Fixed income analytics bonds in high and low interest rate environments

Thenotions of yield, duration, and convexity are referred to confidently and resolutely in the context of single bonds as well as bond portfolios, and the effects of interestrates are ge

Fixed Income Analytics

Wolfgang Marty

Fixed Income Analytics

Bonds in High and Low Interest Rate Environments

Library of Congress Control Number: 2017952064

# Springer International Publishing AG 2017

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission

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The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

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The registered company is Springer International Publishing AG

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In light of an investment environment characterized by low yields and new latory capital regimes, it has become increasingly demanding for investors toachieve sustainable returns Particularly, fixed income investments are called intoquestion There is a solution.

regu-Since the foundation of AgaNola a decade ago, we have put our interest intoconvertibles, and at this point we want to thank our clients for having supported usalso in challenging times—particularly when convertible bonds were considered atmost a niche investment Unjustly!

For being a hybrid, convertible bonds offer the “best of both worlds,” thebenefits of an equity with the advantages of a corporate bond AgaNola is consid-ered a leading provider in this asset class, and to date convertible bonds remain thecore competence of us as a specialized asset manager

As we consider increasingly popular convertible bonds a living and dynamicuniverse, we are placing a great importance on research and the exploration of thenature of this asset class As an internationally renowned expert in the fixed incomeand bond field, Dr Wolfgang Marty has contributed valuable insights to ourwork—making the bridge from theory to portfolio management AgaNola iscommitted to continue to support his fundamental research

We wish Wolfgang Marty lots of success with his latest book

v

Compared to other asset classes, fixed income investments are routinely considered

as a relatively well-understood, transparent, and (above all) safe investment Thenotions of yield, duration, and convexity are referred to confidently and resolutely

in the context of single bonds as well as bond portfolios, and the effects of interestrates are generally believed to be well-understood

At the same time, we live in a world where the amount of private, corporate, andsovereign debt is steadily increasing and where postcrisis stimuli continue to affectand distort investor behavior and markets in an unprecedented way And that iseven before we start contemplating the enormous uncertainties introduced bynegative interest rates

In his book, Dr Wolfgang Marty covers and expands on classic fixed incometheory and terminology with a clarity and transparency that is rare to be found in aworld where computerization of accepted facts often is the norm Wolfganghighlights obvious but commonly unknown conflicts that can be observed, forexample, when applying standard theory outside its default setting or when migrat-ing from single to multiple bond portfolios He also includes the effects of negativeinterest rates into standard theory

Wolfgang’s book makes highly informative reading for anyone exposed to fixedincome concepts, be it as a portfolio manager or as an investor, and it shows thatoften we understand less than we think when studying bond or bond portfolioholdings purely based on their commonly accepted key metrics; Wolfgangencourages to ask questions Anyone building automated software would benefitfrom familiarity with the model discrepancies highlighted as it is to everyone’sdisadvantage if we find these too deeply rooted in commonly and widely appliedtools

In summary, Wolfgang’s book makes interesting reading for the fixed incomenovice as well as the seasoned practitioner

Head of Quantitative Research

Record Currency Management

Dr Jan Hendrik Witte

vii

Computers have become more and more powerful and often are an invaluable aid.But there is a considerable disadvantage: often, the output of a computer program isdifficult to understand, and the end user may be swamped by data In addition,computers solve problems in many dimensions, and, as human beings, we strugglethinking in more than a few dimensions To provide a sound background ofunderstanding to anyone working in fixed income, we intend to illustrate here theessential basic calculations, followed by easy to understand examples.

The reporting of return and risk figure is paramount in the asset managementindustry, and the portfolio manager is often rewarded on performance figures Thefirst motivation for the here presented material were the findings of a working group

of the Swiss Bond Commission (OKS), where we compared the yield for a fixedincome benchmark portfolio calculated by different software providers: we founddifferent yields for the same portfolio and the same underlying time periods Thefollowing questions are obvious: How can a regulating body accept ambiguousfigures? Should there not be a standard?

