Analysing and Interpreting the Yield Curve E-Book

Moorad Choudhry

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 556

Veröffentlichungsjahr: 2019

Beschreibung

Understand and interpret the global debt capital markets Now in a completely updated and expanded edition, this is a technical guide to the yield curve, a key indicator of the global capital markets and the understanding and accurate prediction of which is critical to all market participants. Being able to accurately and timely predict the shape and direction of the curve permits practitioners to consistently outperform the market. Analysing and Interpreting the Yield Curve, 2nd Edition describes what the yield curve is, explains what it tells participants, outlines the significance of certain shapes that the curve assumes and, most importantly, demonstrates what factors drive it and how it is modelled and used. * Covers the FTP curve, the multi-currency curve, CSA, OIS-Libor and 3-curve models * Gets you up to speed on the secured curve * Describes application of theoretical versus market curve relative value trading * Explains the concept of the risk-free rate * Accessible demonstration of curve interpolation best-practice using cubic spline, Nelson-Siegel and Svensson 94 models This advanced text is essential reading for traders, asset managers, bankers and financial analysts, as well as graduate students in banking and finance.

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Analysing and Interpreting the Yield Curve

Second Edition

MOORAD CHOUDHRY

With contributions from Polina Bardaeva, Ken Kortanek, Kevin Liddy, Wolfgang Marty and Vladimir Medvedev

First published 2004

Second Edition published 2019

© 2004, 2019 Moorad Choudhry

John Wiley & Sons Ltd

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Whilst every effort has been made to ensure accuracy, no responsibility for loss occasioned to any person acting or refraining from action as a result of reading any material in this book can be accepted by the author, publisher or any named person or corporate entity. The author may or may not hold, or have held, any security identified in this book.

Library of Congress Cataloging‐in‐Publication Data

Names: Choudhry, Moorad.

Title: Analysing and interpreting the yield curve / Moorad Choudhry.

Other titles: Analysing & interpreting the yield curve | Analyzing and interpreting the yield curve

Description: Second edition. | Chichester, UK : Wiley, 2019. | Series: Wiley finance | Revised edition of the author's Analysing and interpreting the yield curve, c2004. | Includes bibliographical references and index. |

Identifiers: LCCN 2018056454 (print) | LCCN 2018057854 (ebook) | ISBN 9781119141068 (AdobePDF) | ISBN 9781119141051 (ePub) | ISBN 9781119141044 (hardback)

Subjects: LCSH: Bonds—Valuation—Econometric models. | BISAC: BUSINESS & ECONOMICS / Banks & Banking.

Classification: LCC HG4651 (ebook) | LCC HG4651 .C6793 2019 (print) | DDC 332.63/23—dc23

LC record available at https://lccn.loc.gov/2018056454

Cover Design: Wiley

Cover Images: top: © Gen Epic Solutions/Shutterstock, bottom: © hywards/shutterstock

For Lindsay

Ultimate Yummy Mummy

Foreword

It is an honour to be asked to contribute a few words at the beginning of this publication.

Moorad and I first met back in 2010. We were both attending the Group Balance Sheet Management Committee at The Royal Bank of Scotland, our employer at the time. Although I forget the specific theme of the discussion, I remember our desire, incidentally which was not necessarily shared by the other committee members, to investigate and better understand a trend in the balance sheet of our organisation and in the wider economy.

Respecting each other's viewpoint, we quickly became good friends. However, we are like “chalk and cheese”. In Belbin's terminology, Moorad possesses many of the qualities of the “Plant” and “Resource Investigator”, being creative, imaginative and free‐thinking as well as outgoing and enthusiastic, whereas I am most comfortable in the role of the “Completer Finisher”.

Although our shared passion for the financial markets became evident at our first meeting, it was only later I realised Moorad also possesses a burning desire to pass on his knowledge, understanding and perceptive insights to others, by either delivering lectures and presentations or through the written word.

I first became aware of Moorad's teaching skills in 2014 when he invited me to speak to students studying for the Certificate of Bank Treasury Risk Management. This is a practitioner‐oriented professional qualification in bank asset‐liability management, which is delivered to a global audience and was developed by Moorad and the team from WBS Training Ltd. There are also his numerous short course and conference performances I could highlight, all with financial risk management as the enduring theme.

