Fixed income analytics bonds in high and low interest rate environments

Wolfgang Marty Fixed Income Analytics Bonds in High and Low Interest Rate Environments Fixed Income Analytics Wolfgang Marty Fixed Income Analytics Bonds in High and Low Interest Rate Environments Wolfgang Marty AgaNola AG Pfaeffikon Switzerland ISBN 978-3-319-48540-9 ISBN 978-3-319-48541-6 DOI 10.1007/978-3-319-48541-6 (eBook) 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 or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed 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 This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Foreword In light of an investment environment characterized by low yields and new regulatory capital regimes, it has become increasingly demanding for investors to achieve sustainable returns Particularly, fixed income investments are called into question There is a solution Since the foundation of AgaNola a decade ago, we have put our interest into convertibles, and at this point we want to thank our clients for having supported us also in challenging times—particularly when convertible bonds were considered at most a niche investment Unjustly! For being a hybrid, convertible bonds offer the “best of both worlds,” the benefits of an equity with the advantages of a corporate bond AgaNola is considered a leading provider in this asset class, and to date convertible bonds remain the core competence of us as a specialized asset manager As we consider increasingly popular convertible bonds a living and dynamic universe, we are placing a great importance on research and the exploration of the nature of this asset class As an internationally renowned expert in the fixed income and bond field, Dr Wolfgang Marty has contributed valuable insights to our work—making the bridge from theory to portfolio management AgaNola is committed to continue to support his fundamental research We wish Wolfgang Marty lots of success with his latest book Chairman and Founder AgaNola AG Stefan Hiestand v Foreword Compared to other asset classes, fixed income investments are routinely considered as a relatively well-understood, transparent, and (above all) safe investment The notions 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 interest rates are generally believed to be well-understood At the same time, we live in a world where the amount of private, corporate, and sovereign debt is steadily increasing and where postcrisis stimuli continue to affect and distort investor behavior and markets in an unprecedented way And that is even before we start contemplating the enormous uncertainties introduced by negative interest rates In his book, Dr Wolfgang Marty covers and expands on classic fixed income theory and terminology with a clarity and transparency that is rare to be found in a world where computerization of accepted facts often is the norm Wolfgang highlights obvious but commonly unknown conflicts that can be observed, for example, when applying standard theory outside its default setting or when migrating from single to multiple bond portfolios He also includes the effects of negative interest rates into standard theory Wolfgang’s book makes highly informative reading for anyone exposed to fixed income concepts, be it as a portfolio