Mathematical Analysis of Replication by Cash Flow Matching

University of Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany
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Academic Editor: Luca Regis
Received: 17 August 2016 / Accepted: 24 February 2017 / Published: 28 February 2017
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Abstract

The replicating portfolio approach is a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of risk capital. In this note,weelaborateononespecificformulationofareplicatingportfolioproblem. Incontrasttothetwo most popular replication approaches, it does not yield an analytic solution (if, at all, a solution exists andisunique). Further,althoughconvex,theobjectivefunctionseemstobenon-smooth,andhencea numericalsolutionmightthusbemuchmoredemandingthanforthetwomostpopularformulations. Especially for the second reason, this formulation did not (yet) receive much attention in practical applications, in contrast to the other two formulations. In the following, we will demonstrate that the (potential) non-smoothness can be avoided due to an equivalent reformulation as a linear second order cone program (SOCP). This allows for a numerical solution by efficient second order methods like interior point methods or similar. We also show that—under weak assumptions—existence and uniqueness of the optimal solution can be guaranteed. We additionally prove that—under a further similarly weak condition—the fair value of the replicating portfolio equals the fair value of liabilities. Based on these insights, we argue that this unloved stepmother child within the replication problem family indeed represents an equally good formulation for practical purposes. View Full-Text
Keywords: life insurance; replicating portfolio; market consistent valuation; cash flow matching; fair value; stochastic Fermat–Torricelli problem life insurance; replicating portfolio; market consistent valuation; cash flow matching; fair value; stochastic Fermat–Torricelli problem
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Natolski, J.; Werner, R. Mathematical Analysis of Replication by Cash Flow Matching. Risks 2017, 5, 13.

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