An additional complication is linearization, often the first step in analyzing abond portfolio The yield of the bonds in a bond portfolio is routinely added toreport the yield of the total bond portfolio, and different durations of bonds in theportfolio are simply added to indicate the duration of a bond portfolio We foundthat linearization works well for a flat yield curve, but the more the yield deviatesfrom a flat curve, the more the resulting figures become questionable

Also, historically, interest rates have been positive In the present marketconditions, however, interest rates are close to zero or even slightly negative Wefind ourselves confronted with several questions: Does the notion of duration stillmake sense in this new environment? And which formulae can be applied forinterest rates equal or very close to zero? How do discount factors behave? In thefollowing, we attempt to include negative interest in our considerations Forinstance, in the world of convertibles, yield to maturities can easily be negativeand is not problematic

ix

We describe the here presented material in three ways Firstly, we use words andsentences, in order to give an introduction into in the notions, definitions, ideas, andconcepts Secondly, we introduce equations Thirdly, we also use tables and figures

in order to make the outputs of our numerical calculations accessible

Pfaeffikon SZ, Switzerland

July 18, 2017

Wolfgang Marty

This book is based on several presentations, courses, and seminars held in Europeand the Middle East The here presented material is based on a compilation of notesand presentations Presenting fixed income is a unique experiment and I am gratefulfor the many feedbacks from the audience The initial motivation for the book was aseminar held at the education center of the SIX Swiss Exchange I became awarethat many issues in fixed income need to be restudied and revised; moreover, I didnot find satisfying answers to my questions in the pertinent literature The SIXSwiss Exchange Bond Advisory Group was an excellent platform for analyzingopen issues.

Furthermore, the working group “Portfolio Analytics” of the Swiss Bond mission was instrumental for the research activities In particular my thanks go toGeraldine Haldi, Dominik Studer, and Jan Witte They revised part of the manu-script and provided helpful comments

Com-The European Bond Commission (EBC) was very important for my professionaldevelopment The members of the EBC Executive Committee Chris Golden andChristian Schelling gave me continuing support for my activities, and the EBCsessions throughout Europe yielded important ideas for the book

At the moment I am focusing on convertibles My thanks go to Marco Turinelloand Lukas Buxtorf for introducing me into the analytics of convertibles The lastchapter of the book is dedicated to convertibles

The book was written over several years, and I am grateful to my presentemployer AgaNola for the opportunity to complete this book

xi

This book consists of eight chapters The chapters are divided into sections (1.2.3)denotes formula (3) in Sect.1.2 If we refer to formula (2) in Sect.1.2, we only write(2); otherwise we use the full reference (1.2.2) Within the chapters, definitions,assumptions, theorems, and examples are numerated continually, e.g., Theorem 2.1refers to Theorem 1 in Chapter2.

Square brackets [ ] contain references The details of the references are given atthe end of each chapter

xiii

1 Introduction 1

2 The Time Value of Money 5

2.1 The Return Over a Time Unit 5

2.2 Discount Factors 7

2.3 Annuities 12

3 The Flat Yield Curve Concept 17

3.1 The Description of a Straight Bond 17

3.2 Yield Measures 24

3.3 Duration and Convexity 32

3.4 The Approximation of the Internal Rate of Return 55

3.4.1 The Direct Yield of a Portfolio 57

3.4.2 Different Approximation Scheme for the Internal Rate of Return 71

3.4.3 Macaulay Duration Approximation Versus Modified Duration Approximation 81

3.4.4 Calculating the Macaulay Duration 89

3.4.5 Numerical Illustrations 93

References 102

4 The Term Structure of Interest Rate 103

4.1 Spot Rate and the Forward Rate 104

4.2 Discrete Forward Rate and the Instantaneous Forward Curve 107

4.3 Spot Rate and Yield Curve 111

4.4 The Effective Duration 126

References 128

5 Spread Analysis 129

5.1 Interest Rate Spread 129

5.2 Rating Scales 133

5.3 Composite Rating 142

5.4 Optionality 144

References 147

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6 Different Fixed Income Instruments 149

