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Long-term interest rates and consol bond valuation

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Abstract

This article presents a Gaussian three-factor model of the term structure of interest rates which is Markov and time-homogeneous. The model captures the whole term structure and is particularly useful in forward simulations for applications in long-term swap and bond pricing, risk management and portfolio optimization. Kalman filter parameter estimation uses EU swap rate data and is described in detail. The yield curve model is fitted to data up to 2002 and assessed by simulation of yield curve scenarios over the next 2 years. It is then applied to the valuation of callable floating rate consol bonds as recently issued by European banks to raise Tier 1 regulatory capital over the subsequent period from 2005 to 2007.

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Notes

  1. We use boldface throughout the paper to denote random or conditionally random entities.

  2. This assumption implies that the conditional variance of yield changes is constant over time. A number of studies concerned with the relatively short term have found that yield changes are conditionally heteroscedastic, cf. Ball and Torous (1996). Fong and Vasicek (1991) introduced stochastic volatility to represent this situation, whose relevance to the long run is questionable, for pricing (see also Litterman et al, 1991 and Andersen et al, 2004).

  3. Alternatively, De Jong (2000) presents a general way to obtain the exact discrete-time state distributions in affine class models. As the benefits are unclear for our purposes and simulation complexity increases, we have not pursued this approach here.

  4. A par interest rate swap is a standard contract between two counterparties to exchange cash flows. At set time intervals termed reset dates, one pays a predetermined fixed rate of interest on the nominal value, the other a floating rate, until the maturity date of the contract. The floating leg of swap fixes the interest rates for each payment at the rate of a published interest rate. The fixed rate, known as the swap rate, is that interest rate, which makes the fair value of the par swap 0 at inception. Thus, the cash flows of the two legs of a par swap are those of a pair of bonds with face value the swap nominal, one fixed rate and the other floating rate.

  5. But, see the section ‘Pricing consol bonds’ in which more recent data up to 2008 are used.

  6. We also evaluated the quadratic interpolation but deemed the negligible improvement in accuracy not worth the considerable increase in computational burden.

  7. Given the relatively low yield volatilities depicted in Figure 6 and the yield levels in Figure 5 we concluded that the probability of negative yields with our Gaussian model under the market measure is negligible. Using values from the data of Figure 5 and 6, these correspond to a minus 10 standard deviation event. Our decision is borne out by the representative paths in Figures 7 and 8 and, in fact, none of the 500 scenarios simulated produced negative yields over the out-of-sample period. However, see Abu-Mostafa (2001) for a technique for reducing this probability over longer simulation horizons.

  8. For example, in 2005 Deutsche Bank issued a €900 million tranche of bonds at par to face value or nominal. This revives an instrument that has not been in favour since the Russian Revolution, when Tsar Nicholas’ consols became worthless, although UK consols initiated in the eighteenth century are still in existence (with reduced fixed coupon). There is little current literature on their pricing when coupon rates are floating.

  9. We use daily data for consol bond valuation to conform to market practice by issuers who value fixed income instruments incorporating yield curve data on (or just before) the day of sale. Our example here is representative of a number of consol bonds we have valued initially on different dates in the period 2004–2006.

  10. Note that these fits on representative days do not always accurately capture the long end of the yield curve, which might require a fourth factor. They are, however, acceptably accurate up to 10-year maturity and in any event generally err on the conservative side, by producing lower discount rates.

  11. This corresponds to the historical 4-year S&P cumulative default rate for the bond's A rating which suggests that the market was optimistic regarding the bank's possible default on the contract over possibly nearly two and a half centuries.

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Acknowledgements

We thank Drs Yee Sook Yong, Muriel Rietbergen and Giles Thompson for analytical and computational assistance on the research reported herein. We also acknowledge helpful comments from Julian Roberts, Cambridge Finance Seminar participants and anonymous referees, which materially improved the paper.

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  1. Centre for Financial Research, Statistical Laboratory, University of Cambridge, Wilberforce Road, CB3 0WB, Cambridge, UK

    Michael A H Dempster

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  1. Michael A H Dempster
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  2. Elena A Medova
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  3. Michael Villaverde
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Correspondence to Michael A H Dempster.

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Dempster, M., Medova, E. & Villaverde, M. Long-term interest rates and consol bond valuation. J Asset Manag 11, 113–135 (2010). https://doi.org/10.1057/jam.2010.7

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