Abstract
At a given moment, straight bonds1 with different maturities exchanged on the same financial market have different prices.2 In this chapter, we shall examine these differences in terms of yield, in order to make them comparable to each other. The relationship which is supposed to exist between the remaining lifetime of a security and its yield constitutes what is known as ‘the term structure of interest rates’. The aim of this chapter is mainly descriptive and empirical. Of course, we shall see that these differences in price according to maturity are linked to the fluctuations in interest rates on the market and to the resulting risk to bonds. However, we leave it until the next chapter to model this aspect of things explicitly.
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Notes to Chapter 9
By a straight bond we mean a security whose contract unconditionally specifies the future payments which the issuer must make.
Refer to what was said in the introduction to Chapter 8. In exact contrast to what we did in that chapter, we now examine bonds whose signature risk is assumed to be zero. This means that the probability of regular payment of dividends and repayments is equal to 1. For all that, the security is not risk-free over a given period of time, since interest rates on the market fluctuate and produce fluctuations in the prices of bonds. It is this aspect which now requires attention.
As in Section 1.2.
This concept obviously supposes that it is possible to define a unitary holding period for securities to which every investor would conform. This means that all operators intend re-equilibrating their portfolio on the same dates by choosing to resell some securities and buy others. The choice of securities is guided by their respective rates of return, as we saw in Chapter 3, which dealt with portfolio choices. The most natural hypothesis, and the one which gives the greatest freedom to investors, involves reducing this holding period until it is infinitely small. Investors are then free to adjust their portfolio at each instant of the real line. We shall do this in Chapter 10.
The appropriate method of determining the level of risk for the various securities was shown in Chapter 4.
Furthermore, if the period rates of return were different from one security to the next, the present value factors resulting from the composition of the rates of return would also be different and the market values would not be additive: that is to say, a security whose sequence of flows was exactly equal to the sum of the sequences of two other securities would not have a price equal to the sum of the prices of these securities.
Any iterative algorithm must be initialized. A first approximation of yield to maturity which may be used as a starting point for the calculation is given by the coupon plus the average appreciation divided by an average price: \( \frac{{500 + \left( {10500 - 9484.41} \right)/3}} {{\left( {10500 + 9484.41} \right)/2}} = 8.39\% \) On the properties of this approximation, see [Hawawini and Vora 1979].
The reader will recall that a yield is a ratio of cash flows to an initial price, whereas a rate of return incorporates the appreciation produced on a security during a given holding period. It is therefore correct to speak of yield, and not rate of return, to maturity. In fact, definition 2.1 establishes a relationship between the cash flows paid by the security on the one hand, and the initial price on the other.
For this calculation to mean anything, the two securities must be comparable. In the paragraphs which follow, we examine the possibilities of transposition to securities of the same ‘duration’ (see below). We shall see that even then, the procedure is incorrect.
A calculation of this type is described in detail in Section 9.3.
More generally, the forward rate f(t, T, T*) for a period starting at T and ending at T* is calculated as follows: \( f\left( {t,\,T,\,t*} \right) = \left( {\frac{{P\left( {t,\,T} \right)}} {{P\left( {t,\,T*} \right)}}} \right)^{1/\left( {T* - T} \right)} - 1 = \left( {\frac{{\left[ {1 + R\left( {t,\,T*} \right)} \right]^{\left( {T* - T} \right)} }} {{\left[ {1 + R\left( {t,\,T} \right)} \right]^{\left( {T - t} \right)} }}} \right)^{1/\left( {T* - T} \right)} - 1 \).
By ‘forward rates’ we mean the implicit future rates deduced from prices quoted today for bonds with different maturities.’ spot rates’, on the other hand, are the rates covering a period of time of any length but whose starting date of lending or borrowing is today’s date. These’ spot’ rates therefore correspond to the yield to maturity of zero-coupon bonds. In the eyes of investors, the spot rates of interest which will prevail at the various future dates are random variables. They constitute a stochastic process in the sense of Chapter 7. However, the calculation of forward rates, as defined above, yields real numbers and not random variables. Forward rates are, in a sense to be denned, ‘certainty equivalents’ of future rates. Nothing indicates that these are unbiased estimators of these future rates. (An estimator is unbiased if its expected value is equal to the quantity to be estimated.) We return to this question in Chapter 10.
By applying the above definition, we calculate \( f\left( {t,\,t} \right) = \frac{{P\left( {t,\,t} \right)}} {{P\left( {t,\,t + 1} \right)}} - 1 = \frac{1} {{0.9434}} - 1 = 0.6 \) and so on for f(t, t + 1) and f(t, t + 2).
A logarithmic derivative is a derivative in terms of relative variations. It indicates by what percentage the price falls when the discount rate increases by 1% from what it was previously.