6.1 Segmentation of the Yield Curve 149

6.2 Floating Rate Note 150

6.3 Interest Rate Swap 152

6.4 Asset Swap 157

References 158

7 Fixed-Income Benchmarks 159

7.1 Definition and Fundamental Properties 159

7.2 Constructing a Fixed-Income Benchmark 160

7.3 Recent Developments in the Benchmark Industry 162

7.4 Fixed Income as Asset Class 163

7.4.1 Equity Benchmarks 164

7.4.2 Fixed-Income Indices 165

7.4.3 Hedged Fixed-Income Indices 167

7.4.4 Commodity Index 168

References 171

8 Convertible 173

8.1 Basics Notions 173

8.2 The Stock Behavior 174

8.3 The Bond Behavior 176

8.4 The Embedded Call Option 178

References 183

Appendices 185

References 201

Index 203

Wolfgang Marty is senior investment strategist at AgaNola, Pfaeffikon SZ,Switzerland Between 1998 and 2015, he was working with Credit Suisse He joinedCredit Suisse Asset Management in 1998 as head product engineer He specializes

in performance attribution, portfolio optimization, and fixed income in general.Prior to joining Credit Suisse Asset Management, Marty worked for UBS AG inLondon, Chicago, and Zurich He started his career as an assistant for appliedmathematics at the Swiss Federal Institute of Technology

Marty holds a university degree in mathematics from the Swiss Federal Institute

of Technology in Zurich and a doctorate from the University of Zurich He chairsthe method and measure subcommittee of the European Bond Commission (EBC)and is president of the Swiss Bond Commission (OKS) Furthermore, he is amember of the Fixed Income Index Commission at the SIX Swiss Exchange and

a member of the Index Team that monitors the Liquid Swiss Index (LSI)

xvii

Introduction 1

A fixed-income security is a financial obligation of an entity that promises to pay aspecified sum of money at specified future dates The entity can be a government, acompany, or an individual and is called anissuer The investor lends a specifiedamount of money to the issuer Abond is a legal engagement between the issuer and

an investor

Abond is a fixed-income instrument and has usually a finite live Periodic futurecash flows from the issuer to the investor are called the coupon of the bond.Coupons are unaffected by market movements for the live of the bond and reflectthe notion “fixed income.” As depicted in Fig.1.1, astraight bond or a couponpaying bond is a bond that pays a coupon periodically and pays back at the end of itslive the money that was originally invested For precise definitions and analytics,

we refer to Chap.3

The bond markets have grown tremendously, and today there is a large universe

of organizations that issues bonds Together with equities, bonds are the two majortraditional asset classes in financial markets There are much different bonds thanequities For instance, there were 5447 shares traded and admitted to trading on the

EU regulated market (mifiddatabase.esma.europa.eu), and TRAX has data for300,000 government bonds, corporate bonds, medium-term notes, and privatederivative issues (xtrakter.com)

The time to maturity and the coupon are fixed at the issuance of a bond and arethus calledstatic data or reference data, whereas the market price is determined bythe trading activity and is thus calledmarket data

Unlike equities, every bond has potentially special and unique features Acompany has one or two kinds ofequities but many different bonds Bond marketsare very fragmented Figure1.2(seewww.sifma.org/research/statistics.aspx) showsthe development of the four most important segments of the US Bond Market Eversince interest rates began to climb in the late 1960s, the appeal for fixed-incomeinstrument has increased This is due to the fact that interest levels were competitivewith other instruments, and at the same time, the market rates began to fluctuatewidely, providing investors with attractive capital gain opportunities emphasizing