When it comes to writing, Moorad is both versatile and prolific. By my counting, this updated edition of Analysing and Interpreting the Yield Curve is his third book of the year so far. However, there's still a way to go to catch Corin Tellado, a renowned Spanish author who published over 4,000 novels in her lifetime. The comparison is probably unfair as it is doubtless a more difficult challenge to combine risk management theory, practice and current market developments in a captivating read, whilst also pursuing a full‐time career in banking!

Moorad is a past master at simplifying complex ideas and communicating them in an easy going and engaging manner. In these respects, this book is no different to all his other publications. Compared to the first edition, sections have been added for the latest developments in the financial markets, including: the multi‐currency yield curve, the SONIA curve, the interpretation of negative yield curves, a post‐crash discounting technique for the swap curve and how to use the theoretical and observed US Treasury curve as a means of identifying relative value in bond spread trades. These concepts will be of interest to anyone working in the bond markets, be they a trader, sales‐person, fund manager, research analyst, investor or issuer, or, like me, simply a student of the financial markets.

I commend this book to you, hope you enjoy the read and leave you with the words of Benjamin Franklin:

An investment in knowledge pays the best interest.

Chris Westcott

Former Treasurer, Retail and Wealth Division

The Royal Bank of Scotland

1 May 2018

Preface

The yield curve, and everything about it, was my first and most intense love in finance. It probably still is. I could talk about it for hours, at any time, day or night. I think it is the most significant topic in banking, the very foundation of finance. But it's interesting to me to observe how the perception, and indeed the requirement, for technical excellence in banking changes subtly the more senior the level of practitioner. The higher one rises in the profession, the less it would appear that one needs to know about subjects such as the curve, how to interpret it, how best to interpolate it, and how to understand and make sense of what it's trying to tell us. What I thought was the most important and vital issue in finance, something that absolutely everyone had to know about, turns out to be just one more arcane specialism that is not discussed that often at the bank's asset‐liability committee (ALCO), hardly at all at the executive management committee (EXCO), and fewer times still at the Board.

No matter, I still think that it's a very important topic and it's a pity that it isn't viewed in this way by everyone in banking. But that's their look out. The fact that you are reading this book shows that you agree that it's a worthwhile topic to get to grips with!

So, fully 15 years after the first edition, is there anything new to write about on the curve? As it happens, yes, a fair bit. The global financial markets are a very different beast today compared to what they were in 2003. Of course, the fundamentals of yield curve analysis, interpolation, and interpretation remain unchanged. But the behaviours of curves are different in various nuanced ways (and some not so nuanced, and in fact very much “in your face” – for example, the negative interest rate curves that are a commonplace in some countries in the eurozone). That’s why I think this book was worth updating, so that we might cover issues such as multi‐currency curves, the overnight index swap (OIS) curve, and key factors in post‐crash discounting of the swap curve. And speaking of interest‐rate swaps, it is routine to see (for example, in US dollar markets) the swap curve trading through the sovereign bond curve. That would have been inexplicable when I was working as a government bond primary dealer in the 1990s … but then again, negative interest rates as routine would also have been inexplicable. Plenty for us to be getting on with then. And in an era of ever more intensive regulator and compliance burden, having to deal with a purely technical subject may even come across as a breath of fresh air to some practitioners!

I always try to emphasise the practical and the user‐friendly in all my writing. There is such a disconnect between academia and practice in finance that there would be little value in me expounding purely on the theoretical. Unfortunately (or fortunately, depending on your point of view), the yield curve is one of those topics that it is difficult to leave the technical out of. That said, I hope the contents of the book are of relevance and practical value to the practitioner in banking and finance. This is not intended to be a textbook describing nothing that actually takes place in a bank, unlike some finance textbooks I have encountered over the years. Rather, it is meant for those who need to update the curve for use in internal funds transfer pricing, or to estimate the value to be derived from purchasing one bond in preference to another bond, or to price a new issue private placement structured product, or to have an idea of what the market thinks the state of the economy is. In other words, this book is for anyone that is using the yield curve for one or more of its myriad different practical applications in the financial markets.

Hence this second edition, which I hope you find of interest and of some use. As Ian MacDonald said in the preface to his final update of the majestic Revolution In The Head, no further editions will be forthcoming. Or as the last Oi! album proclaimed, “That's yer lot!”.