For our bond (with successive flows of 500, 500 and 10500) and spot rates of 6%, 6.49883% and 6.99688%, the duration calculated by discounting all the flows at the yield-to-maturity (6.9633%) is equal to 2.8554, whereas it is 2.8541 when each flow is discounted at the spot rate corresponding to the date of the flow. The difference may, however, be starker for bonds with more distant maturities.
In this case, duration in the sense of Macaulay.
Another break with logic will be noted here: we have just stated in the previous paragraph that securities having the same level of risk ought to have the same rate of return. The principle stated is curiously based on a comparison of the yields to maturity. We have already said that yields to maturity were not transposable from one security to another. But it is so much more convenient to assume that they are!
This weakness in the principle is of course known to actuaries, who specify that yield to maturity only ‘means anything’ if the hypothesis of a constant rate of reinvestment is adopted. In the example considered, we have seen that the envisaged future rates are 12, 14 and 16% successively.
Cf. Section 9.2.1.
The duration formula is given in Section 9.2.2, Equation 2.3.
This amount is the result of discounting the successive flows 500, 500 and 10500 at the respective spot rates of 7.06%, 7.564% and 8.067%.
Let us repeat: if the yields to maturity were common to all the securities, the rates of return on the various securities during successive periods would differ from one another, which would offer arbitrage opportunities to any investor in a position to buy and sell when the time seemed right.
Although the prices of three securities are sufficient, it would be inefficient not to use the prices of the n securities quoted; as these in practice do not all exactly satisfy a system of n equations with three unknowns, we choose the values of z 1, z 2 and z 3, which minimize the sum of the squares of the differences between the right and left-hand sides of the system.
Another much less insignificant difficulty lies in the fact that the majority of bonds give rise to interim repayment. Furthermore, many issuing firms have in their contracts the option of substituting for repayment by repurchase of their bonds on the Stock Exchange. See Appendix 1 of this chapter.
Cf. [McCulloch 1971, 1975].
That is, in such a way that its derivative is continuous.
Given the hypotheses of continuity, these parameters are not independent of one another.
Used by [Schaefer 1981] and [Bonneville 1983].
The constant terms of the polynomials must be zero since the present value factor corresponding to immediate maturity is equal to 1.
Securities exempt from tax are an exception to this general rule. The valuation of these securities requires, therefore, that they are compared with other securities whose revenues are taxable.
In the presence of taxes, it may be that at equilibrium, even the net flows of taxes of different securities are discounted differently, the gains and losses of an investor not always being treated the same in the calculation of tax. As well as this, fiscal arbitrage often leads the investor to wish to sell bonds short (or to take forward contracts of sale). The tax on short sales (negative holdings of securities) is not symmetrical to the tax which hits pure and simple holdings of securities. This is added to the usual obstacles which are encountered when attempting to sell securities short. These various phenomena may lead certain fiscal categories not to hold any bonds, either positively or negatively, whereas other categories only hold some specific types of security (the ‘corner’ solution).
This is the case at least in the United States.
Remember that we are dealing here exclusively with the risk linked to movements in interest rates. Default risk was dealt with in Chapter 8.
It is important to realize that until now the implicit holding period which we have considered was that of one unit of time, whether in discrete time [t, t + 1] or in continuous time [t, t + dt\. The holding period we are dealing with here [t, t + D] may (and probably will), on the other hand, cover several units of time.
More precisely, the return made on the date of the duration may not be less than the return which would have been made in the absence of a variation in rates, but may be greater.
As in the numerical example, we assume that all the spot rates are equal. The reasoning remains valid in situations other than this, however, on condition that the duration is correctly defined. Cf Note 41 below.
Including reinvested coupons.
For mathematical proofs of the results of this section, see, for example, [Bierwag 1987], Chapter 4.
Remember that this is the price effect measured on the date corresponding to duration, and not at t!.
That is, the yield to maturity of the security when it is acquired.
For other configurations of the range of rates or other types of shock, the analysis of immunization given in this chapter remains equally valid as long as new definitions of duration are used. See, amongst others, [Ingersoll et al. 1978], [Bierwag et al. 1988], [Hawawini 1987], [Schaefer 1984]. [Schaefer 1984] shows, however, that from an operational point of view, conventional immunization strategies (using Macaulay’s duration) generate results practically as good as those based on more sophisticated models.
This section is largely similar to [Cox et al 1981] and [Ingersoll 1987].
Jensen’s inequality: if X is a random variable with an expected value of E(X) and g(.) a convex function, then \( E\left( {g\left( X \right)} \right) \geqslant g\left( {E\left( X \right)} \right) \) Except in special cases, therefore, \( E\left( {g\left( X \right)} \right) \ne g\left( {E\left( X \right)} \right) \).
Either investing in a long security with maturity T, or investing successively in securities with one-period maturities.
It would be equivalent to the expectations hypothesis of the rate of return until maturity, Equation 6.11, if the levels of the future rates were not correlated between themselves, which is not verified empirically.
