Mathematical Analysis of Replication by Cash Flow Matching

Jan Natolski and Ralf Werner
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Jan Natolski: University of Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany
Ralf Werner: University of Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany

Risks, 2017, vol. 5, issue 1, 1-15

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,weelaborateononespeci?cformulationofareplicatingportfolioproblem. 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 ef?cient 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.

Keywords: life insurance; replicating portfolio; market consistent valuation; cash ?ow matching; fair value; stochastic Fermat–Torricelli problem (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (5)

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