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Construct Smith-Wilson risk-free interest rate curves with endogenous and positive ultimate forward rates

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  • Zhao, Chaoyi
  • Jia, Zijian
  • Wu, Lan

Abstract

We propose several methods for obtaining endogenous and positive ultimate forward rates (UFRs) for risk-free interest rate curves based on the Smith-Wilson method. The Smith-Wilson method, which is adopted by Solvency II, can both interpolate the market price data and extrapolate to the UFR. However, the method requires an exogenously-chosen UFR. To obtain an endogenous UFR, de Kort and Vellekoop (2016) proposed an optimization framework based on the Smith-Wilson method. In this paper, we prove the existence of an optimal endogenous UFR to their optimization problem under the condition that the cash flow matrix is square and invertible. In addition, to ensure the positivity of the optimal endogenous UFR during extreme time periods such as the COVID-19 pandemic, we extend their optimization framework by including non-negative constraints. Furthermore, we also propose a new optimization framework that can not only generate endogenous and positive UFRs but also incorporate practitioners' prior knowledge. We prove the feasibility of our frameworks, and conduct empirical studies for both the Chinese government bonds and the EURIBOR swaps to illustrate the capabilities of our methods.

Suggested Citation

  • Zhao, Chaoyi & Jia, Zijian & Wu, Lan, 2024. "Construct Smith-Wilson risk-free interest rate curves with endogenous and positive ultimate forward rates," Insurance: Mathematics and Economics, Elsevier, vol. 114(C), pages 156-175.
    • Handle: RePEc:eee:insuma:v:114:y:2024:i:c:p:156-175
    DOI: 10.1016/j.insmatheco.2023.11.003
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    References listed on IDEAS

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    More about this item

    Keywords

    Ultimate forward rate (UFR); Smith-Wilson method; Risk-free interest rate curve; Endogenous and positive; Solvency II; Chinese government bond; EURIBOR swap;
    All these keywords.

    JEL classification:

    • L51 - Industrial Organization - - Regulation and Industrial Policy - - - Economics of Regulation
    • E4 - Macroeconomics and Monetary Economics - - Money and Interest Rates

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