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

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Abstract

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.

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Notes

  1. See Chapter 2 of the International Monetary Fund Global Financial Stability Report (2013).

  2. As another innovation, EIOPA suggests using the Smith-Wilson (2001) method to fit the zero-coupon yield curve to the market data. Kocken et al. (2012) show that this method is not robust, which is a serious drawback, since the market data usually contains a certain degree of noise and this leads to spurious fluctuations into the spot forward rate [see the figure on page 92 of EIOPA (2014)]. This might lead to uninterpretable results.

  3. The European Bond Commission of the European Federation of Financial Analysts Societies.

  4. The risk-neutral pricing paradigm is based on special assumptions about the asset pricing. Some of them, such as market completeness, can be debated. However, this paradigm is generally accepted, being fairly convenient for practical purposes, especially when dealing with interest rate or credit derivatives. In particular, the corresponding price dynamics is automatically arbitrage-free.

  5. Here we refer again to Chapter 2 of the International Monetary Fund Global Financial Stability Report (2013). The weekly data on CDS transactions are also provided by DTCC Deriv/SERV on the DTCC website.

  6. We adopt some simplifying assumptions about recovery rate, described below.

  7. One can use other liquidity indicators, namely turnover, number of trades, etc., for this purpose provided there is appropriate data available.

  8. ECB methodology sets a maximum bid-ask spread per quote of three basis points as a selection criterion.

  9. Attribution of bond yield to the numerous different factors such as credit risk, liquidity, taxation, and convenience yield is a complicated and generally ill-posed problem, which requires careful model construction and/or regularisation. Furthermore, typically the result of decomposition of yields into several constituents simultaneously is not robust. Thus, in practice the set of constituents is limited to a few of them whereas others are ignored.

  10. In principle it can be done and possibly should be done, in particular to comply with fair value definition by IFRS 13, but it would require a considerable complication of the present paper, which is, however, unnecessary for the explanation of the main idea.

  11. Formally such an information can be described in a standard mathematical language in terms of \(\sigma \)-algebras (namely as a filtration).

  12. See the books by James and Webber (2000) and Andersen and Peterbarg (2010) for a comprehensive review.

  13. From mathematical point of view the default time can be assumed to be a totally inaccessible stopping time.

  14. It is also called the survival probability.

  15. Please refer to the book by Brigo and Mercurio (2006) for the details and the analogies between interest and hazard rates.

  16. Introducing correlations between interest rates and default intensities into the model would be a challenging problem; however, we remind that one of the main goals of the present work is to achieve tractability reasonable for practical purposes together with model independence (within some set of standard assumptions, one of which is the independence of default and interest rates). If we drop the independence assumption, the model independence cannot be achieved; the tractability also suffers considerably

  17. The ISDA CDS Standard Model is administrated by Markit and is used by market participants for CDS pricing and for the conversion of CDS spreads into up-front payments and vice versa. For a rigorous overview of the ISDA CDS Standard Model, see the paper by White (2013).

  18. Here and throughout the article we refrain from considering different day counting conventions and assume that all time variables are expressed in year fractions. Typically, day counting conventions would influence the exact value of the year fractions.

  19. See ISDA Big Bang Protocol (2009a) and Small Bang Protocol (2009b).

  20. The regularisation term, in general, is defined on a narrower functional space of smoother functions than the residual. Please refer to the book by Tikhonov and Arsenin (1977).

  21. In principle, different weights could be assigned to different issuers, for instance, proportional to their outstanding amounts.

  22. We think that the implementation of quantitative easing policy means market manipulation by monetary powers, so it should not be regarded as a market with ‘normal conditions’.

  23. Due to the nature of expressions (5) and (6) for \(P_{j,k}^{bond}\) and \(P_{j,k}^{CDS}\), for convenience reasons the problem should be stated in terms of the risk-free discount function \(D(\cdot )\) and survival probability functions \(Q_k(\cdot )\), rather then the risk-free instantaneous forward rate f and default intensities \(\lambda _k\).

  24. Note that most liquid CDS contracts on euro zone sovereign credits are denominated in US dollars whereas bonds are denominated in euros. However, in order to include USD-denominated CDS in our input data and treat them correctly, we need an additional model, taking into account the dependency between the default risk of the obligor and the exchange rate, resulting in an additional jump in the FX rate at default. This can be done using some advanced modelling, for example, as proposed by Ehlers and Schönbucher (2004), but we consider this as a possible topic for another work in the future.

  25. For more details, please, see: https://www.markit.com/assets/en/docs/products/data/indices/bond-indices/iboxx-rules/documentation/Markit%20iBoxx%20Pricing%20Rules.pdf.

  26. Wrong way risk can also be taken into account for the interpretation issues.

  27. Under the foundation IRB approach, Basel 2 accord prescribes fixed LGD ratios for certain classes of unsecured exposures, in particular, for senior claims on corporates, sovereigns and banks not secured by recognised collateral, a 45 % LGD is applied (subordinated claims correspond to a 75 % LGD level).

  28. For more details on the determinants of euro area sovereign bond yield spreads during the crisis, please see the overview by European Central Bank (2014b).

  29. This also applies to the forward rate parameterisation (20); note that it is impossible to construct a dynamic model, driven by a stochastic differential equation, satisfying no arbitrage condition (as the model is not affine, as long as \(\tau _1\) and \(\tau _2\) are not fixed). Arbitrage free dynamics is a necessary propriety of the model in the context of interest rate derivatives pricing, while in the framework of the present paper we believe that it is not so important; for a more detailed discussion please refer to the book by Diebold and Rudebusch (2013).

  30. Based on these considerations, we get \(\alpha =0.1\) for our particular dataset.

  31. The German bond zero-coupon curves are constructed from Markit data on German government bonds.

  32. ECB curves were downloaded from the official ECB website (see https://www.ecb.europa.eu/stats/money/yc/html/index.en.html).

  33. In order to construct the EIOPA risk-free curves, we replicated the respective methodology (2015).

  34. Note that these alternatives are a Svensson-based fitting method (ECB, built from AAA-rated bonds) and a spline-based (EIOPA, swap-based), so in analyzing our methodology with them, we compare it with the best-practice regulator-approved parametric and spline fitting methods. However, we should point out that for the purposes of this paper, the essence of all considered methodologies is not the fitting technique (parametric or spline), but rather the data choice, data interpretation and pre-processing, economically correct definition of risk-free rates and the approach to their estimation.

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Acknowledgments

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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  1. National Research University Higher School of Economics, Moscow, Russia

    Sergey N. Smirnov, Victor A. Lapshin & Marat Z. Kurbangaleev

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  1. Sergey N. Smirnov
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  2. Victor A. Lapshin
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  3. Marat Z. Kurbangaleev
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Correspondence to Victor A. Lapshin.

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Smirnov, S.N., Lapshin, V.A. & Kurbangaleev, M.Z. Deriving implied risk-free interest rates from bond and CDS quotes: a model-independent approach. Optim Eng 18, 499–536 (2017). https://doi.org/10.1007/s11081-016-9333-2

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