Comments on the text are welcome and should be sent to me via John Wiley & Sons Limited.

All the best.

Moorad Choudhry

Surrey, England

19 December 2018

A Solid Bond In Your Heart

Preface to the First Edition

As Sir Arthur Conan Doyle would have put it, so elementary a form of literature as the textbook on financial economics hardly deserves the dignity of a preface. It is possible though, to bring some instant clarity to the purpose of such a book if we open with a few words here.

In my book The Bond and Money Markets, I try to explain, from first principles, just how important the global debt market is, and describe the various participants that interact with each other in this market. Given the importance of the global bond market, one can never learn too much about it. But this is not a book about the bond market; rather it is about a very specific, and important part of the bond markets. In developed markets, as well as a fair number of developing ones, there is usually a large number of bonds trading at one time, at different yields and with varying terms to maturity. Investors and traders incessantly examine the relationship between the yields on bonds that are in the same class; plotting yields of bonds that differ only in their term to maturity produces what is known as the yield curve. The yield curve represents the bond market. It is sometimes referred to as the term structure of interest rates, but as we shall see later in this book, this expression refers to only one specific type of yield curve. There are lots of different yield curves. We shall examine them all in detail later.

Much of the analysis and pricing activity that takes place in the bond markets revolves around the yield curve. The primary yield curve in any domestic capital market is the government bond yield curve, so for example, in the US market it is the US Treasury yield curve. So in this book we will talk mainly, but not exclusively, about the government yield curve. And because the author spent over five years as a United Kingdom government bond trader (or gilt‐edged market maker), most of the examples will be from the gilt market. But the principles remain the same. It is the importance of the yield curve to just about every aspect of finance that has been the motivation behind writing this book. Our objective is to:

describe what the yield curve is;

explain what it tells us;

try to explain why it assumes certain shapes;

show how we can use it;

introduce how it is modelled;

show how it is fitted from market rates.

We begin with some basic description of bonds and bond mathematics, just to set the scene. We assume a basic knowledge and familiarity with bonds and market institutions, and concentrate on the yield curve. It is an arcane, specialist topic but well worth getting familiar with. We explain term structure theory, describe the most popular mathematical approaches used to model the yield curve, and show how to fit the yield curve using econometric techniques. This knowledge is of great use to just about anyone involved in the bond markets: traders, bond salespersons, fund managers, research analysts, issuers of bonds … in fact issuers, investors, and all the middlemen in between. Investors in the equity markets can also benefit from an understanding of the yield curve, as it enables one to gain a better insight into market sentiment.

We must necessarily be quite focused and specialist in our discussion of the yield curve. Hopefully the more technical material is presented in good order so that it remains accessible. There are any number of textbooks available for the complete beginner, which are recommended in end‐chapter reading lists, along with further reading.

Moorad Choudhry

Surrey, England

30 June 2003

Acknowledgments

Special thanks to The Raynes Park Footy Boys and The Pink Tie Brigade.

Thanks to everyone at Wiley, including Stephen Mullaly, Syd Ganaden, Jean‐Karl Martin, Debbie Scott, Sandra Glue, Banurekha Venkatesan, Elisha Benjamin, Katy Smith and Aida Ferguson.

Big thanks to my co‐authors, Polina Bardaeva, Ken Kortanek, Kevin Liddy, Wolfgang Marty, and Vladimir Medvedev. It's a privilege to work with you.

Thanks to Marc Dodd and everyone at King & Shaxson Ltd, and Martin Ward at One Savings Bank for helping to make my latest return to the markets so enjoyable.

Thanks to everyone at Crown Law and NZIRD, fab people to work with, including Jane Norris, Meghan Nicholson, Lina Worthing, Paul Hale, and Mike Cook.

Thanks to everyone at Alderwick James, including Mr Alderwick and Ms James themselves as well as Liliana Lolata, Jolene Rodrigues, Sally Thurwood, and Sally Baldeh.

Thanks to everyone at The BTRM. What a genuinely great bunch of people I am privileged to work with.

Thanks to Philip Curtis‐Evans at Bloomberg for his assistance and instant responses whenever I requested screen print permissions.

For very kind and very much appreciated comments on Linked In, all the more touching as I have never actually met them, a very special thanks to Brian Twomey, Donald Van Deventer and David Harper. It meant a lot to me, thank you gentlemen.