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that fixed income is not necessarily fixed income Only for the buy and holdinvestor, i.e., the investor who keeps the bond till maturity, cash flows are fixed.The here presented material gives a comprehensive introduction to fixed-incomeanalytics Some of the topics are:

• The transition from a single bond to portfolio of bonds is examined Weinvestigate the nonlinearity of income since just adding characteristics of indi-vidual bonds yields in general wrong results for the overall portfolio

• We consider market-relevant values for interest rates and examine differentshape of the yield curve In particular, we discuss negative interest rates

• We introduce the main ideas for assessing the credit quality of a bond Wecompile different definitions of the default of a bond

• We describe the construction of an income benchmark and give an overview ofdifferent benchmark providers

coupon coupon coupon coupon Original

Investment + coupon

Fig 1.2 The development of the US bond market

We now provide more detail about the different chapters of this book.

Chapter2describes the time value of money This chapter contains the buildingblocks of a fixed-income instrument We introduce the concept of an interest rate

We stress specifically that throughout this book and all its results, we treat negativeand positive interest rates with generality (rather than favoring positive interestrates as has been so common in the literature until now)

In Chap.3, the flat yield curve concept is explained, i.e., every cash flow isdiscounted by the same interest rate This does not mean that the yield curve is flat

If all bonds have the same yield, the yield curve is said to be flat We discussdeviation of the flat yield curve

Theyield to maturity is a well-established measurement for indicating a bond’sfuture yield It is derived from the coupon, the nominal value, and the term tomaturity of the bond

Portfolio analysis frequently refers to the “yield.” The question is which yield?

In the following, we will not focus on a single bond Rather, we will examine the exante yield of an entire bond portfolio, i.e., exclusively future cash flows are factoredinto the calculation The equation for yield to maturity will be generalized to derive

an equation for the bond portfolio (internal rate of return) This equation is notsolved exactly by the programs offered by most software providers; instead, it isconsidered in combination with the yields to maturity of the individual bonds

In Chap.4, we speak about the transition from yield curve to spot curves and spotcurves to forward curves (see Fig.1.3) Figure1.3refers to a specific time and doesnot say anything about the dynamic of the curve Actual prices are measured in themarketplace, and yield, spot, and forward curve are in general calculated orcomputed Duration is a risk measure of bonds and bond portfolios Here, we assessthe durations in the context of a bond and a portfolio of bonds Effective durationversus durations based on the flat yield concept is discussed Modified duration is

forward rate

spot rate

yield to maturies

Fig 1.3 Different interest

rate term structures

used for a sensitivity analysis of a bond portfolio The different durations weintroduced tackle the interest risk and the yield curve risk The duration is thefulcrum of a bond and can be compared to an equilibrium in physics.

In Chap.5, we depart from the assumption that a straight bond is riskless Weconsider credit markets The credit quality of a bond is described by differentspreads We introduce the normal spread and the Z-spread and give the definition

of default of a bond from S&P, Moodys, and Fitch More recent developments ofcredit markets are described We illustrate some figures from a transition matrix anddiscuss composition ratings followed by the description of call and put features of

a bond

In Chap.6, we start withfloat rate notes Unlike fixed coupons, floating rates aretied to the short end of the yield curve We give an introduction in the analytics offloating rate notes We then proceed with theinterest rate swap, which exchangesthe liability of two counterparties Interest swap markets are important for steeringthe duration of a bond portfolio In the last section of the chapter,asset swaps aredescribed

Starting point in Chap 7 are the basic characteristics of a benchmark Anoverview of different benchmark providers is given We describe benchmarksfrom different asset classes and discuss benchmarks for a balanced portfolio Wegive more recent developments in the benchmark industry

In Chap 8, we give an introduction into convertible bonds Convertible iscorporate bond with an option on the stock of the issuing company Convertiblescan behave like a bond as well as a stock We compile the most important notionsdescribing a convertible Difficulties of pricing a convertible are discussed