manager or as an investor, and it shows that often we understand less than we think when studying bond or bond portfolio holdings purely based on their commonly accepted key metrics; Wolfgang encourages to ask questions Anyone building automated software would benefit from familiarity with the model discrepancies highlighted as it is to everyone’s disadvantage if we find these too deeply rooted in commonly and widely applied tools In summary, Wolfgang’s book makes interesting reading for the fixed income novice as well as the seasoned practitioner Head of Quantitative Research Record Currency Management Dr Jan Hendrik Witte vii Preface 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 is difficult to understand, and the end user may be swamped by data In addition, computers solve problems in many dimensions, and, as human beings, we struggle thinking in more than a few dimensions To provide a sound background of understanding to anyone working in fixed income, we intend to illustrate here the essential basic calculations, followed by easy to understand examples The reporting of return and risk figure is paramount in the asset management industry, and the portfolio manager is often rewarded on performance figures The first 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 fixed income benchmark portfolio calculated by different software providers: we found different yields for the same portfolio and the same underlying time periods The following questions are obvious: How can a regulating body accept ambiguous figures? Should there not be a standard? An additional complication is linearization, often the first step in analyzing a bond portfolio The yield of the bonds in a bond portfolio is routinely added to report the yield of the total bond portfolio, and different durations of bonds in the portfolio are simply added to indicate the duration of a bond portfolio We found that linearization works well for a flat yield curve, but the more the yield deviates from a flat curve, the more the resulting figures become questionable Also, historically, interest rates have been positive In the present market conditions, however, interest rates are close to zero or even slightly negative We find ourselves confronted with several questions: Does the notion of duration still make sense in this new environment? And which formulae can be applied for interest rates equal or very close to zero? How discount factors behave? In the following, we attempt to include negative interest in our considerations For instance, in the world of convertibles, yield to maturities can easily be negative and is not problematic ix x Preface We describe the here presented material in three ways Firstly, we use words and sentences, in order to give an introduction into in the notions, definitions, ideas, and concepts 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 Acknowledgments This book is based on several presentations, courses, and seminars held in Europe and the Middle East The here presented material is based on a compilation of notes and presentations Presenting fixed income is a unique experiment and I am grateful for the many feedbacks from the audience The initial motivation for the book was a seminar held at the education center of the SIX