And a very big, big thanks to Mike Kirsopp and everyone at Cambridge & Counties Bank. And I mean everyone! The bank has at least one thing in common with the New Zealand All Blacks…

About the Author

Moorad Choudhry is Head of ALM at Cambridge & Counties Bank in Leicester.

He was previously a gilt‐edged market‐maker and money markets trader at Hoare Govett Securities Limited (later ABN Amro Hoare Govett Limited) and a sterling bond proprietary trader at Hambros Bank Limited. He subsequently traded money markets, asset‐backed commercial paper, and structured finance repo at KBC Financial Products (a subsidiary of KBC Bank N.V.), and was latterly Treasurer, Corporate Banking Division at The Royal Bank of Scotland.

But don't forget the songs

That made you cry,

And the songs that saved your life,

Yes, you're older now

And you're a clever swine,

But they were the only ones who ever stood by you.

______ The Smiths, Rubber Ring (Rough Trade Records, 1985)

PART IIntroduction to the Yield Curve

In Part I we describe the yield curve itself. The bulk of the discussion is in Chapter 1, which looks at the different types of yield curve and, more importantly, introduces the main theories of the yield curve. We also look at interpreting the curve. The language is non‐specialist and should be accessible to anyone with an involvement in the financial markets. This is followed by a look at spot and forward rates, and the derivation of such rates from market yields.

For this second edition we have relegated the introductory chapter on bond yield measurement to the main Appendix.

After a couple of months, his patriotic zeal got on my nerves so much I began to question whether I agreed with him about communism being evil. I agreed it was a bad idea but no longer felt so sure it would ruin the planet. I began to consider the danger of blind faith in, or blind hatred of, a single idea, any idea.

______ Robert Wideman, Unexpected Prisoner: Memoir of a Vietnam POW, 2016

CHAPTER 1The Yield Curve

The main measure of return associated with holding bonds is the yield to maturity (YTM) or gross redemption yield (GRY). In developed markets there is usually a large number of bonds trading at one time, at different yields and with varying terms to maturity. Investors and traders frequently examine the relationship between the yields on bonds that are in the same class. Plotting yields of bonds that differ only in their term to maturity produces the yield curve. The yield curve is an important indicator and knowledge source of the state of a debt capital market.1 It is sometimes referred to as the term structure of interest rates, but strictly speaking this is not correct, as this expression should be reserved for the zero‐coupon yield curve only. We shall examine this in detail later.

Much of the analysis and pricing activity that takes place in the bond markets revolves around the yield curve. The yield curve describes the relationship between a particular redemption yield and a bond's maturity. We should be aware that the GRY of a bond is only ever the actual yield one receives during the period one holds the bond if certain specific, and generally unrealistic, conditions are met. However, we will leave the discussion of this for later.

Plotting the yields of bonds along the maturity term structure will give us our yield curve. It is very important that only bonds from the same class of issuer or with the same degree of liquidity are used when plotting the yield curve. For example, a curve may be constructed for UK gilts or for AA‐rated sterling Eurobonds, but not a mixture of both, because gilts and Eurobonds are bonds from different class issuers. The primary yield curve in any domestic capital market is the government bond yield curve, so for example, in the US market it is the US Treasury yield curve. With the advent of the euro currency in 11 (ultimately 19) countries of the European Union, in theory any euro‐currency government bond can be used to plot a default‐free euro yield curve. In practice, only bonds from the same government are used, as for various reasons different bonds within euro‐currency countries trade at different yields. Outside the government, bond markets yield curves are plotted for Eurobonds, money market instruments, off‐balance sheet instruments, in fact virtually all debt market instruments. Therefore it is always important to remember to compare like‐for‐like when analysing yield curves across markets.

In this chapter, we look at the yield to maturity yield curve as well as other types of yield curves that may be constructed. We also consider how to derive spot and forward yields from a current redemption yield curve. The main emphasis though, is on interpreting the shape of the yield curve, and explaining why it assumes the shapes it does. Later in the book we examine more advanced techniques for fitting, analysing, and interpreting the yield curve.

First though, we introduce the yield curve for beginners, of course experienced practitioners and graduate students may skip this part.