The Time Value of Money 2

In this chapter, we introduce the basic notions and methods for assessing income instruments The subject of this chapter is the connection between time andthe value of money

fixed-2.1 The Return Over a Time Unit

Return measurement always relates to a time span, i.e., it matters whether you earn

a specific amount of money over a day or a month Therefore, return measurementhas to be relative to a unit time period In finance, the most prominent examples are

a day, a month, or a year In Fig.2.1we see a unit time period and a partition intofour time spans of the same length

With a beginning value BV and a yearly or annular interest r, we write

for the ending value EV1 The underlying assumptions of (1) are that:

• We hold the beginning value over one year

• There is no interest payment and no cash flow during the year

Example 2.1 We consider for BV a Coupon C of an annual paying bond Then(1) expresses the ending value EV1after 1 year In the European bond market,coupons are usually paid yearly

The index 1 in EV1says that there is no cash flow during the year and EV1

Next, we assume that one half of the interest is pay out in the middle of the year,which gives

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The question is whether the sequence EVnis bounded or unbounded The answer

is that the sequence is convergent since from calculus we know that

is called the annual effective rate.

Remark 2.1 For discrete compounding, we have

AER¼ er 1:

Example 2.3 We consider a semiannual bond with face value F 1 year beforematuring Furthermore, we assume there are two coupons, i.e., we get C/2 in themiddle of the year and C/2 at the end of the year By using continuous compoundingand prevailing interest r1and r2,we find

We specify N (not necessarily equidistant) knots on the time axis withcorresponding times tkand denote them by

By assuming t0¼ 0, t0is the present or for short t0is now However, in principle,

t0can be in the past (t0< 0) or in the future (t0> 0) For illustration purposes, weuse years as units Then, for equidistant knots of annual cash flows between t0and

t , we have

tk¼ k, 0  k  N: ð2:2:2ÞFor two equidistant knots over 1 year, we have N¼ 2, and the time knots aremarked by

We see that in (3), $1 is discounted by the discount factor d(r, t, tk) We consider

in the following the more general form by considering a cash flow C and abeginning value BV:

Example 2.4 We choose N ¼ 4 in (1, 2) with a cash flow $3 in t ¼ tN¼ 4 With

t ¼ 0 and r(t) ¼ r ¼ 5% in (3), we have for the beginning value BV with (4)

In Fig.2.3, we assume N¼ 10 and show the discount factor for the interest rates

r¼ 0.05, r ¼ 0, and r ¼ 0.05 between the times t0¼ 5 (ex post) and t10¼ 5(ex ante) We see that the behavior of the discount factors is different for positiveand negative discount factors

Remark 2.2 From Eq (2.1.1), we have with C¼ EV after one time unit

C¼ BV 1 þ rð Þ:

On the interval r∈ (1, 0), we see that value is destroyed, i.e., C < BV, and for

r¼ 1, we have complete loss, i.e., C ¼ 0

The following lemma summarizes some fundamental properties about discountfactors:

Lemma 2.1 In (4) we have under the assumption C > 0:

(a) For fixed r ∈ R1with r> 1 and t ∈ R1with t> 0, BV(C, r, t, tk) is amonotonically increasing linear function of C, i.e.,

BVðλC; r; t; tkÞ

¼ λBV C; r; t; tð kÞ, λ ∈ R1: ð2:2:5Þ(5) says that by changing the cash flow by a fixed factor, the value at present ismultiplied by the same factor

0.50 1.00 1.50 2.00 2.50 3.00

(b) For fixed C ∈ R1and t ∈ R1with t> 0, BV(C, r, t, tk) is a monotonicallydecreasing function of r The higher the interest, the less worth is the money atpresent.