Swiss Exchange I became aware that many issues in fixed income need to be restudied and revised; moreover, I did not find satisfying answers to my questions in the pertinent literature The SIX Swiss Exchange Bond Advisory Group was an excellent platform for analyzing open issues Furthermore, the working group “Portfolio Analytics” of the Swiss Bond Commission was instrumental for the research activities In particular my thanks go to Geraldine Haldi, Dominik Studer, and Jan Witte They revised part of the manuscript and provided helpful comments The European Bond Commission (EBC) was very important for my professional development The members of the EBC Executive Committee Chris Golden and Christian Schelling gave me continuing support for my activities, and the EBC sessions throughout Europe yielded important ideas for the book At the moment I am focusing on convertibles My thanks go to Marco Turinello and Lukas Buxtorf for introducing me into the analytics of convertibles The last chapter of the book is dedicated to convertibles The book was written over several years, and I am grateful to my present employer AgaNola for the opportunity to complete this book xi Conventions 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 2.1 refers to Theorem in Chapter Square brackets [ ] contain references The details of the references are given at the end of each chapter xiii Appendices 189 (b) By using the backward difference in the neighborhood x0 of f ðx0 þ hÞ À f ðx0 Þ the first derivative of f is then approximated in the neighborhood x0 by f xị ẳ f x0 ỵ hị f x0 ị ỵ Ohị h D:3ị or analogously f xị ẳ f x0 ỵ hị f x0 hị h 00 ỵ f x0 ị ỵ o h2 : 2h (c) By using the forward difference in the neighborhood of x0 is f ð x0 Þ À f ð x0 À hÞ the first derivative of f is then approximated the neighborhood x0 by f xị ẳ f x0 ị f x0 hị ỵ Ohị, h D:4ị or analogously f xị ẳ f x0 ỵ hị f x0 hị 00 ỵ h f x0 ị ỵ o h2 : 2h The second derivative of f is approximated by (B.3) 00 f xị ẳ f x0 ỵ hị f x0 hị þ OðhÞ: 2h By using (D.1) and (D.2), we have by expressing the second derivative by function value 00 f xị ẳ f x0 hị 2f x0 ị ỵ f x0 ỵ hị ỵ O hị 2h2 or analogously À Á f ðx0 À hÞ À 2f x0 ị ỵ f x0 ỵ hị h 000 þ f ðx0 Þ þ o h2 : 2h 190 Appendices Appendix E (Integral, Riemann Sum) As illustrated in Fig E.1, we consider a function f(x) defined on the interval [a, b] and a partition x0 ¼ a, : , xk < xkÀ1 , : , xN ¼ b of the interval [a, b] By integrating the function, we mean calculating the area beneath and above the function f(x) We define m k f ị ẳ f xị max f xị x ẵxk ;xkỵ1 and M k f ị ẳ x ẵxk ;xkỵ1 and hk ¼ xk À xκÀ1 : We consider the estimation N X b m k f ị hk kẳ1 N X f xịdx M k f ị hk kẳ1 a and the integral is the defined ðb f ðxÞdx ¼ lim hk !0 a N X Mk ðf Þ hk ¼ lim hk !0 k¼1 N X m k f ị hk : kẳ1 The integration is an abbreviation for an infinite sum Fig E.1 Integration of a function f(x) Mk mk x a = x0 xk x k+1 b = x0 Appendices 191 Appendix F (Linear Interpolation) NAVrị ẳ NAVr1 ị ỵ NAVr2 ị NAVr1 ị ðr À r1 Þ: r2 À r1 The condition NAV(r) ẳ leads to ẳ NAVr1 ị ỵ NAVr2 Þ À NAVðr1 Þ ðr À r1 Þ: r2 À r1 which leads to the zero r0 r0 ¼ À NAVðr1 Þ r2 NAVðr2 Þ Appendix G (The Closed Formulae of the Macaulay Duration) Assuming that r > À1, r ∈ R1 a closed formula for