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Mathematical Analysis of Replication by Cash Flow Matching

Author

Listed:
  • Jan Natolski

    (University of Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany)

  • Ralf Werner

    (University of Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany)

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.

Suggested Citation

  • Jan Natolski & Ralf Werner, 2017. "Mathematical Analysis of Replication by Cash Flow Matching," Risks, MDPI, vol. 5(1), pages 1-15, February.
    • Handle: RePEc:gam:jrisks:v:5:y:2017:i:1:p:13-:d:91771
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    References listed on IDEAS

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    Cited by:

    1. Patrick Cheridito & John Ery & Mario V. Wuthrich, 2021. "Assessing asset-liability risk with neural networks," Papers 2105.12432, arXiv.org.
    2. Patrick Cheridito & John Ery & Mario V. Wüthrich, 2020. "Assessing Asset-Liability Risk with Neural Networks," Risks, MDPI, vol. 8(1), pages 1-17, February.
    3. Luca Regis, 2017. "Special Issue “Actuarial and Financial Risks in Life Insurance, Pensions and Household Finance”," Risks, MDPI, vol. 5(4), pages 1-2, December.
    4. Hampus Engsner & Kristoffer Lindensjo & Filip Lindskog, 2018. "The value of a liability cash flow in discrete time subject to capital requirements," Papers 1808.03328, arXiv.org.
    5. Hampus Engsner & Kristoffer Lindensjö & Filip Lindskog, 2020. "The value of a liability cash flow in discrete time subject to capital requirements," Finance and Stochastics, Springer, vol. 24(1), pages 125-167, January.

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