THE YIELD CURVE FOR BEGINNERS

This section is a summary of the importance and application of the yield curve. It was originally written with private investors in mind, so market practitioners may wish to skip this part.

What is the Yield Curve?

The yield curve is a graph that plots the yield of various bonds against their term to maturity. In other words, it is a snapshot of the current level of yields in the market. It is not an historical graph, that is, it does not show the level of yields over time. That would be an historical price (or yield) chart.

Yield curves are like football … it is very easy to grasp the basics, but difficult to become expert at (to continue the football analogy, akin to being good enough to play on a Sunday‐morning park football team and being good enough to play on a team that included David Beckham, Steven Gerrard, and Michael Owen). Let us imagine that we looked up gilt yields in the Financial Times on a day in August 2002 and saw the following:

Table 1.1 shows the yields for gilts of 1‐, 2‐, 5‐, 10‐, 15‐, 20‐, and 30‐year maturity. (The 8% 2021 gilt is slightly under 20 years maturity but it will do for our purposes – there is no gilt that matures in 2022 at the time we are looking at this.)

TABLE 1.1 Gilt yields

Gilt

Red Yield

2

Tr 8% 03

3.79

Tr 5% 04

4.00

Tr 7.25% 07

4.62

Tr 5% 12

4.70

Tr 8.75% 17

4.74

Tr 8% 21

468

Tr 4.25% 32

4.52

We open up Microsoft Excel3 and write down two columns, one for “maturity” and one for “yield”. The years to maturity column forms the x‐axis of the graph, while the yield forms the y‐axis. Then we use the Excel “chart wizard” and it plots our graph for us! The result is as shown in Figure 1.1.

FIGURE 1.1 Creating a yield curve in Excel.

This curve looks about right. Intuitively, we expect that yields increase the greater the maturity. If we lend one person an amount of money for one year and another person the same amount of money for 10 years, we would not charge them both the same rate of interest (assuming both had the same credit risk) – we would most likely charge a higher rate to the 10‐year borrower, for two reasons:

inflation will erode the value of the loan over the longer term, and;

while the longer‐dated borrower may be the same credit quality as the short‐dated borrower, there are other risks. For example, the long‐dated borrower may not be around in 10 years' time. Therefore, as a lender, we need a higher return to compensate for the greater risk the further out into the future we lend money.

So this gives us the positively sloping yield curve we see in Figure 1.1. However, if that is the case, why does the curve not continue to slope upwards, all the way to the 30‐year mark? The rate of upward movement declines after the five‐year mark, and then actually decreases to the 30‐year point. This is a peculiarity of many markets – the 30‐year bond, commonly called the long bond, is usually in such great demand among institutional investors, such as pension funds, that this demand outstrips supply. As a result, the price of this bond is forced upwards, and this moves the yield down to below what it should be.

Constructing a yield curve in the wholesale markets is a little more involved than what we have described above, and a ever‐so‐slightly complicated branch of mathematics is employed to derive the models used to fit yield curves. We consider this in later chapters.

Using the Yield Curve

The yield curve tells us where the bond market is trading now. It also implies the level of trading for the future, or at least what the market thinks will be happening in the future. In other words, it is a good indicator of the future level of the market. It is also a much more reliable indicator than any other used by private investors, and we can prove this empirically.

Let us consider the main uses of the yield curve. All participants in the debt capital markets have an interest in the current shape and level of the yield curve, as well as what this information implies for the future. The main uses are introduced below:

Setting the yield for all debt market instruments.

The yield curve essentially fixes the cost of money over the maturity term structure. The yields of government bonds from the shortest‐maturity instrument to the longest set the benchmark for yields for all other debt instruments in the market, around which all debt instruments are analysed. Issuers of debt (and their underwriting banks) therefore use the yield curve to price bonds and all other debt instruments. Generally the zero‐coupon yield curve is used to price new issue securities, rather than the redemption yield curve.

Acting as an indicator of future yield levels.

As we discuss later in this chapter, the yield curve assumes certain shapes in response to market expectations of the future interest rates. Bond market participants analyse the present shape of the yield curve in an effort to determine the implications regarding the future direction of market interest rates. This is perhaps one of the most important functions of the yield curve, and it is as much an art as a science. The yield curve is scrutinised for its information content, not just by bond traders and fund managers, but also by corporate financiers as part of project appraisal. Central banks and government treasury departments also analyse the yield curve for its information content, not just regarding forward interest rates, but also with regard to expected inflation levels.

