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Note On The Smith–Wilson Interest Rate Curve

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  • FLORIAN GACH

    (Austrian Financial Market Authority, Otto-Wagner-Platz 5, Vienna 1090, Austria)

Abstract

Since the entry into force of Solvency II as of 1 January 2016, all European insurance companies concerned have to use the Smith–Wilson interest rate curve to determine the value of their insurance obligations and thus of a substantial part of their balance sheet. Although Smith & Wilson introduce the underlying discount curve P̲(t) as the sum of a ‘long-term’ discount curve e−f∞t and a linear combination of the so-called Wilson function W(t,u) evaluated at different payment dates uj, that is, P̲(t) = e−f∞t +∑ jβjW(t,uj), a mathematically sound derivation of its shape is missing in the literature. The aim of this paper is to close this gap. To this end, we reformulate the infinite-dimensional optimization problem stated in Smith & Wilson (2000) within an analytically rigorous framework. We prove that it has a unique minimizer and explicitly derive the formula displayed above. In doing so, we show that W(t,u) is in fact the kernel function of a particular reproducing kernel Hilbert space, which is the key result to fully understanding the shape of P̲(t).

Suggested Citation

  • Florian Gach, 2016. "Note On The Smith–Wilson Interest Rate Curve," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(07), pages 1-16, November.
    • Handle: RePEc:wsi:ijtafx:v:19:y:2016:i:07:n:s0219024916500394
    DOI: 10.1142/S0219024916500394
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    References listed on IDEAS

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    1. de Kort, J. & Vellekoop, M.H., 2016. "Term structure extrapolation and asymptotic forward rates," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 107-119.
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    Cited by:

    1. Jørgensen, Peter Løchte, 2018. "An analysis of the Solvency II regulatory framework’s Smith-Wilson model for the term structure of risk-free interest rates," Journal of Banking & Finance, Elsevier, vol. 97(C), pages 219-237.
    2. Lutz Kruschwitz, 2018. "Das Problem der Anschlussverzinsung," Schmalenbach Journal of Business Research, Springer, vol. 70(1), pages 9-45, March.

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