Macaulay duration is [1] Dmac C ỵ 1r ẵ1 ỵ rịn Cn ỵ F n ỵ ịr : ẳ C ỵ rịn 1ị ỵ Fr G:1ị With ẳ we have   C ỵ ẵ1 ỵ rịn Cn ỵ F nr r DMac ẳ n  C ỵ rị 1ị ỵ Fr C C ỵ ẵ1 ỵ rịn n ỵ F n r r r ¼ C n ð ð ỵ r ị 1ị ỵ F r   ! C 1 C F 1ỵ n n nn ỵ r r ỵ r ị r ỵ r ị ỵ r ịn   ẳ : C F ỵ r þ r Þn ð1 þ rÞn We claim   ! n X 1 n j 1ỵ : n n ẳ j r r ỵ r ị r1 ỵ rị jẳ1 ỵ rị G:2ị 192 Appendices and we adopt a proof by induction with respect to n For n ¼ we find ( )( ) (( )( ) ) 1 1 1 1 1ỵ ẳ 1ỵ r r 1ỵr r1 ỵ rị r (( r)( 1)ỵ r 1)ỵ r rỵ1 r ẳ r r 1ỵr 1 ỵ r 1 r ẳ ẳ r 1ỵr r1 ỵ r : ẳ 1ỵr We assume the assertion (2) for n and consider the following algebraic reformulations   ! r n nỵ1 n nỵ r 1ỵr þ rÞ rð1 þ rÞ ð1 þ rÞnþ1 ( )( !) 1 n n 1ỵ ẳ ỵ þ 1À À r r ð1 þ rÞn rð1 þ rịn ỵ rịnỵ1 ỵ rịnỵ1 ( )( !) 1 1 n n 1ỵ ẳ þ n nþ nþ1 r r ð1 þ r Þ ỵ rị ỵ r ị ỵ rịnỵ1 ( ) ! 1 1 1ỵ ẳ þ 1À þ nþ1 r r ð1 þ rÞn ð1 þ r Þ rð1 þ rÞ)nþ1 ( ( ) n n 1 1 ỵ ỵ ẳ n r ỵ rịn r1 ỵ rịnỵ1 ỵrị ỵ rịnỵ1 r 1 þ 1þ ð1 þ nð1 þ rÞ À nrÞ À nỵ1 r ỵ r ị * r1 ỵ rịnỵ1 + ( ) 1 r nỵ1 1ỵ ẳ nỵ nỵ1 r r ỵ rị þ r Þ rð1 þ rÞnþ1 + ( )* 1 ỵ rịnỵ1 r ỵ r nỵ1 1ỵ ẳ nỵ1 r r ỵ r ị r1 ỵ rịnỵ1 * + ( ) 1 nỵ1 1ỵ ẳ : r r þ rÞnþ1 rð1 þ rÞnþ1 Thus (2) is shown for n + and the assertion (2) is verified In order to show expand (2) to < α < 1, we consider with (1) Appendices 193     ! ! 1 n 1 n ỵ 1ỵ 1 ẳ n n r r r r ỵ rịn ỵ rịn ỵ r ị ỵ  ( !) r ị 1 n ỵ 1ỵ r( r )( ỵ rịn !) ỵ rịn 1 n ỵ ỵ r r ỵ rị n ỵ rị n ! n X j 1 α À 1 ẳ ỵ h i j r ỵ rịn jẳ1 ỵ rị n X jỵ1 : ẳ j jẳ1 ỵ rị Thus (1) is shown Appendix H (A Closed Formula for Convexity) The concept of duration and convexity are important for investigating the price behavior of fixed-income instruments They are widely described in literature (see e.g [1]) There are two types of duration Macaulay duration is defined as the average life of a bond and modifies duration DMod Dmac ẳ ỵ rị Dmod : H:1ị We start by dP Dmod ¼ À dr , P Co ẳ d2 P dr2 P H:2ị H:3ị where d denotes the derivative of the price P with respect to r (G.1) is the same as Dmac C ỵ rị ỵ rịn ỵ rị n ỵ ịr ỵ Fn ỵ ị n ỵ ị 1ỵr : ẳ P For r ¼ we have Dmac  À  n nỵ1 C ỵ1 : ẳ Cn ỵ H:4ị 194 Appendices From (2) we have dP ẳ Dmod P dr H:5ị Convexity is defined by Co ẳ d2 P IP dr2 By assuming that IP ¼ P we find (H.1) and (H.5) Co ¼ Dmac À Dmod d dDmod ẳ Dmod ị2 dr Dmod Pị ẳ Dmod ị2 IP dr dr 1ỵr In order to calculate Ddrmac , we consider Dmac ¼ Z P dDmac P ỵ ZDmod P dDmac dDdrmac P Z dP dr ¼ ¼ dr : dr P2 P We find the following algorithm for computing the convexity Co: Calculate DMod by (H.4) and (H.1) Calculate dZ dr we find dZ ỵ þ nÞ þ α ð1 þ rÞn À n ð1 ỵ rị1ỵn n rịr ẳ c dr r2 þ rÞðnÀ1þαÞ 2ðÀ1 À r À ðÀ1 þ α þ nịr ỵ ỵ rịn ỵ rịị r3 ỵ rịn1ỵị ỵ ỵ nị1 ỵ rị11ỵỵnị r ỵ ỵ nị r þ ð1 þ rÞn ð1 þ αrÞÞ r2 11ỵỵnị ỵ ỵ nị ỵ rị c Compute the convexity by d2 P dZ dr P ỵ ZDmod P ẳ D ị Dmod Mod P dr ỵ rị P2 Appendices 195 Appendix I (Cubic Splines) We show that the splines are twice differentiable in the knots for t ¼ tk we have based on (4.3.8)  ðtk À tkÀ1 Þ2 ¼ f j ðtÞÀ t¼tk on (4.3.9) as e ẳ 0, we have  f j tị ( t¼tk ¼ )  c2 ce e2 e3 c2  þ þ À ¼ 6ðtkþ1 À tk Þ t¼tk for the derivative we have (4.3.8) df k  ẳ dt tẳtk df j ỵ  ẳ dt tẳtj ( * 3tk tk1 ị2 6tk tk1 ị +     ẳ tẳtk tk À tkÀ1 )  c 3e2 tk À tkÀ1  ỵe ẳ 6tkỵ1 tk ị tẳtk for the second derivative, we have d2 f k À ¼ d2 t t¼tk * 6ðtk