Measuring and comparing returns across the maturity spectrum.

Portfolio managers use the yield curve to assess the relative value of investments across the maturity spectrum. The yield curve indicates the returns that are available at different maturity points and is therefore very important to fixed income fund managers, who can use it to assess which point of the curve offers the best return relative to other points.

Indicating relative value between different bonds of similar maturity.

The yield curve can be analysed to indicate which bonds are cheap or dear to the curve. Placing bonds relative to the zero‐coupon yield curve helps to highlight which bonds should be bought or sold either outright or as part of a bond spread trade.

Pricing interest rate derivative securities.

The price of derivative securities revolves around the yield curve. At the short end, products such as Forward Rate Agreements are priced off the futures curve, but futures rates reflect the market's view on forward three‐month cash deposit rates. At the longer end, interest rate swaps are priced off the yield curve, while hybrid instruments that incorporate an option feature such as convertibles and callable bonds also reflect current yield curve levels. The “risk‐free” interest rate, which is one of the parameters used in option pricing, is the T‐bill rate or short‐term government repo rate, both constituents of the money market yield curve.

YIELD TO MATURITY YIELD CURVE

The most commonly occurring yield curve is the yield to maturity yield curve. The equation used to calculate the yield to maturity is given in the Appendix. The curve itself is constructed by plotting the yield to maturity against the term to maturity for a group of bonds of the same class. Three different examples are shown at Figure 1.2. Bonds used in constructing the curve will only rarely have an exact number of whole years to redemption, however, it is often common to see yields plotted against whole years on the ‐axis. This is because once a bond is designated the benchmark for that term, its yield is taken to be the representative yield. For example, the then 10‐year benchmark bond in the UK gilt market, the 5.75% Treasury 2009, maintained its benchmark status throughout 1999 and into 2000, even as its term to maturity fell below 10 years. The yield to maturity yield curve is the most commonly observed curve simply because yield to maturity is the most frequent measure of return used. The business sections of daily newspapers, where they quote bond yields at all, usually quote bond yields to maturity.

FIGURE 1.2 Yield to maturity yield curves.

As we might expect, given the source data from which it is constructed, the yield to maturity yield curve contains some inaccuracies. We have already come across the main weakness of the yield to maturity measure, which is the assumption of a constant rate for coupon reinvestment during the bond's life at the redemption yield level. Since market rates fluctuate over time, it is not possible to achieve this (a feature known as reinvestment risk). Only zero‐coupon bondholders avoid reinvestment risk as no coupon is paid during the life of a zero‐coupon bond.

Users of the Bloomberg system can call up a large number of yield curves, for different sectors of the market and different instruments. The main screen is IYC, which presents a menu of government bond redemption yield curves. For example, Figure 1.3 shows screen IYC for the UK gilt curve, as at 18 June 2003. The curves are for benchmark bonds and for gilt zero‐coupon bonds or strips. Figure 1.4 compares the gilt curve and the sterling swap curve. Figure 1.5 shows the corresponding curves for French and German government bonds, on the same screen, and Figure 1.6 shows the benchmark curve for US Treasuries.

FIGURE 1.3 UK gilt yield curves as at 18 June 2003.

Source: © Bloomberg L.P. Reproduced with permission.

FIGURE 1.4 UK gilt yield curve and sterling interest rate swap curve as at 18 June 2003.

Source: © Bloomberg L.P. Reproduced with permission.

FIGURE 1.5 French and German government yield curves, 18 June 2003.

Source: © Bloomberg L.P. Reproduced with permission.

FIGURE 1.6 Treasury yield curve as at 18 June 2003.

Source: © Bloomberg L.P. Reproduced with permission.

Figure 1.7 shows the screen FMC, which presents a menu of different government and corporate bond yield curves. Users select the curve required from this menu. Selecting curves 664, 670, and 673 (see Figure 1.7) gives us composite yield curves for AAA‐, A‐ and BBB‐rated corporate bonds. These curves are shown at Figure 1.8.

FIGURE 1.7 Screen FMC, menu of corporate and government bond yield curves.

Source: © Bloomberg L.P. Reproduced with permission.