(c) For C ∈ R1and for t ∈ R1, BV(C, r, t, tk) is for a fixed r ∈ R1:

• With r> 0 monotonically decreasing

• With r¼ 0 constant

• With1 < r < 0 monotonically increasing function of t

(d) The series of the discount factor

1þ r

ð Þn, n¼ 1, 2, 3, are:

• For r> 0, monotonically decreasing and converging with limit 0

• For r¼ 1, the series is constant with dn¼ 1, n ¼ 1, 2, 3, ,

• For1 < r < 0, the series dnis diverging for n! 1

Proof From (4), we have

1

1þ r

ð Þtk> 0,i.e., in order to show monotonicity, it is enough to consider

BV Cð ; r; t; 0Þ ¼ C

1þ r

ð Þt:The assertions a and b follow from the partial derivatives

Lemma 2.1 discusses the monotonicity of the discount factors We assumedthree independent variables, C, r, and t In the following lemma, we change thethree variables simultaneously, and we see that there is no monotonicity.

Lemma 2.2 For 100C ¼ t (1  t  10) and C ¼ r, the function defined in (4) has aglobal maximum for C¼ 7.259173%, and we have

Figure2.4shows this function, and a numerical method calculates the values

We investigate the behavior of the discount factors in more detail in Example 4.7(Chap.4)

Fig 2.4 Global maximum

Lemma 2.2 A closed formula for the beginning value BVorof an ordinary annuity

in the time span between t0¼ 0 and tNis, for1 < r < 0 or r > 0,

Remark 2.3 EV and BV are related by

Lemma 2.3 (Repayment of Mortgage) We assume that a BV and an interest r > 0are given The periodic payment of ordinary annuity is

Eor¼ EN¼ EV: ð2:3:10aÞWith E0¼ C and (9) with n ¼ 1, .,N, we have for an annuity due in (2b)

We decompose the annuity by the part which is due to increase of the balanceminus the interest rate payment in last period:

C¼ C þ r Eð nÞ  r En:Proof We consider the partial sum

C¼ BV ¼ r 1ð þ rÞ

N

1þ r

ð ÞN 1¼ $742:50:

The amortization schedule is in Table2.1

Table 2.1 Amortization schedule

Month

Beginning of month

mortgage balance

Mortgage payment Interest

Scheduled principal repayment

End of month mortgage balance

The Flat Yield Curve Concept 3

3.1 The Description of a Straight Bond

The financial market consists of the credit market, the capital market, and themoney market The bond market is part of the capital market The financial industrydistinguishes traditional and alternative investments Fixed income instruments aretraditional investments We start with the following definitions

Definition 3.1 A straight bond with price P will pay back the original investment

at its maturity date T and will pay a specified amount of interest on specific datesperiodically

A straight bond is the most basic of debt investments It is also known as a plainvanilla or bullet bond The cash flows illustrated in Fig 1.1 are referring to astraight bond

Example 3.1 (Description of a Bond Universe) Most of the bonds in the Swissbond market are straight bonds

Definition 3.2 The face value F of a bond is the amount repaid to the investorwhen the bond matures The face value is also called the par value of a bond or theprincipal, stated, or maturity value of a bond

Definition 3.3 Coupon C is a term used for each interest payment made to the bondholder

We distinguish between registered and unregistered bonds A bearer bond isunregistered and the investor is anonymous Whoever physically holds the paper onwhich the bond is issued owns the bond Recovery of the value of a bearer bond inthe event of its loss, theft, or destruction is usually impossible The collection of thecoupon is the task of the investor Often, the bank collects the coupon payment onbehalf of the investor If the issuer of the bond kept a record of the investor, we

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speak of a registered bond The issuer of the bond sends the coupon payments to theinvestor.

