NOTE ON THE SMITH–WILSON INTEREST RATE CURVE
Florian Gach ()
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Florian Gach: Austrian Financial Market Authority, Otto-Wagner-Platz 5, Vienna 1090, Austria
International Journal of Theoretical and Applied Finance (IJTAF), 2016, vol. 19, issue 07, 1-16
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).
Keywords: Smith–Wilson; interest rate term structure; ultimate forward rate; EIOPA curve (search for similar items in EconPapers)
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