FIGURE 1.8 Screen FMC, AAA‐, A‐, and BBB‐rated corporate yield curves as at 18 June 2003.

Source: © Bloomberg L.P. Reproduced with permission.

Finally, Figure 1.9 shows screen SWCV, the interest rate swap curve. We show the screen as selected for sterling interest rate swaps.

FIGURE 1.9 Screen SWCV, selected for sterling swap curve as at 18 June 2003.

Source: © Bloomberg L.P. Reproduced with permission.

The yield to maturity yield curve does not distinguish between different payment patterns that may result from bonds with different coupons, that is, the fact that low‐coupon bonds pay a higher portion of their cash flows at a later date than high‐coupon bonds of the same maturity. The curve also assumes an even cash flow pattern for all bonds. Therefore in this case, cash flows are not discounted at the appropriate rate for the bonds in the group being used to construct the curve. To compensate for this, bond analysts may sometimes construct a coupon yield curve, which plots yield to maturity against term to maturity for a group of bonds with the same coupon. This may be useful when a group of bonds contains some with very high coupons – high‐coupon bonds often trade “cheap to the curve”, that is they have higher yields, than corresponding bonds of the same maturity but lower coupon. This is usually because of reinvestment risk and, in some markets (including the UK), for tax reasons.

Now is probably an appropriate time to remind readers that the YTM, and hence the YTM curve, is really just an “assumed” yield. It isn't a true term structure of interest rates. This is because, for the investor to receive the YTM during the period the bond is held, the bond must have been (i) purchased at par, and (ii) all the coupons received during the holding period must have been reinvested at the same YTM. Clearly, these two sets of conditions are rarely, if ever, met during the holding period. So whilst YTM is a useful measure for initial purchase and comparison purposes, one should remember that it is rarely, if ever, the return that is actually received.

THE COUPON YIELD CURVE

The coupon yield curve is a plot of the yield to maturity against term to maturity for a group of bonds with the same coupon. If we construct such a curve, we can see that in general high‐coupon bonds trade at a discount (have higher yields) relative to low‐coupon bonds, because of reinvestment risk and for tax reasons. In the UK, for example, on gilts the coupon is taxed as income tax, while any capital gain is exempt from capital gains tax. Even in jurisdictions where capital gain on bonds is taxable, this can often be deferred whereas income tax cannot. It is frequently the case that yields vary considerably with coupon for the same term to maturity, and with term to maturity for different coupons. Put another way, usually we observe different coupon curves, not only at different levels, but also with different shapes. Distortions arise in the yield to maturity curve if no allowance is made for coupon differences. For this reason, bond analysts frequently draw a line of “best fit” through a plot of redemption yields, because the coupon effect in a group of bonds will produce a curve with humps and troughs. Figure 1.10 shows a hypothetical set of coupon yield curves, however, since in any group of bonds it is unusual to observe bonds with the same coupon along the entire term structure, this type of curve is relatively rare.

FIGURE 1.10 Coupon yield curves.

THE PAR YIELD CURVE

The par yield curve is not usually encountered in secondary market trading, however, it is often constructed for use by corporate financiers and others in the new issues or primary market. The par yield curve plots yield to maturity against term to maturity for current bonds trading at par.4 The par yield is therefore equal to the coupon rate for bonds priced at par or near to par, as the yield to maturity for bonds priced exactly at par is equal to the coupon rate. Those involved in the primary market will use a par yield curve to determine the required coupon for a new bond that is to be issued at par. This is because investors prefer not to pay over par for a new‐issue bond, so the bond requires a coupon that will result in a price at or slightly below par.

The par yield curve can be derived directly from bond yields when bonds are trading at or near par. If bonds in the market are trading substantially away from par, then the resulting curve will be distorted. It is then necessary to derive it by iteration from the spot yield curve. As we would observe at almost any time, it is rare to encounter bonds trading at par for any particular maturity. The market therefore uses actual non‐par vanilla bond yield curves to derive zero‐coupon yield curves and then constructs hypothetical par yields that would be observed were there any par bonds being traded.