See Appendix 2 to this chapter.
Here, risk premiums which are no longer necessarily an increasing function of the time to maturity, as was the case with Hicks.
Hicks’ liquidity preference theory therefore becomes a special case of the preferred habitat theory, one in which the investors’ preferred habitat is the very short term.
This appendix was written with the help of Bertrand Jacquillat, to whom we extend our thanks. Sinking-fund provisions vary greatly from country to country and from bond to bond. A good theoretical treatment is [Ho 1985].
It is in order to diversify this randomness that buyers of bonds, who buy a sufficient portion of the same issue, prefer to buy shares which are spread over several series. Bond valuation theories are therefore allowed to neglect the risk of the draw.
The draw is then made on the shares and no longer on the series. It may often be done on half-series, as the repurchase clause often sets a ceiling on repurchases and permits the issuer to repurchase at maximum a half a repayment block.
We disregard the costs involved in this type of operation.
To check this, define \( Y = \exp \left( { - \int_t^T {r\left( s \right)ds} } \right) \) rewrite the three different versions of the hypothesis and use Jensen’s inequality.
References
G. O. Bierwag, Duration Analysis: Managing Interest Rate Risk. Ballinger, Cambridge, MA, 1987.
[Bierwag et al 1988]_G. O. Bierwag, G. G. Kaufman and G. M. Latta, Duration models: A taxonomy, Journal of Portfolio Management, Autumn 1988.
M. Bonneville, Le prix des obligations sur le marché de Paris. Thèse de Doctorat d’Etat soutenue à l’Université de Lille, 1983.
[Cox et al. 1979]_J. G. Cox, J. E. Ingersoll and S. A. Ross, Duration and the Measurement of Basis Risk, Journal of Business, 52/1, 1979, 51–61.
[Cox et al. 1981]_J. G. Cox, J. E. Ingersoll and S. A. Ross, A Re-examination of Traditional Hypotheses about the Term Structure of Interest Rates The Journal of Finance, 36, No. 4, 1981.
J. Gulbertson, The Term Structure of Interest Rates Quarterly Journal of Economics, 1957.
I. Fisher, Appreciation and Interest, American Economic Association Publications, 1896.
G. Hawawini, Controlling the Interest Rate Risk of Bonds: An Introduction to Duration Analysis and Immunization Strategies, Finanzmarkt und Portfolio Management, No. 4, 1986/87.
G. A. Hawawini and A. Vora, On The Theoretic and Numeric Problem of Approximating The Bond Yield to Maturity, Working Paper No. 160, Salomon Brothers Center for the Study of Financial Institutions, New York University, 1979.
J. Hicks, Value and Capital. Oxford University Press, 1939.
T. S.Y. Ho, The Value of a Sinking Fund Provision under Interest-Rate Risk, in E. I. Altman and M. G. Subrahmanyam, Recent Advances in Corporate Finance. Dow Jones Irwin, 1985.
S. D. Hodges and S. Schaefer, A Model for Bond Portfolio Improvement, Journal of Financial and Quantitative Analysis, 1977, 243-259.
J. E. Ingersoll, Theory of Financial Decision Making. Rowman & Littlefield, 1987.
J. E. Ingersoll, Interest Rates, in The New Palgrave — Finance. Norton, 1989.
[Ingersoll et al. 1978]_J. E. Ingersoll, J. Skelton and R. L. Weil, Duration Forty Years Later, Journal of Financial and Quantitative Analysis, November 1978, 627-650.
R. H. Litzenberger and J. Rolfo, Arbitrage Pricing, Transaction Costs and Taxation of Capital Gains. A Study of Government Bonds with the Same Maturity Date, Journal of Financial Economics, 13, 1984, 337–351.
F. Lutz, The Structure of Interest Rates, Quarterly Journal of Economics, 1940.
F. R. Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the U.S. since 1856, National Bureau of Economic Research, New York: Columbia University Press, 1938.
J. H. McCulloch, Measuring the Term Structure of Interest Rates, Journal of Business, 44, 1971, 19–31.
J. H. McCulloch, The Tax-adjusted Yield Curve, Journal of Finance, 30, 1975, 811–830.
F. Modigliani and R. Sutch, Innovations in Interest Rate Policy, American Economic Review, 1966.
J.-P. Nougier, Méthodes de calcul numérique. Masson, 1983.
S. Schaefer, Measuring a Tax-Specific Term Structure of Interest Rates in the Market for British Government Securities, Economic Journal, 1981, 415-438.
S. Schaefer, Immunisation and Duration: A Review of Theory, Performance and Applications, Midland Corporate Finance Journal, 2, Autumn 1984, 41–58.
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Dumas, B., Allaz, B. (1996). The term structure of interest rates. In: Financial Securities. The Current Issues in Finance Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7116-6_9
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