Throughout this discussion we assume an economist’s world of the per- fect market (also sometimes called the frictionless financial market). Such a perfect capital market is characterized by:
■ Perfect information ■ No taxes
■ Bullet maturity bonds ■ No transaction costs
Of course, in practice markets are not completely perfect. However, assuming perfect markets makes the discussion of spot and forward rates and the term structure easier to handle. When we analyze yield curves for their information content, we have to remember that the mar- kets that they represent are not perfect, and that frequently we observe anomalies that are not explained by the conventional theories.
At any one time it is probably more realistic to suggest that a range of factors contributes to the yield curve being one particular shape. For instance, short-term interest rates are greatly influenced by the availabil-
21See H. Levy, Introduction to Investments, Second Edition (Cincinnati, Ohio: South- Western College Publishing, 1999), pp. 562–564.
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ity of funds in the money market. The slope of the yield curve (usually defined as the 10-year yield minus the 3-month interest rate) is also a measure of the degree of tightness of government monetary policy. A low, upward-sloping curve is often thought to be a sign that an environ- ment of cheap money, due to a looser monetary policy, is to be followed by a period of higher inflation and higher bond yields. Equally, a high downward-sloping curve is taken to mean that a situation of tight credit, due to a stricter monetary policy, will result in falling inflation and lower bond yields.
Inverted yield curves have often preceded recessions; for instance, The Economist in an article from April 1998 remarked that, with one exception, every recession in the United States since 1955 had been pre- ceded by a negative yield curve.22 The analysis is the same: If investors expect a recession they also expect inflation to fall, so the yields on long-term bonds will fall relative to short-term bonds. So the conven- tional explanation for an inverted yield curve is that the markets and the investment community expect either a slow-down of the economy, or an outright recession.23 In this case one would expect the monetary author- ities to ease the money supply by reducing the base interest rate in the near future: hence an inverted curve. At the same time, a reduction of short-term interest rates will affect short-dated bonds and these are sold off by investors, further raising their yield.
While the conventional explanation for negative yield curves is an expectation of economic slow-down, on occasion other factors will be involved. In the UK during the period July 1997–June 1999, the gilt yield curve was inverted.24 There was no general view that the economy was heading for recession; in fact, the newly elected Labour government inherited an economy believed to be in satisfactory shape. Instead, the explanation behind the inverted shape of the gilt yield curve focused on two other factors: (1) the handing of responsibility for setting interest rates to the Monetary Policy Committee (MPC) of the Bank of England and (2) the expectation that the UK would, over the medium term, aban- don sterling and join the euro currency. The yield curve at this time sug- gested that the market expected the MPC to be successful and keep
22The exception was the one precipitated by the 1973 oil shock.
23A recession is formally defined as two successive quarters of falling output in the domestic economy.
24Although the gilt yield curve changed to being positively-sloped out to the 7–8 year maturity area, for a brief period in June–July 1999, it very quickly reverted to being inverted throughout the term structure, and remained so until May–June 2001, when it changed once again to being slightly positive-sloping up to the 4-year term, and inverting from that point onwards. This shape at least is more logical and explain- able.
An Introductory Guide to Analyzing and Interpreting the Yield Curve 89
inflation at a level of around 2.5% over the long term (its target is actu- ally a 1% range either side of 2.5%), and also that sterling interest rates would need to come down over the medium term as part of convergence with interest rates in euroland. These are both medium-term expectations however, and, in the author’s view, are not logical at the short-end of the yield curve. In fact the term structure moved to a positive-sloped shape up to the 6–7 year area, before inverting out to the long-end of the curve, in June 1999. This is a more logical shape for the curve to assume, but it was short-lived and returned to being inverted after the two-year term.
There is, therefore, significant information content in the yield curve, and economists and bond analysts will consider the shape of the curve as part of their policy-making and investment advice. The shape of parts of the curve, whether the short-end or long-end, as well that of the entire curve, can serve as useful predictors of future market condi- tions. As part of an analysis it is also worthwhile considering the yield curves across several different markets and currencies. For instance, the interest-rate swap curve, and its position relative to that of the govern- ment bond yield curve, is also regularly analyzed for its information content. In developed-country economies, the swap market is invariably as liquid as the government bond market, if not more liquid, and so it is common to see the swap curve analyzed when making predictions about, say, the future level of short-term interest rates.
Government policy will influence the shape and level of the yield curve, including policy on public sector borrowing, debt management and open-market operations.25 The market’s perception of the size of public sector debt will influence bond yields; for instance, an increase in the level of debt can lead to an increase in bond yields across the maturity range.
Open-market operations can have a number of effects. In the short-term it can tilt the yield curve both upwards and downwards; longer term, changes in the level of the base rate will affect yield levels. An anticipated rise in base rates can lead to a drop in prices for short-term bonds, whose yields will be expected to rise; this can lead to a (temporary) inverted curve. Finally, debt management policy26 will influence the yield curve.
Much government debt is rolled over as it matures, but the maturity of the replacement debt can have a significant influence on the yield curve in the form of humps in the market segment in which the debt is placed, if the debt is priced by the market at a relatively low price and hence high yield.
25“Open-market operations” refers to the daily operation by the Bank of England to control the level of the money supply (to which end the Bank purchases short-term bills and also engages in repo dealing).