À tkÀ1 Þ2 6ðtk tk1 ị +     ẳ1 tẳtk d2 f j ỵ ẳ 1: d2 t tẳtk and for t ẳ tk+1, we have f j tị ẳ c2 ce e2 ỵ ỵ e3 6tkỵ1 tk ị c ẳ tk tk1 ẳ B A e ẳ t tk ẳ tkỵ1 tk ẳ C B f j t ị ẳ c2 ce e2 e2 c2 ce e2 ỵ ỵ ẳ ỵ ỵ 2 6 196 Appendices This the same to  c2 ỵ ce þ 2e2 ðB À AÞ2 þ 3ðB À AÞðC À Bị ỵ 2C Bị2  ẳ ẳ f j t ị tẳtjỵ1 6 BB 2BA ỵ AA þ 3BC À 3AC À 3BB þ 2CC À 2CB þ 2BB ¼ À2BA þ AA þ BC À 3AC ỵ 3AB ỵ 2CC ẳ  AA ỵ BC 3AC ỵ AB ỵ 2CC  f j tịỵ ẳ tẳtkỵ1 ( ) 2tkỵ1 tk tk1 C Aị2C B Aị ẳ tkỵ1 tk1 ị ẳ 6 ẳ 2CC 2CA BC ỵ AB AC ỵ AA : For the derivative we have (4.3.8) )  c 3e2  ỵe 6tkỵ1 tk ị tẳtkỵ1 1 ẳ tk tk1 ị ỵ tk tk1 ị ỵ tk tk1 ị 2 ẳ tkỵ1 tk1 ị  df j ỵ ẳ tkỵ1 tk1 ị  dt tẳtkỵ1 df k  ẳ  dt tẳtkỵ1 ( for the second derivative, we have ( )  d2 f j À e  ẳ0 ẳ tẳtkỵ1 tkỵ1 tk ị tẳtkỵ1 d t d2 f k ỵ ẳ0  d2 t tẳtkỵ1 Appendix J (See Also [2]) xị f xị ẳ p e 22 , 22 Appendices 197 We use the following notions: • μ: n  vector of expectation values • S: n  n covariance matrix of the historical return • R is a rectangular matrix, i.e., R ∈ RnÂm, with uncorrelated random numbers of normally distributed with μ ¼ and numbers Starting from the definition of the covariance matrix V, À Á V ẳ E XXT EXịE XT : A Cholesky factorization L of the known covariance matrix S is S ¼ LLT : We consider the product of the random matrix R and L Z ¼ LR and calculate the covariance of Z: VLRị ẳ E LRị LRT ELRịELRịT ẳ E LRRT LT EðLRÞE RT LT : Remember that L is a constant matrix in respect to the expectation operator and for constant c R1: EcXị ẳ cEXị: Then we have VLRị ẳ L E RRT ERịE RT LT ẳ LEEịLT : The expected value of E is the identity matrix I: Á L $ N 0; 1ị , LEị ẳ I L is the Cholesky factorization of S: VLRị ẳ LILT ẳ LLT ẳ S: We have shown that the covariance matrix of LE is S: V LRị ẳ S: It is trivial to add expected values μ to Z in an additional step 198 Appendices Appendix K (Derivation of Black-Scholes Differential Equation) We suppose that the stock price denoted by S follows the process dS ẳ dt ỵ dz: S K:1ị The first term contains the expected return μ, and the second term is the risk term represented by the volatility σ It assumed to be a Wiener process Following Ito Lemma there exists a C(S, t) with dC ¼ ! ∂C ∂C ∂ C 2 ∂C μS þ þ σSdz: σ S dt þ ∂S ∂t ∂S ∂S ðK:2Þ The discrete version of (1) and (2) over the time interval Δt is ΔS ¼ μΔt þ σΔz S ΔC ¼ ! ∂C ∂C C 2 C S ỵ ỵ Sdz: S t ỵ S t S2 S K:3ị K:4ị Consider the portfolio that consists of À1 of the call option C and ∂C of shares, ∂S and define the value of the portfolio by ẳ C ỵ ΔC S: S ðK:5Þ Over a interval Δt yields Δπ ẳ C ỵ C S S K:6ị and by (3) and (4) in (6), we have Δπ ¼ ! ∂C ∂ C 2 À À σ S Δt: ∂t ∂S2 ðK:7Þ Appendices 199 No risk term dz [see (1)], that is π, must earn in accordance to the risk free rate rf Δπ ¼ Àrf π Δt: (5) and (7) give !   ∂C C 2 C ỵ S t, t ¼ r σ S C À f ∂t ∂S2 S hence C C C ỵ rf S S ỵ S2 ẳ rf C: ∂t ∂S ∂S ðK:8Þ This is the Black-Scholes partial differential equation The solution is dependent from a boundary condition For a call option, this boundary condition is the PO of the call option (see Fig K.1) With denoting the strike price by K and x Nxị ẳ z2 e dz À1 the price of a European call option in CSị ẳ SNd1 ị KeT Nd2 ị whereby d1 ¼ ln S K D E E   D 2 À r À σ2 T ln KS þ r À σ2 T pffiffiffi pffiffiffi , d2 ¼ : σ T σ T Call Option 70 60 50 40 30 Call option Payoff Call 20 10 0 50 100 Stock Price Fig K.1 The PO of a call option 150 200 200 Appendices Then sensitivities (Greeks) are then ẳ C ẳ Nd1 ị; S p C ¼ S TN0 ðd1 Þ; ∂σ Λ¼ ∂ C N0 d1 ị ẳ p ; S T ẳ ẳ C SN0 d1 ị p rf K eT Nd2 ị; ẳ S T ẳ C ẳ KT erf T Nd1 ị: rf References Fabozzi FJ, Iriving PM (1987) Understanding duration and volatility In: Kopprasch Robert W (ed) The handbook of fixed income securities Chapter 5, 2nd edn Dow Jones-Irwin, Homewood, IL, pp 86–120 Neil R (2003) Currency overlay Wiley, Hoboken, NJ # Springer International Publishing AG 2017 W Marty, Fixed Income Analytics, DOI 10.1007/978-3-319-48541-6 201 Index A Accrued interest, 20–22, 49, 161 Asset swap, 4, 157, 158 B Benchmark bond, 161 Bootstrapping, 113, 115, 118, 119 C Capital market, 17, 20, 140, 149 Clean price, 20, 21, 49 Composite rating, 142–144 Convertible, 4, 149, 173–183 Convexity, 32–55, 145, 180, 181 Current yield, 28 D Direct yield (DY), 28, 29, 57–71, 94, 176 Dirty price, 19 Discount bond, 23, 27, 28, 44, 97, 177 Discount factor, 7–12, 29, 33, 34, 44, 46, 50, 57, 58, 71, 76, 96, 103, 107, 111, 118–120, 125, 152, 155 I Industry-standard benchmark, 159 Interest rate market risk, 112 Interest rate spread, 129–133, 154 Interest rate swap (IRS), 4, 152–157 Internal rate of return, 3, 26, 30, 32, 48, 49, 55–102, 161 Invoice price (IP), 19, 20, 22, 49, 113, 121, 123, 127, 150 M Macaulay Duration, 32–38, 44, 45, 47–50, 55, 77–94, 97, 98, 100, 101, 126, 156 McCulloch equation, 119, 126 Modified duration, 3, 49–51, 55, 77, 81–89, 91, 127, 132, 152, 179 Money market, 17, 24, 112, 149, 150, 161, 163, 164, 169 N Net asset value (NAV), 55–57, 71, 85, 86, 88, 89 E Effective duration, 3, 126–128 P Par bond, 27, 28, 59, 60, 64, 65, 67, 77, 80, 85, 87, 97, 111, 113, 114, 123, 131, 153 Premium bond, 23, 27, 28, 67 F Fisher-Weil duration, 126, 128 Flat price, 21, 22, 127 Flat yield curve, 3, 58, 123 Flat yield curve concept, 3, 17, 99 Forward rate, 103–111 Full price, 19 S Simple price, 21 Spot rate, 26, 32, 49, 101, 103–127, 131, 153 Spread duration, 132, 152 Straight bond, 1, 4, 17–24, 32, 35, 37, 42, 44, 52, 55, 115, 126, 132, 149, 150, 152, 175–177, 179, 181, 183 # Springer International Publishing AG 2017 W Marty, Fixed Income Analytics, DOI 10.1007/978-3-319-48541-6 203 204 T Tailor-made benchmarks, 160 Term structure, Y Yield curve, 2–4, 17, 45, 95, 96, 98, 103, 106, 110–126, 131, 132, 147, 149, 150, 154, 156 Index Yield to maturity (YTM), 3, 25–29, 32, 35–37, 49, 57, 58, 60, 64, 66, 71, 76, 78–82, 89, 90, 101, 103, 104, 106, 114, 115, 118, 126, 129–131, 145, 146, 176, 177 Z Zero coupon bond, 18, 26, 29, 32, 104, 106, 113 .. .Fixed Income Analytics Wolfgang Marty Fixed Income Analytics Bonds in High and Low Interest Rate Environments Wolfgang Marty AgaNola AG Pfaeffikon Switzerland ISBN 978-3-319-48540-9... coupons and the face value Generally, a fixed income instrument is a series of cash flows of coupons and a face value A straight bond is the starting point for studying fixed income instruments Definition... comprehensive introduction to fixed- income analytics Some of the topics are: • The transition from a single bond to portfolio of bonds is examined We investigate the nonlinearity of income since just adding