Optimization and Engineering

, Volume 18, Issue 2, pp 499–536 | Cite as

Deriving implied risk-free interest rates from bond and CDS quotes: a model-independent approach

  • Sergey N. Smirnov
  • Victor A. Lapshin
  • Marat Z. Kurbangaleev


We propose a market-consistent approach to the definition and construction of the implied term structure of the risk-free interest rates which are model-independent with respect to the choice of the fitting method. The main idea consists of the simultaneous fitting of the credit default swap (CDS) and the defaultable bond quotes where the theoretical prices are calculated in the framework of the reduced-form modelling of credit risk under standard assumptions. We obtain not only the implied risk-free zero-coupon yield curve but also the implied issuer-specific hazard rate curves. Prior to fitting, we perform a selection of bond issues and issuers. Next, we check for data consistency via arbitrage-like reasoning. Typically, the initial data needs a consistency adjustment, namely ‘artificial’ widening of the observed bid-ask spreads for the selected financial instruments. We construct feasibility bands representing achievable precision of the fitting procedure depending on maturity. Then we apply this methodology to determine the term structure of the risk-free rates for the euro zone. This generic approach for the calculation of the risk-free reference rates in the euro zone can be helpful for the purposes of insurers and pension funds. In particular, the relevant term structure can be used in the assessment of technical provisions as requested in Article 77 of the Solvency II Level 1 text.


Risk-free interest rates Term structure of credit spreads Sovereign default risk Euro zone Defaultable bond Credit default swap Zero coupon yield Market consistency 

Mathematics Subject Classification

90C90 91B25 

JEL Classification

G12 G15 



The research leading to these results has received funding from the Basic Research Program at the National Research University Higher School of Economics. The authors would like to thank two associate editors and three anonymous referees for their valuable comments. We are also grateful to Chris Golden, Con Keating, Thomas Klepsch, Stavros Zenios, and all participants of the EFFAS-EBC meetings for stimulating discussions and useful suggestions. Finally, we are grateful to Markit for providing bond data.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sergey N. Smirnov
    • 1
  • Victor A. Lapshin
    • 1
  • Marat Z. Kurbangaleev
    • 1
  1. 1.National Research University Higher School of EconomicsMoscowRussia

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