26In the United Kingdom this is now the responsibility of the Debt Management Of- fice.
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The Information Content of the UK Gilt Curve:
A Special Case
In the first half of 1999 various factors combined to increase the demand for gilts, especially at the long-end of the yield curve, at a time of a reduc- tion in the supply of gilts as the government’s borrowing requirement was falling. This increased demand led to a lowering in market liquidity as prices rose and gilts became more expensive (that is, lower-yielding) than government securities in most European countries. This is a relatively new phenomenon, witness 10-year UK government yields at 5.07% compared to U.S. and Germany at 6.08% and 5.10%, respectively, at one point in August 1999.27 At the long-end of the yield curve, UK rates were, for the first time in over 30 years, below both German and U.S. yields, reflecting the market’s positive long-term view of the UK economy. At the end of September 1999, the German 30-year bond (the 4³₄% July 2028) was yielding 5.73% and the U.S. 6.125% 2027 was at 6.29%, compared to the UK 6% 2028, which was trading at a yield of 4.81%.
The relatively high price of UK gilts was reflected in the yield spread of interest-rate swaps versus gilts. For example, in March 1999, 10-year swap spreads over government bonds were over 80 basis points in the UK compared to 40 basis points in Germany. This was historically large and was more than what might be required to account purely for the credit risk of swaps. It appears that this reflected the high demand for gilts, which had depressed the long-end of the yield curve. At this point the market contended that the gilt yield curve no longer provided an accurate guide to expectations about future short-term interest rates.
The sterling swap market, where liquidity is always as high as the gov- ernment market and (as on this occasion) often higher, was viewed as being a more accurate prediction of future short-term interest rates. In hindsight this view turned out to be correct; swap rates fell in the UK in January and February 1999, and by the end of the following month the swap yield curve had become slightly upward-sloping, whereas the gilt yield curve was still inverted. This does indeed suggest that the market foresaw higher future short-term interest rates and that the swap curve predicted this, while the gilt curve did not. Exhibit 4.4 shows the change in the swap yield curve to a more positive slope from December 1998 to March 1999, while the gilt curve remained inverted. This is an occasion when the gilt yield curve’s information content was less relevant than that in another market yield curve, due to the peculiar circumstances resulting from lack of supply to meet increased demand.
27Yields obtained from Bloomberg.
An Introductory Guide to Analyzing and Interpreting the Yield Curve 91
EXHIBIT 4.4 UK Gilt and Swap Yield Curves
Source:Bank of England.
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CHAPTER 5
93
Term Structure Modeling*
David Audley Consultant Richard Chin Consultant Shrikant Ramamurthy Senior Vice President Greenwich Capital
t is the objective of this chapter is to describe the principles and approaches to term structure modeling. Readers familiar with the aca- demic literature addressing the term structure will see that we have adopted an eclectic mixture of ideas from this area (we indicate the sources of these ideas, where appropriate). However, such readers also will note some marked departures from the usual academic assump- tions, necessitating unusual implementations. These are driven by the reality of the markets, often overlooked for the sake of analytic cleanli- ness. We will highlight these and their implications as well.
Computational implementation of anything as complex as the dynamic term structure model described in this chapter naturally engen- ders the rigorous adherence to, yet clever application of, some arcane ideas from software/system engineering. This is beyond the scope of this introduction, but such topics include numerical recipes; mechanisms to ensure internal consistencies during development and build-up; tests for
I
* This chapter is based on a research paper written by the authors while employed by Prudential Securities.
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internal consistency, verification and validation of completed applica- tions (e.g., put-call parity, cash and carry arbitrage, and others); param- eterization of models and applications from the markets; and the utility of advanced computer architectures.
The following division of topics as well as the section flow address theoretical aspects of the term structure and term structure models, fol- lowed by the application of the theory to financial instruments and mar- kets. This is meant to serve only as a “sampler” of how term structure models can be used as strategic tools.
In what follows, we will describe some fundamental concepts of the term structure of interest rates, develop a useful set of static term struc- ture models and describe the usual approaches to extending these into dynamic models. We begin with the familiar, discrete-time modeling approach. That is, units of time quanta are defined (usually in terms of compounding frequency) and financial manipulations are indexed with integer, multiple periods.
We then build on the discussion by introducing the continuous-time analogies to the concepts developed for discrete-time modeling. Continuous- time modeling allows financial manipulations to be freed from discretization artifacts (such as compounding frequency) and provides an algebraic framework that more naturally and rigorously accommodates “rate” as a concept of change. In addition, this approach opens up a huge field of applicable mathematics with the attendant opportunity for abstraction.
For example, continuous-time models free the analyst from artificial a pri- ori assumptions about interest-rate lattices; allowing concentration on the financial analyses at hand; deferring time-step issues to final implementa- tion of an algorithm; and choosing an approach based on convenience, speed, and accuracy.
We next describe the dynamic term-structure model. The assump- tions, derivation, and parameterizations of the general model are described. In the last section we apply the dynamic term structure model to zero-coupon bonds, coupon-paying bonds, and the determination of par-coupon and horizon yield curves. Applications to other fixed- income products are presented in other chapters of this book.