THE ZERO‐COUPON (OR SPOT) YIELD CURVE

The zero‐coupon (or spot) yield curve plots zero‐coupon yields (or spot yields) against term to maturity. A zero‐coupon yield is the yield prevailing on a bond that has no coupons. In the first instance, if there is a liquid zero‐coupon bond market we can plot the yields from these bonds if we wish to construct this curve. However, it is not necessary to have a set of zero‐coupon bonds in order to construct the curve, as we can derive it from a coupon or par yield curve. In fact, in many markets where no zero‐coupon bonds are traded, a spot yield curve is derived from the conventional yield to maturity yield curve. This is of course a theoretical zero‐coupon (spot) yield curve, as opposed to the market or observed spot curve that can be constructed using the yields of actual zero‐coupon bonds trading in the market.5

Interest Rates: Basic Concepts

Spot yields must comply with Equation (1.1). This equation assumes annual coupon payments and that the calculation is carried out on a coupon date so that accrued interest is zero:

(1.1)

where

is the spot or zero‐coupon yield on a bond with years to maturity discount factor.

In (1.1) is the current one‐year spot yield, the current two‐year spot yield, and so on. Theoretically, the spot yield for a particular term to maturity is the same as the yield on a zero‐coupon bond of the same maturity, which is why spot yields are also known as zero‐coupon yields. This last is an important result, as spot yields can be derived from redemption yields that have been observed in the market.

As with the yield to redemption yield curve, the spot yield curve is commonly used in the market. It is viewed as the true term structure of interest rates because there is no reinvestment risk involved – the stated yield is equal to the actual annual return. That is, the yield on a zero‐coupon bond of years maturity is regarded as the true ‐year interest rate. Because the observed government bond redemption yield curve is not considered to be the true interest rate, analysts often construct a theoretical spot yield curve. Essentially this is done by breaking down each coupon bond being observed into its constituent cash flows, which become a series of individual zero‐coupon bonds. For example, £100 nominal of a 5% two‐year bond (paying annual coupons) is considered equivalent to £5 nominal of a one‐year zero‐coupon bond and £105 nominal of a two‐year zero‐coupon bond.

Let us assume that in the market there are 30 bonds all paying annual coupons. The first bond has a maturity of one year, the second bond of two years, and so on out to thirty years. We know the price of each of these bonds, and we wish to determine what the prices imply about the market's estimate of future interest rates. We expect interest rates to vary over time, but that all payments being made on the same date are valued using the same rate. For the one‐year bond, we know its current price and the amount of the payment (comprised of one coupon payment and the redemption proceeds) we will receive at the end of the year, therefore we can calculate the interest rate for the first year – assume the one‐year bond has a coupon of 5%. If the bond is priced at par and we invest £100 today, we will receive £105 in one year's time, hence the rate of interest is apparent and is 5%. For the two‐year bond, we use this interest rate to calculate the future value of its current price in one year's time: this is how much we would receive if we had invested the same amount in the one‐year bond. However, the two‐year bond pays a coupon at the end of the first year and if we subtract this amount from the future value of the current price, the net amount is what we should be giving up in one year in return for the one remaining payment. From these numbers we can calculate the interest rate in year two.

Assume that the two‐year bond pays a coupon of 6% and is priced at £99.00. If the £99.00 is invested at the rate we calculated for the one‐year bond (5%), it will accumulate £103.95 in one year, made up of the £99.00 investment and interest of £4.95. On the payment date in one year's time, the one‐year bond matures and the two‐year bond pays a coupon of 6%. If everyone expected that at this time the two‐year bond would be priced at more than 97.95 (which is 103.95 minus 6.00), then no investor would buy the one‐year bond, since it would be more advantageous to buy the two‐year bond and sell it after one year for a greater return. Similarly, if the price is less than 97.95, no investor would buy the two‐year bond, as it would be cheaper to buy the shorter bond and then buy the longer‐dated bond with the proceeds received when the one‐year bond matures. Therefore the two‐year bond must be priced at exactly 97.95 in 12 months' time. For this £97.95 to grow to £106.00 (the maturity proceeds from the two‐year bond, comprising the redemption payment and coupon interest), the interest rate in year two must be 8.22%. We can check this using the present value formula covered earlier. At these two interest rates, the two bonds are said to be in equilibrium.

This is an important result and shows that (in theory) there can be no arbitrage opportunity along the yield curve – using interest rates available today, the return from buying the two‐year bond must equal the return from buying the one‐year bond and rolling over the proceeds (or reinvesting) for another year. This is known as the breakeven principle, a law of no‐arbitrage.