Figure3.1shows on the horizontal axis the specific dates and the correspondingcash flows denoted with the coupons and the face value Generally, a fixed incomeinstrument is a series of cash flows of coupons and a face value A straight bond isthe starting point for studying fixed income instruments

Definition 3.4 The time to maturity tm ¼ t  tN is the remaining lifetime ofthe bond

Remark 3.1 In Fig.3.1, we assume that t0is the origin, i.e., t0¼ 0 on the time axes.Moreover, in the following, we assume that tk, k ¼ 1, .,N are the times of theanalysis

Definition 3.5 A zero coupon bond is a bond which does not pay interest beforethe maturity date

A straight bond can be considered as a series of zero coupon bonds Bonds havetherefore a so-called linear structure

Definition 3.6 The flat yield curve concept assumes that each coupon and the facevalue of a specific bond are discounted by the same interest rate

We consider an annual paying bond with N Coupons C and face value

F Referring to knots (2.1.2), we specify N equidistant knots on the time axis withcorresponding time tkdenote them

t0= 0

tN = T

t IP,P

We proceed with the following definition:

Definition 3.7 (Invoice Price) A bond pricing with invoice price (IP) quotes theprice of a bond that includes the present values of all future cash flows incurringincluding the interest accruing until the next coupon payment

Remark 3.2 P and IP as a function depend on the variables r, C, F, tN¼ N and thetime and the frequency of the coupon payments In this book, we only considerannual buying bonds, and therefore, the frequency is 1 and C, F are kept constant Inthe following, we consider one point in time Therefore we suppress thesearguments

Remark 3.3 The invoice price is also called the dirtyor full price

For k¼ 0, .,N  1, we have with (1)

IP tð Þ ¼ 1

1þ r

ð Þtkþ1t

XN j¼kþ1

IPð Þ  IPt þð Þ ¼ C, k ¼ 1, , N  1:t

In Fig.3.2, we observe a zigzag line which represents the invoice price of a bond

as introduced in Definition3.7 We distinguish between days where a coupon ispaid and days where no coupon is paid We require the following definition

Definition 3.8 (Accrued Interest) Accrued interest is an accounting method formeasuring the interest rate that is either payable or receivable and has beenrecognized but not yet paid or received It occurs as a result of the difference intiming of cash flows and the measurement of these cash flows If we assumeperiodic coupon payments C, then, for k ¼ 1, , N  1, the accrued interest

AIκ¼ 1  αð κÞC, k ¼ 0, , N  1, t ∈ t½k1; tkÞ ð3:1:3bÞ

is the same as (3a)

As accrued interest is calculated daily, we have to change from the unit year tothe unit days, and we obtain a step function (Fig.3.3)

If the month is calculated with 30 days, the accrued interest is horizontal andstays the same In February, the accrued interest changes vertically, and accruedinterested is cumulated

The International Capital Market Association (ICMA) recommends in its Rule

251 that the number of days accrued should be calculated as the difference between

Invoice price

the date of the last payment inclusive (or the date from which the coupon is due, for

a new issue) up until, but not including, the value date of the transaction

Example 3.2 (Day Counting) We consider a bond that pays a coupon at 20.08.xx,and we assume that the value date of the transaction is 25.01.xx Assuming that themonth is calculating with 30 days, Table3.1gives the number of days

Definition 3.9 (Clean Price) The price of a coupon bond not including anyaccrued interest is called clean price and is denoted by P

Remark 3.6 The flat or simple price is the same as clean price

Fig 3.3 Accrued interest

The price of the bond is based on the evaluation of all cash flows In cal terms expressed, this means that the bond has a linear structure.

mathemati-For continuous compounding, we have

P 0ð Þ ¼XN j¼1

If we spread the coupon over the time t∈ [tk1, tk], i.e., if we consider continuouscompounding by starting an equal distant sample of the interval tj to tj+1(see Appendix E), then

For the difference, we then have

r > 1, the sequence is increasing and decreasing, tively, and for C ¼ r, the difference vanishes Therefore, the assertion (c) is

Example 3.3 We consider a face value F¼ 100 Then, with C ¼ 2% and r ¼ 4%,

we have

P10 ¼ 124:012, P20¼ 159:556,and, with C¼ 4% and r ¼ 2%, we have

P10¼ 78:100, P20¼ 51:405:

Definition 3.10 If the bond price is P¼ 100, then the bond price is said to be at par

If the bond price P is less than 100, then we have a discount bond If the bond price

P is over 100, then we have a premium bond

Corollary 3.1 We consider an annual paying bond with price P, yields r, andCoupon C with C > 0, C ∈ R1 At the times of knots as defined in (2.2.1), thefollowing holds for n¼ 1, 2, 3, ,N  1:

(a) If 0< r < C, then Pn< 100

(b) If r¼ C, then Pn¼100

(c) And if r> C or 1 < r < 0 (negative interest), then Pn> 100

Proof We consider the recursion (5) and we prove the corollary by induction withrespect to n For n¼ 1, we distinguish the following cases:

Assuming that the assertion is true for n, we consider n¼ 1, 2, 3, , N  1 as

Definition 3.11 (Constituents of a Bond Portfolio) For the price Pjof the bond j,

1 j  n, with time of maturities 1  Tj Tn and with cash flows Cj,kand facevalues Fj, we have the price of a bond as a function of r as

Definition 3.12 (Yield to Maturity) Assuming that the price of the bond is given,the yield to maturity (YTM) rjof a bond with price Pjis the solution of (1)

Pj rj

¼XTj1k¼1

In the following, the assumptions are that:

• It is assumed that all coupons are paid, i.e., there are no defaults

• The investor holds the bond until maturity

• We are looking forward, i.e., we consider the cash flow in the future

• The yield to maturity is the solution of this equation written down here, whichsays that the cash flows in the future discounted to today equal to the price paid

in the market The principle is based on an arbitrage relationship, i.e., acondition which avoids a situation with a profit without risk

• It is not clear how yield to maturity is added for different bond

• It is not clear whether the solution is unique

Time Bond

T

n

k j

Fig 3.5 The maturity profile

For the last two points, there is current research being conducted by [1,2].

We assume that a portfolio with n bonds is ordered with decreasing times tomaturity We assume that in this portfolio there are Njof bond j, and n is the number

of bonds that have a cash flow in time tk¼ k, 1  k  Tj Then the portfolio value Pois

Remark 3.7 In this section and the following section, we consider the flat rateconcept This, however, does not mean that the yield curve is flat

In the following, we consider a portfolio consisting of only one bond with agiven price We denote the yield to maturity with YTM

Example 3.5 (YTM of a Zero Coupon Bond) The price of zero coupon bond is

P1:

tr

Definition 3.13 The yield to maturity of a zero coupon bond is called the spotrate

Example 3.6 (YTM in the Last Period) As can be seen in the proof of Theorem2.1, the price of a bond in the last period is

P¼Fþ C

1þ r,and therefore we have

P  1:

Example 3.7 We consider 3 bonds that have a coupon of 3% with YTMs 2%, 3%,

or 4% and time to maturity of 3 years In Table3.2the cash flow analysis can beseen

The second column (t¼ 0) of Table3.2shows the price of the different bonds Inthis example, the YTD is given In practice, the bond is given, and the yield tomaturity has to be computed We illustrate the general principle that if the YTD isbelow the coupon, then the price is above the par value (premium bond) If the YTD

is equal to the coupon, then we have a par bond And that if YTD is beyond thecoupon, the price is below the par value (discount bond)

Example 3.8 We want to determine whether the yield of a semiannual 6% 15-yearbond with face value of $100 selling at $84.25 is 7.2%, 7.6%, or 7.8% We computethe present value PVCof the cash flows of the cash by using the formula

PVC¼C2

1 þ r 2

 n:The price P of the bond is the

P¼ PVCþ PVF:Table3.3below shows the computed values

We see that for the price $84.25, we have YTM¼ 7.8% With YTD ¼ 6.0%,(1) yields P¼ 100%

Table 3.3 Different yield